Haipeng Xing Department of Applied Mathematics and Statistics Outline 1 Types of interest rates 2 Measuring interest rates 3 The n-year spot rate 4 ond pricing 5 Determining treasury zero rates the bootstrap method 6 Forward rates 7 Forward rate agreements 8 Duration 9 Convexity 10 Theories of the term structure of interest rates
Types of interest rates treasury rates Treasury rates are the rates an investor earns on Treasury ill and Treasury bonds. They are instruments used by a government to borrow in its own currency. It is usually assumed that there is no chance that a government will default on an obligation denominated in its own currency. Treasury rates are therefore considered as risk-free rates in the sense that an investor who buys a Treasury bill or Treasury bond is certain that interest and principal payments will be made as promised. Types of interest rates LIOR LIOR (London Interbank O ered Rate) is an unsecured short-term borrowing rate between banks. LIOR rates have traditionally been calculated each business day for 10 currencies and 15 borrowing periods. The borrowing periods range from 1 day to 1 year. LIOR rates are used as reference rates for hundreds of trillions of dollars of transactions throughout the world. In recent years there have been suggestions that some banks may have manupulated their LIOR quotes.
Types of interest rates the federal funds rate In the United States, financial institutions are required to mantain a certain amount of cash with the Federal Reserve. The reserve requirement for a bank at any time depends on its outstanding assets and liabilities. At the end of a day, some financial institutions have surplus funds in their federal account while others have requirement for funds. This leads to borrowing and lending overnight. In the U.S., the overnight rate is called the federal funds rate. The weighted average of the rates in brokered transactions (with weights being determined by the size of the transaction) is termed the e ective federal funds rate. This overnight rate is monitored by the central bank, which may intervene with its own transactions in an attempt to raise or lower it. oth LIOR and the federal funds rates are unsecured borrowing rates. Types of interest rates repo rates A repurchase agreement (repo) is an agreement where a financial institution that owns securities agrees to sell them for a certain price and buy them bank in the future (usually the next day) for a slightly higher price. The financial institution obains a loan in the repo. The interest rate that the financial institution pays is referred to as the repo rate. Repo rates are secured rates, hence a repo rate is generally slightly below the correpsonding fed funds rate.
Measuring interest rates simple interest Interest: In exchange for the use of an investor s money, banks pay a fraction of the account balance back to the investor. The fractional payment is known as interest. The money a bank uses to pay interest is generated by investments and loans that the bank makes with the investor s money. An interest rate, denoted as r, is the fraction of the invested amount used to compute the interest. It is usually expressed as a percentage paid per year. The initially invested amount which earns the interest, denoted as P, is called principal. The sum of the prinicpal amount and earned interest, denoted as A, is called the compound amount. Simple interest: The compound amount after t interest periods (think of them as years) is A = P (1 + r) t. Measuring interest rates compound interest When interest is allowed to earn interest itself, an investment is said to earn compound interest. In this case, part of the interest is paid to the investor more than once a year. Let the number of compounding periods per year be n. Iftheannual interest rate is r, then the interest rate per compounding period is r/n and the formula for compound interest is A = P 1+ n r nt.
Measuring interest rates continuously compounded interest When the number of compunding periods, n, increases such that the procedures of depositing and withdrawing are almost instantaneous, the compound amount A becomes A = lim 1+ P r nt. n!1 n Using the natural logarithm, we obtain a formula for continuously compounded interest, A = Pe rt. The n-year spot rate 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 end of n years, and there are no intermediate payments. The n-year zero-coupon interest rate is also referrred as n-year spot rate, then-year zero rate.
ond pricing Most bonds pay coupons to the holders periodically. The bond s principal (i.e. par value or face value) is paid at the end of its life. The 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. Example 1 (Pricing a coupon bond) Consider a 2-year Treasury bond with a principal of $100 provides coupons at the rate of 6% per annum semiannually. The Treasury zero rates are measured with continuous compounding, as in the table. Then the theoretical price of the bond is Figure 1: Treasury zero raets (Hull, 2014; Table 4.2). $3e 0.05 0.5 + $3e 0.058 1.0 + $3e 0.064 1.5 + $103e 0.068 2.0 = $98.39 ond yield The bond yield is the discount rate that makes the present value of the cash flows on the bond equal to the market price of the bond. Example 2 (Calculating the bond yield) Suppose that the theoretical price of the bond we have been discussing, $98.39, is also its market price. Let y be the yield on the bond, it must satisfy that 3e 0.5y +3e 1.0y +3e 1.5y + 103e 2.0y = 98.39. Solving this equation yields that y =6.76%.
Par yield The par yield for a certain bond maturity is the coupon rate that causes the bond price to equal its par value (or the principal value). Example 3 (Calculating the par yield) Suppose that the coupon on a 2-year bond in our example is c per annum (or c 2 per 6 months). Using the table in Example 4, 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 Solve this equation gives c = 6.87 (with semiannual compounding). Determining treasury zero rates the bootstrap method One way to determine Treasury zero rates is from Treasury bills and coupon-bearing bonds. The most popular approach is known as the bootstrap method. Example 4 (Determining treasury zero rates) To illustrate the bootstrap method, we use the data on the right (Hull, 2014; Table 4.3) to compute the treasury zero rates.
