## Forwatd and future question

vandana.jain
Finance Junkie
Posts: 41
Joined: Tue Jul 24, 2012 4:56 pm

### Forwatd and future question

A fund manager owns a \$50 million USD growth portfolio that has a beta of 1.6 relative to the S&P 500. The S&P 500 Index is trading at 1,190. Calculate the number of futures contracts the fund manager needs to sell to hedge the portfolio. The multiplier of the S&P 500 is 250. Suppose that at the maturity of futures contracts the fund manager experiences a decline in value of his portfolio of 15%. The market index is trading at 1078, and the risk free rate is 3%. Calculate the effectiveness of the hedge.

a. No gain, small loss
b. Gain of 32K
c. Gain of 424K
d. Gain of 1500

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abhishek.chaudhari
Posts: 8
Joined: Tue Jul 17, 2012 11:13 am

### Re: Forwatd and future question

N=1.6*50000000/250*1190=269
Loss=.15%50=7.5 million \$
Gain on Futures=269*250*(112)=7.532 million \$
Net gain-7.532-7.5=.032 million \$=32K gain..
is this the answer kindly confirm?

vandana.jain
Finance Junkie
Posts: 41
Joined: Tue Jul 24, 2012 4:56 pm

### Re: Forwatd and future question

yes, it is the answer
thank you

abhishek.chaudhari
Posts: 8
Joined: Tue Jul 17, 2012 11:13 am

### Re: Forwatd and future question

I actually think the question is flawed, probably because it is trying to query Hull's chapter 3 but it goes astray with imprecision (IMO):

We are assuming that 1,190 is current (at hedge) index spot and 1078 is the future index spot (the future spot, not the current future or future future; there appears to be no futures prices in the question). So no timeframe is given; e.g., is the maturity one year later?
Without a time horizon, we actually do not appear to have the means to infer an index FUTURES price. Please note that 1.5 beta * 50,000,000/(250*1190) assumes incorrectly the current index spot (1190) rather than the current futures price, which is not known? If time = 1.0, for example, we could use 1.5*50 MM/[250*1190*exp(3%*1.0)] = 1226 <-- this would actually be my answer to the first of the compound [sic?] question, already we have an issue!
Lacking that, I think as the only path to finding short futures position gain = (1078 - 1190)/1078 * (1190 * 250 * 269) = \$7.532 million
However, there is a technical problem (IMO): this would be a good approximation, given that delta ~ 1.0 , but for a change in the futures contract where the maturity is not changing. Here the future position goes from F(0) = S(0)*exp(rT) to, at maturity, F(0) = S(0) ... so the question is not aware of this: change in spot does not approximate change in future position over the entire convergence period, it is only an instantaneous approximation.
Lastly, I don't mean to be a purist, but asking for "hedge effectiveness" is not (IMO) the best word choice as that is specifically defined as something else in Geman