# Futures, Forwards and Zeros

### Zero-Coupon Bond Present Value Calculations

The promised cash payment on a pure discount bond is called its face value or par value. The interest earned by investors 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 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 maturity. For a pure discount bond with one-year maturity, we get:

Yield on 1-year pure discount bond = Face Value - price/price

=(1000-950) / \$950 = .0526 or 5.26%

If, however, the bond has a maturity different from one year we would use the present value formula to find its annualized yield. Thus, suppose that we observe a three-year discount bond that promises to pay \$100 each year and a price of ? . We would compute the annualized yield on this bond as the discount rate that makes its face value equal to its price:

 Maturity (in years) Price per \$1 of Face Value Yield (per year) 1 0.95 5.26% 2 0.88 6.60% 3 0.8 7.72% Annualized Yield 6.88%

The prices of money market instruments to arrive at a correct value for the security. The first uses the prices in the second column, and the second uses the yields in the third column. Procedure 1 multiplies each of the three promised cash payments by its corresponding per-dollar price and then add them up.

Present value of first year's cash flow = \$100 x .95 = \$95
Present value of second year's cash flow = \$100 x .88 = \$88
Present value of third year's cash flow = \$100 x .80 = \$80
Total present value = \$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
Present value of second year's cash flow = \$100 / 1.066 = \$88
Present value of third year's cash flow = \$100 / 1.0772 = \$80
Total present value = \$263

It would be a mistake to discount all three cash flows using the same three-year yield of 7.72% per year listed, if we did the value would be:

Present value of year's 1-3 cash flows = \$300 / 1.0772 = \$278

Is there a single discount rate that we can use to discount all three of the payments to get a value of \$263 for the security? Yes. That single discount rate is:

Calculating Bond Yield-to-Maturity (trial and error)
Trial and Error
6.88%
\$263.01
=100*(1 + .0688)-1+100*(1 + .0688)-2+100*(1 + .0688)-3 = \$263.01

The problem is that the 6.88% per-year discount rate appropriate for valuing the three-year annuity is not one of the rates listed anywhere in the chart above. The only way of calculating the YTM is trial and error.

### Yield to Maturity and Spot Rates [viewable here in Excel]

Extract spot rates from the yields to maturity

Spot rate is estimated from the two-year rate as follows:

The following table provides prices and yields on one- to five-year bonds , and extracts spot rates from yields to maturity:

The spot rate is estimated from the two-year rate as follows:

Price of two-year bond = Coupon / (1 + r¹) + (Face value + Coupon²)/(1+r²)²

Assuming the bond is priced at par:

1000 = 42.50/1.04 + 1042.50/(1 + 0r2

Solving for 0r2,

0r2 = sqrt(1042.50/(1000-42.50/1.04))-1 = 4.2553%

 Maturity Yield to Maturity Spot Rate 1 year 4.00% 4.00% 2 year 4.25% 4.26% = SQRT(1042.5/(1000-42.5/1.04))-1 3 year 4.40% 4.41% = SQRT(1044/(1000-44/1.04))-1 4 year 4.50% 4.51% = SQRT(1045/(1000-45/1.04))-1 5 year 4.58% 4.59% = SQRT(1045.8/(1000-45.8/1.04))-1

 Year Spot Rate Verify 1 2 4.26% 1000 = 42.50/1.04 + 1042.50/(1+0r2)² \$1,000 = 42.5/1.04 + 1042.5/((1+0.0426)2) 3 4.41% 1000 = 44/1.04 + 44/1.0426² + 1,044/(1+0r3)³ \$1,000 = 44/1.04 + 44/(1.04262) + 1044/((1+0.0441)3) 4 4.51% 1000 = 45/1.04 + 45/1.0426² + 45/1.044³ + 1,045/(1+0r4)4 \$1,000 = 45/1.04 + 45/1.04262 + 45/1.0443 + 1045/(1+0.0451)4 5 4.59% 1000 = 45.80/1.04 + 45.80/1.0426² + 45.80/1.044³ +45.80/1.04514 + 1,045.80/(1+0r5)5 \$1,000 = 45.8/1.04 + 45.8/1.04262 + 45.8/1.0443 +45.8/1.04514 + 1045.8/(1+0.0459)5

The spot rate is estimate from the two-year rate as follows:

Price of two-year bond = Coupont / (1 + 0r1) = (Face value + Coupon2) / (1 + 0r2)2

Assuming the bond is priced at par,

1000 = 42.50/1.04 + 1042.50/(1+0r2

Solving for 0r2:

 Year Spot Rate Verify 1 2 4.26% 1000 = 42.50/1.04 + 1042.50/(1+0r2)² \$1,000 = 42.5/1.04 + 1042.5/((1+0.0426)2) 3 4.41% 1000 = 44/1.04 + 44/1.0426² + 1,044/(1+0r3)³ \$1,000 = 44/1.04 + 44/(1.04262) + 1044/((1+0.0441)3) 4 4.51% 1000 = 45/1.04 + 45/1.0426² + 45/1.044³ + 1,045/(1+0r4)4 \$1,000 = 45/1.04 + 45/1.04262 + 45/1.0443 + 1045/(1+0.0451)4 5 4.59% 1000 = 45.80/1.04 + 45.80/1.0426² + 45.80/1.044³ +45.80/1.04514 + 1,045.80/(1+0r5)5 \$1,000 = 45.8/1.04 + 45.8/1.04262 + 45.8/1.0443 +45.8/1.04514 + 1045.8/(1+0.0459)5

The difference between yields to maturity and spot rates increases as the bond maturity increases.

### Spot and Forward Rates

The spot rate on a multiperiod bond is an average rate that applies over the periods. The forward rate is one-period rate for a future period, and can be extracted from the spot rates. For instance, if 0S2 is the two-period spot rate, and 0S1 is the one-period spot rate, the forward rate for the second period, 1F2, can be obtained as follows:

1F2 = (1 + 0S2)² / (1 + 0S1) - 1

The forward rate for period 3 can be extracted using the spot rate for periods 2 and 3, and in general, the forward rate for period x can be written as:

n-1Fn = (1 + 0Sn)n / (1 + 0Sn-1)n-1 - 1

The following rates are extracted from the spot rates for one- to five-year bonds. This is illustrated in the following table:

### Spot Rates and Forward Rates

 Maturity Yield to Maturity Spot Rate Forward 1 year 4.00% 4.00% 4.00% 2 year 4.25% 4.26% 4.52% 3 year 4.40% 4.41% 4.71% 4 year 4.50% 4.51% 4.81% 5 year 4.58% 4.59% 4.96%

### Determining Treasury Zero Rates Using the Boot Strap Method [viewable here in Excel]

Another way to determine Treasury zero rates is from Treasury bills and coupon-bearing bonds. The most popular approach is know as the bootstrap method. To illustrate the nature of the method, consider the data below on the prices of five bonds. Because the first three bonds pay no coupons, the zero rates corresponding to the maturities of these bonds can easily be calculated. The 3-month bond provides a return of 2.5 in 3 months on an initial investment of 97.5. With quarterly compounding, the 3-month zero rate is (4 x 100-97.5)/97.5 = 10.256% per annum. With continuous compounding the rate becomes =(1/.25)*LN(1+(0.10256/(1/.25))) = 10.127% per annum.

The 6 month bond provides a return of 5.1% in 6 months 100-94.9 = 5.1% on an initial investment of 94.9. With semiannual compounding the 6-month rate is (2 x 5.1)/94.9 = 10.748% per annum. With continuous compounding it becomes 10.469%.

