Below are links to the following topics:
- Present Value
- Frequency of Compounding
- Nominal Interest vs Real Interest Rate on Bonds
- Saving for College
- Present Value of Annuity
- Present Value Formula for an Annuity
- Using Present Value to Value Cashflows
- Coupon bonds Yield
- Why Yields for the Same Maturities Differ
- Market Price of Two Different Coupon Bonds
- Convertible Bond Conversion
- Estimating Market Value of Debt
- Present Value Examples
When we compute future value, we are asking question like, "How much will we have in 10 years if we invest $1,000 today at an interest rate of 8% per year?
But suppose we want to know how much to invest today in order to reach some target amount at a date in the future. For example, if we need to have $15,000 for a child's college education eight years from now, how much do we have to invest now? To find the answer to this kind of question, we need to calculate the present value of a given future value amount.
Calculating present values is the reverse of calculating future values. That is, it tells us the amount you would have to invest today to accumulate a specified amount in the future. Let's take a look at calculating PV step by step.
Suppose we want to have $1,000 one year from now and can earn 10% interest per year. The amount we must invest now is the present value of $1,000. Since the interest rate is 10%, we know that for every dollar we invest now we will have a future value of $1.10. Therefore, we can write:
Present Value x 1.10 = $1,000
Then the present value is given by:
Present value = $1,000/1.10 = $909.09
So, if the interest rate is 10% per year, we need to invest $909.09 in order to have $1,000 a year from now.
Now instead suppose that the $1,000 is needed two years from now. Clearly, the amount we need to invest today at an interest rate of 10% is less than $909.09, since it will earn interest at the rate of 10% per year for two years. To find the present value, we use our knowledge of how to find future values:
$1,000 = PV x 1.102 = PV x 1.21
or =PV(0.1,2,,-1000) =$826.45
In our example the present value is
PV = 1,000/1.102 = $826.45
Thus, $826.45 invested now at an interest rate of 10% per year will grow to $1,000 in two years.
Calculating present value is called discounting, and the interest rate used in the calculation is often
referred to as the discount rate. Thus, discounting in finance is very different from discounting in
retailing. In retailing it means reducing the price in order to sell more goods; in finance it means
computing the present value of a future sum of money. To distinguish the two kinds of discounting
in the world of business, the calculation of present values is called discounting cash flow (DCF)
The general formula for the present value of $1 to be received n periods from now at a discount
rate of ¡ (per periods) is:
PV = 1/(1 + ¡)n
This is called the present value factor of $1 at an interest rate of ¡ for n periods.
The present value of $1 to be received five years from now at an interest rate of 10% per year is PV = 1/(1.10)5 = 0.62092.
To find the present value of $1,000 to be received in five years at 10%, we just multiply this factor by $1,000 to get $620.92.
Since discounting is just the reverse of compounding (FV), we could use the information that we used for future value factors to find present values. Instead of multiplying by the factor, however, we divide by it. Thus we can find the present value of $1,000 to received in five years at 10% by multiplication or by using Excel to derive the present value factor.
PV = 1/(1.10)5 = 0.62092 *$1,000 = $620.92
or excel =PV(0.1,5,,-1000) = $620.92
Interest rates on loans and saving accounts are usually stated in the form of an annual percentage rate (APR) (e.g., 6% per year) with a certain frequency of compounding (e.g., monthly). Since the frequency of compounding can differ, it is important to have a way of making interest rates comparable. This is done by computing an effective annual rate (EFF), defined as the equivalent interest rate, if compounding were only once per year.
For example, suppose your money earns interest at a stated annual percentage rate (APR) of 6% per year compounded monthly. This means that interest is credited to your account every month at 1/12th the stated APR. Thus, the true interest rate is actually 1/2% per month (or .005 per month as a decimal).
We find the EFF by computing the future value at the end of the year per dollar invested at the beginning of the year. In this example we get:
FV = (1.005)12 = 1.0616778
The effective annual rate is just this number minus one:
EFF = 1.0616778 - 1 = 6.16778% per year
The general formula for the effective annual rate is:
EFF = (1 + APR/m)m - 1
where APR is the annual percentage rate, and m the number of compounding periods per year. Table 4.3 presents the effective annual rates corresponding to an annual percentage rate of 6% per year for different compounding frequencies.
Table 4.3 Effective annual rates for an APR
|Interest rate =||6%||Effective Annual Rate||Verify|
|Annually (m = 1) = 1.101 = 110.00||1||6.1000%||= ((1+0.06/1)1)-1|
|Semiannually (m=2)||2||6.0900%||= ((1+.06/2)2)-1|
|Quarterly (m=4)||4||6.1364%||= ((1+.06/4)4)-1|
|Monthly (m=12)||12||6.1678%||= ((1+.06/12)12)-1|
|Weekly (m=52)||52||6.1800%||= ((1+.06/52)52)-1|
|Daily (m=365)||365||6.1831%||= ((1+.06/365)365)-1|
Effective Annual Interest Rate
Let's say you are earning 6% per year compounded monthly.
The stated interest rate of 6% is the annual percentage rate (APR), the monthly interest rate, .05% (6%/12 = .05%) is the effective annual rate (EFF).
|EFF = FV = (1.005)12 = 1.0616778||6.17%||=1.00512-1|
|or EFF = (1 + APR/n)n - 1||6.17%||=((1+(0.06/12))12)-1|
How much will I earn if I deposit $100 compounded continuously with semi-annual interest at 8% for 3 years?
