# Mortgage

### Fully Amortizing Loan [viewable here in Excel]

The most common loan payment pattern used in real estate finance from the post depression era to the present, and one which is still very prevalent today is the fully amortizing, fixed rate mortgage. This loan payment pattern is used extensively in financing single family residences and is also used in long-term mortgage lending on income-producing properties such a multi-family apartment complexes and shopping centers. This payment pattern simply means that a level, constant, monthly payment is calculated on an original loan amount at a fixed rate of interest for a given term. At the end of the term of the mortgage loan, the original loan amount or principle is completely repaid, of fully amortized, and the lender has earned a fixed rate of interest on the monthly loan balance. However, with this loan the amount of amortization varies each month.

To illustrate how the monthly loan payment calculation is made, we turn to an example using a \$200,000 loan made at 8 percent interest for 30 years. What will be the monthly mortgage payment on this loan, assuming it is to be fully amortized (paid off) at the end of 30 years? Based on our knowledge of discounting annuities, the problem is really no more than finding the present value of an annuity and can be formulated as follows:

Illustration : Calculating The Monthly Payment On A House Loan

Suppose you are trying to borrow \$200,000 to buy a house on a conventional 30-year mortgage with monthly payments. The annual percentage rate on the loan is 8%. The monthly payments on this loan can be estimated using the annuity formula:

Monthly interest rate on loan = APR/ 12 = 0.08/12 = 0.0067

Monthly Payment on Mortgage = \$200,000 x (.0067/(1-(1/(1.0067)365))) = \$1,473.11
=200000*(0.0067/(1-(1/(1.0067)360)))

### Points Added to a Mortgage Loan

To illustrate loan fees and their effect on borrowing costs in more detail, consider the following problem: A borrower would like to finance a property for 30 years at 12 percent interest. The lender indicates that an origination fee of 3 percent of the loan amount will be charged to obtain the loan. What is the actual interest cost of the loan?

We structure the problem by determining the amount of the origination fee or .03 x \$60,000 = \$1,800. Second, we know that the monthly mortgage payments based on 12% will be \$617.17. Now we can determine the effect of the origination on the interest rate being charged as follows:

 Original Loan Amount (less \$1,800 in points paid) \$58,200 What is the monthly loan payment on the \$58,200? (\$598.65) =PMT(0.12/12,360,58200) What is the monthly mortgage payment on a 30 year, \$60,000 loan at 12 percent interest (\$1,800 = 3 points)? \$617.17 =PMT(0.12/12,360,-60000) What is the payoff after 10 years? * number represents years remaining to payoff 360-120=240 \$56,051 =PV(0.12/12,240,-617.17) What is the Annual Percentage Rate (APR) (Use higher monthly pymt and true loan amt (excluding fees))? 12.41% =RATE(360,-617.17,58200,0)*12 What is the payoff after 5 years? * number represents years remaining to payoff 360-60=300 payments due \$58,598 =PV(0.12/12,300,-617.17) Prepayment: If loan is paid off in 5 years what is the cost of remaining points in terms of "yield", or cost? 12.82% =RATE(60,617.17,-58200,58598.16,0)*12

In other words, the amount disbursed by the lender will be \$58,200, but the repayment will be made on the basis of \$60,000 plus interest at 12 percent compounded monthly, in the amount of \$617.17 each month. Consequently, the lender will earn a yield on the \$58,200 actually disbursed, which must be greater than 12 percent.

From the above calculation we can see that the effective cost of the loan assuming it is outstanding until maturity is 12.41 percent.

 Original Loan Amount \$443,750 What is the monthly mortgage payment on a 30 year, \$450,000 loan (1 point added)with a 4.50 percent interest rate? \$2,280 =PMT(0.045/12,360,-450000) What is the payoff after 10 years? * number represents years remaining to payoff 360-120=240 \$360,402 =PV(0.045/12,240,-2280.08) Same loan with 1.40% fees charged = \$6,250 and added to loan, so loan is \$450,000-\$6250= \$443750 (\$2,248) =PMT(0.045/12,360,443750) APR (Use higher monthly pymt and true loan amt (excluding fees)) 4.62% =RATE(360,-2280.08,443750,0)*12 What is the payoff after 5 years? * number represents years remaining to payoff 360-60=300 payments due \$410,209 =PV(0.045/12,300,-2280.08) 5 years Prepayment Penalty: If loan is paid off at or before 5 years, what is the cost of remaining points in terms of "yield" - or cost? 4.83% =RATE(60,2280.08,-443750,410209.93,0)*12

### Points and Prepayment Fees Added to a Mortgage Loan. What is the true APR?

 Loan Amount \$60,000 Mortgage Payment, \$60k loan, 30 years @12% with points added to loan \$617.17 =PMT(0.12/12,360,-60000) What is the payoff after 5 years? * number represents years remaining to payoff 360-60=300 \$58,598 =PV(0.12/12,300,-617.17) Loan Fee (points paid) 3.000% \$1,800 "X" prepayment fee due at month 60 3.000% \$1,758 New Payoff at month 60 with prepayment penalty \$60,356 =58598+1757.94 Monthly payment at true loan amount \$58,200 \$599 =PMT(0.12/12,360,-58200) APR (Use higher monthly pymt and true loan amt (excluding fees)) 12.41% =RATE(360,-617.17,58200,0)*12 What is the payoff after 5 years? * number represents years remaining to payoff 360-60=300 payments due \$60,356 =60355.94 If loan is paid off in 5 years what is the cost of remaining points in terms of "yield" - or cost? 13.25% =RATE(60,617.17,-58200,60355.94,0)*12

### Loan Fees and Early Repayment on a Fixed Rate Mortgage

An important effect of loan fees and early repayment must now be examined in terms of the effect on interest cost. In this section it will be shown that when loan fees are charged and the loan is paid off before maturity, the effective interest cost of the loan increases even further than when the loan is repaid at maturity.

To demonstrate this point, we again assume our borrower obtained the \$60,000 loan at 12 percent for 30 years and was charged \$1,800 (3 percent) loan origination fee. At the end of five years, the borrower decides to sell the property. The mortgage contains a due on sale clause, hence, the loan balance must be repaid at the time the property is sold. What will be the effective interest cost on the loan as a result of both the origination fee and early loan repayment?

To determine the effective interest cost on the loan, we first find the outstanding loan balance after 5 years to be \$58,598 (=PV(0.12/12,300,-617.17)) = \$58,598

To solve for the yield to the lender (cost to borrower), we proceed by finding the rate at which to discount the monthly payments of \$617.17 and the lump-sum payment of \$58,596 after five years so that the present value of both equals \$58,200, or the amount actually disbursed by the lender.

This presents a new type of discounting problem, Here, we are dealing with an annuity in the form of monthly payments for five years and a loan balance, or single lump-sum receipt of cash, at the end of five years. To find the yield on this loan, we proceed as follows:

\$58,200 = \$617.17 (MPVIFA, ?%, 5 yrs.) + \$58,596(MPVIF, ?%, 5 yrs.)

This formulation simply says that we want to find the interest rate (?%) that will make the present value of both the \$617.17 monthly annuity and the \$58,596 received at the end of five years equal to the amount disbursed. We need to take special note that the two interest factors used in the example are different. One factor (MPVIF) is used to discount the single receipt or loan balance. The other factor, (MPVIFA), will be used to discount the payments or monthly annuity. Hence, we cannot use the method of dividing the monthly annuity into the disbursement to find a an interest factor, because we also have the loan balance of \$58,596 to take into account. How do we solve this problem? We find the answer by trial and error (Hard Way); that is, we must begin choosing interest rates and then select the interest factors for five years corresponding to (MPVIFA) and (MPVIF) (if you're using PV charts). We then multiply these factors by the cash payments and determine whether the calculated present value is equal to \$58,200, we have the solution we want.

This trial-and-error process is not as ominous as it may seem. Some careful thought about the problem tells us that because loan fees are being charged, the yield we are seeking must be greater than the contract interest rate of 12 percent compounded monthly. Also, careful thought will lead us to conclude that the yield is going to be greater than 12.43 percent. This is because, as we have seen from the APR, a 12.43 percent yield would be earned by the lender if the loan were repaid at the end of 30 years. Because the loan is being repaid over 5 years, the origination fee of \$1,800 is being earned over 5 years as opposed to 30 years; hence, the effective interest cost to the borrower (yield to lender) will be higher than 12.43 percent. How much higher? Probably not more than 13 percent because, based on the rule of thumb discussed earlier, it would take a fee of about 8 points to increase the yield from 12 percent to 13 percent over 30 years, and since we are dealing with a 3 percent origination fee, it is very unlikely that the yield would be in excess of 13 percent, even after only five years. Therefore, we will use the interest factor at 12 percent and 13 percent and interpolate for the solution as follows.

 Trial and Error 12.82% "MPVIFA" "MPVIF" PV "MPVIFA" =((1-(1+(0.1282/12))-60)/(0.1282/12)), "MPVIF" =1/(1+(0.1282/12))60 44.13 0.5285593 \$58,200 = \$617.17 (MPVIFA, ?%, 5 yrs.) + \$58,596(MPVIF, ?%, 5 yrs.) \$27,235.71 \$30,971.46 \$58,196.63 =617.17*44.13 =58596*0.5285593 =27235.17+30961.46 Prepayment - If loan is paid off in 5 years what is the cost of remaining points in terms of "yield" - or cost? 12.82% =RATE(60,617.16,-58200,58597.93,0)*12

or

 Mortgage Payment, \$60k loan, 30 years @12% \$617.17 =PMT(0.12/12,360,-60000) What is the payoff after 5 years? * number represents years remaining to payoff 360-120=240 \$58,597.93 =PV(0.12/12,300,-\$617.17) Same loan with 3 points charged = 1,800, so loan is \$58,200 w/ \$1,800 in fees - what is the APR? (\$598.65) =PMT(0.12/12,360,58200) APR (Use higher monthly pymt and true loan amt (excluding fees)) 12.41% =RATE(360,-617.17,58200,0)*12 What is the payoff after 5 years? * number represents years remaining to payoff 360-60=300 payments due \$58,597.93 =PV(0.12/12,300,-\$617.17) Original Loan Amount \$58,200.00 Prepayment - If loan is paid off in 5 years what is the cost of remaining points in terms of "yield" - or cost? 12.823% =RATE(60,617.17,-58200,58597.93,0)*12

We have employed a slightly different form of interpolation that was shown in the previous examples. This approach must be used anytime two or more different cash flow patterns, such as an annuity and a single receipt, are encountered in a problem in which the yield must be determined. With Excel, a more precise solution of 12.82 percent is obtained. Also, not that when the cash flows are discounted at 12 percent, the original \$60,000 balance is determined. The detail can be eliminated in future computations.

From the above analysis, we can conclude that the actual yield (or actual interest cost) that we have computed to be approximately 12.82 percent is higher than both the contract interest rate of 12 percent and the 12.43 percent yield computed assuming the loan was outstanding until maturity. This is true because the \$1,800 origination fee over 5 years as opposed to 30 years is equivalent to earning a higher rate of compound interest on the \$58,200 disbursed. Hence, when this additional amount earned is coupled with 12 percent interest being earned on the monthly loan balance, this increases the yield to 12.82%

Another point is that the 12.82 percent yield is not reported to the borrower as being the "annual percentage rate" required under the Truth-in-Lending Act. The reason is that neither the borrow nor lender knows for certain that the loan will be repaid ahead of schedule. Therefore, 12.375 percent will still be reported as the annual percentage rate and 12 percent will be the contract rate, although the actual yield to the lender in this case will be 12.82 percent. It should be remembered that the annual percentage rate under the truth-in-lending requirements never takes into account early repayment of loans. The APR calculation take into account origination fees, but always assumes the loan is paid off at maturity.

### Achieving a Desired Yield by Adding Fees

Let's now consider how fees are determined by lenders when "pricing" a loan. Lenders generally have other alternatives in which they can invest funds. Hence, they will determine available yields on those alternatives against yields and risks on mortgage loans. Similarly, competitive lending terms established by other lenders establish yields that managers must consider when establishing loan terms. By continually monitoring alternatives and competitive conditions, management establishes loan offer terms for various categories of loans, given established underwriting and credit standards for borrowers. Hence a set of terms designed to achieve a competitive yield on categories of loans representing various ratios of loan to value (LTV) (70 percent LTV, 80 percent LTV, etc.) are established for borrowers who are acceptable risks. These terms are then revised as competitive conditions change.

To illustrate, if, based on competitive yields available on alternative investments of equal risk, managers of a lending institution believe that a 14% yield is competitive on 80 percent mortgages with terms of 30 years and expected payment in five years, how can they set terms on all loans made in repayment periods of 5 years, how can they set terms on all loans made in the 80 percent category to ensure a 14 percent yield? Obviously, one way would be to price all loans being originated at a fixed rate of 14%. However, management may also consider pricing loans at 12 percent interest and charging either loan fees or prepayment penalties or both to achieve the required yield. Why would lenders do this? Because (1) they have fixed origination costs to recover, and (2) competitors may still be originating loans at a fixed rate of 12 percent.

 Loan Amount \$60,000 Original term in months 360 Early Payoff in months 60 Mortgage Payment, \$60k loan, 30 years @12% with points added to loan \$617.17 =PMT(0.12/12,360,-60000) What is the payoff after 5 years? * number represents years remaining to payoff 360-60=300 \$58,655 =PV(0.12/12,360-60,-617.77) Loan Fee = 5 points 5.000% =60000*5% = \$3,000 "X"prepayment fee due at month 60 = 5% 5.000% =58655.13 * 5% = \$2,932 New Payoff at month 60 with prepayment penalty \$61,588 =58655.13+2932.76 Monthly payment at true loan amount \$58,200 if fee are added otherwise a no point/no fee loan 586.31 =PMT(0.12/12,360,-57000) APR (Use higher monthly pymt and true loan amt (excluding fees)) 12.70% =RATE(360,-617.17,57000,0)*12 What is the payoff after 5 years? * number represents years remaining to payoff 360-60=300 payments due \$61,588 Original Loan Amount \$57,000 =60000-3000 If loan is paid off in 5 years what is the cost with prepayment penalty in terms of "yield"? 14.11% =IFERROR(RATE(60,617.17,-57000,61587.89,0)*12,0)

If the loan is priced by offering terms of 12 percent interest and a 5% prepayment penalty and the loan is repaid in 5 years, management will have its 14% yield.

### Example 1 Mortgage Payment

 What is the monthly loan payments on a \$200,000 loan over 10 years at .5% interest per month (payment in arrears)? =PMT(0.005,120,200000,0,0) (\$2,220.41) =PMT(0.005,120,200000,0,0) What is the loan amount if monthly payments are \$2,220.41 (after) at .50% interest per month for 10 years? =PV(0.005,120,-2220.41) \$200,000.00 =PV(0.005,120,-2220.41) What is the monthly interest rate if I pay \$2,220.41 (after) per month on a \$200K loan for 10 years? =RATE(120,-2220.41,200000,0,0) 0.5000% =RATE(120,-2220.41,200000,0,0) How long would it take to pay off a \$200k loan if I pay \$2,220.41 at .50% monthly interest? Paid in arrears (after). =NPER(0.5%,-2220.41,200000,0,0) 120 =NPER(0.5%,-2220.41,200000,0,0)

### Example 2 - Value of a Small Office Building [viewable here in Excel]

A small office rents for \$25,000 per year for 25 years with an 8% discount rate (compounded annually), what is the property value today?