 Bond principal (USD) Time to maturity (yrs) Annual Coupon USD Bond price USD Zero-rate w/Normal Compounding Zero-rate (w/continuous compounding) 100 0.25 0 97.5 10.256% 10.127% 100 0.50 0 94.9 10.748% 10.469% 100 1.00 0 90.0 11.111% 10.535% 100 1.50 8 96.0 10.681% 100 2.00 12 101.6 10.808%

 What is the zero-rate for the .75 yr bond? 10.50% What is the zero-rate for the 1.25 yr bond? 10.61% =0.5*0.10535+0.5*0.10681 What is the zero-rate for the 1.75 yr bond? 10.74% =0.5*0.10681+0.5*0.10808

Solving for 1.5 years, we know that the discount rate for the payment at the end of 6 months is 10.469% and that the discount rate for the payment at the end of 1 year is 10.536%. We know that the bond's price, US 96, must equal the present value of all the payments received by the bondholder. Suppose the 1.5 zero rate is denoted by R. It follows (the 4 represent the semiannual interest payment):

 4exp-.10469*.5 + 4exp-.10536*1.0 + 104exp-Rx1.5 = 96 SOLVE for R 96.00 R = ln(.85196)/1.5 = .10681 10.681% 1.5 year Zero rate Trial and error 10.681% =4*exp(-0.10469*0.5) + 4*exp(-0.10536*1) + 104*exp(-.10681*1.5) = 96 or the hard way (discount each payment) 3.796005619 4*exp(-0.10469*0.5) 3.600001856 4*exp(-0.10536*1) 7.396007476 = 3.60000+3.79600 88.604 =96-7.396 Solving for 1.5 years 0.851961538 =(96-(4*exp(-0.10469*0.5) + 4*exp(-0.10536*1)))/104 0.851961467 10.68093% =-LN(B189)/1.5 1.5 year Zero rate 88.6039952 =104*exp(-0.1068093*1.5) 104 is found in the above formula 96.00 = 88.60+3.60000+3.79600 Bond price

Solving for 2.0 years, we know that the discount rate for the payment at the end of 6 months is 10.469% and that the discount rate for the payment at the end of 1 year is 10.536% and the payment at the end of 1.5 years is 10.681%. We know that the bond's price, US 101.6, must equal the present value of all the payments received by the bondholder. Suppose the 2.0 zero rate is denoted by R. It follows:

 Trial and error 6exp-.10469*.5 + 6exp-.10536*1.0 + 6exp-.10681Rx1.5 + 106exp-R*2.0= 101.6 SOLVE for R 101.6 10.808% R = ln(.805605175)/2.0 = .108080757 or 10.808% 2 year Zero rate the hard way (discount each payment) 5.694008429 6*exp(-0.10469*0.5) 5.400002785 6*exp(-0.10536*1) 5.111840264 6*exp(-0.1068*1.5) 16.20585148 85.39414852 0.805605175 10.808% 2 year Zero rate 85.39414852 101.6 Bond price

Calculation of Forward Rates using Zero-Rates (e.g., Spot rate)

F2 = (1 + S2)² / (1 + S1) - 1
r-1Fr = (1 + Sr)² / (1 + Sr-1)r-1 - 1

 Year (n) Zero rate for an n-year investment (% per yr) Forward rate for nth year (% per annum) Verify Verify Forward rate calculated (see below) Equal Zero rate (w C/C) 1 3.00% 3.00% \$103.05 103.05 3.00% Verify Verify Verify Verify 2 4.00% 5.00% \$108.33 108.33 5.00% =4% + (4% - 3%)*(1/(2-1)) 5.00% 4.00% =(0.03+0.05)/2 3 4.60% 5.80% \$114.80 \$114.80 5.80% =4.6% + (4.6% - 4%)*(2/(3-2)) 5.80% 4.60% =(0.03+0.05+0.058)/3 4 5.00% 6.20% \$122.14 \$122.14 6.20% =5% + (5% - 4.6%)*(3/(4-3)) 6.20% 5.00% =(0.03+0.05+0.058+0.062)/4 5 5.30% 6.50% \$130.34 \$130.34 6.50% =5.3% + (5.3% - 5%)*(4/(5-4)) 6.50% 5.30% =(0.03+0.05+0.058+0.062+0.065)/5

### Compounding [viewable here in Excel]

 Zero rate Continuous Compounding Normal Compounding Normal Semiannual Compounding Normal Qtrly Qtrly Compounding Continuous (365 days) This is for year 5 For 1 year - zero rate @ 5.30% \$105.44 \$105.30 \$105.37 \$105.41 \$105.44 This is for year 5 For 2 years - zero rate @ 5.30% \$111.18 \$110.88 \$111.03 \$111.10 \$111.18 This is for year 5 For 3 years - zero rate @ 5.30% \$117.23 \$116.76 \$116.99 \$117.11 \$117.23 This is for year 5 For 4 years - zero rate @ 5.30% \$123.61 \$122.95 \$123.27 \$123.44 \$123.61 This is for year 5 For 5 years - zero rate @ 5.30% \$130.34 \$129.46 \$129.89 \$130.12 \$130.34

 Zero-rates Zero-rates Forward Rates Forward Rates 3.00% 3.00% 4.00% Zero 3% * 4% \$108.33 = 5% for two years 5.00% =4% + (4% - 3%)*(1/(2-1)) Rv = R2 + (R2 - R1)*(T1/(T2 - T1)) 4.60% Zero 4% * 4.6% \$114.80 = 5.8% for three years 5.80% =4.6% + (4.6% - 4%)*(2/(3-2)) Rv = R3 + (R3 - R2)*(T 2/(T3 - T2)) 5.00% Zero 4.6% * 5% \$122.14 = 6.2% for four years 6.20% =5% + (5% - 4.6%)*(3/(4-3)) Rv = R4 + (R4 - R3)*(T 3/(T4 - T3)) 5.30% Zero 5% * 5.3% \$130.34 = 6.5% for five years 6.50% =5.3% + (5.3% - 5%)*(4/(5-4)) Rv = R5 + (R5 - R4)*(T 4/(T5 - T4))

 Forward Rates Zero-rates Zero-rates Forward 3% * 5% \$108.33 = 4% for two years 4.00% =100*exp(0.03*1)*exp(0.05*1) Forward 5% * 5.8% \$114.80 = 4.6% for three years 4.60% =100*exp(0.03*1)*exp(0.05*1)*exp(0.058*1) Forward 5.8% * 6.2% \$122.14 = 5.0% for four years 5.00% =100*exp(0.03*1)*exp(0.05*1)*exp(0.058*1)* exp(0.062*1) Forward 6.2% * 6.5% \$130.34 = 4.6% for five years 5.30% =100*exp(0.03*1)*exp(0.05*1)*exp(0.058*1)* exp(0.062*1)*exp(0.065*1)

Forward interest rates are the rates of interest implied by current zero rates for periods of time in the future. To illustrate how they are calculated, we suppose that a particular set of zero rates are as shown above. The rates are assumed to be continuously compounded. Thus the 3% per annum rate for 1 year means that, in return for an investment of USD \$100 today, an investor receives 100 x exp0.03 x 1 = 103.05 in 1 year; the 4% per annum rate for 2 years means that, in return for an investment of USD \$100 today, the investor receives 100 x exp0.04*2 = 108.33 in 2 years; and so on.

The forward interest rate listed above for year 2 is 5% per annum. This is the rate of interest that is implied by the zero rates for the period of time between the end of the first year and the end of the second year. It can be calculated for the 1-year zero interest rate of 3% per annum and the 2-year zero interest rate of 4% per annum. It is the rate of interest for year 2 that, when combined with 3% per annum for year 1, gives 4% overall for the 2 years. To show that the correct answer is 5% per annum, suppose that USD \$100 is invested. A rate of 3% for the first year and 5% for the second year gives:

100exp.03*1 *exp.05*1 = 108.33

at the end of the second year. A rate of 4% per annum for 2 years gives

100exp.04*2 = 108.33

This example illustrates the general result that when interest rates are continuously compounded and rates in successive periods are combined, the overall equivalent rate is simply the average rate during the whole period. In our example 3% for the first year and 5% for the second year average 4% over the 2 year period. The result is only approximately true when the rates are not continuously compounded.