|FV||Discounted Back||Verify||FV||Discounted Back|
|EXP = exponent for continuous compounding = 2.71828[=100*(EXP((0.08/2)*6))]||$127.125||0.78663||$100.00||=100*(EXP((0.08/2)*6))||=1/(EXP(0.08/2*(3*2)))|
|$100 compounded continuously with semi-annual interest at 8% for 3 years =||$127.125||0.78663||$100.00||=(100*(2.71828)((0.08)*(3)))||=((1/(2.71828)((0.08)*(3)))|
|Normal annual compounding||$125.97||0.79383||$100.00||=100*1.083||=1/(1.083)|
|$100 compounded semi-annually at 8% for 3 years =||$126.53||0.79031||$100.00||=100*((1+(0.08/2))(2*3))||=1/((1+0.08/2)(6))|
|Future Value Continuous Compounding|
|$1000 to be received at the end of 10 years earning 20%, how much must I deposit today?||$135.335||7.38905||$1,000.00||=1000/(2.71828((0.2)*(10)))||=(2.71828((0.2)*(10))))|
Buy or Lease a Copy Machine
|After-Tax Cost Interest Effect 2021||25%||4.50%|
|After-Tax Cost Interest Effect 2022||25%||4.50%|
|Net (lower write off expense)||0.000%|
|Effect of Leverage (approximation) and After-tax Rate of Return||Pretax||After-tax||Leverage|
|Cost of equity||8%||5.760%||50%|
|Cost of debt (based on leverage)||6%||4.320%||7.20%||After-Tax Leverage Return|
|PV of an End-of-the-Period Annuity||0.000%|
|Answer: Pay cash for copy machine or finance it.||
Nominal Interest vs. Real Interest Rate on Bonds
The nominal interest rate (on a bond) is the promised amount of money you receive per unit you lend. The real rate of return is defined as the nominal interest rate you earn corrected for the change in the purchasing power of money. For example, if you earn a nominal interest rate of8% per year and the rate of inflation is 5% per year, then the real rate of return is (8%-5% = 3%). That is almost close. The real rate of return is actually $108/$103 = 2.857%
* inflation = CPI = 5%
The general formula relating the real rate of interest to the nominal rate of interest and the rate of inflation is:
1 + Real interest rate = 1 + Nominal interest rate / 1 + Rate of inflation
Real interest rate = Nominal interest rate - Rate of inflation / 1 + Rate of inflation
Real interest rate = .08 - .05 / 1.05 = .02857 = 2.857%
So what does this mean?
If inflation stayed at 2.857% for the next 45 years, how much would you need in order to buy the same $100 worth of goods in 45 years?
= (1 + .02857)45 = $355 or =FV(0.02857,45,,-100)
= (1+.08)45 = $3,192
Price level in 45 years = (1.05)45 = 8.985
Real FV = Nominal future value / Future price level =3192/8.985 = $355.26
The Balance Sheet
A firm's balance sheet shows its assets (what it owns) and its liabilities (what it owes) as a point in time.
The difference between assets and liabilities is the firm's net worth, also called owners equity. For a corporation, net worth is called stockholders' equity.
The Income Statement
The income statement summarizes the profitability of the firm over a period of time, in this case a year. Income, profit, and earnings all mean the same things: the difference between revenues and expenses.
Typically company expenses are broken down into four categories. The first is the cost of goods sold. This is the expense that a company incurred in producing the products and services is sells during the year, and includes the materials and labor used to manufacture them. The difference between sales or revenues and cost of goods sold is called gross margins.
The second expense category is general, administrative, and selling (SG&A). These represent the expenses incurred in managing the firm (such as the salaries of the managers) and in marketing and distributing the products produced during the year. The difference between gross margins and GS&A is called operating income.
The third category of expenses is the interest expense of the company's debt.
The forth expense is the company's income tax liability.
The Cash-Flow Statement
The statement of cash flows shows all of the cash that flowed into and out of the firm during a period of time. It differs from the income statement, which shows the firm's revenues and expenses.
The cash-flow statement is a useful supplement to the income statement for two reasons. First, it focuses attention on what is happening to the firm's cash position over time. Even very profitable company's can experience financial distress if they run out of cash. Paying attention to the cash flow statement allows the firm's managers and outsiders to see whether the firm is building up or drawing down its cash and to understand why. Often, for example, rapidly growing, profitable firms run short of cash and have difficulty meeting their financial obligations.
The cash-flow statement is also useful because it avoids the judgments about revenue and expense recognition that go into the income statement. The income statement is based on accrual accounting methods, according to which not every revenue is an inflow of cash and not every expense is outflow. A firm's reported net income is affected by many judgments on the part of management about issues such as how to value its inventory and how quickly to depreciate its tangible assets and amortize its intangible assets.
The statement of cash flows is not influenced by these accrual accounting decisions. Therefore, by examining the differences between the firm's cash flow statement and its income statement, one is able to determine the impact of these accounting decisions.
Your daughter is 10 years old, and you are planning to open an account to provide for her college education. Tuition for a year of college is now $15,000. You want to save in equal annual installments over the next eight years to have enough to pay for her first year's tuition eight years from now. If you think you can earn a real rate of interest of 3% per year, how much must you save each year? How much will you actually put into the account each year (in nominal terms) if the rate of inflation turns out to be 5% per year?
You Need to Save for College Tuition Eight Years from Now.
|Tuition today (per year)||$15,000.00|
|Cost of tuition will increase this much annual||5%|
|Interest earned on tuition savings (per year)||8%|
|How much will tuition be in eight years?||$22,161.83||FV = 15000 x 1.058 = $22,162|
|How much must you deposit today to cover the first year of tuition?||$11,973.44||PV = 22162/(1.08)8 = $11,973|
With this plan, the nominal amount saved each year has to be adjusted upward in accordance with the actual rate of inflation. The result will be that the amount accumulated in the account in eight years will be enough to pay for tuition. Thus if the rate of inflation turns out to be 5% per year, then the nominal amount in the account eight years from now will turn out to be $15,000 x 1.058 or $22,162. The tuition required eight years from now is $15,000 in real terms and $22,162 in nominal terms.
Often, we want to compute the present value rather than the future value of an annuity stream. For example, how much would you have to put into a fund earning an interest rate of 10% per year to be able to take out $100 per year for the next three years. The answer is the present value of the three cash flows.
