This Q was presented to us by the lecturer. No matter how hard students asked how to solve c) and d) he didn’t tell us. I tried to trick him by asking him the same Q in a bit different way but he knew what was I really trying to ask. The only thing he said is that risk is not linear. Here’s the question. Focus on part c) and d) only. The following statistics, describing the dynamics (monthly frequency) of the stock market index on the NYSE, are given: mean (continuously compounded) return per month 1%, standard deviation (per month) 2%. Also, the dynamics of the stock market index are assumed to follow a random walk with drift, i.e. ρ = 1. More formally, we can summarize the above statistics as: Xt = c + rho*X(t-1) + e where: Xt - log price of the market index e - shock to index where m = 0; st.dev = 2% per month c = 0,01 Rho = 1 Assume that an agent invests all of his wealth (1,000,000 $) into a fund consisting of a 100% investment in the NYSE market index. Use the above statistics to answer the following questions a) What is/are the source(s) of risk for this agent and what is/are the exposure(s) of the agent to this/these source(s) of risk b) Use the Delta-Normal method to compute the VaR (monthly horizon) of this agent at 97.5% confidence. (Note that the relevant ξ-value for this problem is ξ=1.96). Give the VaR both as a percentage of initial wealth as well as in terms of US dollars. Imagine the agent to rebalance his portfolio. Instead of investing all his wealth in the fund, he now invests 50% in a risk-free (1-month maturity, zero-coupon) bond with an interest rate (or equivalently yield) of 5% in per annum terms. The remaining 50% are invested in the fund, investing in the NYSE market index. Basically, I compute VAR = st.dev (dollar value) * 1,96 = 0,02*1,000,000 * 1,96 = 39,200; c) Compute the VaR (monthly frequency, confidence level 97.5%) for this portfolio both in percentage and in US dollars. (Explain concisely how you compute the VaR). Now here’s the tricky part. If you combine it with the risk free asset, then the new STDEV should be the fraction of the risky portfolio times its STDEV of 0,02%. I multiply the new STDEV by the critical value of 1.96 or in numbers -> 0.5 (weight) * 0,02 * 1,96 or VAR of 14,6. I guess, here risk free asset simply mean gov. bond that can fluctuate with interest rates and it’s not the same as the risk free asset used to derive the Capital Market Line. If that’s the case, we need the STDEV of the bond to find the new STDEV and the correlation with the risky portfolio which are not given. ANY IDEAS ? d) Why is the VaR not exactly equal to half of the value at risk obtained under question b)? Explain briefly. So how should I proceed with c) and d) ? The hint is that Risk is not linear. I think I’m missing something very very obvious.
The variability of the risk-free asset over the 1-month time horizon is 0 (it’s a 1-month zero coupon bond). What is meant here is that you are looking for the portfolio VaR at maturity of the bond (so easy, methinks). Edit: BTW - I think you are supposed to include drift here.
If we assume that it’s zero then I think the VAR of the combined portfolio should be exactly 50% of the 39,200$ which is inconsistent with the d) part - why is the VAR not exactly equal to half the risky VAR. It should be due to diversification but still… Regarding the drift, I wondered why its given. If it is to be included, won’t it become too calculation heavy ?
VaR is not sub additive. to compute you must multiply 50% to the variance not the STDEV
You mean - 0,5 * (0,02)^2 = 0,0002 SQRT of 0,0002 = 0,0141 0,0141*1,000,000*1,96 = 27,636$ ? Agree ? In this case VAR does not detect diversification which is in line with sub-additivity.
2x2equals4 Wrote: ------------------------------------------------------- > You mean - 0,5 * (0,02)^2 = 0,0002 > SQRT of 0,0002 = 0,0141 > > 0,0141*1,000,000*1,96 = 27,636$ ? Agree ? > > In this case VAR does not detect diversification > which is in line with sub-additivity. not sure what you mean by the detect diversification but the equation looks correct now.