Midterm Practice Problems From Review Session:
11/9/2012
Ch 8
From Chapter 8, I went over problems 8.7, 8.9 (Answers in back of the book) and problems 8.16, 8.17, which you have answers from Homework solutions. Make sure you know the duration and convexity equations, and how the bond price changes due to a change in yield/
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Ch 9
From Chapter 9, I went over problems 9.6, 9.9, 9.10 (Answers in the back of the book) and 9.13. VaR problems in my mind break down into three parts: Normally distributed, the “stair cummulative dist function”, and the autoregressive model. Be familiar with how to solve all three.
9.13.
Suppose that daily changes for a portfolio have first-order correlation with correlation parameter 0.12. The 10-day VaR, calculated by multiplying the one-day VaR by , is $2 million. What is a better estimate of the VaR that takes account of autocorrelation?

The correct multiplier for the variance is
10 + 2 × 9 × 0.12 + 2 × 8 × 0.122 + 2 × 7 × 0.123 + . . . + 2 × 0.129 = 10.417
The estimate of VaR should be increased to = 2.229

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Ch 10
From Chapter 10, I went over 10.11, 10.12 (Answers in Book). While many of the formulas here look complicated, when it comes down to it, many parts of these questions in the book are just “plug-and-chug” questions. Here is 10.21 in case you tried it out for practice:
10.21.
Suppose that the parameters in a GARCH(1,1) model are = 0.03, = 0.95 and  = 0.000002.
(a) What is the long-run average volatility?
(b) If the current volatility is 1.5% per day, what is your estimate of the volatility in 20, 40, and 60 days?
(c) What volatility should be used to price 20-, 40-, and 60-day options?
(d) Suppose that there is an event that increases the volatility from 1.5% per day to 2% per day. Estimate the effect on the volatility in 20, 40, and 60 days.
(e) Estimate by how much the event increases the volatilities used to price 20-, 40-, and 60-day options.

(a) The long-run average variance, VL, is The long run average volatility is = 0.01 or 1% per day.
(b) From equation (10.14) the expected variance in 20 days is
0.0001 + 0.9820(0.0152 − 0.0001) = 0.000183
The expected volatility per day is therefore = 0.0135 or 1.35%. Similarly the expected volatilities in 40 and 60 days are 1.25% and 1.17%, respectively.
(c) In equation (10.15) a = ln(1/0.98) = 0.0202. The variance used to price 20-day options is

so that the volatility is 22.61%. Similarly, the volatilities that should be used for 40- and 60-day options are 21.63% and 20.85% per annum, respectively.
(d) From equation (10.14) the expected variance in 20 days is
0.0001 + 0.9820(0.022 − 0.0001) = 0.0003
The expected volatility per day is therefore = 0.0173 or 1.73%. Similarly the expected volatilities in 40 and 60 days are 1.53% and 1.38% per day, respectively.
(e) When today’s volatility increases from 1.5% per day (23.81% per year) to 2% per day (31.75% per year) the equation (10.16) gives the 20-day volatility increase as