# Equitizing cash vs beta formula

Hey guys I report here the quote:

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.

so they say that if you use a futures identical to your portfolio the formulae are the same and they give the same result. However it seems not the case to me.

Let’s suppose I have a 3 bn \$ portoflio in equity that is invested in S&P 500. The risk free rate is 3%. I want to create cash out of my equity portfolio in 5 months. Beta of S&P 500 is 1. Futures on S&P 500 are quoted 1058 and the multiplier is 250. The beta of the futures is 1.

With the beta fomula, the number of contracts would be: Nf = (0-1/1)*3,000,000,000/(250*1,058)=11,342

With the equity out of cash formula: Nf = 3,000,000,000*(1,03^5/12)/(250*1,058) = 11,483

Who can explain to me this difference (or how to reconcile the two things)?

Hedging to get a zero beta isn’t the same as hedging to get the return on cash. Zero beta has unsystematic risk. Cash is considered risk free. The first one is an attempt to change the portfolio beta to zero using futures. The second is an attempt to earn a risk free rate over time using futures. Both are different things.

Yes ok I understand that, but this is a curriculum quote: " 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."

Ok. 11483 contracts is in the future. Discount at the risk free rate to the present you get 11342

So would the two answers in he original post still be the same if the beta of the futures is less than 1, say 0.98?

No if you do not have the exact futures on the underlying you cannot equitize cash (rather, you cannot completely create equity out of cash)

^ oh alright, thanks!