The present value of the annuity is the sum of the present values of each of the three payments of $100:
PV = $100/1.101 + $100/1.102 + $100/1.103
Factoring out the constant payment of $100 per year, we get:
PV = $100 x (1/1.101 + 1/1.102 + $1/1.103
The result is a present value of $248.69. The factor multiplying the $100 payment is the present value of an ordinary annuity of $1 for three years at an interest rate of 10%.
Table 4.5 verifies that, indeed, $248.69 is what you would have to put in the account to be able to take out $100 each year for the next three years.
Column 3 proves that putting in $248.69 allows you to take out $100 per year for 3 years.
|PV = (1-((1+r)-n)) /i)||Years||Deposit/Withdrawals||Compounding Factor||Balance||Subtract each year||Balance|
Present Value Formula for an Annuity
You buy a fixed-income security that promises to pay $100 each year for the next 3 years. What is the 3-year instrument worth today?
|PV Formula for an Annuity||PV = 1 - (1 + i)-n/i|
|i = interest rate|
|n = years|
|A 1% increase in interest rates reduces the price by:||$4.87||=$267.30-$262.43|
|A 1% increase in interest rates reduces the value by:||1.86%||=4.87/$262.42|
Buying an Annuity vs. Bank Certificate of Deposit
You are 65 years old and are considering whether it pays to buy an annuity from an insurance company. For a cost of $10,000, the insurance company will pay you $1,000 per year for the rest of your life. If you can earn 8% per year on your money in a bank account and expect to live until age 80, is it worth buying the annuity? What implied interest rate is the insurance company paying you? How long must you live for the annuity to be worthwhile?
The most direct way to make this investment decision is to compute the present value of the payments from the annuity and compare it to the annuity's $10,000 cost. Assuming it is an ordinary annuity, then it is expected to make 15 payments of $1,000 each, starting at age 80. The present value of these 15 payments at a discount rate of 8% per year is $8,559.48.
|Cost of Annuity||$10,000|
|Number of payments in years||15|
|Bank CD interest rate||8.00%|
|Present value of annuity payments =1000*((1-(1+(0.08))-15)/(0.08)) = $8,559.48||$8,559.48|
|Net present value = $8559 - $10000 =$1,440.52||No, you would not buy this annuity (negative NPV).|
|What is the implied Annuity interest rate? =RATE(15,-1000,10000)||5.56%||=RATE(15,-1000,10000)|
|How many years would I have to live to make this annuity worthwhile earning 8% per year? =NPER(0.08,1000,-10000)||21||=NPER(0.08,1000,-10000)|
In other words, if a bank where offering you an interest rate of 5.56% per year, you could deposit $10,000 now and be able to withdraw $1,000 per year for the next 15 years.
In other words, to generate the same 15 annual payments of $1,000 each, it would be enough to invest $8,559.48 in a bank account paying 8% interest per year. Therefore, the net present value of the investment in the annuity is: NPV = $8,559.48 - $10,000 = -$1,440.52.
In other words, if a bank were offering you an interest rate of 5.56% per year, you could deposit $10,000 now and be able to withdraw $1,000 per year for the next 15 years.
To find the number of years one would have to live to make this annuity worthwhile, we must ask what value of n would make the NV of the investment 0. The correct answer is 21 years.
Looking at it differently, if you live for 21 years rather than 15, the insurance company would wind up paying you an implied interest rate of 8% per year.
A growing annuity is a cash flow that grows at a constant rate for a specified period of time. If A is the current cash flow, and g is the expected growth rate, the time line for a growing annuity appears as follows:
Note that, to qualify as a growing annuity, the growth rate in each period has to be the same as the growth rate in the prior period.
The Process of Discounting
In most cases, the present value of a growing annuity can be estimated by using the following formula:Growing Annuity = A(1 + g)(1-(1+g)n/((1+r)n))/r-g)
Your company is buying a copier, the cost is $10,000. Does it make sense to pay $10,000 cash today or finance the copier for five years at 12% with $3,000 annual payments?
Answer: $10,814 is greater than $10,000 so you would pay $10,000 cash.
|Growing Annuity = A(1 + g)(1-(1+g)n/((1+r)n))/r-g)||$12,755|
|PV =||$10,814||=3000*(1-(1/(1+.12)5))/.12 - .06|
|PV (Calculation End of Period)||$10,814||=PV(.12,5,-3000,0,0)|
|PV (Calculation Beginning of Period)||$10,205||=PV(.12,4,-3000,0,1)|
A gold miner has exclusive mining rights for 20 years, he can extract 5,000 oz. per year, the price of gold is $300/oz., the annual growth rate is 3%, the discount rate is 10%.
|Yr 1||20 yr Total|
|First year the miner can extract 5,000 * 300 = $1,500,000||$1,500,000||$17,760,577.98|
|What is the value of the gold rights today?|
|Gold production annually (oz)||5,000|
|Formula: =(1500000*(1+0.03))*((1-((1+0.03)20)/(1+0.1)20))/(0.1-0.03) = $16,145,980|
Are Thinking of Buying a House or Buying A Car? Use the Following Annuity Formula.
Getting a Mortgage loan
Let's look at an example of a financing decision. You have just decided to buy a house and need to borrow $200,000. One bank offers you a mortgage loan to be repaid over 30 years in 360 monthly payments. If the interest rate is 8% per year, what is the amount of the monthly payment? (Although the interest rate is quoted as an annual percentage rate (APR), the rate is actually 1/12th of 8% or .67% per month). Another bank offers you a 15-year mortgage loan with a monthly payment of $1,100. Which is the better deal?
