# In Chapters 2 (Section 2.8) and 3 (Section 3.7) we analyzed the risk and return of the

## Question:

(a) Parameter Estimation:

i. Using the data in Table 2.6 in Chapter 2, estimate the parameters Î³* and r* of the Vasicek model by nonlinear least squares, as in Equation 15.41.

ii. Download daily data on the 1-month T-bill rate from the Federal Reserve Web site up to December 31,1993, and use these data to estimate Î³, r, and Ïƒ in the Vasicek model. This estimation can be accomplished through a linear regression (see Exercises in Chapter 14). Take literally dt = 1/252 = 1 day and discretize the process into

Where É›t+dt ~ N(0, Ïƒ2dt). This looks similar to a regression

(rt+dt - rt) = Î± + Î²rt + É›t + dt

What are Î± and Î² in terms of the original parameters Î³ and r? How can you estimate Î³?

iii. Compare (e, Î³) with (r*, Î³ *). Interpret the differences and discuss.

(b) Consider now a portfolio with a fraction x = 0.1366 in 1-year T-bills and the remainder (1 - x) in 3-year inverse floaters (see Section 3.7.5 in Chapter 3). We take this portfolio as given. Let r be the continuously compounded interest rate on January 3, 1994.

i. If PIF (r, t; T) denotes the price of the inverse floater discussed in Section 2.8 in Chapter 2, compute its sensitivity to changes in the interest rate (PIF /(r. (You may maintain the assumption that r ii. Compute the dollar convexity of the inverse floater, that is, the second derivative (2PIF/(r2.

iii. If n (r, t; T) denotes the value of the entire portfolio, compute its sensitivity to changes in the interest rate r, ( Î {r,t;T)/dr, as well as its dollar convexity (2 Î (r, t; T) /(r2.

(c) The portfolio Î (r,t;T), as any other security, must satisfy a fundamental pricing equation (Equation 15.24):

i. Write down the equation that Î (r, t; T) must satisfy.

ii. Given your answers above, can you compute the change in value of the portfolio due to the passage of time? That is, what is d Î /dt?

iii. For each day, what then is the capital gain or loss that can be imputed only to the passage of time? What is the intuition behind it? How does this relate to the level of Gamma?

(d) Value-at-Risk: We can use simulations and the Vasicek model to compute a 1-year Value-at-Risk, that is, the maximum loss that the portfolio may incur with a% probability due to the movement in interest rates. Proceed as follows:

i. Given the parameter estimates of the model, simulate M interest rate paths (M large) over a one-year horizon

ii. For each simulated scenario about the interest rate at time t = 1, rt, apply the Vasicek formula and compute the distribution of the Orange County portfolio. Plot the histogram of the portfolio distribution at t - 1.

iii. Compute the 1% and 5% worst cases of the portfolio distribution, and thus obtain the 1% VaR and 5% VaR. Are the ex-post losses suffered by Orange County's portfolio completely unexpected?

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**Related Book For**

## Fixed Income Securities Valuation Risk and Risk Management

**ISBN:** 978-0470109106

1st edition

**Authors:** Pietro Veronesi