The forward rate for year 3 is the rate of interest that is implied by a 4% per annum 2-year zero rate and a 4.6% per annum 3-year zero rate. It is 5.8% per annum. The reason is that an investment for 2 years at 4% per annum combined with an investment for one year at 5.8% per annum gives an overall average return for the three years of 4.6% per annum.

The other forward rates can be calculated similarly and are shown above.

In general, if R1 and R2 are the zero rates for maturities T1 and T2, respectively, and Rv is the forward interest rate for the period of time between T1 and T2, the

Rv = R2 + (R2 - R1)*(T1/(T2 - T1))

### Zero Rate for Year 4

 Rv = R2 + (R2 - R1)*(T1/(T2 - T1)) T = Year T1 = 3 R = Zero rate at time (T) T2 = 4 R1 = 0.046 Zero rate for year 3 R2 = 0.05 Zero rate for year 4 6.20% Forward rate for year 4 =0.05+(0.05-0.046)*(3/(4-3)) = 6.20%

Assume that the zero rates for borrowing and investing are the same (which is close to the truth for a large financial institution), an investor can lock in the forward rate for a future time period. Suppose, for example that the zero rates are as shown above. If an investor borrows USD 100 at 3% for 1 year and then invests the money at 4% for 2 years, the result is a cash outflow of 100*exp(.03x1) = \$103.05 at the end of year 1 and an inflow of 100*exp(.04x2)=108.33 at the end of year 2. Since 108.33 = 103.05 x exp(.05), a return equal to the forward rate (5%) is earned on USD 103.05 during the second year (think compounding of \$103.05).

 Discounted at 3% Continuous Years 4% 1 100.00 104.08 1.00 2 104.08 108.33 4.12 5.12 What's earned on the 103.05 in year 2? 103.05 It is: = 104.08 * .04 = 108.33 - 103.05 = \$5.28 5.2800 5.12 5.0% = (108.33 - 103.05 / 103.05) - 1 = 5% 0.05 108.33 109.42

### Forward Rate Agreement (FRA) Valuation

Suppose that LIBOR zero and forward rate are as in the rows above. Consider an FRA where we will receive a rate of 6%, measured with annual continuous compounding, on a principal of USD \$100 million between the end of year 1 and the end of year 2. In this case, the forward rate is 5% with continuous compounding or 5.127% with annual compounding. From the below formula, it follows that the value of the FRA is:

Forward Rate = 5.0%
With "continuous compounding" =exp(0.05)-1 = 5.127%

 Vr = L(R1 - R2)(T2 - T1)exp-R2T2 100000000*((0.06-0.05127)*(exp(-0.04*2))) Value or cost to "Y" \$805,881 =100000000*((0.06-0.05127)*(exp(-0.04*2))) This is what "Y" would pay "X". 0.923116346 =exp(-0.04*2) Why would "X" agree to this FRA? 0.00873 =(0.06-0.05127) 0.008058805 =0.00873*0.9231163 \$805,880.50 =0.008058805*100000000 \$99,194,119.00 =100000000-805881 1.00812428 =100000000/99194119 \$100,000,000 =1.00812428*99194119

### Duration

The duration of a bond, as it names implies, is a measure of how long on average the holder of the bond has to wait before receiving cash payments. A zero-coupon bond that lasts n years has a duration of n years. However, a coupon bond-bearing bond lasting n years has a duration of less than n years, because the holder receives some of the cash payments prior to year n.

Duration is the ratio of the present value of the cash flow at t1 to the bond price. The bond price is the present value of all payments. The duration is therefore a weighted average of the times when payments are made, with the weight applied to time t1 being equal to the proportion of the bond's total present value provided by the cash flow at t1. The sum of the weights is 1.0. Note that for the purposes of the definition of duration, all discounting is done at the bond yield rate of interest y. (We do not use a different zero rate for each cash flow as described in R1 = m(exprn -1))

When a small change ΔY in the yield is considered, it is approximately true that:

ΔB = dB/dY*ΔY, this can be written as:

ΔB = -BD/dY*ΔY or ΔB/B = -D*ΔY

ΔB/B = -D*ΔY is an approximate relationship between percentage changes in bond price and changes in its yield. It is easy to use and is the reason why duration, which was first suggested by Macaulay in 1938, has become such a popular measure.

Consider a 3-year 10% coupon bond with a face value of USD \$100 with a yield of 12% per annum with continuous compounding. Coupon payments are made semi-annually (\$5). The present value of the bond's cash flow, using the yield as the discount rate are shown in column 3 (e.g. the present value of the first cash flow is 5exp(-0.12*.5) = -4.709.

### Table 4.6 Calculation of Duration

Current yield
12%

 Time (years) Cash flow Present Value Weight Time x Weight Duration 0.5 5 4.7088 0.0500 0.0250 4.7088 1.0 5 4.4346 0.0471 0.0471 4.4346 1.5 5 4.1764 0.0443 0.0665 4.1764 2.0 5 3.9331 0.0417 0.0835 3.9331 2.5 5 3.7041 0.0393 0.0983 3.7041 3.0 105 \$73.2560 0.7776 2.3327 \$73.2560 \$94.213021 1.0000 2.6530 \$94.2130

For the bond in Table 4.6, the bond price B, is \$94.213 and the duration, D, is 2.653, so that:

ΔB = -\$94.213 x 2.653 ΔY

or

ΔB = -249.95 ΔY

When the yield on the bond increases by 10 basis points (=0.1%), ΔY = +0.001. The duration relationship predicts that ΔB = -249.95 x 0.001 = -0.250, so that the bond price goes down to \$94.213 - \$0.250 = \$93.963. How accurate is this? Valuing the bond in terms of its yield in the usual way, we find that, when the bond yield increases by 10 basis points to 12.1%, the bond price is \$93.963 which is the same as that predicted by the duration relationship.

 Change in Yield -12.10% =-(0.12+0.001) -249.9480892 =(-1*\$94.213)*2.653 -0.249948089 =-249.948*0.001 New Bond Price \$93.963 =\$94.213+(-.249948)

Using the same information from above "Duration" and the bond price of \$94.213 and a duration of 2.653. The yield, expressed with semiannual compounding is 12.376%. What is the new yield with a 10 basis points increase using semi-annual interest payments (not continuous compounding)?

 Modified Duration Yield 2.653/1+.123673/2=2.499 12.36% B = -\$94.213 x 2.4985 = -235.39 D = \$94.213 - .235 = \$93.978 This is the "New" YTM if interest rates drop 10 basis points or .01% Trial and error \$94.8421 12.4673% \$94.8421 12.00% 100 Face Value \$100 10% Coupon 10% 3 Maturity 3 6 Number of periods 6 12.4673% Current Yield 12.0000% 1-Jan-17 Purchase date 1-Jan-17 1-Jan-20 Maturity Date 1-Jan-20 \$93.97 \$93.97 Price \$95.08 \$95.08

### Determination of Forward and Futures Prices

Forward contracts are easier to analyze than futures contracts because there is no daily settlement - only a single payment at maturity. Luckily it can be shown that the forward price and futures price of an asset are usually very close when the maturities of the two are the same.

### Investment Assets vs. Consumption Assets

When considering forward and futures contracts, it is important to distinguish between investment assets and consumption assets. An investment asset is an asset that is held for investment purposes by significant number of investors. Stocks and bonds are clearly investment assets. Gold and silver are also examples of investment (Silver, for example, has a number of industrial uses.). However, they do have to satisfy the requirement that they are held by significant number of investors solely for investment. A consumption assets is an asset that is held primarily for consumption. It is not usually held for investment. Examples of consumption assets are commodities such as copper, oil, wheat, corn, coffee and pork bellies.

We can use arbitrage arguments to determine the forward and futures prices of an investment asset from its spot price and other observable market variables. We cannot do this for consumption assets.

