4 The Term Structure of Interest Rates

# 4 The Term Structure of Interest Rates - ACTSC 445:...

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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 P = cF 1 - (1 + y ) - n y + F (1 + y ) - n , 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

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See http://finance.yahoo.com/bonds 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 k —is the ytm for a zero-coupon bond with k periods to maturity, i.e., s 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 year, k = 2, and P (0 , 2) = 0 . 92, then we compute s 2 = (1 / 0 . 92) 1 / 2 - 1 = 0 . 0426 as a semi-annual
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## This note was uploaded on 10/24/2010 for the course ACTSC 445 taught by Professor Christianelemieux during the Fall '09 term at Waterloo.

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4 The Term Structure of Interest Rates - ACTSC 445:...

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