Read the article Orange County Case: Using Value at Risk to Control Financial Riskin your packet (it is also on Philippe Jorions website: http://merage.uci.edu/~jorion/oc/case.html). Please, answer the fol- lowing questions, which represent a modication of the questions on Jorions website. To answer the questions, you will need the data le yields.xls, posted on my website. 2. Let us put ourselves in the position of the Orange County Supervisors, who had to decide in December of 1994 whether to liquidate the port- folio or maintain the strategy (obviously, based on past information only). At that time, interest rates were still on an upward path. A Federal Open Market Committee meeting was looming on December 20, and it was feared that the Fed would raise rates further. To as- sess the possibility of future gains and losses, VAR provides a simple measure of risk in terms that anybody can understanddollars. (a) Duration approximation: The eective duration of the pool was reported by the state auditor as 7.4 years in December 1994. This high duration is the result of two factors: the average duration of individual securities of 2.74 years (most of the securities had a maturity below 5 years), and the leverage of the portfolio, which was 2.7 at the time. In 1994, interest rates went up by about 3%. Compute the loss predicted by the duration approximation and compare your result with the actual loss of $1.64 billion. (b) Computation of portfolio VAR: The yields data le contains 5- year yields from 1953 to 1994. Using this information and the duration approximation, compute the portfolio VAR as of De- cember 1994. Risk should be measured over a month at the 95% level. Report the distribution and compute the VAR: i. using a normal distribution for yield changes (Normal method), and ii. using the actual distribution for yield changes (Historical method). (c) Compare the VAR obtained using the two methods. 3. Interpretation of VAR: (a) Convert the monthly VAR into an annual gure. Is the latter number consistent with the $1.6 billion loss? 1

(b) From December 1994 to December 1995, interest rates fell from 7.8% to 5.25%. Compute the probability of such an event. It seems that both in 1994 and 1995, interest rate swings were par- ticularly large relative to the historical distribution. Can you oer reasons for this unusually high volatile interest rate period. 4. Time Varying Volatility: (a) Compute a time-varying volatility of changes in yields using the following approach to see if the recent volatility is abnormally high. The exponential model (as used in Riskmetrics) is: 2 t = 2 t1 + (1 ) e2 t1 where 2 t is the "conditional," predicted variance for time t and is the "decay" factor, usually selected as 0.97 for monthly data. et is the change in the yield between periods t 1 and t, i.e. et = yieldt yieldt1: The model states that the variance forecast is a combination of the previous month forecast and of the latest squared innovation. For the starting value of the variance (at time t=0), use the average variance over the whole period. (b) Compute the monthly volatility forecast (the square root of 2 t ) and discuss whether recent interest rates swings are explained by elevated volatility. (c) Optional: Check whether the assumption of a conditional normal distribution seems adequate for changes in yields. Compute the number of exceptions at the 1-tailed 95% level, using the monthly volatility forecast just computed and the actual increase in yield. Test whether the number of exceptions is in line with what was expected. 5. Hedging: (a) On December 31, 1994, the portfolio manager decides not to liqui- date the portfolio, but simply to hedge its interest rate exposure. Develop a strategy for hedging the portfolio, using i. interest rate futures ii. interest rate swaps (b) For each strategy, describe the instrument and whether you should take a long or short position. 2

(c) On December 31, 1994, the March T-bond futures contract closed at 99-05. The contract has notional amount of $100,000. Its duration can be measured by that of the Cheapest-To-Deliver (CTD) bond, which is assumed to be 9.2 years. Compute the number of contracts to buy or sell to hedge the Orange County portfolio. (d) This contract has typical trading volume of 300,000-400,000 con- tracts daily. Verify with recent volume data at the Chicago Board of Trade (CBOT). Would it have been possible to put a hedge in place in one day? (e) Assuming that futures can be sold in the required amount, would the resulting portfolio be totally riskless?