Question

As the manager of a large pension fund you have just received $10 million in additional contributions to invest. Currently the S&P 500 is at 275. The six-month futures contract on the index is at 285. The six-month T-bill rate is 8 percent per year, and the dividend yield on the index is 3 percent per year. If you wish to commit $10 million to the market are you better off doing it directly by purchasing the securities that make up the index, or doing it though a long position in the futures contract?   (Differential transactions cost on these two arrangements can be ignored.)

please explain steps

Answer 1

Risk free rate (T-bills rate)=				8	%	or 0.08
6-month rf =				4	%	0r 0.03
Dividend yield				3	%
6-month dividend yield =				1.5	%
Spot rate of index =				275
Future price = Spot rate * (1+ (rf rate-dividend yield))
	275 * (1 + (0.04-0.015)					281.875
	281.875
Theoritical future price is $281.875 and Actual future price is $285.
So, future contract are short instead of investing in securities, payoff will be more.
So, Futures contracts should be short.
Step I : Buy sucurities in market(assume average rate @ 275)				-10000000
no. of index =		36363.64
step II: Short futures at $285. there is no outflow				-
no. of futures =		36363.64
position			+	-10000000
If Market index is at $300						If Market index is at $250
Step III : Short securities in market (assume rate is 300)				10909091		Step III : Short securities in market (assume rate is 300)				9090909
	(300*36363.64)						(250*36363.64)
Step IV : Buy futures at $300				-545454.5		Step IV : Buy futures at $250				1272727
Net outflow (285-300)*36363.64						Net inflow (285-250)*36363.64
Interest foregone net of div. yield (10000000*0.05*1/2)=				-250000		Interest foregone net of div. yield (10000000*0.05*1/2)=				-250000
position				10113636		position				10113636
Net position				113636.36		Net position				10113636
Whether market goes up or down, there is net profit of $113636.36. So futures contract should be short and securities should be long.