Example: Determining treasury zero rates ecause the first three bonds pay no coupons, the zero rates corresponding to the maturities of these bonds can easily be calculated. 97.5 = 100e R 1 0.25 =) R 1 = 10.127% 94.9 = 100e R 2 0.5 =) R 2 = 10.469% 90 = 100e R 3 1.0 =) R 3 = 10.536% The fourth bond lasts 1.5 years, with payments $4, $4, and $104 at the end of year 0.5, 1.0, and 1.5, respectively. 4e R 2 0.5 +4e R 3 1.0 + 104e R 4 1.5 = 96 =) R 4 = 10.681% The 2-year zero rate can be calculated from the 6-month, 1-year, and 1.5-year zero rates. That is, 6e R 2 0.5 +6e R 3 1.0 +6e R 4 1.5 + 106e R 5 2.0 = 101.6 =) R 5 = 10.808% Example: Determining treasury zero rates A chart showing the zero rate as a function of maturity is known as the zero curve. It is usually assume that (1) the zero curve is linear between the points determined using the bootstrap method, and (2) the zero curve is horizontal prior to the first point and horizontal beyond the last point. Figure 2: Zero rates given by the bootstrap method (Hull, 2014; Table 4.4 & Figure 4.1).
Forward rates Forward interest rates are the future rates of interest implied by current zero rates for periods of time in the future. Suppose that the zero rates for time periods T 1 and T 2 are R 1 and R 2 with both rates continuously compounded. The forward rate R F for the period between times T 1 and T 2 is R F = R 2T 2 R 1 T 1 = R 2 +(R 2 R 1 ). (1) T 2 T 1 T 2 T 1 This formula is only approximately true when rates are not expressed with continuous compounding. Letting T 2! T 1 = T, we obtain the instantaneous forward rate for amaturityt,whichistheforwardratethatappliesforaveryshort time period starting at T.Itis R F = R + T @R @T. T 1 Forward rates Remark 1 T 1 R F = R 2 +(R 2 R 1 ). T 2 T 1 If the zero curve is upward sloping between T 1 and T 2 so that R 2 >R 1,thenR F >R 2. If the zero curve is downward sloping between T 1 and T 2 so that R 2 <R 1,thenR F <R 2.
Forward rate agreements (FRAs) If a large investor thinks the rates in the future will be di erent from today s forward rates, one strategy is to enter into a contract known as a forward rate agreement to fix the rate in the future. A FRA is an over-the-counter agreement that a certain rate will apply to a certain principal during a certain future time period. The usual assumption underlying the contract is that the borrowing or lending would normally be done at LIOR. If the agreed fixed rate is greater than the actual LIOR rate for the period, the borrower pays the lender the di erence between the two applied to the principal. If the reverse is true, the lender pays the borrower the di erence applied to the principal. Forward rate agreements (FRAs) The present value of the payment is made at the beginning of the specified period, as illustrated below (Hull, 2014; Example 4.3). Example 5 (FRA) Suppose that a company enters into an 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 LIOR is paid and 4% is received for the 3-month period. If 3-month LIOR proves to be 4.5% for the 3-month period, the cash flow to the lender will be $100, 000, 000 (0.04 0.045) 0.25 = $125, 000 at the 3.25-year point. This is equivalent to a cash flow of 125, 000 (1+ 0.045 0.25) = $123, 609 at the 3-year point.
Forward rate agreements (FRAs) Consider an FRA where company X agrees 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 :TheforwardLIORinterestratefortheperiodbetweentimes T 1 and T 2,calculatedtoday R M : The actual LIOR interest 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 Assume that the rates R K, R F,andR M are all measured with a compounding frequency reflecting the length of the period to which they apply. For company X, the payo at time T 1 is A := L(R K R M )(T 2 T 1 ). 1+R M (T 2 T 1 ) FOr company Y, the payo at time T 1 is A. Values of an FRA An FRA is worth zero when the fixed rate R K equals the forward rate R F. When an FRA is first entered into, R K is set equal to the current value of R F, so that the value of the contract to each side is 0. As time passes, interest rates change, hence the vlaue is no longer 0. The market value of a derivative at a particular time is referred to as its mark-to-market (MTM) value. For an FRA that R K will be received on a principal of L between times T 1 and T 2, its value doay is V FRA = L(R K R F )(T 2 T 1 )e R 2T 2. For an FRA that R F will be paid on a principal of L between times T 1 and T 2, its value doay is V FRA = L(R F R K )(T 2 T 1 )e R 2T 2.
Duration The duration of a bond 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. Consider a bond provides the holder with cash flows c i at time t i (1 apple i apple n). The bond price and bond yield y are related by = nx c i e yt i. i=1 The duration of the bond, D, isdefinedas D = P n i=1 t ic i e yt i = nx i=1 t i h ci e yt i i. (2) Duration When a small change y in the yield is considered, (2) implies that nx = y t i c i e yt i. (3) From equations (2) and (3), we obtain an approximate relationship between percentage changes in a bond price and changes in its yields, = D y. (4) If y is expressed with a compounding frequency of m times a year, then = D y, where the variable D, referred to as the bond s modified duration D D = 1+y/m. i=1
Convexity From Tyalor series expansions, we obtain an expression for as follows, = @ @y y + 1 2 @ 2 @y 2 y 2. Define the measure of convexity for the bond, C = 1 P @ 2 n @y 2 = i=1 c it 2 i e yt i. This leads to = D y + 1 2 C y 2. Figure 3: Two bond portoflios with the same duration (Hull, 2014; Figure 4.2). Theories of the term structure of interest rates Question? What determines the shape of the zero curve? A number of di erent theories have been proposed. Expectations theory: It conjectures that long-term interest rates should reflect expected future short-term interest rates. Market segmentation theory: It conjectures that the short-, medium-, and long-term interest rates are determined by supply and demand in the short-, medium-, and long-term bond markets, respectively. They are independent with each other. Liquidity preference theory: Itassumesthatinvestorspreferto preserve their liquidity and invest funds for short periods of time. orrowers, on the other hand, prefer to borrow at fixed rates for long periods of time. This leads to a situation in which forward rates are greater than expected future zero rates.