### Assumptions and Notation

• - T: Time until delivery date in a forward or futures contract (in years)
• - S0: Price of the asset underlying the forward or futures contract today
• - F0: Forward or futures price today
• - r: Zero-coupon risk-free rate on interest per annum, expressed with continuous compounding, for an investment maturing at the delivery date (i.e., in T years)

The risk-free rate, r, is the rate at which money is borrowed or lent when there is no credit risk, so that money is certain to be repaid. Financial institutions and other participants in derivatives markets assume LIBOR rates rather than Treasury rates are the relevant risk-free rates.

### Forward Price for an Investment Asset

The easiest forward contract to value is one written on an investment asset that provides the holder with no income. Non-dividend-paying stocks and zero-coupon bonds are examples of such investment assets.

Consider a long forward contract to purchase a non-dividend-paying stock in 3 months. Assume the current stock price is \$40 and the 3-month risk-free interest rate is 5% per annum.

Suppose first that the forward price is relatively high at \$43. An arbitrageur can borrow \$40 at the risk-free interest rate of 5% per annum, buy one share, and short a forward contract to sell one share in 3 months. At the end of the 3 months, the arbitrageur delivers the share and receives \$43. The sum of money required to pay off the loan is:

40 * exp(.05*3/12) = \$40.50

By following this strategy, the arbitrageur locks in a profit of \$43.00 - \$40.50 = \$2.50 at the end of the 3-month period.

Suppose next that the forward price is relatively low at \$39. An arbitrageur can short one share at \$40, invest the proceeds of the short sale at 5% per annum for 3 months, and take a long position in a 3-month forward contract for \$39. The proceeds of the short sale grow to 40 * exp(.05*3/12) = \$40.50 in 3 months. At end of 3 months, the arbitrageur pays \$39 takes delivery of the share under the terms of forward contract, and uses it to close out the short position. A net gain of \$40.50 - \$39 = \$1.50 is therefore made at the end of the 3 months.

The first arbitrage works when the forward price is greater then \$40.50. The second arbitrage works when the forward price is less than \$40.50. We deduce that for there to be no arbitrage the forward price must be exactly \$40.50.

### Forward Contract: A Generalization

To generalize this example we consider a forward contract on an investment asset with price S0 that provides no income. Using our notation, T is the time to maturity, r is the risk-free rate, and F0 is the forward price. The relationship between F0 and S0 is:

F0 = S0 * exp(r*T)

If F0 > S0 * exp(r*T), arbitrageurs can buy the asset and short forward contracts on the asset. If F0 < S0 * exp(r*T), they can short the asset and enter into long forward contract on it. In our example, S0 = \$40, r =.05, and T = .25, so that F0 = S0 * exp(r*T)

F0 = 40*exp(.05*.25) = \$40.50

which is in agreement with our earlier calculations.

Consider a 4-month (forward contract to buy a zero-coupon bond that will mature 1 year from today. (This means that the bond will have 8 months to go when the forward contract matures.) The current price of the bond is \$930. We assume that the 4-month risk-free rate of interest (continuously compounded) is 6% per annum. Because zero-coupon bonds provide no income, we can use F0 = S0 * exp(r*T) with T = 4/12, r = .06, and S0 = 930. The forward price, F0, is given by F0 = 930*exp(.06*4/12) = \$948.79 This is the delivery price in a contract negotiated today.

### Forward Contract with Known Income

In this section we consider a forward contract on an investment that will provide a perfectly predictable cash income to the holder. Examples are stocks paying known dividends and coupon-bearing bonds. We adopt the same approach as above. We first look at a numerical example and then review the forward arguments.

Consider a long forward contract to purchase a coupon-bearing bond whose current price is \$900. We will suppose that the forward contract matures in 9 months. We will also suppose that a coupon payment of \$40 is expected after 4 months. We assume that the 4-month and 9-month risk-free interest rates (continuously compounded) are respectively, 3% and 4% per annum.

Suppose first that the forward price is relatively high at \$910. An arbitrageur can borrow \$900 to buy the bond and short a forward contract. The coupon payment has a present value of 40*exp(-.03*4/12) = \$39.60. Of the \$900, \$39.60 is therefore borrowed at 3% per annum for 4 months so that it can be repaid with the coupon payment. The remaining \$860.40 is borrowed at 4% per annum for 9 months. The amount owing at the end of the 9-month period is 860.40*exp(.04*.75) = \$886.60. A sum of \$910 is received for the bond under the terms of the forward contract. The arbitrageur therefore makes a net profit of:

910 - 886.60 = \$23.40

Suppose next that the forward price is relatively low at \$870. An investor can short the bond and enter into a long forward contract. Of the \$900 realized from shorting the bond, \$39.60 is invested for 4 months at 3% per annum so that it grows into an amount sufficient to pay the coupon on the bond. The remaining \$860.40 is invested for 9 months a 4% per annum and grows to \$866.60. Under the terms of the forward contract, \$870 is paid to buy the bond and the short position is closed out. The investor therefore gains:

886.60 - 870 = \$16.60

The first strategy produces a profit when the forward price is greater than \$886.60, whereas the second strategy produces a profit when the forward price is less than \$886.60. It follows that if there are no arbitrage opportunities then the forward price must be \$886.60.

Arbitrage opportunities when 9-month forward price is out of line with spot price for asset providing known cash income. (Asset price = \$900; income of \$40 occurs at 4 months; 4-month and 9-month rates are, respectively, 3% and 4% per annum).

 Forward price = \$910 Forward price = \$910 Action now: Action now: Borrow \$900: \$39.60 for 4 months and \$860.40 for 9 months. Short 1 unit of asset to realize \$900. Buy 1 unit of asset Invest \$39.60 for 4 months and \$860.40 for 9 months. Enter into forward contract to sell asset in 9 months for \$910. Enter into forward contract to buy asset in 9 months for \$870. Action in 4 months: Action in 4 months: Receive \$40 of income on asset. Receive \$40 of income on asset. Use \$40 to repay first loan w/interest. Pay income of \$40 on asset. Action in 9 months: Action in 9 months: Sell asset for \$910. Receive \$886.60 from 9-month investment. Use \$886.60 to repay second loan Buy asset for \$870 with interest. Close out short position Profit realized = \$23.40 Profit realized = \$16.60

### Forward Price: A Generalization

We can generalize from the information above, when an investment asset will provide income with a present value of I during the life of a forward contract, we have:

F0 = (S0 - I) * exp(r*T)

In our example, S0 = 900, I = 40*exp(.03*4/12) = \$39.60, r = .04, and T = .75, so that

F0 = (900-39.60)*exp(.04*3/12) = \$886.60

Consider a 10-month forward contract on a stock when the stock price is \$50. We assume that the risk-free rate of interest (continuously compounded) is 8% per annum for all maturities. We also assume that dividends of \$.75 per share are expected after 3 months, 6 months, and 9 months. The present value of the dividends, I is:

I = .75*exp(-.08*3/12) + .75*exp(-.08*6/12) + .75*exp(-.08*9/12) = 2.162

The variable T is 10 months so that the forward price, F0, is given

F0 = (50-2.162)*exp(.08*10/12) = \$51.41

If the forward price were less than this, an arbitrageur would short the stock and buy forward contracts. If the forward price were greater than this, an arbitrageur would short forward contracts and buy the stock in the spot market.

### Forward Contract with Known Yield [viewable here in Excel]

We now consider the situation where the asset underlying a forward contract provides a know yield rather than a know cash income. This means that the income is know when expressed as a percentage of the asset's price at the time the income is paid. Suppose that an asset is expected to provide a yield of 5% per annum. This could mean that income is paid once a year and is equal to 5% of the asset price at the time it is paid, in which case the yield would be 5% with annual compounding. Alternatively, it could mean that income is paid twice a year and is equal to 2.5% of the asset price at the time it is paid in which case the yield would be 5% per annum with semi-annual compounding. Earlier, we explained that we will normally measure interest rates with continuous compounding. Similarly, we will normally measure yields with continuous compounding. Formulas for translating a yield measured with one compounding frequency to a yield measured with another compounding frequency are the same as those given for interest rate.

