Creating Syntheic Cash vs Hedging Equity Exp w/ Futures

Synthethic cash formula = (Vp*(1+Rf)^t ) / Pf

Hedging Equity Exposure = Beta Adj * (Vp / Pf)

My question is, why do you take into account the Rf rate when you are creating systhetic cash, but not when you are hedging equity exposure. The goal of both formulas seems to be the same, to neutralize equity exposure, but the formulas are slightly different.

Time period, and other factors make them different.

For synthetic cash, the argument is as follows.You are holding stocks in your portfolio, but for the next 6 months, you do not want the equity exposure, instead you are seeking exposure to cash for the next 6 months or so. You may be anticipating a steep decline in the stock market for the next 6 months.

One way to convert your equity exposure to cash for the next 6 months is to do the following. Sell off all the stocks in your portfolio and keep the proceeds (Vp) in T-Bills which will grow at the risk free rate for the next 6 months to Vp * (1+rf)^0.5 and at the end of 6 months sell the T-bills and buy equity back again. But, selling all of your stocks now and investing in Tbills and after 6 months selling off T-bills and buying the stocks again can be expensive because of transaction costs etc.

A better way to accomplish the cash exposure is to use the synthetic cash route. This is accomplished by keeping all the stocks in your portfolio (Don’t sell any stocks) and sell 6 month futures instead. Your position is now long in stock portfolio and short in 6 month equity futures. This is a risk neutral strategy. This risk neutral strategy should earn risk free return over the 6 month period. That is why you should multiply the current portfolio value Vp by (1+Rf)^0.5.

On the other hand, with regards to hedging equity exposure, we are not trying to create synthetic cash and thus we are not trying to earn the risk free rate. We are only trying to ensure that our portfolio value does not go down. So, we are long in stock and simultaneously short in equity futures so that any loss in the long position can be neutralized by the gain in short futures position.

After writing all this, I get the feeling that to we probably ought to subtract the dividend yield from the portfolio value to arrive at the futures value: Eg. use Vp minus dividends instead of Vp in the numerator of your formula for hedging equity exposure. Any way I will leave it to the experts to answer my question.

Remember your Level II Futures price calculation…

PV (Dividends) is removed from the St - to arrive at the Futures Price.

Thanks for the thoughtful replies.

Believe the answer to my question is just different objectives. With equitizing cash, you are looking to earn Rf. With hedging you are reducing market exposure.

@psriniva - I believe the general formula to figure # of contracts needed ignores dividends (or lets them cancel out) as you are just trying to isolate the Rf return.

You do need to use the dividend rate if you want to calculate the actual equivlant number of share purchased. See page 361 of CFAI book 5 for the formula there.

Isn’t the difference that the time value of money is embedded in beta?

IMHO, Time value of money is not embedded in beta. As @ftwcfa says, may be dividends are ignored. Somebody who has read Cai material, please chime in. P

I suggest re-read the example 5 and the last part of exhibit 3. The futures’ exipration, and underlying secuties are different. You “lock in” risk free rate in synthetic cash for a short term (like 3 months). In hedging with futures, it doesn’t give the time horizon. Otherwise, I don’t see the difference if the underlying securities are the same and pays dividend. Goog question, though.

when hedging with futures - you are provided a futures price and that includes the effects of the dividends. Remember in Level II -> when we valued a Futures Contract

Ft = St - PV(Dividends)

So Stock/Index paid dividends - so the value of your portfolio grew.

There is a opposite effect on the futures side.

and this gets reflected in the final effective beta you arrive at for your portfolio.

Ok. Thanks cpk123. I see it now after I started paying more attention to some of the responses here. Futures price already reflects the dividends. Cool. P