The month payment on the 30-year mortgage is computed by using a monthly period (n = 360 months) and a monthly interest of .67%. The payment is $1,468 per month.
|Monthly Rate = 8%/12 = .67%||0.67%|
|Number of payments||360|
|PV or monthly payments: =PMT(.08/12,360,-200000,0,0)||$1,468|
|PV or monthly payments: =200000*((.08/12)/((1-(1/(1+(.08/12))360))))||$1,468|
|PV or monthly payments: =200000*((.67/(1-(1/1.067)360)))||$1,468|
At first glance, it might seem as if the 30-year mortgage is a better deal, since the $1,028.61 monthly payment is less than the $1,100 for the 15 year mortgage. But the 15-year mortgage is finished after only 180 payments. The monthly interest rate is .8677, or an annual percentage rate of 10.4%
What is the value of a one year $1,000 console bond with a 6% coupon, today's interest rate is 9%?
A very important special type of annuity is a perpetual annuity of perpetuity. A perpetuity is a stream of cash flow that continues forever. The classic example is the "consol" bonds issued by the British government in the nineteenth century which pay interest each year on the stated face value of the bonds but have no maturity date. Another example, and perhaps a more relevant one today, is a share of preferred stock that pays a fixed cash dividend each period (usually every quarter year) and never matures.
A disturbing feature of any perpetual annuity is that you cannot compute the future value of its cash flows because it is infinite. Nevertheless, it has perfectly well-defined and determinable present value. It might at first seem paradoxical that a series of cash flows that last forever can have a finite value today. But consider a perpetual stream of $100 per year. If the interest rate is 10% per year, how much is this perpetuity worth today?
The answer is $1,000. To see why, consider how much money you would have to put into a bank account offering interest of 10% per year in order to be able to take out $100 every year forever. If you put in $1,000, then at the end of the first year you would have $1,100 in the account. You would take out $100, leaving $1,000 for the second year. Clearly, if the interest rate stayed at 10% per year, and you had a fountain of youth nearby, you could go on doing this forever.
More generally, the formula for the present value of a level perpetuity is:
PV = C/i
where C is the periodic payment and ¡ is the interest rate expressed as a decimal fraction.
Coupon Console Bond
|Coupon Console Bond||6%|
|Current interest rate||9%|
|Value of bond||$667|
|Formula||=(1000 *.06)/.09 = $666.67|
Using present value formulas to value known cash flows (PV = 1 - (1 + i)-n/i).
In a world with a single risk-free interest rate, computing the present value of any stream of cash flows is relatively uncomplicated. It involves applying a discounted-cash-flow formula using the risk-free interest rate as the discount rate.
For example, suppose you buy a fixed-income security that promises to pay $100 each year for the next three years. How much is this three-year annuity worth if you know that the appropriate discount rate is 6% per year? The answer - $267.30 - can be found easily using excel or a financial calculator.
Recall that the formula for the present value of an ordinary annuity of $1 per period for n periods at an interest rate of ¡ is:
PV = (1-((1+r)-n))/i)
Now suppose that an hour after you buy the security, the risk-free interest rate rises from 6% to 7% per year, and you want to sell it. How much can you get for it?
The level of market interest rates has changed, but the promised future cash flows from you security has not. In order for an investor to earn 7% per year on your security, its price has to drop. How much? The answer is that it must fall to the point where its price equals the present value of the promised cash flows discounted at 7%.
At a price of $262.43, a fixed-income security that promises to pay $100 each year for the next three years offers its purchaser a rate of return of 7% per year. Thus, the price of any existing fixed-income security falls when market interest rates rise because investors will only be willing to buy them if they offer a competitive yield.
Thus, a rise of 1% in the inters rate causes a drop of $4.87 in the market value of your security. Similarly, a fall in interest rates causes a rise in its market value.
This illustrates the most basic principle in valuing know cash flows:
A change in market interest rates causes a change in the opposite direction in the market values of all existing contracts promising fixed payments in the future.
You buy a fixed-income security that promises to pay $100 each year for the next 3 years. What's that 3-yr instrument worth today?
Bond Price Change if Interest Rates Change
|I = interest rate|
|n = years||Value|
|A 1% increase in interest rates reduces the price by:||$4.87||=$267.30-$262.43|
|A 1% increase in interest rates reduces the value by:||1.86%||=$4.87/$262.43|
Discount Bond Price Change if Interest Rates Change
The Basic Building Blocks of a Zero-Coupon Bond
In valuing contracts promising a stream of know cash flows, the place to start is a listing of the market prices on pure discount bonds. (Also called pure discount bonds.) These are bonds that promise a single payment of cash at some date in the future, called maturity date.
Pure discount bonds are the basic building blocks in valuing all contracts promising streams of know cash flows. This is because we can always decompose any contract - no matter how complicated its pattern of certain future cash flows - into its component cash flows, value each one separately, and then add them up.
The promised cash payment on a pure discount bond is called its face value or par value. The interest earned by investor on pure discount bonds is the difference between the price paid for the bond and the face value received at the maturity date. Thus, for a pure discount bond with a face value of $1,000 maturing in one year and a purchase price of $950, the interest earned is the $50 difference between the $1,000 face value and the $950 purchase price.
The yield (interest rate) on a pure discount bond is the annualized rate of return to investors who buy it and hold it until it matures. For a pure discount bond with a one-year maturity like the one in our example we get:
Yield on 1-year pure discount bond = Face value - price / price
= 1000 - 950 / 1000 = .0526 or 5.26%
Coupon Bond Yields
|If you pay $950 for $1,000 bond maturing in one year, what is the yield (not YTM)?|
|Face Value =||$1,000|
|Purchase price =||$950|
|Yield =RATE(1,,-950,1000) = 5.26%||5.26%|
|Formula = 1000-950/950 = .0526 or 5.26%|
|What if you paid $1,050 for the same bond, what is the yield (not YTM)?|
|Yield =RATE(1,,-1050,1000)*-1 = 4.76%||4.76%|
|Formula: = 1050-1000/1050 = 4.76%||4.76%|
If, however, the bond has a maturity different from one year we would use the present value formula to find its annualized yield on this bond as the discount rate that makes its face value equal to its price.