 Value \$266,869 =PV(0.08,25,-25000,0,0) Rate 8.00% =RATE(25,25000,-266869.4,0,0) Perpetuity \$312,500 =PV(0.08,1000,-25000,0,0) calculated \$312,500 =25000/0.08 Paid in advance, perpetuity \$337,500 =PV(0.08,1000,-25000,0,1)

### Example 3 - What was the small office building discount rate?

A property is currently worth \$2 million, it was purchase for \$1.75 million with a 5 year lease attached, what was the discount rate?

 Rate 2.707% =RATE(5,,-1750000,2000000,0) Value \$2,000,000 =FV(E270,5,,-1750000)

### Example 4 - What should I pay for a leasehold interest?

A leasehold interest sold for \$230k, the lease had 4 yrs remaining and rent was \$6k per month.
 If we accept a monthly yield of .75%, what "profit rent"(?) is shown by the transaction? \$5,681 =PMT(0.0075,48,-230000,0,1) If the interest rate is changed to .75% (9.0%/yr) per month (compounded monthly), how much should I pay for a property yielding \$25k per month in advance (property is worth \$5 million in five years)? (\$4,406,865) =PV(0.0075,60,25000,5000000,1) What is the rate? 0.75% =RATE(60,25000,-4406865,5000000,1)

### Example 5 - What is the Internal Rate of Return?

An investor purchased a property for \$1.2 million, the rent is \$12,000 per month. If the property is sold in 5 years for \$1.5 million, what is the return?
 Investment -\$1,200,000 Cash flow 1 \$144,000 Cash flow 2 \$144,000 Cash flow 3 \$144,000 Cash flow 4 \$144,000 Cash flow 5 \$144,000 Sum \$1,500,000 IRR 13.54%

### Example 6 - Property Value in 5 years?

An investor purchased a property for \$1.6 million, the rent is \$10,000 per month (paid in advance), and the discount rate is 1% per month (compounded monthly), what would the property be worth in 5 years?
 Value in 5 years = \$2,081,851 =FV(0.01,60,10000,-1600000,1) Monthly interest rate = 1.0% =RATE(60,10000,-1600000,2081851) The initial investment = (\$1,600,000) =PV(0.01,60,10000,2081851,1)

### Example 7 - Mortgage Loan Payment

 You can afford loan payments of \$2,500 per month at .45% (per month) over 20 years, what is the max. loan? \$366,434 =PV(0.0045,240,-2500) What is the monthly mortgage interest rate if you borrow \$366,433.74 for 20 years? 0.45% =RATE(240,-2500,366434) Annual interest rate 5.40% =0.0045*12 How long would it take to pay off a \$366,433.74 loan if you pay \$2,500 at .45% monthly interest? Paid in arrears. 240 =NPER(0.0045,-2500,366434) What is the monthly loan payment on a \$366,433.74 mortgage loan at .45% per month for 20 years? (\$2,500) =PMT(0.0045,240,366434)

### Example 8 - What is the remaining mortgage balance?

 What is the monthly loan payment on a \$150,000 mortgage at .45% per month for 97.75 months? (\$1,900) =PMT(0.0045,97.75552389,150000) What is the monthly mortgage interest rate if you borrow \$190,000 for 97.75 months? 0.45% =RATE(97.7555,-1900,150000) What is the loan amount if you pay \$1,900 (after) per month at .45% interest per month for 97.75 months? \$150,000 =PV(0.0045,97.7555,-1900)

### Example 9 - Interest Rate on a Consumer Loan

 Consumer loan of \$1,000 for 12 months with upfront monthly payments of \$100, what is the interest rate? 3.50% =RATE(12,-100,1000,0,1) Annual interest rate 42.00% =0.035*12 What is the monthly payment on a \$1,000 loan at 3.50% per month for 12 months, paid upfront? \$100 =PMT(0.035,12,-1000,0,1) How long would it take to pay off a \$1,000 loan if you pay \$100 at 3.50% monthly interest (Paid upfront)? 12.00 =NPER(0.035,-100,1000,0,1) What is the loan amount if you pay \$100 (upfront) per month at 3.50% interest per month for 12 months? \$1,000 =PV(0.035,12,-100,0,1)

### Example 10 - 30/15 Mortgage with Balloon Payment

You have a \$300k balloon mortgage (30/15) loan with monthly payments on \$100k, \$200k is due at the end of year 15.

 Monthly interest is .4% (4.8% annual), payments are in arrears? What are the monthly mortgage payments. \$1,580.41 =PMT(0.004,180,-300000,200000,0) What is the monthly mortgage interest rate if you borrow \$300,000 for 180 months (15 yr balloon at \$200k)? 0.40% =-RATE(180,-1580,-200000,300000,0,0) What is the annual interest rate? 4.80% =0.004*12 What is the remaining loan amount if you pay \$1,580.41 per month at .40% interest per month for 180 months? 202,933 =PV(-0.0040081,-180,1580,0,0) How long would it take to pay off a \$300,000 loan if you pay \$1580.41 at .40% monthly interest for 15 years with a \$200,000 balloon payment at the end of the term? 179 =NPER(-0.004,1580,300000,-200000) How much can you borrow if you can afford \$3,000 per month for 10 years at .4% monthly (4.8% annual) interest? \$285,468 =PV(0.004,120,-3000,0,0)

### Example 11 - Typical Mortgage Loan with Points

 Loan Amount \$375,000 Points 2% Annual Rate 7% or monthly = 0.005833333 =0.07/12 Term in months 120 Actual Loan Amount \$367,500 =375000-(375000*0.02) Monthly Service Charge -\$25 Monthly payment on \$150,000 (\$3,708) =PMT(0.00291667,120,375000,0,0) Total monthly payment (\$3,733) =-3708-25 Effective cost of loan (per month) 0.339% =RATE(120,-3733,367500,0,0) Annual cost of loan (monthly rate *12) 4.068% =0.339%*12 Annual Effective Cost of the Loan (APR) 4.145% =(1 + (0.04068 / 12)) 12 - 1 Effective Rate (7%) 7.23% =EFFECT(0.07,12) Nominal Rate 7.00%

### Estimating Market Value of Loan [viewable here in Excel]

We have considered several problems in which the balance of a loan was determined after payments had been made for a number of years. The balance of the loan represents the amount that the borrower must repay the lender in order to satisfy the loan contract. (Any prepayment penalties would of course have to be added to the loan balance.) The loan balance may be interpreted as the "contract" or "book value" of the loan. However, if interest rates have changed since the origination of the loan, the loan balance will probably not represent the "market" value of the loan.

The market value of a loan is the amount on the loan. It can be thought of as the amount that could be loaned so that the remaining payments on the loan would give the lender a return equal to the current market rate of interest.

Finding the market value of a loan simply involves calculating the present value of the remaining payments at the market rate of interest. For example, suppose a loan was make 5 years ago for \$80,000 with an interest rate of 10 percent and monthly payments over a 20-year loan term. Payments on the loan are \$772.02 per month. As we know, one way of finding the current balance of the loan is to compute the present value of the remaining loan payments, at the contract interest rate of 10 percent. We have

### Estimating Market Value of Loan

 Loan balance = PV(.10/12,15*12,-772.02) = \$71,842 \$71,160 General approach first Original Loan Market Value of a Loan \$60,000 \$80,000 Coupon 10.00% 10.00% Current Cost of Debt 10.00% 10.00% Periods 30 20 Value \$59,431 \$79,241 Value Value =(10%*10%)*(1-(1/(1+10%)30))/10%+(60000/(1+10%)10) Value = (10%*10%)*(1-(1/(1+10%)20))/10%+(80000/(1+10%)10) Principal and Interest Payment \$522 \$765 Principal and Interest Payment =PMT(10%/12,30*12,-59431) =PMT(10%/12,20*12,-79241) If rates increase change from 10% to 15%. \$48,576 \$71,189 If rates increase change from 10% to 15%. =PV(10%/12,15*12,-522) =PV(10%/12,15*12,-765)

### Estimating Market Value of Bonds

 General approach first If rates increases from 10% to 15%. Market Value of Bonds \$60,000 \$80,000 \$71,189 Coupon 10.00% 10.00% Current Cost of Debt (Change in % rates) 8.00% 15.00% Periods 10 20 Value \$68,052 \$54,963 =(10%*60000)*(1-(1/(1+8%)10))/8%+(60000/(1+8%)10) =(10%*80000)*(1-(1/(1+15%)20))/15%+(80000/(1+15%)20) Loss or Gain Due to Changes in Interest Rates \$8,621 -\$24,278 -\$16,552 -23.25% 1% change in interest rate changes price by: 4.65%

To find the market value of the loan, we compute the present value of the remaining payments at the market interest rate. Suppose that rate is currently 15 percent. We have:

Loan balance = PV(.15/12,15*12,-772.02) = \$55,160

Thus, the market value of the loan is \$55,160, as compared to the loan balance of \$71,842. The market value of \$55,161 is the amount that the lender would receive if the loan were sold to another lender, investor or the secondary market. We could say that the above loan is selling at a "discount." The amount of the difference in this case would be \$71,842 - \$55,161 = \$16,681. We could also say that the mortgage is selling at a discount of 23 percent of its "face" value.

The market value of the loan is lower than the contract loan balance in this example because interest rates have risen relative to the interest rate (10 percent) at which the loan was originated some 10 years ago. However, the borrower is required to make payments based on 10 percent even though market rates have risen to 15 percent. This is one reason why adjustable rate mortgages have become attractive to lenders. With an adjustable rate mortgage, the market value of the outstanding loan will not differ as much when compared to a new loan originated at market rates of interest. In fact, if the interest rate on the outstanding loan could be adjusted at each payment interval and there were no limitation (caps) on the amount of the adjustment, then the contract rate on the loan would always equal the market rate. In this event, the loan balance and market value for such a loan would always be equal because future payments would be based on current rates of interest.

### Incremental Mortgage Borrowing Costs [viewable here in Excel]

We begin by considering the question of how to evaluate two loan alternatives where one alternative involves borrowing additional funds relative to the other alternative. For example, assume a borrower is purchasing a property for \$100,000 and faces two possible loan alternatives. A lender is willing to make an 80 percent first mortgage loan for \$80,000, for 25 years at 12 percent interest. The same lender is also willing to lend 90 percent, for \$90,000, for 25 years at 13 percent. Both loans will have fixed interest rates and constant payment mortgages. How should this borrower compare these alternatives?

To analyze this problem, emphasis should be placed on a basic concept call the incremental or marginal cost of borrowing. Based on the material presented earlier, we know how to compute the effective cost of borrowing for one specific loan. However, it is equally important in real estate finance to compare financing alternatives, or situation in which the borrower can finance the purchase of real estate in more than one way or under different lending terms.

 Loan Amount Loan Constant Monthly Payments Alt. II at 13% \$90,000 x 0.0112784 x \$1,015.06 Alt. I at 12% \$80,000 x 0.0105322 x \$842.58 Difference \$10,000 Difference \$172.48

We want to find the annual rate of interest, compounded monthly, that makes the present value of the difference in mortgage payments, or \$172.47, equal to \$10,000, or the incremental amount of loan proceeds received. As previously discussed, one approach is to solve directly for the interest factor. We have:

 Scenario 1 Purchase \$100,000 Loan \$80,000 Term in yrs 25 Rate 12% Monthly Payment \$842.58 =PMT(12%/12,25*12,80000) Scenario 2 Purchase \$100,000 Loan \$90,000 \$10,000 Term in yrs 25 Computed 90% loan rate Rate 13% 13% =(80000/90000*12%)+(10000/90000*20.57%) Monthly Payment -\$1,015.05 =PMT(13%/12,25*12,90000) Difference (What is the true cost of borrowing an additional \$10,0000?) It's not 1%! -\$173.00 =-1015+842 Cost is 20.635% =RATE(25*12,-173,10000)*12

As seen from above, we have computed the monthly interest factor for the \$172.47 annuity to be 57.997. If you're looking at a present value annuity factor table, you will see that this factor is close to the 25-year factor in the 20 percent interest rate table (otherwise excel solves this using the Rate formula.) Hence, if our borrower desires to borrow the additional \$10,000 with the \$90,000 loan, the cost of doing so will be over 20 percent, a rate considerably higher than 13 percent. This cost is referred to as the marginal or incremental cost of borrowing. The 13 percent rate on the \$90,000 loan can be thought of as a weighted average of the 12 percent rate on The \$80,000 loaned the and the 20.57 percent rate on the additional \$10,000. That is,

(80/90 x 12%) + (10/90 x 20.57%) = 12.95% or 13% rounded

The borrower must consider this cost when evaluating the decision as to whether the additional \$10,000 should be borrowed. If the borrowing has sufficient funds so that the \$10,000 would not have to borrowed, it tells the borrower what rate of interest must be earned on funds not invested in a property because of the larger amount borrowed. In other words, by obtaining a larger loan (\$90,000 versus \$80,000), this means that \$10,000 less will be required as a down payment from the borrower than would have been the case had the \$80,000 loan been made. Hence, unless the borrower can earn 20.57 percent interest or more on a \$10,000 investment of equal risk on funds not invested in the property, he would be better off with the smaller loan of \$80,000.

If the borrower does not have sufficient down payment for an \$80,000 loan and needs to borrow \$90,000, the incremental borrowing costs indicates the cost of obtaining the extra \$10,000 by obtaining a larger first mortgage. There may be an alternative way of obtaining the extra \$10,000. For example, if the borrower could obtain a second mortgage for \$10,000 at a rate less than 20.57 percent, this may be a better alternative than a 90 percent loan. Therefore, the marginal cost concept is also an opportunity cost concept in that it tells the borrower the minimum rate of interest that must be earned, or the maximum amount that should be paid, on any additional amounts borrowed.

It should be noted that the 20.57 percent figure we calculated above also represent the return that the lender earns on the additional \$10,000 loaned to the borrower. That is, the cost of a loan to the borrower will reflect the return on the loan to the lender. Of course it should be kept in mind that the figures we are calculating do not take federal income tax consideration into account, which are also important in determining returns and costs. For example, if the borrower is in a higher tax bracket than the lender, then the after-tax cost to the borrower will be less than the after-tax return to the lender.

### Early Mortgage Repayment

We should note that in this example, the incremental cost of borrowing will depend on when the loan is repaid. For example, if the loan is repaid after five years, instead of being held the entire loan term, the incremental borrowing cost increases from 20.57 to 20.83 percent. To see this, we modify the above analysis to consider the fact that if the loan is repaid after five years, the amount that would be repaid on the \$80,000 loan will differ from the amount that would be repaid on the \$90,000 loan. Thus, in addition to considering the difference in payments between the two loans, we must also consider the difference in the loan balances at the time the loan is repaid. We can find the incremental borrowing cost as follows:

Computing the marginal cost, we have:

### Early Repayment

 Loan Amount Difference (\$90,000 - \$80,000) \$10,000 =90000-80000 Rate on 80% Loan 12% Monthly Payment on 80% Loan \$842.58 =PMT(12%/12,25*12,-80000) Rate on 90% Loan 13% Monthly Payment on 90% Loan -\$1,015.05 =PMT(13%/12,25*12,90000) Monthly Payment Difference -\$172.47 =-1015.05+842.58 Payoff 80% loan in 5 yrs - balance due \$76,523 =PV(12%/12,20*12,-842.58) Payoff 90% loan in 5 yrs - balance due \$86,640 =PV(13%/12,20*12,-1015.05) Difference \$10,117 =86640-76523 Marginal cost if paid off in 5 years vs. 25 years: 20.39% =RATE(60,-173,10117,-10000)*12

In this case we cannot simply solve for an interest factor and use the tables to find interest rate. This is because there are two present value factors in the above equation. To find the answer we must find the interest rate that makes the present value of the monthly annuity and lump sum equal to \$10,000. The method for value of the monthly annuity and lump sum equal to \$10,000, The method for doing this has been discussed earlier. The reader should be able to verify that the incremental borrowing cost is now 20.83 percent. Thus, early repayment has increased the incremental cost of borrowing from 20.57 percent to 20.83 percent. As we will see next, the impact of early repayment may be greater when there are also points involved on one or both of the loans.

