a) The current price of a non-dividend-paying stock is $30 and the stock is expected to pay a dividend of $0.5 in one months' time and then 0.5 again in four months' time. The risk-free rate of return is 6.00% p.a. (Continuously compounded). If the price on a six-month forward contract is F0,0.5=28, is there an arbitrage opportunity? If arbitrage is possible, design a strategy to take advantage of it. What is the arbitrage profit earned from this strategy?

b) An Australia fund manager is expecting a rising stock market over the next 6 months and wants to gain an increased exposure to this trend. Their portfolio currently valued at $50.000.000 has a beta of 0.8. How many S&P200 futures contracts (with nominal value of F0,t x $25 each where F0,t = 4,000) does the fund have to transact in to increase the beta of the portfolio to 1.4? should the fund take a long or short position in these contracts? Why would the fund manager fail to get the target beta by simply investing additional money into the S&P200 stocks?

In the given problem we have two sections which are being attended below

a) Arbitrage opportunity

In order to determine the Arbitrage Opportunity we need to find out Theoritical Forward Price & Compare it with Actual Forward Price.

For calculating the Theoritical Forward Price we need to adjust the Spot Price bt subtracting the Present value of Dividends to be received in two periods.

PV of Dividend ($0.5) to be received in one month's time is calcuted below

PV of Dividend = $0.5 *e^{-0.06 *1/12} (Given Dividend = $0.5 ,time = 1month or 1/12 of year & Interest = 6%=0.06)

PV of Dividend = $0.5*e^-0.005 =0.5/1.005 =$ 0.4975

PV of Dividend($0.5) to be received in 4months time is calculated as under

=$0.5 * e^-0.06*4/12 =$0.5*e^-0.02 = $0.5 /1.0202= $0.49

Given Spot rate = $30 & hence

Adjusted Spot rate = $30 -$0.4975-$0.49 = $29.0125

Theoritical Forward Price = Adj Spot * e^rt where Adj Spot = $29.0125 ,r= 0.06 & t=6months =0.5 on substituting we get

= $29.0125*e^0.06*0.5 =$29.0125 *e^0.03 = $29.0125*1.03045 =$29.8961 Therefore

Thoritical Forward rate for 6months = $29.8961 while

Actual 6months Forwward rate = $28

As Actual Forward rate  is lower than the Theoritical Forward rate & hence forward rate is undervalued & hence there is an Arbitage opportunity. It can be availed by buying Forward & selling Spot.

Strategy & CashFlows

Step 1 Short Sell the Stock & Realize $30

Step 2 Subtract the PV of Dividend (Opportunity loss of Dividend receipt) =$0.9875 (as computed 0.49+0.4975)

Step 3 : Invest the balance amount ($30-0.9875) for 6months $6%p.a.

Step 4 : Receive the maturity amount( $29.0125*e^0.06*0.5) after 6months = $29.8961

Step 5 : Buy the stock $28 & hand over the share to the broker for Short Sale at the begining

Step 6 : Book Profit = Amount Received - Amount paid = $29.8961 - $28 = $1.8961

b) Number of Future Contracts to be purchase fo achieve the Target Beta & Strategy to be adopted

Formula for Number of Contracts to be purchased

Number of Contracts = Portfolio Value *(Desired Beta - Beta of Portfolio) / Value of Future Contract

In the given case Portfolio Value = $50,000,000 Current Beta of Portfolio = 0.8 &Desired portfolio =1.4  

Value of Future Contract = $25*4000 =$100,000. Substituting these values in the above formula we get

Number of Future Contracts to be purchased = $50,000,000 * (1.4 -0.8) /$100,000 = 500*0.6 =300 Contracts

Number of Future Contracts to be purchased to get the Target Beta = 300

When the Desired Beta is more than the Existing Beta ,the Outlook is Bullish. When the Outlook is Bullish ,Future Contracts should be bought .It means Long Position need to be taken.

As the beta of S&P Stocks are normally shown as 1 & hence investing the entire money on S&P Stock may not help the Fund Manageer to get the desired Beta.