Homework
Assignment 3
This is a simple two-part homework assignment designed to show you how you might use the Minimum Variance Hedge Ratio for cross hedges. To complete this homework you must download the Excel workbook mvhr.xls. You should also review the material in Lecture 3 and the treatment of Cross Hedging in chapter 3 of Hull.
In both the lecture and in Hull’s treatment of the MVHR, we used the delta (the first difference, or in other words the change in the value of the observation from one time period to the next ... see the sample slide where that was done) because that is the traditional approach. Using the continuous growth rate (CGR), though, will give a less biased result for certain kinds of problems.. For that reason in Part 1 below we will use the delta, but in Part 2 we will use the CGR.
Part 1:
Open your MVHR workbook to the page labeled MVHR.. You will see 16 observations for a gasoline futures price (assume it to be six months out, although you don’t need to know this to complete the homework). Based upon what you see, and using the delta for each observation, calculate the MVHR for gasoline and JP4. (You should get an answer close to one because, of course, because it turns out that gasoline is a good hedge for JP4 – these data are based upon real data from a few years back, tweaked for this homework).
Part 2:
Suppose you run a large long portfolio of stocks. Suppose that you don’t want to liquidate but you are concerned about a sudden and sharp downturn in the market. You decide that you want to hedge one or two months into the future by shorting with S&P500 futures contracts (such a hedge would not normally cover a long period of time, it would be a rolling short hedge that would continue as long as you were concerned).
Review Hull’s treatment of Hedging an Equity Portfolio. Now return to your Excel workbook and select the page labeled Equity Hedge. What you will see there are historical data for the S&P500 and the Net Asset Value (NAV) per share for a large portfolio of stocks similar to those that might be held by a mutual fund (these are real data for a real portfolio). Let us say that the portfolio consists of 3,012,000 shares so at a current NAV of $34.11 it is worth almost $103 millions.
Using the same technique used in Part 1 except here using CGRs calculated with natural logs rather than deltas, calculate the MVHR. (The PNAV should be represented in the numerator of the calculation). Because this is a real portfolio with many stocks that are in the S&P500, you should get a value close to one, but not one. As you can see, the are closely but not perfectly correlated and their variances are a little different.
Note that we are using the spot S&P500 and the spot portfolio NAV rather than a futures price. Not much would be gained by using the futures price (as Hull points out, futures settlement can be used but if you go to the trouble of using that, you should consider using a rolling hedge adjusted each day).
Given that the portfolio value is $103 millions and each futures contract is equal to 250 times the S&P500, using the 9/8/2008 value for the S&P500, (a) how many contracts would we short if we use our MVHR calculation, (b) given the current initial margin for this contract of $22,500, how much cash will this hedge require?