### Mortgage Origination Fees

It should be apparent that the incremental borrowing cost concept is extremely important when deciding how much should be borrowed to finance a given transaction. In the preceding section, the two alternatives consider were fairly straightforward with the only differences between them being the interest rate and the amount borrowed. As discussed earlier, in most cases financing alternatives under consideration will have different interest rates as the amount borrowed increases and, possibly different maturities. Also, loan origination fees will usually be charged on the loan alternatives. This section considers differences in loan fees on two loan alternatives. Differences in loan maturities are consider later.

The first case we wish to consider is the incremental cost of borrowing when loan origination fees are charged on the two-25 year loan alternatives (listed above). For example, if a \$1,600 origination fee (two points) is charged on the \$80,000 loan and a \$2,700 fee (three points) is charged on the \$90,000 loan, how does this affect the incremental cost of borrowing? These differences can be easily included in the cost computation as follows.

### Differences in Amounts Borrowed and Monthly Mortgage Payments

Same as above but you pay 2 points origination

 Monthly Payment on \$80,000 loan at 12% for 25 years \$842.58 =PMT(12%/12,25*12,80000) Monthly Payment on \$90,000 loan at 13% for 25 years \$1,015.05 =PMT(13%/12,25*12,90000) Difference (What is the true cost of borrowing an additional \$10,0000?) It's not 1%! -\$173.00 =-1015+842 Payoff 80% loan in 5 yrs - balance due \$76,523 =PV(12%/12,20*12,-842.58) Payoff 90% loan in 5 yrs - balance due \$86,640 =PV(13%/12,20*12,-1015.05) Difference \$10,117 =86640-76523 80% loan with 2% origination \$1,600 =80000*0.02 90% loan with 3% origination \$2,700 =90000*0.03 80% true loan \$78,400 =80000-1600 90% true loan \$87,300 =90000-2700 Difference \$8,900 =87300-78400 APR 23.26% =RATE(25*12,-173,,8900)*-12 If paid off at year 5 19.10% =RATE(60,-173,-8900,10117)*-12

As we did previously, we want to find an annual rate of interest, compounded monthly, that makes the present value of the difference in mortgage payments, or \$172.47, equal to \$8,900, or the incremental amount of loan proceeds received. Using excel, we can find that the exact answer is 23.19 percent. Hence, the marginal cost increases to about 23.3 percent when the effects of origination fees are included in the analysis. This results from the fact that \$1,100 in additional fees are charged on the \$90,000 loan. Thus, the borrower only benefits from an additional \$8,900 instead of \$10,000.

As before, the marginal or incremental cost of borrowing increases if the loan is repaid before maturity. For example, if in the above problem, the loan were repaid after five years, the incremental cost would increase to about 24.67 percent.

### Incremental Borrowing Cost Versus a Second Mortgage

The incremental borrowing cost obviously depends on how much the interest rate increases with the loan-to-value ratio. In the example considered previously, the interest rate increased from 12 percent to 13 percent (a difference of 1 percent) when the loan-to-value ratio increased from 80 percent to 90 percent. When no points where charged and the loan was held until maturity, the incremental borrowing cost was 20.57 percent. The incremental borrowing cost would increase if the differential between the rate on the 80 percent loan and the 90 percent loan were greater than 1 percent. Conversely, the incremental borrowing cost would decrease if the differential were less than 1 percent.

Because borrowers have a choice between obtaining a 90 percent loan or obtaining an 80 percent loan plus a second mortgage for the remaining 10 percent, we would expect the incremental borrowing cost to be competitive with the rate of a second mortgage with the same maturity. In the above example, if a second mortgage with a maturity of 25 years can be obtained with an effective borrowing cost that is much less than 20.57 percent, then the 90 percent loan would not be competitive. This would imply that the 1 percent yield differential between 90 percent loan and the 80 percent loan is too great. Lenders have to adjust the differential (or the second mortgage rate) so that the incremental borrowing cost is about the same as the effective cost of a second mortgage.

Figure 9.1 Incremental borrowing cost versus interest rate differential

To illustrate, in Table 9.1 we calculate the incremental borrowing cost for the alternatives discussed earlier, which assume that the loan is prepaid after five years. The exhibit shows how the incremental borrowing cost is affected by the interest rate differential between the rate on the 90 percent loan versus the 80 percent loan. A zero percent interest rate differential means that the contract interest rate is the same for the 90 percent loan as the80 percent loan, which is 12 percent. A 1 percent differential means the contract rate is 1 percent higher, e.g., 13 percent for the 90 percent loan.

When the interest rate differential is zero, the incremental cost is the same as the effective cost of the loan. For example, with no points the incremental cost is exactly 12 percent, the same as the interest rate for the 80 percent loan. As the interest rate differential increases, the incremental borrowing cost increases. The incremental cost increases by about the same rate for each loan.

Suppose that a second mortgage for 10 percent of the purchase price (on top of an 80 percent first mortgage) can be obtained with an effective cost of 20 percent with a 25-year maturity. This is added to Table 9.1 This implies that, to be competitive, the 90 percent loan should be priced such that its incremental cost over an 80 percent loan is 20 percent. For example, suppose lenders expect the loan to be prepaid on average after five years and that they want to charge two points on an 80 percent loan and three points on a 90 percent loan as we have assumed in the previous examples. Referring to Table 9.1, this implies that the interest rate differential should be about .50 percent or 50 basis points. Alternatively, if they do not want to charge any points on either loan, the interest rate differential would have to be about 90 basis points.

### Differences in Mortgage Maturities

In "Incremental borrowing cost versus a second mortgage" the loan alternatives considered had the same maturities (25 years). How does one determine the incremental cost of alternatives that have different maturities as well as different interest rates? Do differences in maturities materially change results? We examine these questions by changing our previous example and assuming that the \$90,000 alternative has a 30-year maturity as well as a higher interest rate. How would the analysis be changed? We first compute the following information:

 Loan Amount Monthly Payments Years 1 - 25 Monthly Payments Years 25 - 30 Alt. III at 13%, 30 years - =PMT(0.13/12,360,-90000,0) = \$995.58 \$90,000 \$995.58 \$995.58 Alt. I at 12%, 25 years - =PMT(0.12/12,12*25,-80000,0) = \$842.58 \$80,000 \$842.58 \$842.58 Difference \$10,000 \$153.00 \$153.00

In this case we compute the monthly payment for a \$90,000, 30 year loan at 13 percent interest, which is \$995.58. However, there are two differences in the series of monthly payments relevant to our example. For the first 25 years, should alternative III be chosen over alternative I, the borrower will pay an additional \$153.00 per month. For the final five-year period, or years 26 through 30, the difference between payments will be the full \$995.58 payment on alternative III because the \$80,000 loan would be repaid in 25 years. Hence the incremental cost must be computed by considering the payment difference as two annuities or grouped cash flows as follows:

 Trial and Error 18.864% "MPVIFA" =((1-(1+(0.18864/12))-300)/(0.18864/12)), "MPVIFA" =((1-(1+(0.18864/12))-60)/(0.18864/12)) , "MPVIF" =1/(1+(0.18864/12))300 "MPVIFA" "MPVIFA" "MPVIF" 63.02 38.66 0.0092854 \$153.00(MPVIFA, ?%, 25 yrs) + \$995.58(MPVIFA, ?%, 5 yrs)(MPVIF, ?%, 25 yrs) = 10,000 9642.06 357.39 \$10,000 =153*63.02 =995.58*(38.66*0.0092854) =9643+357

Because the desired present value is \$10,000, the answer must be slight less than 19 percent. Using Excel and solving for IRR with uneven or grouped cash flows, we can find the solution is 18.86 percent. Hence the marginal or incremental cost of borrowing the additional \$10,000 given the interest rate increases from 12 to 13 percent, but the loan term increases from 25 years to 30 years will be about 18.86 percent. This compares to the incremental cost of 20.57 percent in the first example where there also were no fees charged but both maturities were 25 years. The reason the marginal cost is lower in this case is that although a higher rate must be paid on the \$90,000 loan, it will be repaid over a longer maturity period, 30 years. Even though the borrower pays a higher rate for the \$90,000 loan versus the \$80,000 loan, there is a benefit of a longer amortization period (and thus lower monthly payments) on the \$90,000 loan.

We should point out that if the loan is expected to be repaid before maturity, both the difference in monthly payments and loan balances in the year of repayment must be taken into account when computing the marginal borrowing cost. Also, should any origination fee be charged, the incremental funds disbursed by the lender should be reduced accordingly.

### Mortgage Refinancing [viewable here in Excel]

On occasion, an opportunity may arise for an individual to refinance a mortgage loan at a reduced rate of interest. For example, during 2017 - 2020, interest rates fell sufficiently to encourage many borrowers to refinance their home mortgage.

The fundamental relationship that must be known in any refinancing decision include at least three factors: (1) terms on the present outstanding loan, (2) new loan terms being considered, and (3) any charges associated with paying off the existing loan or acquiring the new loan (such as prepayment penalties on the existing loan or origination and closing fees on the new loan). To illustrate, assume a borrower took out a mortgage loan 5 years ago for \$80,000 at 15 percent interest for 30 years (monthly payment). After 5 years, interest rates fall, and a new mortgage loan is available at 14 percent for 25 years. The loan balance on the existing loan is \$78,976.50. Suppose that the prepayment penalty of 2 percent must be paid on the existing loan, and the lender who is making the loan available also requires an origination fee of \$2,500 plus \$25 for incidental closing costs if the new loan is made. Should the borrower refinance?

In answering this question, we must analyze the costs associated with refinancing and the benefits or savings that all accrue due to the reduction in interest charges, should the borrower choose to refinance.

The costs associated with refinancing are as follows:

 Cost to refinance Prepayment penalty: 2% x \$78,976.50 = \$1,580 Origination fee, new loan \$2,500 Recording, etc, new loan \$25 \$4,105

Benefits from refinancing are obviously the interest savings that results from a lower interest rate. Hence, if refinancing occurs, the monthly mortgage payment under the new terms will be lower than payments under the existing mortgage. Monthly benefits would be \$60.88 per month as shown:

 Paying the loan off in 10 years (currently loan is 25 yrs) Effective cost of Refinancing Adding refinance fees to new loan. Scenario 2 Scenario 2 Scenario 2 Payoff refinance loan in 10 yrs Refinance APR Refinance Fees Existing Loan \$80,000 Rate 15% Term in years 30 Prepayment Penalty =PMT(15%/12,30*12,-80000) = 2% 2% Monthly Payment =PMT(15%/12,30*12,-80000) = \$1,011.56 \$1,011.56 Payoff =PV(15%/12,12*25,-1012.56) = \$79,055 \$79,055 \$72,276 New Loan New Loan \$79,055 \$71,485 \$74,949 \$83,161 Rate 14.00% Term in years 25 Loan Origination Fees =2500/79055 = 3.162% 3.162% Other closing costs \$25 Monthly Payment =PMT(14%/12,25*12,-79055)=\$952 \$952 \$951.63 \$989.02 Monthly Savings by Refinancing \$59.56 \$195.56 \$23 Cost to Refinance Prepayment fee @ 2% (300 months still owed) \$1,581 Loan Origination Fee \$2,500 Other Closing Costs \$25 Total Refinance Costs (Paid out of pocket) \$4,106 Return on monthly saving over refinance cost (if greater then existing loan rate, refinance) 17.16% 14.86% 14.62% =RATE(25*12,59.56,-4106)*12 = 17.16% =RATE(12*25,952,-74949)*12 - 14.86% =RATE(25*12,989.02,-79055)*12 = 14.62% This calculation must be greater then 15% to refinance Lower then 15% is a cost savings, make sense to refinance Lower then 15% is a cost savings, make sense to refinance

One way to approach this problem is to ask whether it is worth "investing" or paying out, \$4,105 (cost to refinance) to save \$60.87 per month over the term of the loan. Perhaps the \$4,105 could be invested in a more profitable alternative? To analyze this question, we should determine what rate of return is earned on the investment of \$4,105 for 25 years, given that \$60.57 per month represents a savings. We find that the yield on the \$4,105 investment, with savings of \$60.87 per month over 25 years, would be equivalent to earning 17.57 percent per year. If another alternative equal in risk cannot be found which provides a 17.57 percent annual return, the refinancing should be undertaken. This return appears to be attractive because it is higher than the market rate of 14 percent that must be paid on the new loan. Thus, refinancing is probably desirable.

### Early Loan Repayment - Loan Refinancing

If the property is not held for the full 25 years the monthly savings of \$60.87 do not occur for the entire 25-year term, and therefore the refinancing is not as attractive. To demonstrate, if we assume the borrower plans to hold the property for only 10 more years after refinancing, is refinancing still worthwhile? To analyze this alternative, note that the \$4,105 cost will not change should the refinancing be undertaken; however, the benefits (savings) will change. The \$60.87 monthly benefits will be realized for only 10 years. In addition, since the refinanced loan is expected to be repaid after 10 years, there will be a difference between loan balances on the existing loan and the new loan, due to different amortization rates.

 Loan Amount Loan balance, 15th year - existing loan* \$72,275 Based on \$80,000, 15 percent, 30 years, prepaid after 15 years. Loan balance, 10th year - new loan \$71,386 Based on \$78,976, 14 percent, 25 years, prepaid after 10 years. Difference \$889

The new calculation comparing loan balances under the existing loan and under the new loan terms shows that if refinancing occurs, the amount saved because of a lower loan balance is \$889, should the new loan be made. Hence, total savings in the event of refinancing would be \$60.87 per month for 10 years, plus \$889 at the end of 10 years. Do these savings justify an outlay of \$4,105 in refinancing costs? To answer this question, we compute the return on the \$4,105 outlay as follows:

 \$60.87(MPVIFA, ?%, 10 yrs) + \$889(MPVIF, ?%, 10 yrs) = \$4,105 Trial and Error 14.210% "MPVIFA" =((1-(1+(0.1421/12))120)/(0.1421/12)), "MPVIF" =1/(1+(0.1421/12))120 "MPVIFA" "MPVIF" 63.88 0.2434966 \$60.87(MPVIFA, ?%, 10 yrs) + \$889(MPVIF, ?%, 10 yrs) = \$4,105 3888.38 216.47 \$4,105 =60.87*63.88 =889*0.2434966 =3888.83+216.47

Figure 9.3 IRR from Savings when Refinancing

Because the loan is repaid early and the monthly savings of \$60.87 will not be received over the full 25-year period, the yield must be below the 17.57 percent yield computed in the refinancing example. The yield earned due to refinancing in this case will be 14.21 percent per year for the 10-year period.

Obviously, this return is lower than the 17.57 percent computed by assuming the loan was repaid after 25 years. This is true because the refinancing cost of \$4,105 remained the same, while the savings stream of \$60.87 was shortened from 25 years to 10 years. Although an additional \$889 was saved because of differences in loan balances, it did not offset the reduction in monthly savings that will have occur from the 10th through the 25th year. The relationship between the IRR and the number of years the loan is held after refinancing is illustrated in Table 9.3. Note that the returns from refinancing are negative if the loan is held for only five years after prepayment. The return rises sharply for each additional year the loan is held after prepayment until it is held for about 15 add itional years. In analyzing refinancing decisions, then, not only must costs and benefits (savings) be compared, but the time period one expects to hold property also enter the decision.

