# Synthetic cash and equity and the discount rate

Three scenarios: 1. You’ve got idle cash \$10 now and you want to convert that to synthetic equities. You plan to use equity futures contracts that will expire in 3 months. 2. You know you will get \$10 cash in 3 months’ time and you want to convert that to synthetic equities. You plan to use equity futures contracts that will expire in 3 months. 3. You’ve got a \$10 equity position and want to convert it to cash for 3 months. You plan to use equity futures contracts that will expire in 3 months. I know that in calculating for the number of futures contracts required, we need to start with the future value of the \$10 at t=3 months. Something like \$10 x (1+r)^(3/12) Question is: what discount factor should be used in each of these scenarios? Risk-free? Or the dividend yield of the equity index? Many thanks! - sticky

sticky Wrote: ------------------------------------------------------- > Three scenarios: > > > 1. You’ve got idle cash \$10 now and you want to > convert that to synthetic equities. You plan to > use equity futures contracts that will expire in 3 > months. > > 2. You know you will get \$10 cash in 3 months’ > time and you want to convert that to synthetic > equities. You plan to use equity futures > contracts that will expire in 3 months. > > 3. You’ve got a \$10 equity position and want to > convert it to cash for 3 months. You plan to use > equity futures contracts that will expire in 3 > months. > > > I know that in calculating for the number of > futures contracts required, we need to start with > the future value of the \$10 at t=3 months. > Something like \$10 x (1+r)^(3/12) > > Question is: what discount factor should be used > in each of these scenarios? Risk-free? Or the > dividend yield of the equity index? > > Many thanks! > > - sticky 1) Maybe you are overthinking this a bit. If you have \$10 you want to convert into synthetic equity, you just buy \$10 notional of an equity futures contract and deposit the \$10 in risk-free stuff. 2) Same as #1 except you really have \$10 discounted by the three month rate. 3) Just short \$10 notional worth of futures contracts. In none of these calculations does the rate come up or the dividend yield.

Thanks JoeyD. Questions below. JoeyDVivre Wrote: > 1) Maybe you are overthinking this a bit. If you > have \$10 you want to convert into synthetic > equity, you just buy \$10 notional of an equity > futures contract and deposit the \$10 in risk-free > stuff. Interesting. On p.107 of CFAI vol 5, FV of \$10 in 3 months’ time is found first, using Rf, before that FV is divided by (futures price x multiplier) to arrive at the required number of futures contract. This is different from what you said (no need to get FV). Did I miss anything? > 2) Same as #1 except you really have \$10 > discounted by the three month rate. Will come back to you on this later. > 3) Just short \$10 notional worth of futures > contracts. Hmmm… example 5 on p.110, CFAI vol 5 seems to be different again. It also gets the FV of \$10 over 3 months, using Rf, before that FV is divided by the futures contract price to arrive at the required number of futures contracts. Comments? - sticky

sticky Wrote: ------------------------------------------------------- > Three scenarios: > > > 1. You’ve got idle cash \$10 now and you want to > convert that to synthetic equities. You plan to > use equity futures contracts that will expire in 3 > months. Cash should never be idle. It should be invested in Rf, thus you have to buy buy 10*e^rt/f*n Futures > > 2. You know you will get \$10 cash in 3 months’ > time and you want to convert that to synthetic > equities. You plan to use equity futures > contracts that will expire in 3 months. > you have to buy 10/f*n Futures > 3. You’ve got a \$10 equity position and want to > convert it to cash for 3 months. You plan to use > equity futures contracts that will expire in 3 > months. > sell 10*e^rt/f*n Futures Hedged position should appreciate at rf > > I know that in calculating for the number of > futures contracts required, we need to start with > the future value of the \$10 at t=3 months. > Something like \$10 x (1+r)^(3/12) > > Question is: what discount factor should be used > in each of these scenarios? Risk-free? Or the > dividend yield of the equity index? > > Many thanks! > > - sticky

comp_sci_kid Wrote: ------------------------------------------------------- > > 1. You’ve got idle cash \$10 now and you want > Cash should never be idle. It should be invested > in Rf, thus you have to buy > > buy 10*e^rt/f*n Futures Fine. Now you want to invest in equities RIGHT NOW (t=0, not 3 months). Should we expect our exposure to grow like an equity, ie. at dividend yield, instead of Rf? (this is actually what I want to bring up. Which rate?) > > 2. You know you will get \$10 cash in 3 months’ > > time and you want to convert that to synthetic > > equities. You plan to use equity futures > > contracts that will expire in 3 months. > > > > you have to buy 10/f*n Futures I also agree with you. No need to calc for FV. (\$10 is already future upcoming) > > 3. You’ve got a \$10 equity position and want > to > > convert it to cash for 3 months. You plan to > use > > equity futures contracts that will expire in 3 > > months. > > > > sell 10*e^rt/f*n Futures > > Hedged position should appreciate at rf interesting. In (1), we convert from cash and we use Rf (in a sense to grow the cash) to calc FV. Here we convert from equity and again we use Rf to calc FV. Should we use cash rate to grow equity? - sticky

steps. future value of the synthetic equity future valued using the Rf. pv that to see what you have to invest now - Rf. now many units of stock? rounded contracts times multiplier - that’s the future stock units. discount that using the divident rate to get the present value of the stock units.

strikershank Wrote: ------------------------------------------------------- > steps. steps for (1), I suppose? Questions below. > future value of the synthetic equity future valued > using the Rf. If your target is to virtually invest in equities NOW, should your position be growing at Rf or equity rate (ie dividend yield), over the 3 months? > pv that to see what you have to invest now - Rf. I don’t understand this. Is this “what you have to invest now” at t=0 or t=3 months? > now many units of stock? rounded contracts times > multiplier - that’s the future stock units. > > discount that using the divident rate to get the > present value of the stock units. this part makes sense to me. - sticky

futures by the nature of the contract grow at Rf so to answer your question above you use the Rf for futures. the part wher eyou said “i don’t understand this” referes to time = 0

strikershank Wrote: ------------------------------------------------------- > futures by the nature of the contract grow at Rf > so to answer your question above you use the Rf > for futures. hmm … are u saying that, say, EQUITY index futures, is also growing at Rf, not the dividend yield? Mind if you could give an example? I seem to be seeing the light in the dark tunnel now. - sticky

i could give an example but its best to follow through with something in the text. less chance for an error or oversight…try again now that you are seeing the light and the questions likely will make sense. equity index futures grow at Rf, not dividend yield - correct.

strikershank Wrote: ------------------------------------------------------- > i could give an example but its best to follow > through with something in the text. less chance > for an error or oversight…try again now that you > are seeing the light and the questions likely will > make sense. ok > equity index futures grow at Rf, not dividend > yield - correct. then, how come the “units of stocks” growing at dividend yield? - sticky

because if you own a stock you recieve dividends on it. what they are showing is the equivalent units of stock your futures position translates into. and the pv of that is the future stock value less the continously compounded dividend rate.