Question

AXE stock is currently trading at $50 per share. