Futures Question

[ABFRM]

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

i think question has some data missing...

but i need to confirm....

 

[ABFRM]

will you please help me david with this question....

 

[ShaktiRathore]

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?

 

[ABFRM]

okay... but it was a trick to give you risk free rate then

 

[ShaktiRathore]

my question is 1190 is not a futures price so u cant use that in the denominator...

Actually the manager is taking futures position on the index.Initially if spot price is S0 than futures price is same which can change in future. As F=S0*e^rt. we can use this price as futures because futures are derivatives whose value depends on the value of the underlying which in this case is index.Here the futures has same price as index so that its price depends on the price of index. And kindly tell if answer is correct.

 

[ABFRM]

yeah answer is correct but i am attaching a file which is an excerpt from HULL where it is specifically mentioned abt S&P 5oo futures price

 

[ShaktiRathore]

yeah answer is correct but i am attaching a file which is an excerpt from HULL where it is specifically mentioned abt S&P 5oo futures price

Well,
IN the example given in Hull. There is no final value of index given, So the example provides with risk free rate to know the final position of the index which is already provided in the above Question. Also in hull's example futures price is explicitly mentioned which we should take but in above Q we need to assume the futures price as same as that of index. I the presentation of Question does matter.

 

[ShaktiRathore]

Also the Hull's example is calculating the expected portfolio return that's why its considering the risk free rate. Don't get confused

 

[David Harper CFA FRM]

I can't really improve on ShaktiRathore's answer (star awarded!), it seems to me to be the best you do with the 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 like ShakhtiRathore's 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 as ShakhtiRathore says (exactly!), 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

I totally agree with ShaktiRathore's implications about the riskfree rate. Questions contain red herrings often, although in this case, the question is not sufficiently precise to earn such a scrutiny, thanks,

 

[David Harper CFA FRM]

abchaudh I was posting and missed your interim: I do totally agree with your Hull citation that we want the futures price, but the question doesn't give a term to maturity (3 months? one year?), so the question seems unaware of that, or has mis-labeled the prices, as otherwise i don't see how you can access it (Shaktirathore's idea to use delta is inspired, but alas i think that requires term to be exact also), thanks,

 

[ShaktiRathore]

Yeah I agree with David that we need to make an assumption of delta equal to one also that the futures and spot price at any instant are almost the same otherwise an arbitrage is possible. If no arbitrage condition is not violated than assumption of almost same futures and spot price is valid. So it is fair to assume futures price almost equal to spot price of index for the above question.

 

[David Harper CFA FRM]

yes, just to illustrate the problem with delta here, I was thinking ... if the time horizon to maturity were not an issue (ie, if we wanted just an estimate of instantaneous futures price change), we could use:
LN[F(t)/F(0)] = LN[S(t)*exp(rT)/(S(0)*exp(rT))] ~= LN[S(t)/S(0)]; i.e., consistent with delta = exp(rT) = 1.0, because assumes r = 0.

But this question concerns the change in futures price from today until maturity = 0, so would need, I think:
LN[F(t)/F(0)] = LN[S(t)/(S(0)*exp(rT))] ~= LN[S(t)/S(0)] - rT; i.e.,
the return from F(0) = S(0)*exp(rT) growing not to F(t) = S(t)*exp(rT) but rather, at maturity, F(t) = S(t); even if not convergence, still captures (T) as the reduced time interval