synthetic T-Bill

CFA Level III Derivatives
#1

hey this q is from schweser, i don’t understand why they haven’t multiplied the portfolio value by the risk free rate??

this is all the information given: $100m portfolio, 1.1 beta, wants to hedge using s&p500 futures trading at 596.7 with a $250 multiplier.

then it asks how many futures you need to buy/sell to create a synthetic t-Bill.

i thought you needed to do V(1 + r)^t to get the future value of the synthetic cash. any advice please? thanks so much.

#2

I would agree to you, but have they given time frame for the quity exposure. Simple thing made tricky…

Any one else plse…

#3

theoretically you’d invest the theoretical proceeds from the short futures into a risk free asset , so you need :

Vt(final value ) / (1+rf)^t = V ( present value of position )

Therefore you don’t need to use the future value of cash. the real-bond should provide the multiplier

#4

I’m confused – looking in the CFA textbook they definitely do it the way I posted. Argh, thought I knew this, now I don’t know if I’m being tripped up by a lazy Schweser question, or if I need to revise it some more…

Janakisri I’m sorry but I don’t follow. there are no proceeds from going short futures…

#5

1

Risk-free rate is already built in the futures price. It will accrue as future price converges with spot.

Easier way to think about it is in terms of a hedge for a portfolio that consists of 1 share of an S&P500-tracking ETF.

Say, S&P spot at t=0 = $1000, Rrisk-free = 3%, dividend yield = 0%.

Initial price of an S&P futures contract expiring at t=1 = $1000*(1.03)/(1.00) = $1030

Scenario 1: S&P remains unchanged , i.e. S&P(0) = S&P(1)

Profit = Long Spot Payoff + Short Futures Payoff = 0 + (1030 - 1000) = 30 => 30 / 1000 = 3% = Rrisk-free

Scenario 2: S&P gains 10%, i.e. S&P(1) = S&P(0) * 1.1

Profit = 1000*(1.1 - 1) + (1030 - 1000*1.1) = 100 + (-70) = 30 => 3% = Rrisk-free

Of course, when you hedge using contracts that are based on a different portfolio it adds uncertainty (basis risk), but the reasoning remains the same. At least that’s how I understand it (which quite possibly is wrong)

#6

Given the information we are given, aren’t we essentially be asked to adjust the Beta to 0. I simply understood it to be the formula from LOS36d and solved it accordingly.

(New Beta - Old Beta)/(Beta of futures) * (Vp / (Pf * multiplier)

= ((0-1.1) / 1) * (100m / (597.6 * 250)) = -737 future contracts

#7

yes I agree with the information given, that’s what you’d do…I guess my question is, if they had given the risk free rate, should it have been incorporated?

the more i think about it, the more I think it should be. The idea of the transaction is that you lock in the current value of your stock portfolio (ie give up any gains, don’t suffer any losses) and end up with an investment that grows at the risk free rate.

if you have $1000 of stocks (say) and the risk free rate is 5%, you need to end up with someone paying you $1050 at the end. the only way to do that with selling futures is to sell $1050 of futures in the first place.

#8

Futures price adjusts to the premium necessary to get the risk free rate on a hedged investment

There is a time-value-of-money difference between futures price and spot price . So you do not have to incorporate risk free rate

#9

Thanks for your reply! – I see what you’re saying, but the CFA textbook does include the risk free rate. Hence my confusion.

#10

explain the word “include” in the context . Quote or describe the passage a bit moe.

#11

You are right. The ‘time-value-of-money’ at the risk free discount rate.

#12

There is a question like this in the CFA books.

Number of futures contracts = V(1+r)^(t) / q*f

so…

100m * (1+r)^(t) / 596.7*250 = # contracts.

I think there are points from the question you are leaving out in your post.

#13

nope that is all the information given! if you have Q-bank, it’s question 92463