 # Equitizing cash vs # contracts needed

># of contracts to create synthetic equity = (Tbills Held)(1+RF)t / (PF)(multiplier) >#of equity index contracts for altering beta = (BT - BP / Bf) (VP / (PF)(multiplier) Anyone know why the first equation accounts for the increase in the tbills underlying the contract but the second one doesnt? You could make an arguement that the second equation could be substituted for the first by using 0 as a value of existing portfolio beta. >>Just to provide additional context around the question, the schewser practice exam #2 question 19.1 in p.m Increases the equity position value by the risk free rate to determine the contracts needed, then in question 19.4, which is almost the same question but for a Japanese index, they use the portolio value of the equity without increasing by RFR.

this a very good question, with number 2 you’re weighting beta by the market val of portfolio and futures, but obv mkt val of the futures is the pv of price as settlement occurs in the future (ie if you sell today the money comes later vs sell equity today and youre paid today). The only thing i can assume to make this make sense, is that you buy/sell very shortdated futures so the pv is almost equal to the futs price anyway. Although this means you achieve your beta for only a short time, but given the questions on this give no indication of timeframe, it could be a fair assumption to make.

Because you’re creating a synthetic position in the 1st one but not the 2nd one.

bpdulog Wrote: ------------------------------------------------------- > Because you’re creating a synthetic position in > the 1st one but not the 2nd one. Well with the 1st one you invest enough cash to settle the futures hence use PV. But with the second one you’re still going to purchase/sell futures, and given this cash exchange happens at settlement, your total portfolio value changes by the PV of the futures price*contracts NOT the futures price*contracts so to be exactly correct you would have to use PV of the futures price in the second. Synthetic has nothing to do with it, in either case you are purchasing/selling futures

kurupt1 Wrote: ------------------------------------------------------- > bpdulog Wrote: > -------------------------------------------------- > ----- > > Because you’re creating a synthetic position in > > the 1st one but not the 2nd one. > > > Well with the 1st one you invest enough cash to > settle the futures hence use PV. But with the > second one you’re still going to purchase/sell > futures, and given this cash exchange happens at > settlement, your total portfolio value changes by > the PV of the futures price*contracts NOT the > futures price*contracts so to be exactly correct > you would have to use PV of the futures price in > the second. > > Synthetic has nothing to do with it, in either > case you are purchasing/selling futures Then why does formula 6 instruct us to raise the value by the risk free rate on page 356 of Volume 5?

bump

i’m not disagreeing with that formula, i’m disagreeing with the # of contracts needed to change beta formula. i agree that when equitising, you need to invest enough money at the rf rate to settle the futures, and as such, the value you invest (V* in that formula) is the PV of the number of contracts * multiplier (as in formula 7) If i was to go through the target beta formula and strategy in a bit more detail, i’m sure the reason for not using the PV of the futures price will become apparent, but it’s such a tiny thing, that i’m just going to accept it for now

If u see the word synethic or convert to cash, then 1+r

Page 363 CFAI Text " 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 identical 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." (Level III Volume 5 Alternative Investments, Risk Management, and the Application of Derivatives , 4th Edition. Pearson Learning Solutions p. 363).

I see, so it looks like the difference depends on if the risk you want to remove. To remove the entire risk, both systematic and non systematic, use the formula which utilizes the fv of treasuries, to just remove systematic risk, use the beta formula which uses the portfolio value…

What I got out of it was that if the stock portfolio is identical to the index, use the = (Tbills Held)(1+RF)t / (PF)(multiplier) formula. If it is not identical, use the other = (BT - BP / Bf) (VP / (PF)(multiplier) It further goes on to say: “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.” (Level III Volume 5 Alternative Investments, Risk Management, and the Application of Derivatives , 4th Edition. Pearson Learning Solutions p. 363). and the footnote: A key element in this statement is that the futures beta is the beta of the underlying index, multiplied by the present value interest factor using the risk-free rate. This is a complex and subtle point, however, that we simply state without going into the mathematical proof (Level III Volume 5 Alternative Investments, Risk Management, and the Application of Derivatives , 4th Edition. Pearson Learning Solutions p. 363).

Bump I’m still not really getting it. I don’t see what the FV factor has to do w. whether your portfolio’s beta is identical to the index or not. Are they simply saying that if you already hold a portfolio of stock, the PV of the FV of that portfolio is the current value? Meaning, if I own 100 stocks, these 100 stocks represent the PV of the 100 stock portfolio at any point in time in the future. However, if I hold cash and want to equitize to stock at some point in the future, given F(0,t) = S0 x e^(r t), I need to invest enough s.t. the future value of my position matches the future value of the stock. So, I invest enough to accumulate to the future value?