Suppose that we observe the set of pure discount bond prices in Table 8.1. Following standard practice, the bond prices are quoted as a fraction of face value.
Table 8.1 Price of pure discount bonds and yields
|Maturity (in years)||Price per $1 of Face Value||Yield (per year)|
There are two alternative procedures that we can use to arrive at a correct value for the security. The first procedure uses the prices in the second column of Table 8.1, and the second procedure uses the yields in the last column Procedure 1 multiples each of the three promised cash payments by its corresponding per dollar price and then adds them up:
Present value of first year's cash flow = $100 x .95 = $95.00
Present value of second year's cash flow = $100 x .88 = $88.00
Present value of third year's cash flow = $100 x .8 = $80.00
Total = $263
The resulting estimate of the security's value is $263
Procedure 2 gets the same result by discounting each year's promised cash payment at the yield corresponding to that maturity:
|Present value of first year's cash flow = $100/1.0526 = $95.00||$95.00||=100/1.0526|
|Present value of second year's cash flow = $100 /1.0660 2 = $88.00||$88.00||= 100 /1.066022|
|Present value of third year's cash flow = $100 /1.0772 3 = $80.00||$80.00||= 100 /1.07723|
|Total = $263||$263.00||=95+88+80 = 263|
Note, however, that it would be a mistake to discount all three cash flows using the same three-year yield of 7.72% per year listed in the last row of Table 8.1. If we did so, we get a value of $259, which is $4 too low.
Is there a single discount rate that we can use to discount all three of the payments the way we did using PV = (1-((1+r)-n)) /i) to get a value of $263 for the security? The answer is yes: the single discount rate is 6.88% per year. To verify this, substitute 6.88% for ¡ in the formula for the present value on an annuity:
=100*(1-((1+0.0688)-3))/0.0688 = $263.01.
The problem is that the 6.88% per-year discount rate appropriate valuing the three-year annuity is not one of the rates listed anywhere in Table 8.1. We derived it from our knowledge using trial & error or a calculator and solving for ¡.
We can summarize the main conclusion from this section as follows: When the yield curve is not flat i.e., when observed yields are not the same for all maturities), the correct procedure for valuing a contract or a security promising a stream of known cash payments is to discount each of the payments at the rate corresponding to a pure discount bond of its maturity and then add the resulting individual-payment values.
A coupon bond obligates the issuer to make periodic payments of interest - called coupon payments- to the bondholder for the life of the bond, and then to pay the face value of the bond when the bond matures (i.e., when the last payment comes due.) The periodic payments of interest are called coupons because at one time most bonds had coupons attached to them that investors would tear off and present to the bond issuer for payment.
The coupon rate of the bond is the interest rate applied to the face value to compute the coupon payment. Thus, a bond with a face value of $1,000 that makes annual coupon payments at a coupon rate of 10% obligates the issuer to pay .10 x $1,000 = $100 every year. If the bond's maturity is six years, then at the end of six years, the issuer pays the last coupon of $100 and the face value of $1,000.
The cash flows from this coupon bond are displayed in Figure 8.2. We see that the stream of promised cash flows has an annuity component (a fixed per period amount) of $100 per year and a "balloon" or "bullet" payment of $1,000 at maturity.
|Coupon Bonds||Current Yield||YTM|
|Current yield = Coupon/Current Bond Price = 100/950, 10% rate||10.53%||15.79%|
|Current yield = Coupon/Current Bond Price = 100/1050, 10% rate||9.52%||4.76%|
|Current yield = Coupon/Current Bond Price = 100/950, 10% rate, 2 year maturity||13.00%|
|Current yield = Coupon/Current Bond Price = 100/1050, 10% rate, 2 year maturity||7.23%|
The $100 annual coupon is fixed at the time the bond is issued and remains constant until the bond's maturity date. On the date the bond is issued, it usually has a price (equal to its face value) of $1,000.
The relation between prices and yields on coupon bonds is more complicated than for pure discount bonds. As we will see, when the prices of coupon bonds are different from their face value, the meaning of the term yield is itself ambiguous.
Bond Yield to Maturity
Coupon bond - the right of a bond buyer to receive interest payments based on the bonds face value throughout the life of the bond at the coupon rate (e.g. coupon rate = 5% per year for 10 years).
Current yield = Coupon/Current Bond Price = $100/$1,000 = 10%
Coupon bonds with a market price equal to their face value are called par bonds. When a coupon bond's market price equals its face value, its yield is the same as its coupon rate. For example consider a bond maturing in one year that pays an annual coupon at a rate of 10% of its $1,000 face value. This bond will pay its holders $1,100 a year from now - a coupon payment of $100 and the face value of $1,000.
Thus, if the current price of our 10%-coupon bond is $1,000, its yield is 10%.
Current yield = Coupon/Current Bond Price = 100/950, 10% rate
Current yield = Coupon/Current Bond Price = 100/1050, 10% rate
Current yield = Coupon/Current Bond Price = 100/950, 10% rate, 2 year maturity
Current yield = Coupon/Current Bond Price = 100/1050, 10% rate, 2 year maturity
Bond Pricing Principle #1: Par Bonds
If a bond's price equals its face value, then its yield equals its coupon rate.
Often the price of a coupon bond and its face value are not the same. This situation would occur, for instance, if the level of interest rates in the economy falls after the bond is issued. So, for example, suppose that a one-year 10% coupon bond was originally issued as a 20-year-maturity bond 19 years ago. At that time, the yield curve was flat at 10% per year. Now the bond has one year remaining before it matures, and the interest rate on one-year bonds is 5% per year.
Although the 10%-coupon bond was issued at par ($1,000), its market price will now be $1,047.62. Since the bond's price is now higher than its face value, it is called a premium bond.
What is its yield?
There are two different yields that we can compute. The first is called the current yield, the annual coupon divided by the bond's current price.