### Effective Cost of Refinancing

The refinancing problem can also be analyzed by using an extension of the effective concept discussed previously. From this previous discussion, we know that points increase the effective cost of a loan. In our problem, the borrower would be making a new loan for \$78,976.50, but must pay \$4,105 in "fees" to do so. Although these fees include the prepayment penalty on the old loan, this can be thought of as a cost of making a new loan by refinancing. Thus, the borrower in effect receives \$78,976.50 less \$4,105 or \$74,871.50. Payments on the new loan when made at 14 percent for 25 years would be \$950.69. To find the effective cost for the case where the loan is held to maturity, or 25 years, we proceed as follows:

 Trial and Error \$950.69(MPVIFA, ?%, 25 yrs) = \$74,871.50 14.8544% "MPVIFA" MPVIFA =((1-(1+(14.8558%/12))300)/(14.8558%/12)) 78.7617 =950.69*78.7617 74877.96 =RATE(12*25,950.61,-74871.5)*12 = 14.8544% 14.8544%

We obtain an interest rate of 14.86%. This can be interpreted as the effective cost of obtaining the new loan by refinancing. Since this cost is less than the rate on the old loan (15%), refinancing would seem to be desirable. Thus, we arrive at the same conclusion as we did when we calculated the return on investing in refinancing.

### Borrowing the Refinancing Costs

In "Effective Cost of Refinancing" we assumed that the borrower had to pay (as a cash outlay) the refinancing cost of \$4,105. But it is likely that if the borrower were going to go to the trouble of refinancing, he may also be able to borrow the refinancing costs. How does this affect our analysis?

The borrower now gets a loan for the loan balance of \$78,976.50 plus the fees of \$4,105.00 for a total of \$83,081.50. Payments at the 14 percent rate (assuming the interest rate is still the same) would be \$1,100.10. What do we compare this to now that the borrower has no cash outlay when refinancing? The answer is simple. These payments are still less than those on the old loan (\$1,011.56). Given that the borrower has lower payments (\$11.46) for 300 months without any cash outlay, it is desirable to refinance.

We could, of course, also compute the effective cost of refinancing as we did in the previous section. In this case, the total amount of the loan is \$83, 081.50; however, the borrower only benefits from \$78,976.50 (the loan amount less the refinancing costs). Using the payment of \$1,000.10 and assuming the new loan is held for the full loan term, we can calculate the effective cost as follows:

 Trial and Error \$1000.01(MPVIFA, ?%, 25 yrs) = \$78976.50 14.81% "MPVIFA" MPVIFA =((1-(1+(14.81%/12))-300)/(14.81%/12)) 78.98 =1000.1*78.97 78977.90 =RATE(12*25,1000.1,-78976.5)*12 = 14.81% 14.81%

Solving for the effective interest rate, we obtain an answer of 14.81%, which is virtually the same as we obtained in the previous section. In fact, the only reason the answer is slightly lower is that the origination fee on the new loan was assumed to remain at \$2,500 even though the amount of the loan was increased to cover the refinancing costs.

Note that whether we calculate a return on investing in refinancing, or we calculate the effective cost of refinancing, we arrive at the same conclusion. Often there are many ways of considering a problem that lead to similar conclusions. It is informative to look at problem several ways to gain skill in handling the wide variety of financial alternatives one may encounter. Knowing alternative ways of analyzing a problem also reduces the chance of applying an incorrect solution technique.

### Mortgage Assumption: The Effective Cost of Two or More Loans [viewable here in Excel]

There are many situations where the buyer of a home may be considering a combination of two or more loans (e.g., a first and second mortgage to finance the home). One situation where this could arise is when a loan is being assumed because a favorable rate of interest exists on the first mortgage. However, the amount of cash necessary for the buyer to assume a mortgage may be prohibitive. This can occur because the seller has already paid down the balance of the loan, and because the home has appreciated in value since it was originally financed by the seller. Thus, the buyer must use a second mortgage to bridge the gap between the amount available from the loan assumption and the desired total loan amount.

Suppose an individual bought a \$100,000 property and made a mortgage loan 5 years ago for \$80,000 at 10 percent interest for a term of 25 years. Due to price appreciation the market value of the property has risen in value over the past five years to \$115,000. The amount of cash equity required by the buyer to assume the sellers loan would be \$39,669, determined as follows:

### Mortgage Assumption - The Effective Cost of Two or More Loans.

 Existing property owner bought home 5 yrs ago \$100,000 Interest Rate 10% Loan amount \$80,000 Loan term in years 25 Monthly Payments \$727 =PMT(10%/12,25*12,-80000) Current Loan Balance after 5 years (20 yrs remaining) \$75,335 =PV(10%/12,20*12,-727) Value of property \$115,000 =100000*1.15 What cost is lower, a new 80% first mortgage with 20% second or assume the existing loan and include a second? Assume Purchase price \$115,000 Assume sellers mortgage balance \$75,333 Buyers cash equity or second mortgage? \$39,667 =115000-75333 Buyer has only this much cash -\$23,000 =(115000*0.8)-115000 Buyer selects 1st @80% LTV \$92,000 =115000*0.8 Term in years 20 Interest rate 12% Monthly Payments - 1st mortgage \$1,013 =PMT(12%/12,20*12,-92000) Combine existing seller loan with 2nd mortgage with 20% down payment. Buyer needs a second mortgage to close the deal. \$62,667 =115000-(75333-23000) Second term in years 20 Second mortgage interest rate 14% Monthly Payment - Second Mortgage \$779 =PMT(14%/12,20*12,-62667) Monthly Payment - Assumed 1st mortgage (20 years remaining) \$727 Monthly payment 1st & 2nd \$1,506 =779+727 What is the combined interest rate of 1st & 2nd loans (weighted cost of capital)? 19.21% =RATE(20*12,1506,-92000)*12

If the buyer does not have \$39,669 in cash, even though he desires a loan assumption, he may be unable to complete the transaction. One alternative available to the buyer who could not make the large cash outlay in the above example may be to secure a second mortgage. However, using a second mortgage will be justified in the case only if the terms of the second mortgage, when combined with the terms of the assumed mortgage, will make the borrower as well or better off if the entire purchase had been financed with a new \$92,000 loan (80 percent of value) at 12 percent for 20 years, so we must know how to combine a second mortgage with the assumed first mortgage to determine whether or not the assumption would be as attractive as the new mortgage loan. Suppose a second mortgage for \$16,669 (\$92,000 - \$75,331) could be obtained at a 14 percent rate for a 20-year term. To analyze this problem, we compute the combined mortgage payments on the assumed loan and a second mortgage loan made for 20 years at 14 percent.

The combined monthly payments equal \$934.24. We now want to compute the effective cost of the combined payments that are made on the combined loan of \$92,000. We find an answer of 10.75 percent. This is the cost of obtaining \$92,000 with the loan assumption and second mortgage. Since this is less than the cost of obtaining \$92,000 with a new first mortgage at a rate of 12 percent, the borrower is still better off with the loan assumption and a second mortgage. It is important to note, however, that the above analysis does not consider the fact the seller of the home may have raised the price of the home to capture the benefit of his below market rate loan which could be assumed.

### Second Mortgages with a Shorter Maturity

What if the second mortgage is for only 5 years? Does it make sense to do the assumption? In most cases second mortgages may not be available for a 20-year period. If a five-year term were available on a second mortgage loan at 14 percent interest, would the borrow still be better off by assuming the existing mortgage and making a second mortgage? To answer this question, we must determine the combined interest cost on the assumed mortgage, which carries a rate of 10 percent for 20 remaining years, and the second mortgage, which would carry a rate of 14 percent for 5 years. This combined rate can then be compared to the current 12 percent rate for 20 years presently available, should the property be financed with an entirely new mortgage loan.

 To combine terms on the assumable mortgage and second mortgage, we add monthly payments together as follows: Monthly payments Assumed loan -- Based on original terms: \$80,000, 10 percent, 25 years \$726.96 Second loan -- Based on \$16,669, 14 percent, 5 years \$387.86 Total \$1,114.82

The sum of the two monthly payments is equal to \$1,114.82. However, the combined \$1,114.82 monthly payments will be made for only five years. After five years, the second mortgage will be completely repaid, and only the \$726.96 payments on the assumed loan will be made through the 20th year.

Whether or not the combined mortgages should be used by the borrower can now be determined by again solving for the combined cost of borrowing. This cost is based on the monthly payments under both the assumed loan and second mortgage, for the respective number of months payments must be made, in relation to the \$92,000 amount being financed. This can easily be seen to be the monthly payments of \$387.86 on the second mortgage for 5 years and the \$726.96 payments on the assumed mortgage for 20 years, both discounted by an interest rate that results in the present value of \$92,000.

 Trial and Error 10.2879% "MPVIFA" "MPVIFA" "MPVIFA" =((1-(1+(0.102879/12))-60)/(0.102879/12)), "MPVIFA" =((1-(1+(10.2879%/12))-240)/(10.2879%/12)) 46.75 101.61 \$387.86(MPVIFA, ?%, 5 yrs) + \$726.97(MPVIFA, ?%, 20 yrs) = \$92,000 \$18,132.46 \$73,867.42 \$91,999.88 =387.86*46.75 =726.97*101.61 =18132.46+73867.42

We must find the interest rate that makes the present value of the combined monthly mortgage payments (grouped cash flows) equal to \$92,000. Using a financial calculator or Excel, we find that the combined interest cost on the existing mortgage if assumed for 20 years and the second mortgage made for 5 years is 10.29 percent. This combined package of financing must again be compared to the 12 percent interest rate currently available on an \$80,000 mortgage for 20 years. Because the effective cost of the two combined loans is less than the market rate, this is the best alternative. It should be noted, however, that for the first 5 years the combined monthly payments of \$1,114.82, should the assumption and second mortgage combination be made, would be higher than payments that would be made with a new mortgage for \$92,000 at 12 percent for 20 years, which would be \$1,013.00 per month. Although this is offset by the lower \$726.96 payments after 5 years, the borrower must decide which pattern of monthly loan payments fits his income pattern, in addition to simply choosing the loan alternative with the lower effective borrowing cost. A borrower can be willing to pay a higher effective cost for a loan (or combination of loans) that has lower monthly payments.

### Effects of Below Market Financing on House Prices [viewable here in Excel]

There are many situations where a home buyer may have an opportunity to purchase a home and obtain financing at a below market interest rate. One case which we have already discussed occurs when the seller of the house has a below market rate loan that can be assumed by the buyer. Below market financing might also be provided by the seller of the home with a purchase money mortgage. In this case, the seller provides some or all of the financing to the buyer at an interest rate lower than currently available in the market. Indeed this type of financing is quite common during periods of tight credit and high interest rates.

It should be obvious to the reader that below market rate loans have value to the buyer. However, because the informed seller of the home also recognizes the value of such financing, we would expect the seller to increase the price of the house to reflect this value. That is, the "price" of the house would be higher when accompanied with below market financing than it would be with market rate financing.

We now consider how a buyer would analyze whether to purchase a house with below market financing if the house price is higher than that of an otherwise comparable home that does not have below market financing. To illustrate, suppose a home could be purchased for \$105,000 subject to an assumable loan at 9 percent interest rate with a 15-year remaining term, a balance of \$70,000, and payments of \$709.99 per month. A comparable home without any special financing cost \$100,000, and a loan for \$70,000 could be obtained at a market rate of 11 percent with a 15-year term. Which alternative is best for the buyer? Note that we are assuming that the two loan amounts are the same. In analyzing this problem, we must consider whether it is desirable for the buyer to pay an additional \$5,000 in cash for the home (additional equity invested) in order to receive the benefit of lower payments on the below market loan. The calculations are as follows:

### Below Market Financing on Prices: Assuming a Mortgage Loan With Below Market Interest Rate

Will the higher house price be reflected in asking price? Let seeâ€¦

 Purchase price \$105,000 Assumed interest rate 9% Term (years remaining on loan) 15 Loan balance \$70,000 Loan payments \$710 Comparable home asking price \$100,000 Loan amount \$70,000 Interest rate (market rate) 11% Term in years 15 Down Payment Payment Scenario 2 Balance is \$50 k Borrower needs Second loan for \$20 k, 15 years at 14% Market rate loan \$30,000 \$796 \$796 Loan assumption \$35,000 \$710 \$773 Difference \$5,000 \$87 \$24 Paying the extra \$5,000 is the equivalent of earning 19.41% vs. the 11% market rate loan or a net 8.41% gain. 19.78% -1.89% Go with tradition loan At what price should the property be if loan is to be assumed and priced in parity with the market interest rate? \$7,533 \$107,535 \$101,990 Less then \$105k, go w/traditional financing Payment at higher value (verification/check) -\$1,222 Interest rate (market rate) (verification/check) 11%

We find that making the additional \$5,000 down payment would result in earning the equivalent of 19.41 percent because of the lower loan payments. Alternatively, should the buyer decide not to pay the additional \$5,000, he would have to find a return of 19.41 percent on the \$5,000 in an investment comparable in risk. Because the 19.41 percent rate is higher than the 11 percent market rate, it appears desirable.

### Wraparound Mortgage Loan [viewable here in Excel]

Wraparound loans are used to obtain additional financing on a property while keeping an existing loan in place. The wraparound lender makes a loan for a face amount that is equal to the existing loan balance plus the amount of additional financing. The wraparound lender agrees to make the payments on the existing loan as long as the borrower makes payments on the wraparound loan. Rather than make payments on the original loan plus payments on a second mortgage, the borrower makes a payment only on the wraparound loan.

To illustrate, suppose a homeowner named Smith has an existing loan with a balance of \$90,000 and monthly payments of \$860.09. The interest rate on the loan is 8 percent and the remaining loan term is 15 years. From the time Smith originally obtained this loan, the home has risen in value such that it is now worth \$150,000. Smith's current loan balance is 60 percent of the current value of the property. He would like to borrow an additional \$30,000, which would increase his debt to \$120,000 or 80 percent of the property value.

Assume that the current effective interest rate on a first mortgage with an 80 loan-to-value ratio is 11 1/2 percent with a term of 15 years, and the current effective interest rate on a second mortgage for an additional 20 percent of value (\$30,000) would be 15 1/2 percent for 15 years.

A lender, different from the holder of Smith's existing loan, is willing to make a wraparound loan for \$120,000 at a 10 percent rate for 15-year term. Payments on this loan would be \$1,289.53 per month. If Smith makes this loan, the wraparound lender will take over the payments on Smith's current loan. That is, Smith will pay \$1,289.53 to the wraparound lender, and the wraparound lender will make the \$860.09 payment on the original loan. Thus, Smith's payment would increase by \$1,289.53 less \$860.09 or \$429.44 per month as compared to what he would make before obtaining the wraparound loan. Because the wraparound lender is taking over the payments on the old loan, Smith will actually only receive \$30,000 in cash (the \$120,000 amount for the wraparound loan less the \$90,000 balance of Smith's current loan).

Is the wraparound loan a desirable alternative for Smith to obtain an additional \$30,000? The rate on the \$120,000 wraparound loan (10 percent) is less than the market rate (11 1/2 percent) on a new first mortgage for the entire \$120,000. Thus, the wraparound loan would be more desirable than refinancing with a new first mortgage. Why would the wraparound lender make a loan that has a lower rate than a new first mortgage? The answer is that the wraparound lender is primarily concerned with earning a competitive rate of return on the incremental funds loaned, i.e., the additional \$30,000. It is the effective cost of the incremental funds loaned that the borrower also should be concerned about.

