SOLUTION Fama Bliss and Risk Premiums in the Term Structure

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1 SOLUTION Fama Bliss and Risk Premiums in the Term Structure Question (i EH Regression Results Holding period return year 3 year 4 year 5 year Intercept (std err Slope (std err R sq Comment here on whether your results reject or fail to reject the EH: The slope coefficients are all around 1 and statistically different from 0. EH implies slope coefficients equal to 0, i.e. no signal, including the forward spot spread, should be able to give us information about the return on long bonds vs the return on short bonds. Therefore, the results suggest there is money to be made. Note that, of course, this is risk involved. The R^ of this regression are around Question (ii Cumulative P&L Account year 3 year 4 year 5 year

2 Annual P&L year 3year 4year 5year P&L Long/Short Long Trading Strategy ($ SINGLE GRAPH HERE Date year 3year 4year 5year

3 Question (iii Sharpe Ratio S&P Fama French diversified equity 0.48 Portfolio based FX carry trade 0.54 U.S. treasury (UST carry trade 0.4 year year year 0.6 5year

4 Solution First, a couple of general things. If inflation is defined as π t+1 = log(p t+1 /P t, then the nominal kernel must be related to the real kernel as and the process for inflation is m t+1 = n t+1 e π t+1, π t+1 = log m t+1 ( log n t+1. Also, since the nominal bond price must satisfy b 1 t = E t n t+1 e π t+1, some lognormal algebra gives us that the short rate, i t = log b 1 t, is i t = r t + E t π t+1 1 Var t π t+1 + Cov t ( log nt+1, π t+1. (1 The intuition behind the covariance term is as follows. If the covariance is positive, then inflation tends to be high when marginal utility is high, or when consumption is low. When inflation is high, the holders of nominal bonds lose purchasing power. Therefore, if the covariance is positive, this means that nominal bonds tend to pay off badly in real units when consumption is low, so that investors demand an inflation risk premium in the form of a higher nominal yield in order to compensate them against this risk. 1. Since γ m = γ n = 0, log m t+1 = µ m + λ m ε t+1 log n t+1 = µ n + λ n u t+1 The short nominal and real rates are: i t = µ m λ m/ r r = µ n λ n/ The process for inflation is π t+1 = µ m µ n + λ m ε t+1 λ n u t+1. The inflation risk premium is, therefore, i t r t E t π t+1 = ( λ n λ m /. You can confirm that 1 Var ( ( t π t+1 + Cov t log nt+1, π t+1 = λ n λ m /. 6

5 . The calibrated values are: θ z = σ z = ϕ z = λ m = The calibrated values are: θ x = σ x = ϕ x = λ n = Inflation: π t+1 = µ m + z t + λ m ε t+1 ( µ n + x t + λ n u t+1 = µ m + i t + λ m ε t+1 ( µ n + r t + λ n u t Expected inflation is E t π t+1 = µ m µ n =.0315, or 3.15%. The inflation risk premium is i t r t E t π t+1 = = , or 0.85%. If the yields are the same, at.0%, we get E t π t+1 = % and the risk premium is (still i t r t E t π t+1 = = The point is that the inflation risk premium represents a wedge between the the UST TIPS spread and expected inflation. You can t just subtract the TIPS yield from the UST yield and get expected inflation, as people sometimes do. When the yields are the same at %, for example, expected inflation is negative, not zero. 6. The unconditional risk premium is E ( i t r t E t π t+1 = θz θ z E [ µ m + i t + λ m ε t+1 ( µ n + r t + λ n u t+1 ] = θ z θ x µ m θ z + µ n θ x = µ n µ m = ( λ n λ m / =

6 In this model, since the conditional variances and covariances don t vary with time (unlike the conditional mean, the unconditional risk premium is the same as the conditional risk premium. A better model would have timevariation in the risk premium. See Backus, Foresi, and Telmer (1999 for some simple examples of how to build-in time-variation in the conditional variance (e.g., ala Cox-Ingersoll-Ross. 7. The most obvious thing is that inflation is ridiculously volatile. If σ εu = 0 for instance, the volatility is Var(πt+1 = ( γmσ z 1 ϕ + λ m + γ mσz z 1 ϕ + λ 1/ n z which is about 450% when evaluated at our calibrated values! We could, therefore, choose this parameter to get more reasonable inflation volatility, but we won t be able to match the data without a richer model. This model is just way to simple (no stochastic vol, for instance. Note that, interestingly, identifying this parameter by going beyond UST and TIPS data is a nice example of how we can bring macroeconomic information to bear on a model that s not identified by interest rate data alone (see answer to next question. 8. The answer (as far as I can tell is that it s not identified. To understand this, note that: (a Our calibration scheme involved separately identifying the 4 nominal kernel parameters and the 4 real kernel parameters. So, in this sense, we didn t need to know the only interaction parameter between the stochastic processes for n and m. (b But, it still might be the case that, for any numerical specification of the 8 parameters, there is an implied value for σ εu that is required for consistency. For instance, the nominal short rate must satisfy equation (1 above. If you work out the variance and the covariance terms you ll see that they equal: Var t π t+1 = λ m + λ n λ n λ m σ εu Cov t ( log nt+1, π t+1 = λ n λ n λ m σ εu So you see that, once subbed into equation (1 the two terms involving σ εu cancel. Similarly, if you work out the price of a nominal two-period bond using this: b ( t = E t nt+1 e π t+1 b 1 t+1 ( = E t nt+1 e π t+1 e z t+1 8

7 You ll see that it doesn t depend on σ εu, as it won t for bond prices of any maturity. Note that if we wanted to use data on inflation, things would be different. Just look at the expression above for the conditional variance. The volatility of inflation can identify σ εu. 9