Mathematics of Financial Derivatives Lecture 9 Solesne Bourguin bourguin@math.bu.edu Boston University Department of Mathematics and Statistics
Table of contents 1. Zero-coupon rates and bond pricing 2. Forward rates 3. Forward rate agreements 1
Zero-coupon rates and bond pricing
Zero-coupons Definition: zero-coupon rates The n-year zero-coupon interest rate is the rate of interest earned on an investment that starts today and lasts for n years. All the interest and principal is realized at the ends of n years. There are no intermediate payments. The n-year zero coupon rate is sometimes referred to as the n-year sport rate or the n-year zero rate. Concretely, suppose a 5-year zero rate with continuous compounding is 5% per annum. This means that $100, if invested for 5 years, grows to 100 e 0.05 5 = 128.40. Most interest rates we observe on the market are not zero rates (think of a government bond with a 6% coupon). 2
Bond pricing How do we price a bond? Most bonds pay coupons to the holder periodically. The bond s principal (also known as par value or face value) is paid at the end of its life. The theoretical price of a bond can be calculated as the present value of all the cash flows that will be received by the owner of the bond. As cash flows happen at different times, what interest rate should be used to discount? We use the zero rates. 3
Bond pricing Example Consider the situation where Treasury zero rates (continuously compounded) are given in the following table. Maturity (years) Zero rate (%) 0.5 5.0 1.0 5.8 1.5 6.4 2.0 6.8 Suppose that a 2-year Treasury bond with a principal of $100 provides coupons at the rate of 6% per annum semiannually. To compute the price of the bond, we discount each cash flow with the corresponding zero rate: 3e 0.05 0.5 + 3e 0.058 1.0 + 3e 0.064 1.5 + 103e 0.068 2.0 = 98.39. 4
Bond yield Remark Observe that the price of a zero coupon bond with maturity T and a principal of $1 is given by P(0, T ) = e R T T, where R T is the zero rate for maturity T. Definition: bond yield A bond s yield is the single discount rate that, when applied to all cash flows, gives a bond price equal to its market price. Suppose that the theoretical price of the bond we have been considering, $98.39, is also its market value. If y is the ield on the bond, expressed with continuous compounding, it must be true that 3e y 0.5 + 3e y 1.0 + 3e y 1.5 + 103e y 2.0 = 98.39. This equation can be solved using a iterative numerical procedure to give y = 6.76%. 5
Par yield Definition: par yield The par yield for a certain bond maturity is the coupon rate that causes the bond price to equal its par value. Usually the bond is assumed to provide semiannual coupons. Suppose that the coupon on a 2-year bond in our example is c per annum. The value of the bond is equal to its par value of 100 when c 2 e 0.05 0.5 + c 2 e 0.058 1.0 + c ( 2 e 0.064 1.5 + 100 + c ) e 0.068 2.0 = 100. 2 This equation can be solved in a straightforward way to give c = 6.87. The 2-year par yield is therefore 6.87% per annum. 6
Par yield Definition: par yield (continued) More generally, if d is the present value of $1 received at the maturity of the bond, A is the value of an annuity that pays 1 dollar on each coupon payment date, and m is the number of coupon payments per year, then the par yield c must satisfy so that 100 = A c m + 100d c = (100 100d)m. A In our example, m = 2, d = e 0.068 2, and A = e 0.05 0.5 + e 0.058 1.0 + e 0.064 1.5 + e 0.068 2.0 = 3.70027. The formula confirms that the par yield is 6.87% per annum. 7
Determining zero rates Determining zero rates One way of determining zero rates is to observe the yields on strips. These are zero-coupon bonds that are synthetically created by traders when they sell coupons on a Treasury bond separately from the principal. Another way to determine zero rates is from Treasury bills and coupon-bearing bonds. The most popular approach is known as the bootstrap method. Consider the following table of bond prices. Bond principal ($) Time to maturity (years) Annual coupon ($) Bond price ($) 100 0.25 0 97.5 100 0.50 0 94.9 100 1.00 0 90.0 100 1.50 8 96.0 100 2.00 12 101.6 Half the stated coupon is assumed to be paid every 6 months. 8
Determining zero rates Determining zero rates (continued) Because the first 3 bonds pay no coupons, the zero rates corresponding to the maturities of these bonds can easily be calculated. The 3-months bond has the effect of turning an investment of 97.5 into 100 in 3 months. So the (cont. comp.) 3-months zero rate R is therefore given by solving 100 = 97.5e R 0.25. We get 10.127% per annum. The 6-month zero rate is similarly given by solving 100 = 94.9e R 0.5, which gives us 10.469% per annum. Similarly, the 1-year zero rate is 10.536% per annum. 9
