Question

Consider a 3-year forward contract on stock. The stock will pay $5-dividend every year in years 1 through 3 and currently sells for $80. The forward will expire right after the stock’s dividend payment in year 3. The forward price is $78, and the risk-free interest rate is 4% per annum. We want to make an arbitrage such that net cash flow in year 3 is positive and net cash flows from year 0 through year 2 are zero. In this arbitrage, what position do we need regarding 3-year zero-coupon bonds?

(a)buy 3-year bond such that we pay $70.58 now

(b) buy 3-year bond such that we pay $73.61 now

(c) sell 3-year bond such that we receive $70.58 now

(d) sell 3-year bond such that we receive $73.61 now

Answer 1

For creating arbitrage opportunity, we need to spot mispricing in markets. For finding mispricing here, we need to compare the prevailing forward price and what should be the forward price as per given information.

Stock Price: $80

Dividend Per Year = $5

Interest Rate = 4%

Time to Maturity = 3 Years

Forward Price in market = $78.

Now, for finding the should be forward price, we need to find out the present value of dividend.

Year Dividend Present Value Factor    Present Value

1 $5 0.962 $4.81

2 $5 0.925 $4.62

3 $5 0.889 $4.44

Total Present Value of Dividends $13.87

For calculating forward price,

Forward Price =

Forward Price =  

Forward Price = $74.56

As we can see that expected forward price is more than what is prevailing in the market, so it is overpriced at the time. So, for taking the benefit of this arbitrage opportunity we should sell the overpriced thing and buy the underpriced asset. For doing that,

1. We should see Forward Price, thus we would recieve $78 in the Year 3.

2. For buying the stock, we need to take the loan as we want our cashflow to be 0 in imtermediate years. So, a loan needs to be taken.

3. Now we would recieve $5 in year 1 as dividends and we want to keep cashflows as 0, so we need to use this to payback the loan which we would need to buy the stock. So, present value of this $5 i.e. $4.81 would be used to pay back the loan. So, a loan of $4.81 for one year would be taken.

4. Similarly for Year 2 dividends, we would do that. We will take a loan of $4.62 for 2 years which we will pay using $5 as dividend.

5. Now rest of the stock price would be funded through 3 year loan which would be serviced through forward price and 3rd year dividend.

3 Year Loan = $80 (Stock Price) - $4.81 (1 Year Loan Cashflow) - $4.62 (2 Year Loan Cashflow) = $70.57.

Thus we need to sell 3 year bond such that we would receive $70.58 now. So, option (c) is correct.