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unit4 the term structure of interest rates

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University of Waterloo
Actuarial Science
Jiahua Chen

ACTSC 445: Asset-Liability Management Department of Statistics and Actuarial Science, University of Waterloo Unit 4 – The Term Structure of Interest Rates Different factors affect the value of interest rates associated with fixed income securities like bonds. In what follows, we will focus on the term structure of interest rates, that is, how rates change with the maturity of these securities. Note that each security leads to a different set of interest rates. To study the term structure of interest rates, it is useful to focus on one type of security. So which one should we choose? A natural choice is to focus on the Treasury market (i.e., securities issued by the government (T-bills, notes and bonds)). There are two reasons for this choice: (1) treasury securities are considered default-free, so differences in assessment of the creditworthiness of the issuer cannot affect the yield estimates for these securities; (2) as the most active bond market, the Treasury market offers no illiquidity problems, and prices can readily be observed. Types of interest rates In this section, we discuss four different ways of representing interest rates: 1. yield-to-maturity 2. spot rate 3. forward rate 4. short rate. Yield-to-maturity (ytm) • This measure is widely used for bonds (see Unit 2). • Given by the constant interest rate that equates the discounted value of the future cash flows under the bond and its current market price. • Also called internal rate of return. • More precisely, using the same notation as in the notes for Unit 2, the ytm y (measured on the same type of period as the coupon-paying period) is the value such that −n P = cF 1 − (1 + y) + F(1 + y)−n , y where we assume for simplicity that cF is the actual value of the coupon. • The yield curve refers to the graphical depiction of the yield level as a function of time. 1 • See for current and historical data on US Treasury Securities. • Important to note: ytm is cash-flow dependent... Another problem is that it does not reveal the year by year information. • When using ytm for the term structure of interest rates, presumably the same coupon rate is used for all different maturities. Spot rates First, we need to introduce some notation. Let P(t,t + k) be the price at time t of a zero-coupon (or pure discount) bond with k periods until maturity (and face value F = 1). Let t = 0. The spot rate for k periods to maturity—denoted s —iskthe ytm for a zero-coupon bond with k periods to maturity, i.e., k is such that P(0,k) = (1 + s k −k (Note: just like in the above discussion of ytm, for simplicity we assume here that the spot rates are measured on the same type of period as the coupon-paying period. Later on, we’ll work with annual rates compounded at the same frequency as the coupons. E.g, right now if coupons are paid twice a 1/2 year, k = 2, and P(0,2) = 0.92, then we compute s = (120.92) − 1 = 0.0426 as a semi-annual rate; later on, we’ll instead work with the annual rate, which here would be 2 × 0.0426 = 0.0851.) Figure 1 shows the spot-rate curve from 0 to 30 years for different maturities of Canadian Treasury securities. (The data has been obtained from the Bank of Canada website Figure 1: Canadian Treasury Spot Rates from 0 to 30 years at selected times Example: an initial investment of 100$ accumulates to 106$ after 1 year, and to 113.42$ after two years. What are the corresponding one-year and two-year spot rates? 2 −1 −2 Solution: we have that1s is such that 100 = 106(1 + s⇒ s1= 0.06, and s2is such that 100 = 113.42(1 2 s )⇒ s2= 0.06499. Note: with a term structure of interest rates based on spot rates, we can find the price P of a bond by using X n P = cF(1 + s ) −k + F(1 + s ) −n , (1) k n k=1 and then find its ytm. Question: using the same data as in the previous example, assume you have the following alternative strategy. Invest 100$ at time 0; at time 1, use the proceeds from this investment to reinvest at a rate f, where f is determined at time 0. What should be f so that there is no arbitrage opportunity? Solution: the two strategies should produce the same outcome at time 2, otherwise there would be an arbitrage opportunity. Hence we must have that f satisfies 106(1 + f) = 113.42 ⇒ f = 113.42/106 − 1 = 7%. Forward rates Forward rates are the interest rates that would be used for contracts made today covering transactions in future periods. At t = 0, we denote by f j,kthe forward rate covering period j to k. For instance, if the unit of time is years, then f corresponds to the forward rate between year 2 and year 4. Can think of the forward 2,4 rate fj,kas the rate agreed upon at time 0 for borrowing money at time j that will be repaid at time k. The relationship with spot rates is as follows: (1 + s )k (1 + f )k−j = k j,k (1 + sj)j or in terms of prices for zero-coupon bonds, we have P(0,j) (1 + fj,kk−j = . P(0,k) The one-period forward rates f k,k+1 for k = 0,1,..., are denoted as f (wekdrop the k + 1 in the notation). Using this notation, we can give yet another relationship between forward rates and spot rates: 1/k 1 + sk= ((1 + f )01 + f )1..(1 + f k−1)) . In other words, the spot rates are geometric averages of the forward rates. For example, previously we had f 0 = s 1 = 0.06 and f 1 = 0.07 (denoted f above), corresponding to a two-year spot rate 1/2 s2= (1.06 × 1.07) = 1.06499. Figure 2 shows how the spot rates and the one-period forward rates compare in two different scenarios. As shown by these relationships, spot rates uniquely determine forward rates and vice-versa. 3 f w a d t p s t a p s t a s f r a d a s Figure 2: One-period forward rates versus spot rates Short rates Short rates refer to the one-period interest rates that may turn out to be available in the market during a future period. More precisely, the short rate r is the rate that will apply between time k and k + 1. k The difference with the forward rate f is that khe latter is the rate agreed upon at time 0 that applies between time k and k + 1, whereas r is the ackual rate (unknown at time 0) that will apply during this period. Hence short rates are treated as random variables. Continuous models are often used to represent s
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