Current yield = Coupon / Price = $100 / $1047 = 9.55%
ALWAYS KEEP IN MIND TWO THINGS - THE ORIGINAL COUPON RATE, AND THE BONDS FINAL PRICE EQUALS $1,000.
The current yield overstates the true yield on the premium bond because it ignores the fact that at maturity you will receive only $1,000 - $47.62 less than you paid for the bond.
To take account of the fact that a bond's face value and its price may differ, we compute a different yield called the yield-to-maturity. The yield-to-maturity is defined as the discount rate that makes the present value of the bond's stream of promised cash payments equal to its price.
The yield-to-maturity takes account of all the cash payments you will receive from purchasing the bond, including the face value of $1,000 at maturity. In our example, because the bond is maturing in one year, it is easy to compute the yield-to-maturity.
Figure 8.2 Cash flows for 10% $1,000 Coupon Bond
Yield-to-maturity = Coupon + Face Value at Maturity - Today's Price / Today's Price
Yield-to-maturity = $100 + $1000 - $1,047.62 / $1,047.62 = 5%
Thus, we see that if you used the current yield of 9.55% as a guide to what you would be earning if you bought the bond, you would be seriously misled.
When the maturity of a coupon bond is greater than a year, the calculation of its yield-to-maturity is more complicated than just shown. For example, suppose that you are considering buying a two-year 10% coupon bond with a face value of $1,000 and a current price of $1,100. What is its yield?
Its current yield is 9.09%.
Current yield = Coupon /Price = $100/$1100 = 9.09%
But as in the case of the one-year premium bond, the current yield ignores the fact that at maturity, you will receive less than the $1,100 that you paid. The yield-to-maturity when bond maturity is greater than one year is the discount rate that makes the present value of the stream of cash payments equal to the bond's price:
PV = ΣPMT/(1+ i)t + FV / (1 + i)t;
where n is the number of annual payment periods until the bond's maturity , i is the annual yield-to-maturity, PMT is the coupon payment, and FV is the face value of the bond received at maturity.
The yield-to-maturity on a multi-period coupon bond can be computed easily on most financial calculators by entering the bond's maturity as n, its price as PV (with a negative sign), its face value as FV, its coupon as PMT, and computing i.
Current Bond Price
|Current Bond Price||($1,100.00)||($1,100.00)||0|
|Year 1 10% Coupon||$100.00||$90.91||1|
|Year 2 10% Coupon||$100.00||$82.64||2|
|Bond price at maturity||$1,000.00||$826.45||2|
|YTM =RATE(2,100,-1100,1000) = 4.65%||4.65%|
Thus, the yield-to-maturity on this two-year premium bond is considerably less than its current yield.
These examples illustrate a general principle about the relations between bond prices and yields:
Bond Pricing Principle #2: Premium Bonds
If a coupon bond has a price higher than its face value, its yield-to-maturity is less than its current yield, which is in turn less than its coupon rate. For a premium bond:
Yield-to-maturity < Current yield < coupon rate
Now let us consider a bond with a 4% coupon rate maturing in two years and ten years. The price is $950, since the price is below the face value of the bond, we call it a discounted bond. (Note it is not a pure discount bond since it does pay a coupon.)
What is its yield? As in the previous case of a premium bond we can compute two different yields - the current yield and the yield to maturity:
The current yield = Coupon/Price = $40/$950 = 4.21%
The current yield understates the true yield in the case of the discount bond because it ignores the fact that at maturity you will receive more than you paid for the bond. When the bond matures, you will receive the $1,000 face value, not the $950 price that you paid for it.
The yield-to-maturity takes account of all of the cash payments you will receive from purchasing the bond, including the face value of $1,000 at maturity.
Bond Pricing Principle #3: Discount Bonds
If a coupon bond has a price lower than its face value, its yield to maturity is greater than its current yield, which is in turn greater than its coupon rate.
For discount bonds:
Yields-to-maturity > Current yield > Coupon rate
The current bond price is $950, the number of years remaining is 6, the principal amount is $1,000 and the annual coupon rate is 10%. Calculate the bond Yield-to-Maturity using trial and error.
|10% bond maturing in 6 yrs, current price is $950||Coupon 10%||4% bond maturing in 2 yrs, current price is $950|
|Cash Flows on a 10% $1,000 Coupon Bond||($950.00)||($950.00)||0||($950.00)||($950.00)||0|
|Calculating Bond Yield-to-Maturity (trial and error)||$1,000.00||$564.47||6||$1,000.00|
Often we observe that two US Treasury bonds with the same maturity have different yields to maturity. Is this a violation of the Law of One Price? The answer is no. In fact, for bonds with different coupon rates, the Law of One Price implies that, unless the yield curve is flat, bonds of the same maturity will have different yields to maturity.
The Effect of the Coupon Rate
For example, consider two different two-year coupon bonds - one with a coupon rate of 5% and the other with a coupon rate of 10%. Suppose the current market price and yields of one - and two-year pure discount bonds are as follows:
The Effect of the Coupon Rate
|Current Market Price per $1 of||Yield||Yield||Market Price at||Market Price at||Market Price at|
|Maturity||Face Value = 32nds)||(per year)||Coupon||32nd's||(per year)||4% & 6%||5%||10%|
|1 year||$.961538 =(100/1.04)||4.00%||5.00%||4.9216||96 4.92/32||4.00%||$961.54||$952.38||$909.09|
|2 years||$.889996 =(100/1.06)2||6.00%||10.00%||31.9872||88 31.98/32||6.00%||$890.00||$907.03||$826.45|
According to the Law of One Price the first-year cash flows from each coupon bond must have a per-dollar price of $.961538, and the second-year cash flows must have a per-dollar price of $.88996. Therefore the market prices of the two different coupon bonds should be:
Market Price of Two Different Coupon Bonds
|Price||Yield to Maturity|
|1 year 5% bond at .961538 = .961538 x $50 + .88996 x $1,050 = $982.53||= .961538 x $50 + .88996 x $1,050 = $982.53||982.53||5.95%||=RATE(2,50,-982.53,1000)|
|2 year 10% bond at .889996 = .961538 x $100 + .88996 x $1,100 = $1,075.11||= .961538 x $100 + .88996 x $1,100 = $1,075.11||$1075.11||5.91%||=RATE(2,100,-1075.11,1000)|
|Yield to Maturity -- Two Year Bond||Cash flow||Maturity||YTM (trial an error)||Verified Price|
|$982.57 =50*(1+0.595)-1+50*(1+0.0595)-2+1000*(1+0.0595)-2 = $982.57||50||1000||5.95%||$982.57||=50*(1+0.595)-1+50*(1+0.0595)-2+1000*(1+0.0595)-2 = $982.57|
|$1075.15 =100*(1+0.0591)-2+100*(1+0.0591)-2+1000*(1+0.0591)-2||100||1000||5.91%||$1,075.08||=100*(1+0.0591)-1+100*(1+0.0591)-2+1000*(1+0.0591)-2 = $1075.08|
Thus, we see that in order to obey the Law of One Price, the two bonds must have different yields to maturity. As a general principle:
When the yield curve is not flat, bonds of the same maturity with different coupon rates have different yields to maturity.