Define q as the average yield per annum on an asset during the life of a forward contract with continuous compounding. It can be shown that:

F0 = S0 * exp(r-q)T

Consider a 6-month forward contract on an asset that is expected to provide income equal to 2% of the asset price once during a 6-month period. The risk-free rate of interest (with continuous compounding) is 10% per annum. The asset price is \$25. In this case S0 = 25, r = .10, and T = 0.5. The yield is 4% per annum with semiannual compounding. Using R1 = m*Ln(1 + Rv/m)

 4*exp(-.10469*.5) + 4*(exp(-.10536*1.0) + 104*exp(-Rx1.5) = 96 SOLVE for R R = ln(.85196)/1.5 = .10681 Trial and error 10.68% =4*exp(-0.10469*0.5) + 4*exp(-0.10536*1) + 104*exp(-.1068*1.5) = 96 or 0.851961467 =(96-(4*exp(-0.10469*0.5) + 4*exp(-0.10536*1)))/104 = .851961467 10.68% =-(LN(0.851961467)/1.5) = 10.68%

You will need to discount the 2% x 2 = 4% using R1 = m*Ln(1 + Rv/m)

The answer is 3.96% per annum with semiannual compounding. It follows:

q = .0396, so that from F0 = (S0 *exp(r-q)*T the forward price, F0 is given by:

F0 = 25 * exp(.1-.0396)*.5 = \$25.77

### Valuing Forward Contracts with Known Yield

Suppose K is the delivery price for a contract that was negotiated some time ago, the delivery date is T years from today, and r is the T-year risk-free interest rate. The variable F0 is the forward price that would be applicable if we negotiated the contract today. We also define:

f: Value of forward contract today

It is important to be clear about the meaning of the variables F0, K, and f. At the beginning of the life of the forward contract, the delivery price, K, is set equal to the forward price. F0, and the value of the contract, f, is 0. As time passes, K stays the same (because it is part of the definition of the contract), but the forward price changes and the value of the contract becomes either positive or negative.

A general result, applicable to all long forward contracts (both those on investment assets and those on consumption assets) is:

f (value of the contract) = (F0 - K (delivery price))*exp(-rt)

To see why this is correct, we use an argument analogous to the one we used for forward rate agreements. We compare a long forward contract that has a delivery price of F0 with an otherwise identical long forward contract that has a delivery price of K. The difference between the two is only in the amount that will be paid for the underlying asset at time T. Under the first contract, this amount is F0; under the second contract it is K. A cash outflow difference of F0 - K at time T translate to a difference if (F0 - K)*exp(-rt) today. The contract with a delivery price F0 is therefore less valuable than the contract with delivery price K by an amount (F0 - K)*exp(-rt). The value of the contract with a delivery price of F0 is by definition zero. If follows that the value of the contract with a delivery price of K is (F0 - K)*exp(-rt). This proves f = (F0 - K)*exp(-rt). Similarly, the value of a short forward contract with delivery price K is:

(K - F0)*exp(-rt)

A long forward contract on a nondividend-paying stock was entered into some time ago. It currently has 6 months to maturity. The risk-free rate of interest (with continuous compounding) is 10% per annum, the stock price is \$25, and the delivery price is \$24. In this case, S0 = \$25, r = .10, T = .5 and K =\$24. Using the formula for Forward Price for an Investment:

F0 = S0 * exp(r*T), F0 = 25 *exp(.1*.5) = \$26.28

From equation f = (F0 - K)*exp(-r*T) above, the value of the forward contract is:

f = (26.28-24)*exp(-.1*.5) = \$2.17

Using f = (F0 - K)*exp(-r*t) in conjunction with F0 = S0 * exp(r*T) gives the following expression for the value of a forward contract on an investment asset that provides no income:

f = (S0 - K)*exp(-r*T) =25-24*exp(-0.1*0.5) = \$2.17

Similarly, using f = (F0 - K)*exp(-rt) with F0 = (S0 - I) * exp(r*T) gives the following expression for the value of a long forward contract on an investment asset that provides a know income with present value I:

f = (S0 - I - K)*exp(-r*T) =(25-0.10-24)*exp(-0.10*0.5) = \$0.86

Finally, using f = (F0 - K)*exp(-r*t) in conjunction with F0 = S0 * exp(r-q)T gives the following expression for the value of a long forward contract on an investment asset that provides a known yield at rate q:

f = S0*exp(-q*T) - K*exp(-r*T) = (25)*exp(-0.10*0.5) -(24)*exp(-0.10*0.5) = \$2.05

### Are Forward Prices and Futures Prices Equal?

The forward price for a contract with a certain delivery date is in theory the same price as the futures price for a contract with that delivery date.

When interest rates vary unpredictably (as they do in the real world), forward and futures prices are in theory no longer the same.

### Futures Prices of Stock Indices

Consider a 3-month futures contract on the Nifty. Suppose that the stocks underlying the index provide a dividend yield of 1% per annum, that the current value of the index is 2,700, and that the continuously compounded risk-free interest rate is 5% per annum. In this case, r =.05, S0 = 2,700, T = .25 and q = .01. Hence, the futures price, F0 is given by:

F0 = S0 * exp(r-q)T

F0 = 2,700 * exp(.05-.01)*.25 = 2,727.16

In practice, the dividend yield on the portfolio underlying an index varies week by week throughout the year. The chosen value of q should represent the average annualized dividend yield during the life of the contract. The dividends used for estimating q should be those for which the ex-dividend date is during the life of the futures contract.

### Forward and Futures Contracts on Currencies

The underlying asset is one unit of the foreign currency (from the perspective of a US investor.) We will therefore define the variable S0 as the current spot price in dollars of one unit of the foreign currency and F0 as the forward or futures price in dollars of one unit of the foreign currency. This is consistent with the way we have defined S0 and F0 for other assets underlying forward and futures contracts. However, it does not necessarily correspond to the way spot and forward exchange rates are quoted. For major exchange rates other than the British pound, euro, Australian dollar, and New Zealand dollar, a spot or forward exchange rate is normally quoted as the number of units of the currency that are equivalent to one US dollar.

A foreign currency has the property that the holder of the currency can earn interest at the risk-free interest rate prevailing in the foreign currency. For example, the holder can invest the currency in a foreign-denominated bond. We define rx as the value of the foreign risk-free rate when money is invested for time T. The variable r is the US dollar risk-free rate when money is invested for this period of time. The relationship between F0 and S0 is:

F0 = S0 * exp(r-rx)*T

This is the well-known interest rate parity relationship from international finance. Suppose that an individual starts with 1,000 units of the foreign currency. There are two ways it can be converted to dollars at time T. One is by investing it for T years at rx and entering into a forward contract to sell the proceeds for dollars at time T. This generates 1,000*exp(rx*T)*F0 dollars. The other is by exchanging the foreign currency for dollars in the spot market and investing the proceeds for T years at r. This generates 1,000 * S0 x exp(r*T) dollars. In the absence of arbitrage opportunities, the two strategies must give the same result. Hence

1,000*exp(rx*T)*F0 = 1,000*S0exp(r*T)

so that

F0 = S0exp(r-rx)*T

Suppose that the 2-year interest rates in Australia and the United States are 5% and 7% respectively and the spot exchange rate between the Australian dollar (AUD) and the US dollar (USD) is 0.6200 USD per AUD. Using

F0 = S0 * exp(r-rx)T

the 2-year forward exchange rate should be:

0.62*exp(.07-.05)*2 = .6453

Suppose first that the 2-year forward exchange rate is less than this, say 0.6300. An arbitrageur can:

• - Borrow 1,000 AUD at 5% per annum for 2 years, convert to 620 USD and invest the USD at 7% (both rates are continuously compounded).
• - Enter into a forward contract to buy 1,105.17 * 0.63 = 696.26 USD (e.g., 1,000 * exp(.05x2) = 1,105.17).