Determining zero rates Determining zero rates (continued) The fourth bond lasts 1.5 years. The payments are as follows: $4 after 6 months, $4 after 1 year, and $104 after 1.5 years. We already have the 6-months and 1-year zero rates. We are thus only missing the 1.5-year zero rate. Call it R. It is given by solving 4e 0.10469 0.5 + 4e 0.10536 1.0 + 104e R 1.5 = 96. We get R = 10.681%. This is the only zero rate that is consistent with the 6-months and 1-year rate deduced previously. Remark Note that we couldn t have calculated the 1.5-year zero rate without having previously computed the 0.5-year and 1-year zero rates first: the bootstrap method is a recursive procedure. 10
Determining zero rates Determining zero rates (continued) The 2-year zero rate can be calculated similarly based on the 6-months, 1-year, and 1.5-year zero rates already calculated. I R is the 2-year zero rate, it is given by solving 6e 0.10469 0.5 + 6e 0.10536 1.0 + 6e 0.10681 1.5 + 106e R 2.0 = 101.6. We get R = 10.808%. Compiling all of the zero rates we calculated, we get Maturity (years) Zero rate (%) 0.25 10.127 0.50 10.469 1.00 10.536 1.50 10.681 2.00 10.808 11
Determining zero rates Determining zero rates (continued) We can plot this data to get a feel of how zero rates evolve with the maturity. Zero rate (% per annum) 11 10 0 0.5 1.0 1.5 2.0 Maturity (years) 12
Determining zero rates Remark In practice, we do not usually have bonds with maturities equal to exactly 1.5 years, 2 years, 2.5 years, and so on. The approach is then to interpolate between the bond price data before it is used to calculate the zero curve. For example, if it is known that a 2.3-year bond with a coupon of 6% sells for 98 and a 2.7-year bond with a coupon of 6.5% sells for 99, it might be assumed (interpolated)that a 2.5-year bond with a coupon of 6.25% would sell for 98.5. 13
Forward rates
Forward rates Definition: forward rates Forward rates are the future rates of interest implied by current zero rates for periods of time in the future. Example Suppose the zero curve is as follows. Maturity (years) Zero rate (%) 1 3.0 2 4.0 If we invest $100 for 2 years, we get 100e 0.04 2 = $108.33. But we could equivalently invest $100 for 1 year, and reinvest the proceeds for 1 more year. 14
Forward rates Example (continued) In the absence of arbitrage opportunities, these two approaches have to yield the same result. So we can compute the 1-year forward rate in 1 year by solving 100e 0.03 1 e R F 1 = 100e 0.04 2, which yields R F = 5%. 4% T=0, $100 T=1, $103.05 T=2, $108.33 3% R F =? 15
Forward rates Remark Observe that the overall rate (4%) is just the average of the intermediate rates over the whole period (only valid is the compounding is continuous). General case In general, if R 1 and R 2 are the zero rates for maturities T 1 and T 2, respectively, and R F is the forward rate for the period of time between T 1 and T 2, then it must hold that e R1 T1 e R F (T 2 T 1) = e R2 T2, so that R F = R 2T 2 R 1 T 1 T 2 T 1. 16
Forward rates General case (continued) The general situation corresponds to the following diagram. R 2 T 0 = 0 T 1 T 2 R 1 R F Observe that we can rewrite the equation for R F as T 1 R F = R 2 + (R 2 R 1 ). T 2 T 1 17
Forward rates General case (continued) Taking the limit as T 2 goes to T 1, we get R F = R + T R T, where R is the zero rate for maturity T. The value of R F obtained this way is known as the instantaneous forward rate for a maturity T. This is the forward rate that is applicable to a very short future time period that begins at time T. Recall that P(0, T ) = e RT, so that we also have R F = ln P(0, T ). T 18
Forward rates Locking forward rates If a financial institution can borrow or lend at the zero rates, it can lock in the forward rates. Suppose once again that we have the following zero curve. Maturity (years) Zero rate (%) 1 3.0 2 4.0 For example, it can borrow $100 at 3% for 1 year and invest these $100 at 4% for 2 years, the result is a cash outflow of 100e 0.03 1 = $103.05 at the end of year 1 and an inflow of 100e 0.04 2 = $108.33 at the end of year 2. 19
Forward rates Locking forward rates (continued) Since 108.33 = 103.05e 0.05 1, a return equal to the forward rate (5%) is earned on $103.05 during the second year. If an investor thinks that rates in the future will be different from today s forward rates, there are many trading strategies that the investor will find attractive. One of these involves entering into a contract known as a forward rate agreement. 20