The Behavior Of Bond Pricing Over Time
In this section we examine how bond prices change over time as a result of the passage of time and changes in interest rates.
If the yield curve were flat and interest rates did not change, any default-free discount bonds would rise with the passage of time, and any premium bonds would fall. This is because eventually bonds mature, and their price must equal their face value at maturity. We would therefore expect the prices of discount bonds and premium bonds to move toward their face value as they approach maturity. This implied price pattern is illustrated for the case of 20-year pure discount bond shown to the right.
Let us illustrate the calculation assuming the face value of bond is $1,000 with 6% yield with a 20 year maturity.
Note the price at year 20 and the coupon and how both change year by year.
The Effect of the Passage of Time
|=1/(1.06)20 x 1,000 = $311.80|
|Price (19 years to maturity) =|
|=1/(1.06)19*1000 = $330.51|
|Years||Present Value||Price if bond matures at Years -1||6%||Proportional change in price|
Interest Rate Risk
Normally, we think of buying U.S. Treasury bonds as a "conservative" investment policy because there is no risk of default involved. However, an economic environment of changing interest rates can produce big gains or losses for investors in long-term bonds.
Figure 8.4 illustrates the sensitivity of long-term bond prices to interest rates. It shows the magnitude of the changes that would occur in the prices of 30-year pure discount bonds and 30-year 8% par bonds if the level of interest rates moved to a value different from 8% immediately after the bonds are purchased. Each curve in Figure 8.4 corresponds to a different bond. Along the ordinate we measure the ratio of the bond's price computed using the indicated interest rate to its price computed at a discount rate of 8%.
Figure 8.4 Sensitivity of Bond Price to Interest Rates
For example, at an interest rate of 8% per year, the price of a 30-year 8% coupon bond with a face value of $1,000 would be $1,000, while at an interest rate of 9% per year its price is $897.26. The ratio of its price at a 9% interest rate to its price at 8% interest rate is therefore 897.26/1,000 = .89726. We can therefore say that if the level of interest rates were to rise from 8% to 9%, the price of the par bond would fall by roughly 10%.
The figure shows the magnitude of the changes that would occur in the prices of 30-year pure discount bonds and 30-year 8% coupon par bonds if the level of interest rates moved to a value different from 8% immediately after the bonds are purchased. The ordinate measures the ratio of the bond's price computed at the indicated interest rate to its price computed at a discount rate of 8%. Thus at an interest rate of 8%, the price of the pure discount bond would fall by roughly 23%.
Note in Figure 8.4 that the curve corresponding to the pure discount bond is steeper than the par bond's curve. This greater steepness reflects its greater interest-rate sensitivity.
Convertible Bond Conversion
A convertible bond is a bond that can be converted into a predetermined number of shares, at the option of the bondholder. While it generally does not pay to convert at the time of the bond issue, conversion becomes a more attractive option as stock prices increase. Firms generally add conversion options to bonds to lower the interest rate paid on the bonds.
In a typical convertible bond, the bondholder is given the option to convert the bond into a specified number of shares of stock. The conversion ratio measures the number of shares of stock for which each bond may be exchanged. The market conversion value is the current value of the shares for which the bonds can be exchanged. The conversion premium is the excess of the bond value over the conversion value of the bond.
Thus a convertible bond with a par value of $1,000, which is convertible into 50 shares of stock, has a conversion ratio of 50. The conversion ratio can also be used to compute a conversion price - the par value divided by the conversion ratio, yielding a conversion price of $20. If the current stock price is $25, the market conversion is $1,250 (50 x $25). If the convertible bond is trading at $1,300, the conversion is $26.