The 620 USD that are invested at 7% grow to 620*exp(0.072*2) = 713.17 USD in 2 years. Of this, \$696.26 USD (=1,105.17*0.63) are used to purchase 1,105.17 AUD (.63 conversion) under the terms of the forward contract. This is exactly enough to repay principal and interest on the 1,000 AUD that are borrowed (1,000 * exp(.05x2) = 1,105.17). The strategy therefore gives rise to a riskless profit of 713.17 - 696.26 = 16.91 USD. (Try this with 100 million AUD!)

Suppose next that the 2-year forward rate is 0.6600 (greater than the .06453 value given by F0 = S0 * exp(r-rx)T. An arbitrageur can:

- Borrow 1,000 USD at 7% per annum for 2 years, convert to 1,000/0.6200 = 1,612.90 AUD,

and invest the AUD at 5%.

- Enter into a forward contract to sell 1,782.53 * 0.66 = 1,176.47 USD (e.g., 1,612.90*exp(0.05*2) = 1,782.53).

The 1,612.90 AUD that are invested at 5% grow to 1,612.90*exp(0.05*2) = 1,782.53 AUD in 2 years. The forward contract has the effect of converting this to 1,176.47 USD. The amount needed to payoff the USD borrowings is 1,000*exp(0.07*2)=1,150.27 USD. This strategy therefore gives rise to a riskless profit of 1,176.47 - 1,150.27 = 26.20 USD.

Keep in mind, a foreign currency can be regarded as an investment asset paying a know yield. The yield is the risk-free rate of interest in the foreign currency.

### Income and Storage Costs

Hedging strategies of gold producers leads to a requirement on the part of investment banks to borrow gold. Gold owners such as central banks charge interest in the form of what is know as the gold lease rate when they lend gold. The same is true of silver. Gold and silver can therefore provide income to the holder. Like other commodities they also have storage costs.

F0 = S0*exp(r*T) shows that the absence of storage costs and income. Storage costs can be treated as negative income. If U is the present value of all the storage costs, net of income during the life of a forward contract, it follows that:

F0 = (S0 + U)*exp(r*T)

Consider a 1-year futures contract on an investment asset that provides no income. It costs \$2 per unit to store the asset, with the payment being made at the end of the year. Assume that the spot price is \$450 per unit and the risk-free rate is 7% per annum for all maturities. This corresponds to r = 0.07, S0 = 450, T = 1, and U = 2*exp(-0.07*1) = 1.865

Using F0 = (S0 + U)*exp(r*T), the theoretical futures price is given by:

F0 = (450+1.865)*exp(0.07*1) = \$484.63

If the actual futures price is greater than 484.63, an arbitrageur can buy the asset and short 1-year futures contracts to lock in a profit. If the actual futures price is less than 484.63, an investor who already owns the asset can improve the return by selling the asset and buying the futures contracts.

If the storage costs net of income incurred at any time are proportional to the price of the commodity, they can be treated as negative yield. In this case:

F0 = S0 *exp(r+u)T

where u denotes the storage costs per annum as a proportion of the spot price net of any yield earned on the asset.

### Consumption Commodities

Commodities that are consumption assets rather than investment assets usually provide no income, but can be subject to significant storage costs. We now review the arbitrage strategies used to determine futures prices from spot prices carefully. Suppose that, instead of F0 = (S0 + U)*exp(r*T), we have:

F0 > (S0 + U)*exp(r*T)

To take advantage of this opportunity, an arbitrageur can implement the following strategy:

• - Borrow an amount S0 + U at the risk-free rate and use it to purchase one unit of the commodity and to pay storage costs.
• - Short a forward contract on one unit of the commodity.

If we regard the futures contract as a forward contract, this strategy leads to a profit of F0 - (S0 + U)*exp(r*T) at time T. There is no problem implementing the strategy for any commodity. However, as arbitrageurs do so, there will be a tendency for S0 to increase and F0 to decrease until F0 > (S0 + U)*exp(r*T) is no longer true. We conclude F0 > (S0 + U)*exp(r*T) cannot hold for any significant length of time.

Suppose next that

F0 < (S0 + U)*exp(r*T)

When the commodity is an investment asset, we can argue that many investors hold the commodity solely for investment. When they observe the inequality in F0 < (S0 + U)*exp(r*T) they will find it profitable to do the following:

• - Sell the commodity, save the storage costs, and invest the proceeds at the risk-free interest rate.
• - Take a long position in a forward contract.

The result is a riskless profit at maturity of (S0 + U)*exp(r*T) - F0 relative to the position the investor would have been in if they held the commodity. It follows that F0 < (S0 + U)*exp(r*T) cannot hold for long. Because neither F0 > (S0 + U)*exp(r*T) nor F0 < (S0 + U)*exp(r*T) can hold for long, we must have F0 = (S0 + U)*exp(r*T).

This argument cannot be used for a commodity that is a consumption asset rather than an investment asset. Individuals and companies who own a consumption commodity usually plan to use it in some way. They are reluctant to sell the commodity in the spot market and buy forward or futures contracts, because forward and futures contracts cannot be consumed (for example, oil futures cannot be used to feed a refinery!). There is therefore nothing to stop F0 = (S0 + U)*exp(r*T) from holding, and all we can assert for a consumption commodity is:

F0 <= (S0 + U)*exp(r*T)

If storage costs are expressed as a proportion U of the spot price, the equivalent result is:

F0 <= S0 *exp(r+u)T

### Convenience Yield

Users of a consumption commodity may feel that ownership of the physical commodity provides benefits that are not obtained by holders of futures contracts. For example, an oil refiner is unlikely to regard a futures contract on crude oil in the same way as crude oil held in inventory. The crude oil in inventory can be an input to the refining process, whereas a futures contract cannot be used for this purpose. In general, ownership of the physical asset enables a manufacturer to keep a production process running and perhaps profit from temporary local shortages. A futures contract does not do the same. The benefits from holding the physical asset are sometimes referred to as the convenience yield provided by the commodity. If the dollar amount of storage costs is know and has a present value U, then the convenience yield y is defined such that:

F0*exp(y*T) = (S0 + U)*exp(r*T)

If the storage costs per unit are a constant proportion, u, of the spot price, the y is defined so that

F0*exp(y*T) = S0 *exp(r+u)T

or

F0 = S0 *exp(r+u-y)T

The convenience yield reflects the market's expectations concerning the future availability of the commodity. The greater the possibility that shortages will occur, the higher the convenience yield. If users of the commodity have high inventories, there is very little chance of shortages in the near future and the convenience yield tends to be low. If inventories are low, convenience yields are usually high.

### The Cost to Carry

The relationship between futures prices and spot prices can be summarized in terms of the cost of carry. This means the storage cost plus the interest that is paid to finance the asset less the income earned on the asset. For a non-dividend-paying stock, the cost of carry is r, because there are no storage costs and no income is earned; for a stock index, it is r-q, because income is earned at rate q on the asset. For a currency, it is r-rx; for a commodity that provides income at rate q and requires storage costs at rate u, it is r-q+u; and so on.

Define the cost of carry as c. For an investment asset the futures price is:

F0 = S0 *exp(c*T)

For a consumption asset, it is

F0 = S0 *exp(c-y)T

where y is the convenience yield.