What is the cost of the incremental \$30,000? This is analogous to determining the incremental borrowing cost of a loan as discussed at the beginning of this section. That is, we want to know the incremental cost of the 80 percent wraparound loan versus the 60 percent existing loan. In order to get the additional \$30,000 on the wraparound loan, the borrower must pay a 10 percent interest rate on the entire \$120,000, not just the additional \$30,000. Because the rate on the existing \$90,000 is only 8 percent, the incremental cost of the additional \$30,000 is greater than 10 percent. The question is whether the incremental cost is more or less than the 15.5 percent rate for a second mortgage for \$30,000.

The incremental borrowing cost of the wraparound loan can be determined by finding the interest rate that equates the present value of the additional payment with the additional funds received. We have:

### Wraparound Loan

 Wraparound Loans Existing loan \$90,000 monthly payment \$860 rate 8% term 15 Value \$150,000 LTV 60% Current Market Rate, first loan, 15 years, 80% LTV 11.50% \$120,000 \$1,401.83 =PMT(0.115/12,15*12,-120000) Current Market Rate, second loan, 15 years, 100% LTV 15.50% \$30,000 \$430.20 =PMT(0.155/12,15*12,-30000) Lender offers borrower a "Wraparound" loan with the following terms: Loan \$120,000 Rate 10% Term 15 Monthly payment =PMT(0.1/12,15*12,-120000)=\$1,289.53 \$1,289.53 What is monthly mortgage payment on \$30k? =1289.53-860=\$430 \$430 \$30,000 Rate paid on (incremental cost) on "cash out" (i.e., second mortgage) 15.49% Does this "Wraparound" loan make sense (borrower is netting \$30K) or should lender just get a second loan? Yes Lower payment and lower interest rate on second \$112 0.01% =1401.83-1289.53 =0.155-0.1549

We find that the interest rate is 15.46 percent or about 15.5 percent. This is the same as the rate for a second mortgage, which is what we would expect. The wraparound lender can charge a lower rate on the wraparound loan and still earn a competitive rate on the incremental funds loaned because the existing loan is at a below market rate. The wraparound rate of 10 percent is, in effect, a weighted average of the rate on the existing loan (8 percent) and the rate on a second mortgage (15.5 percent). If the existing loan were at the market rate for a 60 percent loan, then the wraparound rate would have to be equal to the rate on an 80 percent loan so that the wraparound lender would earn a rate of return on the incremental funds equal to a second mortgage rate.

Is there any reason why the wraparound lender should be willing to make the loan at a rate that is more attractive than a second mortgage? The wraparound loan is, in effect, a second mortgage because the original loan is still intact. Furthermore, the loan-to-value ratio is increased by the same amount with the wraparound loan as it would be with a second mortgage. There is one advantage with the wraparound loan, however, due to the fact that the wraparound lender makes the payments on the first mortgage loan. Hence, control is retained over default in its payment, whereas if a second mortgage was made, the second mortgage lender would not necessarily be aware of a default on the first mortgage loan and may not be included in foreclosure action resulting from such a default. In a typical wraparound mortgage agreement, the wrap lender obligates itself to make payments on the original mortgage only to the extent that payments are received from the borrower, and the borrower agrees to comply with all of the covenants in the original mortgage except for payment. Any default by the borrower will be realized by the wraparound lender who not want to see the property go into foreclosure. The wrap lender may make advances on the first mortgage and add them to the balance on the wrap loan, foreclose on its mortgage, or negotiate for the title to the property in lieu of foreclosure, while still making payments on the first lien. Thus, the wraparound lender may be willing to earn an incremental return that is slightly lower than a second mortgage rate.

It should be noted that the original mortgage may contain a prohibition against further encumbrances or a due-on-sale clause that may preclude use of a wraparound loan to access equity in, or finance the sale of, property. In the absence of such restrictions, the original lender may also be willing to work out a deal with Smith that would be attractive to both of them. For example, this lender might offer Smith a new first mortgage at the same 10 percent rate as the wraparound loan (rather than the 11.5 percent market rate on a first mortgage) if Smith agrees to borrow the additional \$30,000 from the bank. Again, because the 10 percent rate applies to the entire \$90,000 (not just the additional \$30,000) the original lender can earn an incremental return of 15.5 percent on the incremental funds advanced. Thus, the existing lender can earn a competitive rate of return on the new funds and keep the existing borrower as a customer. The lender still earns 8 percent on the existing loan, but this would also be true if the borrower gets a second mortgage or a wraparound loan from a different lender. Thus, they may be willing to, in effect, offer the same deal as a wraparound lender by charging a rate on a new first mortgage that is equal to the wraparound rate of 10 percent.

### Buydown Mortgage Loan (seller pays upfront "discount points" to lower the buyers monthly mortgage payments)

With a buydown loan, the seller of the seller of the home (frequently a builder) pays an amount to a lender to buy down or lower the interest rate on the loan for the borrower for a specified period of time. This may be done in periods of high interest rates in an attempt to help borrowers qualify for financing. For example, suppose interest are currently 15 percent and a purchaser of a builder's home only has enough income to qualify for financing based on payments that would result from a loan at a 13 percent fixed rate. Let's assume that the loan will be for \$75,000 with monthly amortization based on a 30-year term. Payments based on the market rate of 15 percent would be \$948.33 per month. Payments at a 13 percent rate would only be \$829.65 per month. The buyer's income would qualify him at \$829.64 but not at \$948.33. Suppose the builder wanted to buy down the interest rate from 15 to 13 percent, thereby enabling the bank to make the loan such that payments are only \$829.65 per month for the first five years of the loan term, increasing to 948.33 for the remaining loan term. To accomplish this the builder would have to make up the difference in payments (\$118.68 per month for the five-year period). If this difference were paid by the builder to the lender at the time the loan closed, the amount paid would be the present value of the difference in payments, discounted at the market rate of 15 percent. Thus we have:

 Buydown Loan amount \$75,000 Term 30 Rate 15% Monthly payment \$948.33 =PMT(0.15/12,30*12,-75000) Borrower can only afford \$75,000, 30 year loan at 13% and monthly payments ofâ€¦ \$829.65 =PMT(0.13/12,30*12,-75000) Difference \$118.68 =948.33-829.65 Seller will "buydown" the loan for 5 years, what is the cost to seller/builder? \$4,989 =PV(0.15/12,5*12,-118.68)

In short, the property price should reflect this cost. Buyer should shop loans for rate at or below 13%.

The builder would therefore pay \$4,988.67 to the lender to buy down the loan. When coupled with the payments received from the buyer, the lender would earn a market rate of 15 percent and be willing to qualify the buyer.

As indicated above, one advantage of the buydown is that it may allow borrowers to qualify for the loan whose current income might not otherwise allow them to meet the lender's payment-to-income criteria. However, the reader should also realize, based on our discussions of cash equivalent value, that the builder will probably have added the buydown amount to the price of the home. Thus, the borrower might be better off bargaining for a lower price on the home and obtaining his or her own loan at the market rate. We would expect that the same home or similar one could be obtained for \$4,988.67 less without a buydown. The borrow is, in effect, paying \$4,988.67 in "points" to lower the interest rate to 13 percent to 15 percent.

It should also be noted that many buydowns are executed with graduated payments for three or five years.

That is, they may be initiated with monthly payments of \$826.65 and step up each year by a specified amount until \$948.33 is reached in the fifth year.

Some buydown programs are also used I conjunction with ARM's, where the initial rate of interest will be bought down. Because initial rates on ARMs are typically lower than those on FRMs, this results in even lower initial payments, thereby allowing more buyers to qualify. However, this latter buydown practice has been discouraged because payments may increase considerably, particularly if there is an increase in the market rate of interest. In these cases, payments would rise because of higher market rates and because future payments have not been bought down.

### Convertible Mortgage [viewable here in Excel]

A convertible mortgage gives the lender an option to purchase a full or a partial interest in the property, at the end of some specified period of time. The purchase option allows the lender to convert its mortgage to equity ownership, hence the term convertible mortgage. This approach also may be viewed by the lender as a combination of a mortgage loan and purchase of a call option, or right to acquire a full or partial equity interest for a predetermined price of the option's expiration date.

To illustrate, we assume that the property evaluated in the previous examples will be financed with a \$700,000 (70 percent of value) convertible mortgage that allows the lender to acquire 65 percent of the equity ownership in the property at the end of the fifth year. The loan will be amortized over 30 years with monthly payments. The interest rate on the loan is assumed to be 8.5 percent versus 10 percent for the conventional loan. The lender is willing to accept the lower interest rate in exchange for the conversion option. The 150 basis points difference in interest rates between the conventional mortgage and the convertible mortgage represents the "price" that the lender must pay for the call option.

Table 12.16 illustrates the after-tax cash flows for the investor under the assumption that the property is financed with the convertible mortgage described above and that the lender exercises the option to purchase a 65 percent interest in the property at the end of the fifth year. We would expect the lender to exercise this option because 65 percent of the estimated sale price (\$753,528) is greater than the mortgage balance at the end of the fifth year (\$668,432). That is, the option is "in the money" at the time it can be exercised. For comparison with the previous examples, we also assume that the investor will sell the remaining 35 percent interest in the property.

### Lender's Yield on Convertible Mortgage

The lender's yield on a convertible mortgage depends on the interest rate charged on the mortgage as well as any gain on conversion of the mortgage into an equity position. If the mortgage is not converted, the lender's yield will equal the interest rate on the loan. This is the lower limit of the yield, assuming that the borrower does not default on the mortgage. In the above example the lender's yield on the convertible mortgage is greater than the 8.5 percent interest rate on the loan because of the gain on conversion of the mortgage balance into an equity position. This gain occurred because the conversion option included with the mortgage was assumed to be "in the money" on its exercise date. Thus, the mortgage lender receives mortgage payments of \$5,382.39 per month, plus a 65 percent interest in the property worth \$753,528 at the end of the fifth year. The lender's effective yield is calculated as follows:

 Loan \$700,000 Market rate interest rate 10% Convertible mortgage loan rate 8.50% Loan to value 70% Term in years 30 Conversion as % of sales price at end of year 5 65% Mortgage payment (monthly) \$5,382 =PMT(0.085/12,30*12,-700000) Mortgage balance end of year 5 \$668,383 =PV(0.085/12,300,-5382) Estimated value of 65% stake at end of yr. 5 \$753,528 Lender's yield 10.39% =RATE(60,64589/12,-53528,700000)

We obtain a yield of 10.40 percent. This is the lender's before-tax rate of return on the convertible mortgage. It can also be interpreted as the borrower's effective borrowing cost for the convertible mortgage (before tax).

### Valuation of a Leased Fee Estate

In the office building valuation example above (pg 449) it was assumed that property was not encumbered by any existing leases and that it could be leased at market rate. Thus, the value would be considered an estimate of the value of a "fee simple estate." Properties are often purchased subject to existing leases, and the terms in these leases must be honored by the investor. In these cases, "leased fee" estates are acquired by investors. The existing leases may have above- or below-market rental rates that affect the value of the leased fee estate.

To illustrate, assume that instead of earning \$500,000 per year, the office building evaluated in the previous example has an existing net lease with a remaining term of five years at a flat rate of only \$400,000 per year. How does this affect the value? The value will clearly be lower because the rents will be less for the first five years. The value at the end of the five-year lease term should not be affected, however, because the property may be leased at the market rate at that time. That is, the income for the remaining economic life of the property should be the same as if it had not been encumbered by the below market lease. Thus, the resale price would be \$5,796,370 as for the fee simple value estimate made earlier. We can now estimate the value of the leased fee estate as the present value of \$400,000 per year NOI plus the present value of the estimated value at the end of the lease of \$5,796,370.

Before we calculate the present value, one additional question must be considered. Should the discount rate still be 13 percent? The answer depends on whether the leased fee estate is considered more or less risky than a fee simple estate. This depends in part on the creditworthiness of the tenant. The lease represents a contract between the lessor and the leasee. The riskiness of the NOI collected during the term of the lease is dependent on the riskiness of this financial contract. On course, if the tenant defaults, the owner has the right to attempt to lease the property to someone else at the market rate. The point is that by leasing the property, some of the risk of owning the property and collecting current market rents is exchanged for the risk of collecting on the lease contract. Because the existing lease is at a below-market rate, it is likely that there will be default on the lease, especially if the property could be subleased. Furthermore, it is more likely that the owner could lease the property at a rate higher than its current lease rate rather than lower. Thus, a discount rate slightly lower than 13 percent might be justified. We will assume that a 12.5 percent discount rate is appropriate. The value of the leased fee estate would therefore be as follows:

### Leasehold Estate with Below Market Lease Rates

 Leasehold Estate with below market lease rates Building NOI for five years with leases at market rate (fee simple estate) \$500,000 Property value end of year 5 \$5,000,000 Term 5 Another building with same loan term and interest rate NOI \$400,000 Term 5 What's the value with below market lease rates? \$4,640,801 =PV(0.125,5,-400000,-5796370) \$359,199 =5000000-4640801

In this case, the difference between the value of the fee simple estate (\$5,000,000) and the value of the leased fee estate (\$4,640,801) should reflect the value of the "leasehold estate," which is \$359,199. That is, the below market lease results in the value of the fee simple estate being divided among the lease-hold and leased fee interests.

### Selling and Renovation of Income Properties

An investor purchases a real estate investment based on the benefits expected to be received over an anticipated holding period. That is, the investor computes the various measures of investment performance based on expectation at the time the property is purchased. After the property is purchased, however, many things can change that affect the actual performance of the property. These same things may affect the investor's decision as to whether the property continues to meet his investment objectives. For example, market rents may not be increasing as fast as expected, thus reducing the investor's cash flow. Tax laws may have changed, as they did in 2016, thus changing the benefit for some investors more than others. The point is that a periodic evaluation should be made to determine whether properties should be sold.

Even if the investor's projections for a property are accurate, there may be factors that will influence the investor to sell after a specified number of years.

One factor in particular relates to the potential benefits associated with leverage that we have discussed in other sections. Assuming that the mortgage on the property has positive amortization, the outstanding mortgage balance decreases each year and the investor's equity position increases. Although this "equity buildup" may appear desirable in the sense that the investor will get more cash from the property when it is sold, it also means that each year the investor has more funds "tied up" in the property. Any increase in the value of the property over time, whether anticipated or not, will also contribute to an increase in the investor's equity buildup.

Equity buildup represents funds that the investor could place in another investment if the current property were sold. This is the opportunity cost of not selling the property. The proceeds that the investor could have received if the property were sold can be thought of as the amount of equity investment made to keep the property for an additional period of time. But unless the property is refinanced, a greater portion of equity capital remains invested in relation to the cash flow being received from continuing to operate the property. Further, while the total mortgage payment (debt service) remains the same, the interest portion of the payment decreases each year, resulting in lower tax deductions. Hence, the investor is also losing the benefits of financial leverage each year.

### A Decision Rule for Property Disposition

The following is a discussion of the factors that should be considered by investors to determine whether a property should be sold or whether ownership should be retained. It is based on an incremental, or marginal, return criteria that should be utilized by investors when faced with such decision making.