|Bond (at maturity)||$1,000|
|Current interest rate (for bond)||5.25%|
|Years to maturity in years||19|
|Convertible Bond Value Today||$1,064|
|Value of Bond (note negative exponent) (=(0.0213*1000)*((1-(1+0.0525)-19)/0.0525)+(1000/((1+0.0525)19)))||$631|
|Equity (Value of Equity Component = Price of Convertible Bond - Value of Straight Bond Component= $1,064 - 629.91 = $434.09)||$433|
|Straight Bond Component = $64 (PVA, 19 years, mkt % rate 5.25%) = 1000/1.0519 = 629.19|
|PVA=Present Value of Annuity|
Estimating Market Value of Debt
The market value of debt can be difficult to obtain directly from a publicly traded company, since very few firms have all their debt in the form of bonds outstanding trading in the market. Many firms have nontraded debt, such as bank debt, which is specified in book value terms but not market value terms. A simple way to convert book value debt into market value debt is to treat the entire debt on the books as one coupon bond, with a coupon set equal to the interest expense on all the debt and the maturity set equal to the face-value weighted average maturity of the debt, and then to value this coupon bond at the current cost of debt for the company. Thus, the market value of $1 billion in debt, with interest expenses of $60 million and a maturity of six years, when the current cost of debt is 7.5 percent, can be estimated as follows:
|Market Value of Bonds||$1,000|
|Current Interest Rates (Cost of Debt)||7.50%|
|Periods (in years)||6.00|
|Estimating Market Value of Debt||$930|
|Formula =(.06*1000)*(1-(1/(1+.075)6))/.075+(1000/(1+.075)6) = $930|
Example 1 Current Market Value of a Bond
|Market Value of Bonds||$1,000|
|Current Interest Rates (Cost of Debt)||7.50%|
|Periods (in years)||6.00|
|Estimating Market Value of Debt||$930|
|Formula =(.06*1000)*(1-(1/(1+.075)6))/.075+(1000/(1+.075)6) = $930|
Example 2 Annual Compounding
|You borrow $1,000 for 3 yrs at 7% (compounded annually), how much do you have to pay back?||$1,225||=FV(0.07,3,0,-1000,0)|
|What is the present value of $1,225 at 7.00% (compounded annually) for 3 years?||($1,000)||=PV(0.07,3,,1225,1)|
|What's the interest rate (compounded annually) if you borrow $1,000 and after 3 years you end up paying back $1,225?||7.000%||=RATE(3,,-1000,1225,1)|
|You borrow $1,000 at 7.00% (compounded annually) per year, how many years will it take to pay back $1,225?||3||=NPER(.07,,-1000,1225,0)|
Example 3 Future Value
|$1,000 has accumulated to $3,000 in 8 yrs, what is the annual growth rate (compounded annually)?||14.720%||=RATE(8,0,-1000,3000,0)|
|What is the future value of $1,000 at 8.00% (compounded annually) for 8 years?||3,000||=FV(0.1472,8,,-1000)|
|What's the interest rate (compounded annually) if you borrow $1,000 and after 8 years you end up paying back $1,472?||14.720%||=PV(0.1472,8,,-3000)|
|You borrow $1,472 at 8.00% (compounded annually) per year, how many years will it take to pay back $3,000?||8||=NPER(0.1472,0,-1000,3000)|
Example 4 Future Value
|Let's say you have a $5,500 investment and you add $500 each month (at the end of each month) earning .75% per month (compounded monthly), how much will you have earned after 3 years?||$27,774||=FV(0.0075,36,-500,-5500,0)|
Example 5 Average Annual Return
|You had an account balance five years ago of $25,000, you deposit $4,500 at the end of each year. Current balance is $70K, what is the average annual return (compounded annually)?||10.94%||=RATE(5,-4500,-25000,70000,0)|
Example 6 Annuity
You deposit $1,000 per month (end of month [compounded monthly]) earning 2.06% annually for the next 10 years with the intent of accumulating a $1 million, how much should you deposit now if the account earns .1717% rate per month (2.06% annually)?
|Annual Compounding Rate||2.06%||Deposit|
|Monthly Compounding Rate = 2.06/12 = 0.1717%||0.1717%||=0.0206/12|
|With monthly compounding: (=PV(0.001717,120,-1000,1000000,0))||$1,000,038||=FV(0.1717%,120,-1000,-705614,0)||($705,583)||=PV(0.1717%,120,-1000,1000000,0)|
|With annual compounding: (=PV(0.0206,10,-1000,1000000,0))||$1,000,000||=FV(0.0206,10,-1000,-806584,0)||($806,584)||=PV(0.0206,10,-1000,1000000,0)|
Example 7 Present Value
|What is the present value of $25,000 in 5 years at 6.5% per year (compounded annually)? =PV(0.065,5,,-25000,0) = $18247||$18,247|
|What is the future value of $18,247 in 5 years at 6.5% per year (compounded annually)? =FV(0.065,5,,-18247.02) = $25,000||$25,000|
Example 8 Monthly Annuities - Investment Yields
Assume that an investor makes an investment of $51,593 to receive $400 at the end of each month for the next 20 years (240 months). What annual rate of return, compounded monthly, would be earned on the $51,593?
|How much will you receive each month if you deposit $51,592 earning 7.0% interest for the next 20 years? =PMT(0.07/12,240,-51593,0,0) = $400||$400||=PMT(0.07/12,240,-51593,0,0)|
|What is the initial investment required to receive $400 per month for the next 20 years earning 7% interest? =PV(0.07/12,240,-400) = $51,593||$51,593||=PV(0.07/12,240,-400)|
|What annual rate of return, compounded monthly, would be earned on the $51,593? =RATE(240,-400,51593,0,0)*12 = 7.0%||7.00%||=RATE(240,-400,51593,0,0)*12|
|How many months will it take an initial deposit of $51,592 earning 7.0% annually with $400 monthly withdrawals until it reaches zero? =NPER(0.07/12,-400,51593,0,0)||240||=NPER(0.07/12,-400,51593,0,0)|
|What is the total interest payment (20 years) minus the $400 monthly dividend? =FV(0.07/12,240,-400,51593) = 1.05%||1.05%||=FV(0.07/12,240,-400,51593)|
We find the interest factor for the present value of an ordinary annuity of $1 per month for 20 years for an interest rate compounded monthly. The corresponding interest rate is 7 percent. Hence, the IRR is 7 percent compounded monthly on the $51,593 investment. Both the recovery of $51,593 plus $44,407 in interest was embedded in the stream of $400 monthly cash receipts (dividend) over the 20-year period.
|Using "Rate", invest $51,593 and receive $400 per month for 20 years, what is the compounding rate? =RATE(20*12,-400,51593,,0)*12 = 7.00%||7.00%|
|Effective yield 6% compounded monthly =EFFECT(0.06,12) = 6.17%||6.17%|
|Nominal yield 6% compounded monthly =NOMINAL(0.06,12) = 5.84%||5.84%|
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