### Day Count and Quotation Conventions

Number of days between dates/Number of days in reference period x Interest earned in reference period

Three day count conventions that are commonly used in the United States are:

• - Actual/actual (in period)
• - 30/360
• - Actual/360

The actual/actual (in period) day count is used for Treasury bonds in the US. This means that the interest earned between two dates is based on the ratio of the actual days elapsed to the actual number of days in the period between coupon payments. Suppose that the bond principal is \$100, coupon payment dates are March 1 and September 1, the coupon rate is 8%, and we wish to calculate the interest earned between March 1 and July 3. The reference period is from March 1 to September 1. There are 184 (actual) days in this period, and interest of \$4 is earned during the period. There are 124 (actual) days between March 1 and July 3. The interest earned between March 1 and July 3 is:

124/184 *\$4 = 2.6957

The 30/360 day count is used for corporate and municipal bonds in the US. This means that we assume 30 days per month and 360 days per year when carrying out calculations. With the 30/360 days count, the total number of days between March 1 and September 1 is 180. The total number of days between March 1 and July 3 is 122 days. The interest earned is:

122/180*\$4 = 2.7111

The actual/360 day count is used for money market instruments in the US. This indicates that the reference period is 360 days. The interest earned during part of a year is calculated by dividing the actual number of elapsed days by 360 and multiplying by the rate. The interest earned in 90 days is therefore exactly one-forth of the quoted rate, and the interest earned in a whole year of 365 days is 365/360 times the quoted rate.

### Treasury Bond Price Quotations

The prices of money market instruments are sometimes quoted using a discount rate. This is the interest earned as a percentage of the final face value rather than as a percentage of the initial price paid for the instrument An example is Treasury bills in the United States. If the price of a 30-day Treasury bill is quoted as 1.70, this means that the annualized rate of interest earned is 1.70% of the face value. Suppose that the face value is \$100. Interest of \$.1416 is earned over the 30-day life. This corresponds to a true rate of interest of .1416/(100-.1463) = 1.416% for the 30-day period.

=100*0.017*(30/360)
\$0.1417
p = 360/n*(100 - Y)
1.4190%

### US Treasury Bonds

Treasury bond prices in the US are quoted in dollars and thirty-seconds of a dollar. The quote price is for a bond with a face value of \$100. Thus, a quote of 90-05 indicates that the quoted price for a bond with a face value of \$100,000 is \$90,156.25 (90 5/32).

The quoted price, which refers to as the clean price, is not the same as the cash price paid by the buyer of the bond, which traders refer to as the dirty price.

In general,

Cash price = Quoted price + Accrued interest since last coupon date

To illustrate this formula, suppose that it is March 5, 2019, and the bond under consideration is an 11% coupon bond maturing on July 10, 2028, with a quoted price of 95-16 or \$95.50. Because coupons are paid semi-annually on government bonds (and the final coupon is at maturity), the most recent coupon date is January 10, 2020 and the next coupon date is July 10, 2020. The number of days between January 10, 2020 and March 5, 2020 is 55 days, whereas the number of days between January 10, 2020 and July 5, 2020, is 182. On a bond with \$100 face value, the coupon payment is \$5.50 on January 10 and July 10. The accrued interest on March 5, 2010, is the share of the July 10 coupon accruing to the bondholders on March 2, 2020. Because actual/actual in period is used for Treasury bonds in the United States, this is:

55/182 x \$5.50 = \$1.66

The cash price per \$100 face value for the bond is therefore:

\$95.50 + \$1.66 = \$97.14

Thus, the cash price of a \$100,000 bond is \$97,140.

### Eurodollar Futures

The most popular interest rate futures contract in the US is the 3-month Eurodollar futures contract traded on the Chicago Mercantile Exchange (CME). A Eurodollar is a dollar deposited in a US or foreign bank outside the US. The Eurodollar interest rate is the rate of interest earned on Eurodollar deposited by one bank with another bank. It is essentially the same as the LIBOR rate.

Three-month Eurodollar futures contracts are futures contracts on the 3-month (90-day) Eurodollar interest rate. They allow an investor to lock in an interest rate on \$1 million for a future 3-month period. The 3-month period to which the interest rate applies starts on the third Wednesday of the delivery month. The contracts have delivery months of March, June, September and December for up to 10 years into the future. This means that in 2019 an investor can use Eurodollar futures to lock in an interest rate for 3-month periods that are as far into the future as 2028 (now 2021). Short-maturity contracts trade for months other than March, June, September and December.

To understand how Eurodollar futures contracts work, consider a June 2019 contract where the quoted settlement price on January 8, 2019 is 94.79. The contract ends on the third Wednesday of the delivery month. In the case of this contract, the third Wednesday of the delivery month is June 19, 2019. The contract is marked to market in the usual way until that date. However, on June 19, 2019, the settlement price is set equal to 100 - R, where R is the actual 3-month Eurodollar interest rate on that day, expressed with quarterly compounding and an actual/360 day count convention. (Thus if the 3-month Eurodollar interest rate on June 19, 2019, turned out to be 4%, the final settlement price would be 100 - 4 = 96.) There is a final settlement reflecting this settlement price and all contracts are declared closed.

The contract is designed so that a 1 basis point (=.001) move in the futures quote corresponds to a gain or loss of \$25 per contract. When a Eurodollar futures quote increases by 1 basis point, a trader who is long one contract gains \$25 and a trader who is short one contract loses \$25. Similarly, when the quote decreases by 1 basis point a trader who is long one contract loses \$25 and a trader who is short one contract gains \$25. Suppose, for example, that the settlement price changes from 94.70 to 94.90 between January 8 and January 9. Traders with long positions lose \$275 per contract. The \$25 per basis point rule is consistent with the point made earlier that the contract locks in an interest rate on \$1 million dollars for 3-months. When an interest rate per year changes by 1 basis point, the interest earned on 1 million dollars for 3 months changes by:

1,000,000 x 0001 x .25 = 25

or \$25. Since the futures quote is 100 minus the futures interest rate, an investor who is long gains when interest rates fall and one who is short gains when interest rates rise.

On January 8, 2019, an investor wants to lock in the interest rate that will be earned on \$5 million for 3 months starting on June 19, 2019. The investor buys five June 19 Eurodollar futures contracts at 94.79 (~ 5.21%). On June 19, 2019, the 3-month LIBOR interest rate is 4%, so that the final settlement price proves to be 96.00. The investor gains 5 x 25 x (9,600 - 9,470) = \$15,125 on the long futures position. The interest earned on the \$5 million for 3 months at 4% is:

5,000,000 x 0.25 x 0.04 = 50,000

or \$50,000. The gain on the futures contract brings this up to \$65,125. This is the interest that would have been earned if the interest rate had been 5.21% (5,000,000 x 0.25 x 0.0521 = 65,125). This illustration shows that the futures trade has the effect of locking in an interest rate equal to (100-94.79)%, or 5.21%.

The exchange defines the contract price as:

10,000 x (100-0.25 x (100 - Q))

where Q is the quote. Thus, the settlement price of 94.70 for the June, 2019 contract corresponds to a contract price of:

10,000 x (100-0.25 x (100 - 94.79)) = \$986,975

The final contract price is:

10,000 x (100-0.25 x (100 - 96.00)) = \$990,000

and the difference between the initial and final contract price is \$3,025 per contract, so that an investor with a long position in five contracts gains 5 x 3,025 = \$15,125.

### Forward vs. Futures Interest Rates

The Eurodollar futures contract is similar to a forward rate agreement (FRA) in that it locks in an interest rate for a future period. For short maturities (up to a year or so), the two contracts can be assumed to be the same and the Eurodollar futures interest rate can be assumed to be the same as the corresponding forward interest rate. For longer-dated contracts, differences between the contracts become important. Compare a Eurodollar futures contract on an interest rate for the period between times T1 and T2 with an FRA for the same period. The Eurodollar futures contract is settled daily. The final settlement is an time T1 and reflects the realized interest rate for the period between T1 and T2. By contrast the FRA is not settled daily and the final settlement reflecting the realized interest rate between times T1 and T2 is made at time T22.

There are therefore two differences between a Eurodollar futures contract and an FRA. These are:

1. The difference between a Eurodollar futures contract and a similar contract where there is no daily settlement. The latter is a forward contract where a payoff equal to the difference between the forward interest rate and the realized interest rate is paid at time T1.

2. The difference between a forward contact where thee is settlement at time T1 and a forward contract where there is settlement at T2.