To illustrate the criteria that should be applied when making a decision to keep a property or to sell it, we assume that an investor acquired a very small retail property five years ago at a cost of \$200,000. The Apex Center was 15 years old at the time of purchase and was financed with a 75 percent mortgage made at 11 percent interest for 25 years. Depreciation is being taken on a straight-line basis with 80 percent of the original cost (\$160,000) allocated to the building and 20 percent allocated to land. We assume that the property was purchased prior to the 2016 tax reform act. Thus, straight-line depreciation is used over a depreciable life of 31.9 years. The reader should also note that the marginal tax of 25 percent is being used. This was the maximum rate in effect from 2016-2020. Results during the past five years of operation are shown in Table 14.1.

 Table 14.1 Past Operating Results, Apex Center Year 1 2 3 4 5 A. Before-tax cash flow: 39,000 40,560 42,182 43,870 45,624 Rents 19,500 20,280 21,091 21,935 22,812 Less operating expenses 19,500 20,280 21,091 21,935 22,812 Net operating income (NOI) 17,642 17,642 17,642 17,642 17,642 Before-tax cash flow 1,858 2,638 3,449 4,293 5,170 B. Taxable income or loss: Net operating income (NOI) 19,500 20,280 21,091 21,935 22,812 Less interest 16,441 16,302 16,146 15,973 15,780 Depreciation 8,421 8,421 8,421 8,421 8,421 Taxable income (loss) (5,362) (4,443) (3,476) (2,459) (1,389) Tax (2,681) (2,222) (1,738) (1,230) (694) C. After-tax cash flow: Before-tax cash flow (BTCF) 1,858 2,638 3,449 4,293 5,170 Less tax (2,681) (2,222) (1,738) (1,230) (694) After-tax cash flow (ATCF) 4,539 4,860 5,187 5,522 5,865

If Apex were sold today, it is estimated that the property could be sold for \$250,000. Selling costs equal to 6 percent of the sale price would have to be paid. The cash flows from sale of the property (if sold today) are shown in Table 14.2. Note that it is assumed that the capital gain from sale of the property is taxed as ordinary income in accordance to today's tax act. This is because the property is assumed to have been sold after the tax act was passed. These assumptions are made in this illustra tion for two reasons. First, it represents the typical situation facing investors at the time of this writing. Second, the reader should understand the implications of purchasing a property under one tax law, then facing a decision to sell it under a different tax law.

 Table 14.2 Estimates of Cash Flows from Sale Today Sale Price 250,000 Less sale costs (at 6%) 15,000 Less mortgage balance 142,432 Before-tax cash flow (BCTF) 92,568 Taxes in year of sale: Sale price 250,000 Less selling expenses 15,000 Original cost basis 200,000 Less accumulated depreciation 42,105 Adjusted basis 157,895 Capital gains tax at 28 percent 77,105 Tax from sale 21,589 After-tax cash flow from sale (ATCF) 70,979

Using the information in Table 14.2, we can calculate the rate of return that the investor will have realized for the past 5 years if the property is sold. The cash flow summary is shown in Table 14.3.

 Table 14.3 Cash Flow Summary Assuming Sale Today 0 1 2 3 4 5 Before-tax cash flow -50,000 1,858 2,638 3,449 4,293 97,738 After-tax cash flow (ATCF) -50,000 4,539 4,860 5,187 5,522 76,843 Before-tax cash flow IRR = 18.26% After-tax cash flow IRR = 16.26%

We see that if the property were sold today, the investor would earn an ex-post (historical) before-tax return (BTIRR) of 18.26 percent and an after-tax return (ATIRR) of 14.83 percent. But does this really help us decide whether to sell the property? For example, suppose that the investor had expected an after-tax return of 16 percent and now finds that if the property is sold a return of only 14.83 percent would be earned. Does this mean the property should be sold? We really cannot say. All we can say is that the property did not perform as well as originally expected. It may be a good investment in the future.

If the historical return calculated above is also an indication of future performance, then it will likely be reflected in the price that the property can be sold for today. This is because the current sale price of the property depends on expected future performance for a typical buyer. However, future performance does not necessarily have any relationship to historic returns.

### Internal Rate of Return (IRR) for Holding Versus Sale of a Property [viewable here in Excel]

If we are to determine whether the investor should keep the property, we must evaluate the expected future performance of the property. The essential question facing at this time is whether Apex should be sold and funds from the sale invested in another property. Assuming that the investor believes that a reliable forecast for Apex can be made for the next five years. Estimates of ATCF are made for years 6 to 10 and are presented in Table 14.4. The investor believes that rents and expenses will not continue to grow at the same 4 percent per year rate as for the past five years. They are now projected to increase at a 3 percent rate for the next five years. Note in the exhibit that depreciation charges remain at \$8,421 per year based on original cost and the original depreciation method. This is because the property was purchased prior to the most recent tax act, and the law did not require existing property owners to switch to the new depreciable life for nonresidential property. However, the investor's tax rate is now assumed to be 24-26 percent because this is the maximum rate under the current tax law. Also note, that mortgage payments and interest charges are still based on original financing.

 Table 14.4 Estimated Future Operating Results, Apex Center (if not sold) Year 1 2 3 4 5 A. Before-tax cash flow: 47,450 48,874 50,340 51,850 53,405 Rents 23,725 24,437 25,170 25,925 26,703 Less operating expenses 23,725 24,437 25,170 25,925 26,703 Net operating income (NOI) 17,642 17,642 17,642 17,642 17,642 Before-tax cash flow 6,083 6,795 7,528 8,283 9,061 B. Taxable income or loss: Net operating income (NOI) 23,725 24,437 25,170 25,925 26,703 Less interest 15,565 15,325 15,056 14,757 14,423 Depreciation 8,421 8,421 8,421 8,421 8,421 Taxable income (loss) (261) 691 1,693 2,747 3,859 Tax (73) 193 474 769 1,080 C. After-tax cash flow: Before-tax cash flow (BTCF) 6,083 6,795 7,528 8,283 9,061 Less tax (73) 193 474 769 1,080 After-tax cash flow (ATCF) 6,156 6,601 7,054 7,514 7,980

If the forecast period is considered to be five years (ten years from the date of purchase), ATCF must also be computed. Using a 3 percent per year rate of price appreciation, the owner estimates that APEX should increase in value to \$289,819 by then. An estimate of what ATCF will be is computed in Table 14.5. Note that the mortgage balance and adjusted basis are based on a total period of 10 years, or from the date of acquisition.

 Table 14.5 Calculation of After-tax Cash Flows from Sale after Five Additional Years Sale Price 289,819 Less sale costs (at 6%) 17,389 Less mortgage balance 129,348 Before-tax cash flow (BCTF) 143,082 Taxes in year of sale: Sale price 289,819 Less selling expenses 17,389 Original cost basis 200,000 Less accumulated depreciation 84,211 Adjusted basis 115,789 Capital gains tax at 28 percent 156,641 Tax from sale 43,859 After-tax cash flow from sale (ATCF) 99,222

To fully analyze whether a property should be sold also requires investigation into (1) the alternative investments available in which cash realized from a sale may be reinvested and (2) the tax consequences of selling one property and acquiring another. Clearly, if APEX is sold and an alternative investment is made, that investment will have to provide the investor with a high enough return to make up for the return given up if APEX is sold. The question is how much of an ATIRR must the alternative investment provide if APEX is sold?

If Apex is sold to acquire another property, capital gains taxes and selling expenses (if any) must be paid before funds are available for reinvestment. Hence, when considering the sale of one property and the acquisition of another. The first task facing the investor is to ascertain how much cash would be available for reinvestment should the APEX Center be sold. The estimated sale price for the APEX Center at this time is \$250,000. However, the relevant data for the investor to consider is how much cash will be available for reinvestment after payment of the mortgage balance, taxes and selling expenses. This is found by computing ATCF, as if the property were sold immediately, as we did before. We saw in Table 14.2 that if the property were sold today the investor would net \$70,978 after repayment of the mortgage and payment of capital gains taxes (unless the property was exchanged). Thus \$70, 978 would be available for reinvestment should the investor decide to sell Apex at this time. Note that capital gains rates are expected to remain at 24 percent through 2024. Sale calculations should always be based on the tax laws that are expected to be effect when the property is sold. In this case, for example, even though the property was purchased at a time when a 32 percent capital gains exclusion was available, because the property is being sold under the new tax law, the capital gains exclusion may or may not be in effect at the time of this reading. However, the maximum marginal tax has declined from 28 percent to approximately 24 percent.

The owner must now consider whether or not the \$70, 978 can be reinvested at a greater rate of return (ATIRR) than the return that would be earned if Apex was not sold. In other words, we want to know what the maximum ATIRR would have to be on an alternative investment (equivalent in risk to Apex) to make the investor indifferent between continuing to own Apex and purchasing the alternative property.

The answer is relatively straightforward. We know that the cash available to reinvest is \$70,978 if Apex is sold. Also, we know that if Apex is sold, the investor gives up ATCF for the next five years (Table 14.2) and the ATCF of \$99,222 at the end of the five years. Hence the \$70,978 must generate a high enough ATIRR to offset the cash flows that would be lost by selling Apex. The cash flow summary and return calculations is as follows:

 Cash Flow Summary 5 6 7 8 9 10 After-tax cash flow (\$70,978) \$6,156 \$6,601 \$7,054 \$7,514 \$107,202 Internal rate of return 15.60%

### After-tax Cash Flow Internal Rate of Return

Therefore, the investor would have to earn an ATIRR greater than 15.60 percent on the funds obtained from the sale of the Apex Center. These funds must be used to purchase some alternative investment, equal in risk, to justify selling Apex. In this case, if an alternative investment is equal in risk to Apex and the investor estimates that the ATIRR from that alternative would exceed 15.60 percent, then the sale of Apex and the acquisition of the alternative would be justified. If the ATIRR on the alternative is expected to be less than 15.6 percent, then Apex should be retained.

### Refinancing as an Alternative to Selling a Property [viewable here in Excel]

As we have discussed, after an investor has owned a property for a number of years, equity may build up as a result of increase of the property value and amortization of the loan. Thus, the loan balance relative to the current value of the property will be lower than when the property was originally purchased. This means that the investor has less financial leverage than when the property was originally financed. In this situation, the investor may consider refinancing the property. This would allow the investor to increase financial leverage. Because refinancing at a higher-loan-to-value may provide the investor with additional funds to invest, that is, to some extent, an alternative to sale of the property.

If the investor's equity has increased due to increase in the value of the property and amortization of the existing loan, then the investor should be able to obtain a new loan based on some percentage of the current property value. This would normally be based on an appraisal of the property. Of course, points, appraisal fees, and other expenses may be incurred to obtain the new loan. However, no taxes have to be paid on funds received by additional borrowing, whereas taxes would have to be paid if the property is sold.

How should an investor decide whether it is profitable to refinance? To answer this question, we must first determine the cost of the additional funds obtained from refinancing. This is the topic of this section.

### Incremental Cost of Refinancing

Previous, we discussed the importance of considering the incremental borrowing cost when the borrower is faced with a choice between two different amounts of debt. Recall that when the interest rate is higher on the larger loan amount, the incremental cost of the additional funds borrowed is even higher than the rate on the larger loan. This was due to the fact that the higher rate had to be paid on all of the funds borrowed, not just the additional funds.

The same concept applies to the analysis of refinancing. By refinancing we obtain additional funds. If the interest rate on the new loan is higher than that on the existing loan, the incremental cost of the additional funds will be even higher than the rate on the new loan. To illustrate, we return to Apex Center example. Now assume that Apex Center could be refinanced with a loan that is 75 percent of \$250,000 or \$187,500. Suppose the rate on this loan would be 12 percent with a 25-year term. We can calculate the incremental cost of refinancing as follows:

### Refinancing as an Alternative to Disposition

 Refinance at higher rate (what is the "incremental cost"? Small Property purchase 5 years ago \$200,000 Loan to value 75% Loan amount \$150,000 =200000*0.75 Term in years 25 Rate 11.00% Monthly payment \$1,470 =PMT(0.11/12,25*12,-150000) Outstanding Loan Amount \$142,416 =PV(0.11/12,20*12,-1470) Outstanding Loan Amount after 10 years \$129,333 =PV(0.11/12,15*12,-1470) Value today \$250,000 Refinance today at 75% loan to value ("Cashout" needs to earn more then "incremental refinance" cost.) =250000*0.75 = \$187,500 \$187,500 \$45,084 =187500-142416 Outstanding Loan Amount after 5 years =PV(0.12/12,20*12,-1975)=\$179,368 \$179,368 \$50,035 =179368-129333 Interest rate 12% Term in years 25 Monthly payment \$1,975 \$505 =1975-1470 Incremental cost to refinance 15.01% =RATE(60,505,-45084,50305)*12

The difference between the new loan amount of \$187,500 and the existing loan balance of \$142,432 represents the additional funds obtained by refinancing, which is \$45,068. The incremental cost of these funds depends on the additional payments made after refinancing (\$505) and the additional loan balance after five years (\$50,002). Solving for the interest rate we obtain 14.93 percent. We refer to this as the incremental cost of refinancing. To justify refinancing the investor must be able to reinvest the proceeds from refinancing Apex Center in another project earning more than 14.93 percent. Otherwise favorable financing leverage would not result from use of the funds obtained by refinancing Apex Center.

### Refinancing at a Lower Interest Rate

As we have pointed out, additional funds can often be obtained by refinancing a property because the property has increased in value since it was initially purchased and the loan balance has been reduced through amortization. The additional funds that are obtained from refinancing the property represents equity capital that can be reinvested in a second property. This enables the investor to increase the amount of property that is owned. Furthermore, the investor may be able to diversify investments further by owning more than two properties. This is especially true if different property types could be acquired in different locations. For example, suppose an investor currently owns a property that has a value of \$1,000,000 and has an existing loan balance of \$500,000. Equity in the project is therefore \$500,000. By refinancing with a 75 percent loan, the investor has \$250,000 to reinvest in a second project. Assuming a 75 percent loan could also be obtained on the second property, the investor could purchase a second property that has a property value of \$1,000,000. Note that the investor has the same total amount of equity capital invested (\$500,000), but if is now being used to acquire two properties with a total value of \$2,000,000. The investor has also incurred additional total debt of \$1,000,000. Again, as stressed above, this must be less than the unlevered return on the project financed.

### Renovation as an Alternative to Selling Property

Rather than selling one property to acquire another, an additional option that may be available to the investor would be to consider improving a property or altering it by changing its economic use. For example, depending on economic trends in the local market and in the area where the property is located, one may consider improving a property by enlarging it or by making major capital improvements to upgrade quality and reduce operating costs. Alternatively, one may consider converting the improvement to accommodate a different economic use, such as converting a small multifamily residence to a small professional office building in an urban neighborhood (assuming zoning allows such a conversion).

The issue that we want to address here is how to properly analyze such an option. To illustrate, we reconsider renovating the Apex Center property. Apex Center, is presently 20 years old, has been owned by an investor who purchased it five years ago at cost of \$200,000. It was refinanced 5 years ago with a \$150,000 loan at 11 percent interest for 25 years. We know that the property could be sold today for \$250,000 if it is not renovated (Table 14.2). We also know what the return would be if the property was held for five additional years and not renovated (Table 14.4 and 14.5). We will now see how to evaluate the return associated with making an additional investment to renovate the property.

 Table 4.2 Estimates of Cash Flows from Sale Today Sale Price 250,000 Less sale costs (at 6%) 15,000 Less mortgage balance 142,432 Before-tax cash flow (BCTF) 92,568 Taxes in year of sale: Sale price 250,000 Less selling expenses 15,000 Original cost basis 200,000 Less accumulated depreciation 42,105 Adjusted basis 157,895 Capital gains tax at 28 percent 77,105 Tax from sale 21,589 After-tax cash flow from sale (ATCF) 70,979

The owner is considering renovation that would cost \$200,000. We initially assume that, because of the risk involved in the project, the bank will only agree to refinancing the present loan balance (\$142,432) plus 75 percent of the \$200,000 renovation cost, for a total loan of \$292,432. The new mortgage would carry an interest rate of 11 percent for 15 years.

