Confused about synthetic positions

CFA Level III Derivatives
#1

Hi everyone,

Could someone help me out with this please? Thank you.

My questions

For Scenario 1, I do not understand why the future value is computed using the risk-free rate. Given that you want stock exposure, doesn’t it make more sense to use dividend yield of 2.5%?

For Scenario 2, I think it makes sense to use risk-free rate in computing future value because you are getting a synthetic cash position so your future value should be based on risk-free rate.

Scenario 1

Text: Manager A holds $25,000,000 market value of 3-month Treasury bills yielding 1% and wishes to create $20,000,000 of synthetic S&P 500 stock exposure for three months. The S&P contract is priced at 1,750, the dollar multiplier is 250, and the underlying stocks have a dividend yield of 2.5%.

Question: Calculate the number of contracts to buy or sell and the zero coupon position to take. Assume the desired beta is the same as the futures beta at 1.07.

Answer: Nf = [(1.07 – 0)/1.07] × [($20,000,000 × 1.01^3/12) / (1,750 × $250)]

Scenario 2

Text: Manager B has a large position in U.K. stocks that are similar to a major U.K. stock index. She wishes to create GBP 15,000,000 of synthetic cash earning 2.0% for a six-month period. The futures index contract is priced at 3,700 with a multiplier of 10. The stocks have a dividend yield of 3.0%.

Question: Calculate the number of contracts to buy or sell and the amount of synthetic cash created. Assume the desired beta is the same as the futures beta at 0.95.

Answer: Nf = [(0.0 – 0.95) / 0.95] × [(15,000,000 × 1.02^6/12) / (3,700 × 10)]

#2

There are some details missing here. Where are you getting these questions?

#3

Hi S2000magician,

This is one of the blue box questions in schwesser reading 28. It was in Los 28.b

Thank you

#4

Scenario 1: A synthetic stock position is created by going long the futures contract and investing cash at the risk free rate (i.e. 1% by investing in 3-month Treasury bills). The dividend yield is only being used to calculate the number of initial shares that are being replicated.

Scenario 2: Same explanation as Scenario 1. You should never use the dividend yield to approximate the risk free rate.

#5

I forgot to add for Scenario 2 that a synthetic cash position is created by owning the underlying stock and selling the futures contract. Nevertheless, you want to achieve the same end result as investing cash at the risk free rate.

#6

Thanks for responding but what’s the rationale of using the risk free rate? If you are trying to replicate cash, then using the risk free rate makes sense to me as you want your investment to act like cash and therefore it should grow at the risk free rate.

if you want it to replicate stock, then shouldn’t you want it to return the dividend yield? Why would you still want it to return cash?

thank you

#7

You want to replicate a synthetic stock position using the end result of a cash position.

To calculate the end result of the cash position, this should be invested in a risk free asset earning the risk free rate.

#8

So by this logic you would then want to replicate a cash position using the end result of a stock position.

So to calculate the end result of the stock position, you should be using the dividend yield?

Thanks for helping me out. I am real bad at derivatives.

#9

Ending position of a futures contract is the stock return in this case. No further adjustments required using the dividend yield.