These two components to the difference between the contracts cause some confusion in practice. Both decrease the forward rate relative to the futures rate, but for long-dated contracts the reduction caused by the second difference is much smaller than that caused by the first. The reason why the first difference (daily settlement) decreases the forward rate follows from the arguments in "Are Forward Prices and Futures Prices Equal?" Suppose you have a contract where the payoff is RM - RF at time T1 where RF is a predetermined rate for the period between T1 and T2, and RM is the realized rate for this period, and you have the option to switch to daily settlement. In this case daily settlement tends to lead to cash inflows when rates are high and cash outflows when rates are low. You would therefore find switching to daily settlement to be attractive because you tend to have more money in your margin account when rates are high. As a result the market would therefore set RF higher for the daily settlement alternative (reducing your cumulative expected payoff). To put this the other way round, switching from daily settlement to settlement at time T1 reduces RF.

To understand the reason why the second difference reduces the forward rate, suppose that the payoff of RM - RF is at time T2 instead of T1 (as it is for regular FRA). If RM is high, the payoff is positive. Because rates are high, the cost to you of having the payoff that you receive at time T2 rather than T1 is relatively high. If RM is low, the payoff is negative. Because rates are low, the benefit to you of having the payoff you make at time T2 rather than T1 is relatively low. Overall, you would rather have the payoff at time T1. If it is at time T2, rather than T1, you must be compensated by a reduction in RF.

Analysts make what is known as convexity adjustments to account for the total difference between the two rates. One popular adjustment is4

Forward rate = Futures rate - 1/2σ2T1T2

where, as above, T1 is the time to maturity of the futures contract and T2 is the time to maturity of the rate underlying the futures contract. The variable σ is the standard deviation of the change in the short-term interest rate in 1 year. Both rates are expressed with continuous compounding. A typical value for σ is 1.2% or 0.012.

### Example

Consider the situation where σ = 0.012 and we wish to calculate the forward rate when the 8-year Eurodollar futures price quote is 94. In this case T1 = 8, T2 = 8.25, and the convexity adjustments is

(Equation 6.3) 1/2 x 0.0122 x 8 x 8.25 = 0.00475

or 0.475% (47.5 basis points). The futures rate is 6% per annum on an actual/360 basis with quarterly compounding. This corresponds to 1.5% per 90 days or an annual rate of (365/90) ln 1.015 = 6.038% with continuous compounding and an actual/365 day count. The estimate of the forward rate given by equation (6.3), therefore, is 6.038 - 0.475 = 5.563% per annum with continuous compounding. The table below shows how the size of the adjustment increases with the time to maturity.

 Maturity of Futures (years) Convexity adjustments (basis points) 2 3.2 4 12.2 6 27.0 8 47.5 10 73.8

We can see from this table that the size of the adjustment is roughly proportional to the square of the time to maturity of the futures contract. Thus the convexity adjustment for the 8-year contract is approximately 16 times that for a 2-year contract.

### Using Eurodollar Futures to Extend the LIBOR Zero Curve

The LIBOR zero curve out to 1 years is determined by the 1-month, 3-month, 6-month, and 12-month LIBOR rates. Once the convexity adjustment has been made, Eurodollar futures are often used to extend the zero curve. Suppose that the ith Eurodollar futures contract matures at time Ti (i= 1,2,3…). It is usually assumed that the forward interest rate calculated from ith futures contract applies to the period Ti to Ti+1. (In practice this is close to true.) This enables a bootstrap procedure to be used to determine zero rates. Suppose that F0 is the forward rate calculated from the ith Eurodollar futures contract and Ri is the zero rate for a maturity Ti. From equation Rv = (R2 * T2) - (R1 * T1)/(T2 - T1)) so that:

Fi = (Ri+1 * Ti+1) - (Ri * Ti)/(Ti+1 - Ti))

so that

Ri+1= F0(Ti+1 - Ti) +(Ri * Ti)/Ti+1

The 400-day LIBOR zero rate has been calculated at 4.80% with continuous compounding and, from Eurodollar futures quotes, it has been calculated that (a) the forward rate for a 90-day period beginning in 400 days is 5.30% with continuous compounding, (b) the forward rate for a 90-day period beginning in 491 days is 5.50% with continuous compounding, and (c) the forward rate for a 90-day period beginning 589 days is 5.60% with continuous compounding. Using the formula just above, the 491 day forward rate is

.053 x 91 + .048 x 400 /491 = .04893

Similarly, we can use the second forward rate to obtain the 589-day rate as

=(.055 x 98 + .04893 * 491)/589 = .04994

The next forward rate of 5.60% would be used to determine the zero curve out to the maturity of the next Eurodollar futures contract. (Note that, even though the rate underlying the Eurodollar futures contract is a 90-day rate it is assumed to apply to the 91 or 98 day elapsing between Eurodollar contract maturities.)

=(.056 x 98 + .04893 * 491)/589 = 5.13%

### Duration-Based Hedging Using Futures

Consider the situation where a position in an asset that is interest rate dependent, such as a bond portfolio or a money market security, is being hedged using an interest rate futures contract. Defined:

• - Fa: Contract price for the interest rate futures contract
• - Dx: Duration of the asset underlying the futures contract at the maturity of the futures contract
• - P: Forward value of the portfolio being hedged at the maturity of the hedge (in practice, this is usually assumed to be the same as the value of the portfolio today)
• - De: Duration of the portfolio at the maturity of the hedge

If we assume that the change in the yield, Δy, is the same for all maturities, which means that only parallel shifts in the yield curve can occur, it is approximately true that:

ΔP = -PDeΔY

It is also approximately true that:

The number if contracts required to hedge against an uncertain ΔY, therefore, is

This is the duration-based hedge ratio. It is sometimes also called the price sensitive hedge ratio. Using it has the effect of making the duration of the entire position zero.

When the hedging instrument is a Treasury bond futures contract, the hedger must base Dx on an assumption that one particular bond will be delivered. This means that the hedger must estimate which of the available bonds is likely to be the cheapest to deliver at the time the hedge is put in place. If, subsequently, the interest rate environment changes so that it looks as though a different bond will be cheapest to deliver, then the hedge has to be adjusted and its performance may be worse than anticipated.

When the hedges are constructed using interest rate futures, it is important to keep in mind that interest rates and futures prices move in opposite direction. When interest rates go up, an interest rates and futures price goes down. When interest rates go down, the reverse happens, and the interest rate futures price goes up. Thus, a company in a position to lose money if interest rates drop should hedge by taking a long futures position. Similarly, a company in a position to lose money if interest rates rise should hedge by taking a short futures position.

The hedger tries to choose the futures contract so that the duration of the underlying asset is as close as possible to the duration of the asset being hedged. Eurodollar futures tend to be used for exposures to short-term interest rates, whereas Treasury bond and Treasury note futures contracts are used for exposures to longer-term rates.

It is August 2 and a fund manager with \$10 million invested in government bonds is concerned that interest rate are expected to be highly volatile over the next 3 months. The fund manager decides to use the December T-bond futures contract to hedge the value of the portfolio. The current futures price is 93-02, or 93.0625. Because each contract is for the delivery of \$100,000 face value of bonds, the futures contract is \$93,062.50.

Suppose that the duration of the bond portfolio in 3 months will be 6.80 years. The cheapest-to-deliver bond in the T-bond contract is expected to be a 20-year 12% per annum coupon bond. The yield on this bond is currently 8.80% per annum, and the duration will be 9.20 years at maturity of the futures contract.

The fund manager requires a short position in T-bond futures to hedge the bond portfolio. If interest rates go up a gain will be made on the short futures position, but a loss will be made on the bond portfolio. If interest rates decrease, a loss will be made on the short position, but there will be a gain on the bond portfolio. The number of bond futures contracts that should be shorted can be calculated from the above equation as:

10,000,000/93062.50 x 6.80/9.20 = 79.42

To the nearest whole number, the portfolio manager should short 79 contracts.

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

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