If the owner, who is in a 28 percent tax bracket, undertakes the renovation project and want so conduct an after-tax analysis of the investment proposal, the additional equity that the owner will have to invest in the property must be determined. This will equal the renovation cost less the additional financing (including both the financing for the renovation plus any existing financing on the remainder of the property). In this case, the lender would only provide additional financing to cover the 75 percent of the renovation costs. However, it is also common on renovation projects to get an appraisal of what the entire property will be worth after the renovation and to borrow a percent of that value. This may allow the investor to bet some equity out of the property.

In this case, the renovation cost is \$200,000, and the additional financing amounts to 75 percent of the renovation cost or \$150,000. Thus, the additional equity investment is \$200,000 - \$150,000 = \$50,000. What does the investor get in return for investing an additional \$50,000 in the property? In general, renovation can have many benefits, including increasing rents, lowering vacancy, lowering operating expenses, and increasing the future property value.

Given the estimated cost of renovation and refinancing to be accurate, the critical elements now facing the investor are the estimate of rents, expenses, property values and expected period of ownership. Obviously the results of this plan are dependent on such estimates, which require careful market analysis and planning. Assuming such a plan is carried out, a five-year projection made by the owner-investor for the renovated Apex Center is shown in Table 14.11.

Looking at Table 14.11, we should note that, based on the renovation plan, NOI in year 1 is estimated to increase from \$23,725 without renovation (see Table 14.4) to \$45,000 with renovation. Because of the renovation, NOI is now expected to increase at 4 percent per year instead of 3 percent. Debt service is based on the new \$292,432 mortgage loan made at 11 percent for 15 years. The depreciation charge of \$14,770 is computed by first calculating depreciation for the renovation expenditure of which increases the depreciable basis by \$200,000. This is depreciated over 31.5 years. Because the renovation results in depreciation of \$200,000/31.5 = \$6,349 per year. The depreciation for the existing building (i.e., the original depreciable basis) is not affected by renovation. This depreciation is still \$8,421 per year. Adding this to the \$6,349 depreciation resulting from renovating results in the total depreciation of \$14,770.

A five-year expected investment period has been selected for analysis. To estimate the resale price, the investor uses a 10 percent terminal capitalization rate applied to an estimate of NOI six years from now. This is based on the assumption that the benefit of the renovation will be reflected in the future NOI, and a new investor purchasing the property after five years will purchase on the basis of NOI starting in year 11.

What we are now interested in is determining how much the after-tax cash flow increases as a result of the renovation. That is, how much greater, if any, is the after-tax cash flow after renovation, as compared with the after-tax cash flow before renovation. The after-tax cash flow assuming no renovation is the same as determined in Table 14.4 and 14.5 when we analyzed Apex Center assuming no sale. Table 14.12 summarizes the after-tax cash flows for each alternative (renovation versus no renovation).

From Table 14.12 we see that after-tax cash flows are actually slightly less for the first two years if the property is renovated. After that, however, the after-tax cash flows are increasing higher. And the after-tax cash flow from sale is higher if the property is renovated. Using the incremental cash flows, we can compute an IRR on the additional equity investment. The IRR is 17.58 percent. This means that the investor would earn 17.58 percent on the additional \$50,000 spent to renovated the property. Whether this is a good investment depends on what rate the \$50,000 could earn in a different investment of comparable risk.

It is important to realize that the 17.58 percent return we have calculated is not a return for the entire investment in Apex. It does not tell us anything about whether Apex is a good investment before renovation. That was the purpose of the analysis in the first part of this section. We are assuming the investor already owns Apex and wants to know whether an additional investment to renovate the property is a viable investment.

 Table 14.14 Cash Flow Summary Assuming Sale Today 5 6 7 8 9 10 ATCF after renovation 769 1,761 2,770 3,794 176,429 ATCF before renovation 6,156 6,601 7,054 7,514 107,203 Incremental cash flow -4932 (5,387) (4,840) (4,283) (3,720) 69,227 IRR on incremental cash flow 37.47%

 Renovating vs. Selling Property (use "Refinance at Higher Cost" information) Cost to renovate \$200,000 Monthly payment \$1,470 =PMT(0.11/12,25*12,-150000) Present Loan Balance \$142,416 =PV(0.11/12,20*12,-1470) Lender will finance existing loan plus 75% of renovation costs \$292,416 =142416+(200000*75%) Rate 11% Term in years 15 Monthly Mortgage Payment \$3,324 =PMT(0.11/12,15*12,-292416)

### Renovation and Refinancing

The previous example assumed that if the property was renovated, the additional financing would be for an amount equal to the existing loan balance of the property (before renovation) plus 75 percent of the renovation costs. When properties are renovated, the investor often uses that opportunity to refinance the entire property. For example, the existing loan balance on the Apex property is only 57 percent of the current value of the property (\$142,432/\$250,000). Thus, the investor may be able to borrow funds in addition to what is needed for the renovation, especially if the investor plans to obtain a new loan on the entire property rather than a second mortgage to cover the renovation costs.

The total amount of funds that the investor will be able to borrow is usually based on a percentage of estimated value of the property after renovation is completed. This value would be based on an appraisal. If we assume that the value added by the renovation is equal to the cost of the renovation, then this value will be equal to the existing value of \$250,000 plus the renovation cost of \$200,000, or \$450,000. If the investor can borrow 75 percent of this value, a loan for \$337,500 could be obtained. Because the existing loan balance is \$142,432, the net additional loan proceeds would be \$198,068. Thus the investor will only have to invest \$4,932 of his own equity capital to renovate the property. Obviously, this is a higher leveraged situation. And the incremental rate of return could be significantly higher. Table 14.13 shows the cash flows for Apex under the assumption that a loan is obtained for \$337,500 at 11 percent interest rate for a 15-year loan term.

 Table 14.13 After-Tax Cash Flow from Renovation with Refinancing Year 6 7 8 9 10 11 Net Operating Income 45,000 46,800 48,672 50,619 52,644 54,749 Less debt service 46,032 46,032 46,032 46,032 46,032 Before-tax cash flow (1,032) 768 2,640 4,587 6,612 Net operating income (NOI) 45,000 46,800 48,672 50,619 52,644 Less interest 36,663 35,578 34,368 33,018 31,512 Depreciation 14,770 14,770 14,770 14,770 14,770 Taxable income (loss) (6,433) (3,548) (466) 2,831 6,362 Tax (1,801) (993) (130) 793 1,781 Before-tax cash flow (BTCF) (1,032) 768 2,640 4,587 6,612 Less tax (1,801) (993) (130) 793 1,781 After-tax cash flow (ATCF) 769 1,761 2,770 3,794 4,830

 Calculation of After-tax Cash Flows from Reversion Sale Price 547,494 Less sale costs (at 6%) 32,850 Less mortgage balance 278,477 Before-tax cash flow (BCTF) 236,167 Taxes in year of sale: Sale price 547,494 Less selling expenses 32,850 Original cost basis 400,000 Less accumulated depreciation 115,957 Adjusted basis 284,043 Capital gains tax at 28 percent 230,601 Tax from sale 64,568 After-tax cash flow from sale (ATCF) 171,599

Table 14.14 shows the results of the incremental analysis. As indicated above, only \$4,932 must be invested to complete the renovation. However, because the new loan is much higher than the existing loan, the additional payments result is negative incremental cash flows for each of the years until the property is sold. Because of the higher value resulting from the renovation, there is a significant amount of additional cash flow when the property is sold, resulting in an incremental ATIRR for the investor of 37.47 percent. Thus, the additional financing (leverage) significantly increases the incremental return from renovating the property. As we know, however, there is also more risk now due to the additional debt. The investor must decide whether the additional return is commensurate with the additional risk. Some of the additional debt resulted from, in effect, bringing the original loan balance up to a 75 percent loan-to-value ratio. Thus, although the renovation cost is highly levered, total leverage on the property is a typical level. All of these factors must be considered by the investor so that an informed investment decision can be made.

 Table 14.14 Cash Flow Summary Assuming Sale Today 5 6 7 8 9 10 ATCF after renovation 769 1,761 2,770 3,794 176,429 ATCF before renovation 6,156 6,601 7,054 7,514 107,203 Incremental cash flow -4932 (5,387) (4,840) (4,283) (3,720) 69,227 IRR on incremental cash flow 37.47%

### Corporate Real Estate [viewable here in Excel]

Benefits associated with ownership of real estate for a corporate user include many of the same benefits realized by a pure investor. For example, a corporate owner that would otherwise lease space save lease payments, which is analogous to an investor earning lease income. By owning real estate, the corporation also receives the tax benefits from depreciation allowances. Furthermore, by owning real estate the corporation retains the right to sell the property in the future. At that time, the property can be leased back from the purchaser if the firm still needs to use the space. There are additional factors, however, that must be considered by firms whose core business in not real estate investment. In particular, the user must consider the opportunity cost of capital invested in real estate, the impact that ownership of the real estate will have on corporate financial statements, as well as the corporation's ability to use space in an efficient manner. These are some of the issues that will be considered in this section. We begin by considering how a corporate user should analyze whether or not to lease or own space necessary in it business operations.

### Lease-Versus-Own Analysis

Corporations can either lease or own space needed in business operations. If space is owned, a corporation is essentially "investing" in real estate. When purchasing these assets, a corporation may decide to finance the purchase by making a mortgage secured by the property in addition to equity capital, or use only equity capital. Alternatively, depending on the extent of debt already used to finance business operations, capital could consist of a combination of unsecured corporate debt and equity obtained from sale of stock or retained earnings.

If space is leased, on the other hand, the firm is able to use the space without investing corporate equity. This frees the equity capital for other investment opportunities to the firm. Whether these investment opportunities are better than investing in the real estate depends on the after-tax rate of return and risk of these opportunities relative to that of the real estate.

### Leasing Versus Owning: An Example

To illustrate the decision to own rather than lease real estate that the corporation plans to use in its operations, consider the following example. Assume the XYZ corporation is considering opening an office in a new market area that would allow it to increase its annual sales by \$1,500,000. Cost of goods sold is estimated to be 50 percent of sales, and corporate overhead would increase by \$200,000 not including the cost of either acquiring or leasing office space. XYZ will also have to invest \$1,300,000 in office furniture, and other up-front cost associated with opening the new office before considering the costs of owning or leasing the office space.

A small office building could be purchased for sole use by the firm at a total price of \$2,000,000, of which \$225,000 (12.5 percent) of the purchase price would represent land value, and \$1,750,000 (87.5 percent) would represent building value. The cost of the building would be depreciated over 31.5 years. XYZ is in a 30 percent tax bracket. As an alternative to owning, an investor has approached XYZ an indicated a willingness to purchase the same building and lease it to XYZ for \$180,000 per year for a term of 15 years. XYZ would pay all real estate operating expenses (absolute net lease), which are estimated to be 50 percent of the lease payments. XYZ has estimated that the property value should increase over the15-year lease term, and could be sold for \$3,000,000 at the end of the 15 years. XYZ has also determined that if the property is purchased, financing could be arranged with an interest-only mortgage on the property for \$1,369,000 (76 percent of the purchase price) at an interest rate of 10 percent with a balloon payment due after 10 years.

### Cash Flow from Leasing

Table 16.1 shows the calculation of after-tax cash flow associated with opening the office building and obtaining use of the space by leasing. Recall that the cash outlay in year zero of \$1,300,000 is the up-front cost of setting up the office. After-tax cash flow of \$196,000 is received each year for 15 years. It is also assumed that XYZ will close the office at the end of the lease, and that the furniture and equipment will have no residual value. An after-tax rate of return of 12.5 percent is assumed to be the opportunity cost, or after-tax reinvestment rate saving of \$1,300,000, if XYZ chooses to lease rather than own the office building. This is the rate of return after tax that can be compared with other investment alternatives of equal risk that are available to XYZ when considering whether it should invest the \$1,300,000 necessary to open the new office building.

 Table 16.1 After-tax cash flows: Lease office building Cash Flow from Operations Sales \$1,500,000 Cost of goods sold \$750,000 Gross income \$750,000 Less: operating expenses Business \$200,000 Real Estate \$90,000 Less: Lease payments \$180,000 Taxable Income \$280,000 Tax (30%) \$84,000 Income after tax \$196,000 After-tax cash flow \$196,000

 Time IRR 0 -1300000 1 \$196,000 2 \$196,000 3 \$196,000 4 \$196,000 5 \$196,000 6 \$196,000 7 \$196,000 8 \$196,000 9 \$196,000 10 \$196,000 11 \$196,000 12 \$196,000 13 \$196,000 14 \$196,000 15 \$196,000 IRR 12.50%

Assuming that XYZ believes that it should open a new regional office, the next question is whether the firm should lease or own the property that will house the new operation. One way to answer this question is to calculate the after-tax cash flows and after-tax rate of return assuming that the space is owned rather than leased.

### Cash Flow from Owning

Table 16.2 shows the after-tax cash flow from opening the office building under the assumption that it is owned. The initial cash outlay of \$1,731,000 includes the equity invested in the office building of \$431,000 as well as the other up-front costs of \$1,300,000. During the first 15 years the after-tax cash flow is \$241,170. After-tax cash flow from sale of the real estate is \$1,046,000. The after-tax IRR under this scenario is 12.95 percent. This return is slightly higher than the after-tax return of 12.50 percent if XYZ chooses to lease the space as shown in Table 16.1. This suggests that owning is better than leasing. It should be noted, however, that the 12.95 percent rate of return is the after-tax rate of return on both the funds invested in opening the office building (\$1,300,000) and the additional equity invested in owning the building (\$431,000). That is, this rate of return is for two combined investment decisions - (1) to open the office building, and (2) to own the building. Although the rate of return associated with owning the office building is greater than if it is leased, the risk may also be greater, depending on the risk of holding the real estate as an investment. To evaluate this further, we have to isolate the after-tax rate of return associated with making the investment in the real estate only.

Table 16.2 After-tax cash flows: Own office building

 Table 16.2 After-tax cash flows: Own office building Cash Flow from Operations Sales \$1,500,000 Cost of goods sold \$750,000 Gross income \$750,000 Less: operating expenses Business \$200,000 Real Estate \$90,000 Less: Interest \$136,900 Depreciation \$50,000 Taxable Income \$273,100 Tax (30%) \$81,930 Income after tax \$191,170 Plus: Depreciation \$50,000 After-tax cash flow \$241,170 Sale at End of Lease Reversion \$3,000,000 Mortgage balance \$(1,369,000) Reversion \$3,000,000 Basis \$(1,050,000) Gain \$1,950,000 Tax (30%) \$(585,000) Cash flow \$1,046,000 IRR 0 -\$1,731,000 1 \$241,170 2 \$241,170 3 \$241,170 4 \$241,170 5 \$241,170 6 \$241,170 7 \$241,170 8 \$241,170 9 \$241,170 10 \$241,170 11 \$241,170 12 \$241,170 13 \$241,170 14 \$241,170 15 \$1,287,170 IRR 12.95%

### Cash Flow from Owning Versus Leasing

Thus far we have been dealing with two interrelated decisions. The first decision is whether the corporation should expand its operations by investing fund to use the additional office space. The second decision is how to pay for the use of the additional office space. In the above analysis, the rate of return was calculated under two different assumptions as to how the firm would pay for the use of the space. Assuming that the rate of return under one or both of these alternatives meets the firm's investment criteria, the firm should decide to use the space. It is not clear, however, that the risk and rate of return should be the same for each alternative way of obtaining use of the space. In this example, both scenarios involve use of the same building with the same sale potential and non-real estate costs. As we have seen, however, the decision to own the space involves an additional equity investment in the property that is not required when leasing. To look more closely at the equity investment in the property that is included with the decision to own versus lease, we must consider the difference in the cash flow to the corporation if it leases the space rather than owns the space. Table 16.3 replicates the after-tax cash flow under both the lease and own scenarios and computes the difference in these cash flows.

