Lecture 14_ The Fixed Income Market Part 2_ Time Varying Interest Rates and Yield Curves.pdf

15.433 INVESTMENTS Class 14: The Fixed Income Market Part 2: Time Varying Interest Rates and Yield Curves Spring 2003 Time-Varying Interest Rates T - B il l R a te s ( m o n th y ly , % ) 0 2 4 6 8 10 12 Ju n -8 5 Ju n -8 6 Ju n -8 7 Ju n -8 8 Ju n -8 9 Ju n -9 0 Ju n -9 1 Ju n -9 2 Ju n -9 3 Ju n -9 4 Ju n -9 5 Ju n -9 6 Ju n -9 7 Ju n -9 8 Ju n -9 9 Ju n -0 0 Ju n -0 1 US0003M US0001M Figure 1: Time varying interest rates, Source: Bloomberg. A Model for Stochastic Interest Rates Let rt be the time-t one-period interest rate: rt+1 = rt = k (r? ? rt) + σ · εt+1 (1) To be consistent with our earlier notation, rt = rt,1! Important: ? εt is the random shock that occurs at time t. ? The shocks follow a standard normal distribution. ? The shocks are independent across time. ? k, r, and σ are constant coefficients. ? The Coefficients and the Moments r is the long-run mean of the interest rate. E (rt) =? (2) σ is related to the volatility of the interest rate. var(rt) =? (3) k captures the rate at which the interest rate reverts to its long-run mean, r?, 0 k : cov(rt, rt+1) =? (4) The Risk of Rolling Over At time t, let rt be zero-coupon interest rate with maturity n. Consider the value of dollar by following the two investment strategies: 1. Long-term: invest in the (2) two-period bond. e 2·rt,2 (5) 2. Short term: invest in the (1) one-period bond and roll over at time t + 1. e rt,1·rt+1,1 (6) The long-term investment is riskless, while the short-term investment is risky. In particular, it is exposed to the random shock εt that is unknown at time t. ? ? Implication for the Yield Curve The time-t value of $ 1 to be collected at t + 2: 1. The short-term strategy: E e ?[rt,1·rt+1,1] (7) 2. The long-term strategy: ?2·rt,2 (8)e If investors are risk-neutral, then they are indifferent between the two strategies, implying rt,2 ? rt,1 = 1 k (r? ? rt,1) ? 2 1 4 σ2 (9) When do we have an upwa