Formula 1 (LOS 36.a) # of contracts = (Bt-Bp)/Bf x Vp/(Pf x multiplier) Formula 2 (LOS 36.c) # of contracts = -(Vp x (1+Rf)^t) / (Pf x multiplier) Formula 2 is described as offsetting an equity position with equity futures (assumes that the portfolio beta is equal to the futures beta) to earn the risk-free rate over a time period. But formula 1 should also earn the risk-free rate if beta is neutralized in the first component (portfolio with beta of zero should earn risk-free rate). But formula 1 lacks the (1+Rf)^t factor. In my mind, when paired up with an equity portfolio, the number of futures contracts determined by formula 1 should zero out the beta and earn the risk-free rate. What is the purpose of the (1+Rf)^t factor then? What am I missing?
part 2: you have a portfolio which you want to create synthetic cash out of.
This is explained in the book:
You might be wondering about the relationship between the number of futures contracts given here and the number of futures contracts required to adjust the port- folio beta to zero. Here we are selling a given number of futures contracts against stock to effectively convert the stock to a risk-free asset. Does that not mean that the portfolio would then have a beta of zero? In Section 3.2, we gave a different formula to reduce the portfolio beta to zero. These formulas do not appear to be the same. Would they give the same value of Nf? In the example here, we sell the precise number of futures to completely hedge the stock portfolio. The stock portfolio, however, has to be identi- cal to the index. It cannot have a different beta. The other formula, which reduces the beta to zero, is more general and can be used to eliminate the systematic risk on any portfolio. Note, however, that only systematic risk is eliminated. If the portfolio is not fully diversified, some risk will remain, but that risk is diversifiable, and the expected return on that portfolio would still be the risk-free rate. If we apply that formula to a portfolio that is identical to the index on which the futures is based, the two formulas are the same and the number of futures contracts to sell is the same in both cases.
basically contracts*Pf*multiplier/(1+RF)^t = Vp , if the futures matches the portfolio very closely in terms of their beta.
now plug the above into formula 1 ( i.e. set Bt=0 and Bp=Bf and substitute Pf*multiplier with Vp*(1+RF^t)/contracts )
and you will get formula 2 .
Thanks guys. Per cpk123’s response, I set formula 1 equal to formula 2 and removed what was common between both.
Basically, when you set formula 1 = formula 2, you get the below simplified: Bp/Bf = (1+Rf)^t
Index futures and an index portfolio should have *very* similar betas, except for the fact that index futures represent a FUTURE VALUE price, whereas the index portfolio does not. So the ratio of return between the two (and hence, the Beta) should equal (1+Rf)^t.
Formula 1 glosses over this and formula 2 forces you to assume that the portfolio beta and index beta are identical, which would actually be difficult to find in reality.
That said, there’s no way CFAI goes into this level of depth on the exam. Just an indulgence on my part.
thanks ducati , that is good simplification.