IV FixedIncome Securities A Bond Prices and Yields
IV. Fixed-Income Securities • A. Bond Prices and Yields • B. The Term Structure of Interest Rates 1. 2. 3. 4. Definitions Theories of the Yield Curve Empirical Evidence Misc. Items • C. Managing Bond Portfolios
B. The Term Structure of Interest Rates • 1. Definitions Yield as a function of the time to maturity of a zerocoupon bond is called the term structure of interest rates or the yield curve. Spot rates: interest rates for different maturities observed at any point in time. Short rates: interest rates for a given time interval. Forward rates: the interest rates at which one can contract today for loans in the future (future short rates).
B. The Term Structure of Interest Rates • 1. Definitions Example: Short rates and the yield curve Suppose that people have expectations about short rates for the next, say, 4 years. These expectations of future short rates are what generate the yield curve we observe.
B. The Term Structure of Interest Rates • 1. Definitions Example: Short rates and the yield curve Year 0 1 2 3 Interest Rate 8% 10% 11% Bond prices: P 1 = 1000/(1. 08) = 925. 93 P 2 = 1000/[ (1. 08)(1. 10) ] = 841. 75 Pn = 1/[ (1+r 1)(1+r 2)(1+r 3)…(1+rn) ] The yield curve: 925. 93 = 1000/(1+y 1) y 1 = 0. 08 841. 75 = 1000/(1+y 2)2 y 2 = 0. 08995 758. 33 = 1000/(1+y 3)3 y 3 = 0. 09660…etc.
B. The Term Structure of Interest Rates • 1. Definitions Example: Implied forward rates Spot rates are interest rates for different maturities observed at any point in time. Forward rates are the rates at which one can contract today for loans in the future (future short rates). How can we determine the forward rates implied by the termstructure?
B. The Term Structure of Interest Rates • 1. Definitions Example: Implied forward rates Time Spot rates (annual) . 5 yr 1 yr 2 yrs 5. 25 5. 50 6. 02 Objective: invest funds over a one year horizon Strategy A: Strategy B: Buy and hold a one-year bond: return = (1+z 2)2 Roll over strategy: invest in a 6 -month bond and roll over the investment 6 months from now at the 6 -month forward rate. Return = (1+z 1)(1+f 1) In the absence of arbitrage, the two strategies should yield exactly the same returns. Or, (1+z 1)(1+f 1) = (1+z 2)2 (1. 02625)(1+f 1) = (1. 0275)2 f 1 = 0. 028752 (5. 75% annually)
B. The Term Structure of Interest Rates • 2. Theories of the Yield Curve Expectations Theory: the shape of the yield curve can be explained by investors’ expectations about future interest rates. ( expected short rate = forward rate implied by the term structure. ) Formally, this hypothesis implies that the long rate is an average of current and expected short rates. I. e. , liquidity premiums are zero. Thus, (1 + yt, 2)2 = (1 + rt, 1)(1 + ft+1, 1) = (1 + rt, 1)(1 + E(rt+1, 1) ) Implications: A downward (upward) sloping yield curve implies that interest rates are expected to fall (rise) in the future. Intuition?
B. The Term Structure of Interest Rates • 2. Theories of the Yield Curve Liquidity Preference Theory: (1) Expectations influence the shape of the yield curve. (2) Short-term issues are more desirable to investors than long-term issues because the former are more liquid. Formally, (1 + yt, n)n = (1 + rt, 1)[1 + ft+1, 1]…[1 + ft+n-1, 1] = (1 + rt, 1)[1 + E(rt+1, 1) + L 2]…[1 + E(rt+n-1, 1) + Ln] Implications: If Ln > Ln-1 > … L 2 > 0, then the yield curve will be upward sloping even when no changes in interest rates are expected. A downward sloping yield curve implies that interest rates are expected to decline.
B. The Term Structure of Interest Rates • 2. Theories of the Yield Curve Market Segmentation Theory: Long- and short-term bonds are traded in distinct or segmented markets, each of which finds its own equilibrium independently. Preferred Habitat Theory: Investors prefer specific maturity ranges, but can be induced to switch is premiums are sufficient. Implications: There is no reason for term premiums to be positive or to be an increasing function of maturity.
B. The Term Structure of Interest Rates • 3. Empirical Evidence Chief difficulty: joint hypothesis problem General conclusions: Expectations are important, but pure form of expectations hypothesis is rejected. Positive term premiums do appear to exist. However, they do not appear to increase monotonically over the whole span of forward rates. They also vary over time. There are some seasonal patterns.
B. The Term Structure of Interest Rates • 4. Misc. Items Q: Should an investor buy long-term bonds because they have higher yields? Q: What can account for a rising yield curve? Q: What can account for a higher forward rate? Q: Why might interest rates move? Remark: The yield curve is a good predictor of the business cycle. It reflects the collective wisdom of market participants whose decisions and expectations help determine bond yields.
B. The Term Structure of Interest Rates • Homework, ch 15 1 -6, 10 -11, 13, 20 -21
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