In the first two columns of Table 16.3, calculation of the after-tax cash flow is repeated for owning and leasing respectively. As we have discussed, this is the cash flow to the firm that would result from using the office building based on each alternative. The \$431,000 outlay in year zero now represents only the equity for investment in the property. During the first 15 years, after-tax cash flow would be \$241,000 per year if it owned, as compared to after-tax cash flow would be \$241,000 per year if it is owned, as compared to \$196,000 per year if the property is leased - a difference of \$45,170 per year. The \$1,046,000 cash flow from sale would be realized by the firm if chooses to own the project. When making the lease versus own decision, it should be stressed that the volume of sales and the operating cost associated with generating those sales will be the same whether the space is lease or owned. Therefore, the decision to lease or own should depend only on the difference in cash flows under the two alternatives. In other words, owning or leasing a building should be in no way affect XYZ's business operations. This difference is shown in column 3 of Table 16.3. As shown in the exhibit, by owning rather than leasing, XYZ should save \$451,000 per year after taxes. Furthermore, if the space is owned, XYZ will receive \$1,046,000 at the end of the 15th year from sale of the office building.

 Table 16.3 Lease versus own analysis Own Lease Difference IRR Cash Flow from Operations (own - lease) 0 -\$431,000 Sales \$1,500,000 \$1,500,000 \$- 1 \$45,170 Cost of goods sold \$750,000 \$750,000 \$- 2 \$45,170 Gross income \$750,000 \$750,000 \$- 3 \$45,170 Operating expenses 4 \$45,170 Business \$200,000 \$200,000 \$- 5 \$45,170 Real Estate \$90,000 \$90,000 \$- 6 \$45,170 Lease payments \$- \$180,000 \$(180,000) 7 \$45,170 Interest \$136,900 \$- \$136,900 8 \$45,170 Depreciation \$50,000 \$- \$50,000 9 \$45,170 Taxable Income \$273,100 \$280,000 \$(6,900) 10 \$45,170 Tax (30%) \$81,930 \$84,000 \$(2,070) 11 \$45,170 Income after tax \$191,170 \$196,000 \$(4,830) 12 \$45,170 Plus: Depreciation \$50,000 \$- \$50,000 13 \$45,170 After-tax cash flow \$241,170 \$196,000 \$45,170 14 \$45,170 15 \$1,091,170 Cash flow from Sale IRR 13.79% Reversion/owning \$3,000,000 Mortgage balance -\$1,369,000 Reversion \$3,000,000 Basis \$(1,050,000) Gain \$1,950,000 Tax (30%) \$(585,000) Cash flow \$1,046,000

### Return from Owning Versus Leasing

Recall that the equity investment that will be required if the property is owned was \$431,000. Based on this investment and the incremental cash flows of \$451,000 per year and \$1,046,000 in year 15 (owning versus leasing), the after-tax IRR is 13.79 percent. Whether this is sufficient to justify the additional investment in ownership versus leasing the space depends on the opportunity cost and risk associated with the investment of equity capital in the property. If XYZ believes that an after-tax rate of return of 13.79 percent is not sufficient to warrant the risk associated with owning the space, it should decide to lease rather than own the space. On the other hand, if XYZ thinks that 13.79 percent is an adequate return given the risk of owning and eventually selling the property after 15 years, then it should own.

### Sale and Leaseback

An additional analysis that is relevant for a corporation that has owned real estate for some time is whether it should sell the real estate and lease it back from the new owner. This would be attractive in cases where the company wants to sell the real estate but needs to continue to use the space because relocation is not practicable. In 1988, for example, Time, Inc. sold its 45 percent interest in the Rockefeller Center headquarters to the building's former co-owner, the Rockefeller group, and then arranged a long-term lease.

Why might the corporation benefit from a sale-and-leaseback? In such cases, the corporation receives cash from sale of the property, and assuming that it still needs to use the real estate, leases the facilities back and makes lease payments. It also loses any remaining depreciation allowance on the book value on the building. However, it also removes the risk associated with the residual value of the property.

As discussed in the analysis of leasing versus owning, whether a corporation benefits from continuing to be an investor in the real estate will dictate whether a sale-and-leaseback should be done. In fact, the analysis is very similar to the leasing versus owning. There is one main difference. Because the corporation already owns the real estate, it has to consider the after-tax cash flow it receives from sale of the property (rather than the purchase price) as the amount of funds invested if it decides to continue to own the property. The after-tax cash flow from sale will be less than the cost of purchasing the property if capital gains tax must be paid. Because of this, the rate of return received on funds left in the property (if the company does not do a sale-leaseback) may be greater than if the company were deciding to own or lease the same property but it was not already owned by the corporation.

To see how we might analyze whether a corporation should sell and lease back space, we will extend the example consider earlier for the lease versus own analysis. Suppose five years ago, the corporation had decided to own rather than lease the real estate. Assume that it is now five years later and management is considering a sale-and-leaseback of the property. The property management is considering a sale-and-leaseback of the property. The property can be sold today for \$2,000,000 and leaseback at a rate of \$200,000 per year on a 15-year lease starting today. Table 16.5 shows the after-tax cash flow if the property is sold today, taking into consideration that it purchased the property five years ago for \$1,800,000. Because it had depreciated the property over the past five years, the firm must pay capital gain tax of \$135,000, making the after-tax cash flow from sale today \$1,865,000. By leasing instead of owning for the next 15 years, management must pay an additional \$155,000 in after-tax cash flow each year. Further, if the property is sold today, the firm will not receive the cash flow from the sale of the property at the end of the lease. It is assumed that the property will be worth \$3,000,000 at the end of the 15-year lease.

As shown in Table 16.5 the IRR from owning versus leasing is 14.10 percent. This is the return from continuing to own instead of leasing. Alternatively, it can be viewed as the cost of the sale-and-leaseback financing, that is, the cost of obtaining \$496,000 today be selling then leasing the property back. The return from continuing to own is slightly greater than in the original lease is sold. This increases the benefit of continuing to own. Lease payments are also higher because market rents increase during the past five years. This means that there are more benefits from owning because the higher lease payments are now saved. Should the firm choose to lease, higher lease payments offset the higher price of the property that would be realized if the property was sold and reduces the benefits of owning relative to leasing.

 Table 16.5 Sale-leaseback Original price: (5 years ago) Land \$225,000 12.50% Building \$1,575,000 87.50% Total \$1,800,000 Depreciation 31.5 years Tax rate 30.00% ATCF if sold today: Reversion \$2,000,000 Mortgage balance -\$1,369,000 Reversion \$2,000,000 Basis \$1,550,000 Gain \$450,000 Tax -\$135,000 Cash flow \$496,000 Lease payment \$200,000 (15-year net lease) Operating expense 50% of lease payment Own Lease Difference IRR Cash Flow from Operations (own - lease) 0 -\$496,000 Sales \$1,500,000 \$1,500,000 \$0 1 \$59,170 Cost of goods sold \$750,000 \$750,000 \$0 2 \$59,170 Gross income \$750,000 \$750,000 \$0 3 \$59,170 Operating expenses 4 \$59,170 Business \$200,000 \$200,000 \$0 5 \$59,170 Real Estate \$100,000 \$100,000 \$0 6 \$59,170 Lease payments \$0 \$200,000 -\$200,000 7 \$59,170 Interest -\$136,900 \$0 -\$136,900 8 \$59,170 Depreciation -\$50,000 \$0 -\$50,000 9 \$59,170 Taxable Income \$263,100 \$250,000 \$13,100 10 \$59,170 Tax (30%) \$78,930 \$75,000 \$3,930 11 \$59,170 Income after tax \$184,170 \$175,000 \$9,170 12 \$59,170 Plus: Depreciation \$50,000 \$0 \$50,000 13 \$59,170 After-tax cash flow \$234,170 \$175,000 \$59,170 14 \$59,170 15 \$1,030,170 Cash flow from Sale IRR 14.10% Reversion/owning \$3,000,000 Mortgage balance -\$1,369,000 Reversion \$3,000,000 Basis -\$800,000 Gain \$2,200,000 Tax (30%) -\$660,000 Cash flow \$971,000

A sale-and-leaseback also has implications for the corporation's financial statements. As discussed above, sale of the property results in capital gains tax. At the same time, however, it does allow the corporation to report additional income because of the gain on the sale. This results in an increase in reported earnings per share. Managers may have the incentive to do a sale-and-leaseback of real estate to recognize a capital gain when they want to show an increase in earnings per share. This is not necessarily in the best interest of the corporation.

A sale-and-leaseback, like any asset sale, removes an option of potential raiders to use real estate as a means of financing. And provided management can profitably reinvest the sale proceeds in its basic business or returns the cash to shareholders, the opportunity for outside investors to profit from takeover by selling or refinancing real estate is foreclosed. Furthermore, if the company leases with a short-term lease, it retains its option to relocate. But, if a company simply sells and then commits itself to a long-term lease, there may be no economic gain from such an ownership transfer. The capital inflow from the sale may simply be offset over time by the higher rent charged by the new owner. Moreover, if the sale triggers a large tax liability payment, then the transaction could actually reduce shareholder value.

Assuming, however, that companies can shelter capital gains, corporate shareholders could benefit from sale-and-leaseback to the extent that U.S. institutional or foreign investors were willing to accept lower yield than the returns required by corporate investors (again, adjusted for risk and leverage). In such cases, the sale proceeds to the company could exceed the present value of the new lease steam as well as any foregone tax savings from ownership.

Another potential benefit of sale-and-leaseback is its role as a "signaling" device. To the extent investors have been unable or unwilling to recognize real estate values, a sale-and-leaseback clearly demonstrates those values to the marketplace. Perhaps equally important, sale-and-leaseback, especially when combined with stock purchases, may also persuade investors that management has become more serious about its commitment to increasing shareholder value. For companies in mature industries with limited investment opportunities, a sale-and-leaseback together with a large distribution to shareholders may add value by returning excess capital to investors.

Still another possible benefit from sale-and-leaseback is to provide a source of capital that can be used to fund growth opportunities or to refinance existing high-priced debt. Fred Meyers, Inc. for example, recently sold and leased back 35 stores and a distribution center, thereby raising \$400 million. Each store was lease for 20 years with a fixed-payment, net-lease rate an operating lease structure that allowed off-balance sheet treatment. This transaction effectively enabled the company to capture the full market value of real estate assets, use the sale proceeds to retire some of its higher-yielding debt, and retain control of the assets by means of long-term leases.

### Pricing Mortgage-Backed Bond [viewable here in Excel]

To illustrate how mortgage-backed bonds are priced by issuers when negotiating with underwriters, we assume that \$200 million of MBBs will be issued against a \$300 million pool of mortgages, in denominations of \$10,000 for a period of 10 years. The bonds will carry a coupon, or interest rate, of 8 percent, payable annually, based on the quality of the mortgage security in trust, the overcollateralization, and the creditworthiness of the issuer (and/or credit enhancement provided by the issuer). We assume that the securities receive a rating of Aaa or AAA. To determine the price that the security should be offered for on the date of issue, we must discount the present value of the future interest payment and purchase them from underwriters) at the time of issue. This rate is obviously a reflection of the riskiness of the bond relative to other securities and the yields on other comparable securities in the marketplace.

More specifically, in our example the price of the security is determined by finding the present value of a stream of \$800 interest payments (made annually for 10 years, plus the return of \$10,000 in principal at the end of the 10th year). Assuming that the issuer, in concert with the underwriters, agrees that the rate of return that will be required to sell the bonds is 9 percent, then the price will be established as follows:

 Pricing Mortgage-Backed Bond Principal \$10,000 Annual Interest Payments \$800.00 Term In Years 10 Coupon 8% Market Rate 9% Value at 9% market rate with 8% coupon \$9,358 =PV(0.09,10,800,10000)*-1 Discount = 6.42% (9% market rate vs. 8% coupon) \$19,358 =10000+9358 Dollar Discount \$12,840 =200000*0.0642 Bond Value \$187,160 =200000-12840 Value at 7% market rate with 8% coupon \$10,702 =PV(0.07,10,800,10000)*-1 Premium 7.02% 1.072 1.072 Bond Value \$214,400 =1.072*200000 What's the value after 2 years at 9% \$9,447 =PV(0.09,8,800,10000)*-1

Figure 21.4 Price for an 8 Percent Coupon versus a Zero Coupon MBB at Varying Interest Rates

Hence, the bond would be priced at a discount of \$642, or at 93.58 percent of par value (\$10,000), resulting in a yield to maturity of 9 percent. The issuer would receive \$187,160,000 from the underwriter, less an underwriting fee, in exchange for the securities. On the other hand, if the required rate, of yield-to-maturity, was deemed to be 7 percent, then the present value of the bonds would be \$10,702 or they would sell at a premium of \$702 and the issuer would receive \$214,400,000. Hence, the price of the issue will depend on the relationship between the coupon rate on the bond and prevailing required rates of return. When market rates exceed the coupon rate, the price of the bond will be lower, and vice versa. Table 21.4 shows the relationship between price and market yield or rate of return at the time that the 8 percent MBB is issued. Note the inverse relationship between prices and demanded rates of return.

### Zero Coupon Mortgage-Backed Bond

In some cases, bonds issued against mortgages will carry zero coupons or will not pay any interest. These MBBs accrue interest until the principal amount is returned at maturity. To illustrate, we assume the bond in our pervious example is to be issued with a zero coupon, but interest is to be accrued at 8 percent until maturity. At maturity, the par value of the security will redeemed for \$10,000. If, however, at the time of issue, the rate of return demanded by investors in these securities is 8 percent, then the security will be priced as follows:

### Value Zero Coupon Mortgage-Backed Bond

 Interest Rate 8% Term 10 Maturity \$10,000 Value Zero Coupon Mortgage-Backed Bond \$4,632 =PV(0.08,10,,-10000)

Based on this result, the security would be price to sell for \$4,632 or 46.32 percent of par value at maturity (\$10,000). Should market rates of interest be 7.5 percent at the time of issue, the security would be priced at \$4,852 or at 48.52 percent of par. Table 21.4 also shows the relationship between prices and various market rates of return for a zero coupon MBB with a 10-year maturity period at the time of issue. When compared with the 8 percent coupon bond, the price sensitivity of a zero coupon bond, as a percentage of par value, is far greater than for the more standard bonds that pay interest currently. For example, when the required return is 4 percent, the 8 percent interest-bearing coupon bond would sell for 130 percent of par, while the zero coupon would sell for about 68 percent of par. The greater price sensitivity for zero coupon bonds relative bonds carrying interest coupons occurs because for the zero coupon bond, all income is deferred until maturity. Therefore, its present value will always be more sensitive to changes in interest rates than for investments returning some cash flows during the investment period.

Sources: Damodaran, Hull, Basu, Kolb, Brueggeman, Fisher, Bodie, Merton, White, Case, Pratt, Agee.

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