Forward rate agreements
Forward rates agreements Definition: forward rate agreement (FRA) A forward rate agreement (FRA) is an over-the-counter transaction designed to fix the interest rate that will apply to either borrowing or lending a certain principal during a specified future period of time. The usual assumption underlying the contract is that the borrowing or lending would normally be done at LIBOR. If the agreed fixed rate is greater than the actual LIBOR rate for the period, the borrower pays the lender the difference between the two applied to the principal. If the reverse is true, the lender pays the borrower the difference applied to the principal. 21
Forward rates agreements Example Suppose that a company enters into a FRA that is designed to ensure it will receive a fixed rate of 4% on a principal of $100 million for a 3-month period starting in 3 years. The FRA is an exchange where LIBOR is paid and 4% is received for the 3 months period. If the 3-month LIBOR proves to be 4.5% for the 3-month period, the cash flow to the lender will be 100, 000, 000 [ ( 1 + 0.04 4 ) 4 1 4 (1 + 0.045 4 = 100, 000, 000 (0.04 0.045) 0.25 = $125, 000 at the 3.25-year point. ) 4 1 4 ] 22
Forward rates agreements Example (continued) This is equivalent to a cash flow of 125, 000 = $123, 609 1 + 0.045 0.25 at the 3-year point. The cash flow to the party on the opposite side of the transaction will be +$125, 000 at the 3.25-year point or +$123, 609 at the 3-year point. Remark Observe that all interest rates in this example are expressed with quarterly compounding. For simplifying calculations, we will always assume the compounding is at the same frequency as the specified future period of time for which the FRA applies. 23
Forward rates agreements General case Consider a FRA where company X is agreeing to lend money to company Y for the period of time between T 1 and T 2. Define: R K : The fixed rate of interest agreed to in the FRA. R F : The forward LIBOR interest rate for the period between times T 1 and T 2, calculated today. R M : The actual LIBOR rate observed in the market at time T 1 for the period between times T 1 and T 2. L: The principal underlying the contract. We will leave aside our usual assumption of continuous compounding and assume the rates R K, R F and R M are all measured with a compounding frequency reflecting the length of the period to which they apply. This means that compounding frequency = T 2 T 1. This is the usual market practice for FRA). 24
Forward rates agreements Remark We will leave aside our usual assumption of continuous compounding and assume the rates R K, R F and R M are all measured with a compounding frequency reflecting the length of the period to which they apply. This means that if T 2 T 1 = 0.5, they are expressed with semiannual compounding, if T 2 T 1 = 0.25, they are expressed with quarterly compounding, and so on. This is the usual market practice for FRA. This allows to simplify formulas and calculations, and is not really restrictive as we know that we can always compute an equivalent rate from any frequency to any other frequency. 25
Forward rates agreements General case (continued) Normally, company X would earn R M from the LIBOR loan. The FRA means that it will earn R K. The extra interest rate (which may be negative) that it earns as a result of entering into the FRA is R K R M. The interest rate is set at time T 1 and paid at time T 2. The extra interest rate therefore leads to a cash flow to company X at time T 2 of L(R K R M )(T 2 T 1 ). Similarly, there is a cash flow to company Y at time T 2 of L(R M R K )(T 2 T 1 ). 26
Forward rates agreements Remark We see that there is another interpretation of the FRA. It is an agreement where company X will receive interest on the principal between T 1 and T 2 at the fixed rate R K and pay interest at the realized LIBOR rate of R M. Company Y will pay interest on the principal between T 1 and T 2 at the fixed rate of R K and receive interest at R M. This interpretation of a FRA will be very important when we talk about interest rate swaps later. 27
Forward rates agreements General case (continued) FRA are usually settled at time T 1, even though interest rate payment normally occur at time T 2. This means that we must discount the payoff from time T 2 to time T 1. So, for company X, the payoff at time T 1 is L(R K R M )(T 2 T 1 ) 1 + R M (T 2 T 1 ) and, for company Y, the payoff at time T 1 is L(R M R K )(T 2 T 1 ). 1 + R M (T 2 T 1 ) 28