Take Home Mid-Term Examination

RPRD 801 Fall 2015 Take Home Mid-Term Examination

This examination is due on noon Friday November 6. The answers should be typed.

There are three questions. Attempt all three questions.

1. Consider the basic Merton model that is the foundation of the KMV structural credit model. One of the weaknesses of the KMV model is that for many corporations in Canada, share and bond markets are so illiquid it is difficult to obtain reliable prices. As a consequence you have been asked to construct a Monte Carlo simulation of the cash flows for a mining company Big Gold. Your bank is considering lending to a company financing a new gold mine in Rurutania. You have talked to your fundamental credit analysts who specialize in gold mining loans, and to your quants. You have to meld both views into one model. (a) Recall the model in Section 3 of Crean Milne where there is a large investment followed by a stochastic net revenues generated by gold price demand shocks. Assuming a simple binomial demand shock under the statistical measure P derive the stochastic net revenue as a binomial stochastic difference equation. (b) Assuming that all agents are truly risk neutral (P=Q), and assuming a flat non- stochastic term structure with rate r, derive the binomial stochastic difference equation for the value of the firm. Compare with the Merton sde model. Is the drift term equal to r? Why? (c) If agents were not truly risk neutral (P≠Q) then what is the standard arbitrage pricing method for deducing Q? If that method cannot be used because the share and bond markets for Big Gold are illiquid then then what method does KMV use? See Crouhy (2001) Ch. 9 Section 6. Notice the caveats on reliability in pricing. (d) As the Merton model assumes a stochastic differential equation (sde) based on the firm’s value, then what implicit assumptions are being made about the financial market valuation of cash flows? (e) Given that the gold mine has only one loan from your bank, and no other debt, then how would you calculate the PD and Recovery Rates (RR)? (f) The basic Merton model assumes that default only occurs at the horizon T. Could you revise your model to allow default before T? Show how in a simple binomial example. (g) Consider a Crean Milne model where there are two iid stochastic factors, a binomial normal demand shock and a downward Poisson jump demand shock (over the last few years Gold prices have had some occasional major price falls). Redo the analysis above deriving the sde for the value of the firm. How many stochastic factors do you have? How would you estimate the coefficients on the factor(s)? Is there an estimation problem? Is there a valuation problem? Are your estimates on PD and RR reliable? Why? (h) Now you are informed by the bank’s country credit risk analyst that there is a possibility that the Ruritanian government could impose a tax on the mine if the price of gold goes high and the mine becomes very profitable. Give a simple modification of the model to adapt to that risk. Could you estimate the coefficients on the sde from past data? How would you do a Monte Carlo? Would it be reliable? (i) Now you are told that the gold mining company has a contract with Gliderman Sox to lobby the Ruratanian government and improve the chances that the tax will not be imposed. The contingent contract will be very expensive and cost USDX. Can you incorporate that into your PD and RR simulation? How?

2. You have just been appointed as bond portfolio manager to Fast Food Corp the biotech food company. You have been told to use duration measures to hedge Fast Food’s bond portfolio. (a) Define a bond’s duration and convexity. See J&T section 13.4 (b) Use duration to hedge a bond portfolio of 1 and 2 year bonds. (c) Can you use duration and convexity to hedge a 1 and 10 year bond? How many bonds would you need for the hedge and what would determine your hedge portfolio? (d) What are the major limitations of duration measures to hedge bond portfolios? (e) Are the Ho-Lee or Black, Derman and Toy models better models for hedging bonds? Or would you require more stochastic factors. Hint J&T Ch. 15. (f) If you had more stochastic factors, would their coefficients be easy to estimate? Would you expect them to be stable? See Duffee (2012).

3. Consider two assets with hourly returns as given below: Return to Asset 1 Return to Asset 2

State 1 1 - 1 Prob(state 1) = 0.45

State 2 - 1 1 Prob(state 2) = 0.45

State 3 - 5 - 5 Prob (state 3) = 0.09

State 4 - 20 - 20 Prob (state 4) = 0.01

(a) Given this process, compute VaR for each asset over two horizons: 10 trading hours ahead and 100 trading hours ahead. Assuming an initial portfolio of $1000, compute and graph the frequency distribution for an equally weighted portfolio. (For simplicity in calculations and given the short horizon we have ignored compounding.) (b) Use a Monte Carlo simulation to create a histogram for 10 and 100 day samples drawn from each asset and the portfolio. Compare these histograms with their theoretical frequencies. Note the importance of the number of simulation runs in approximating the true distribution. (c) Assume that the true process is as above, but you do not know the true process, and you observe only 100 hours of data. How confident are you identifying the “true” process and estimating its parameters? Why? (d) Can you extend these examples to illustrate other properties discussed in Dowd Chs.4 and 5 (e.g. modeling non-constant volatilities)? Construct an example to illustrate “ghost” effects. Illustrate the dangers in predicting risks around the time of a change in volatility.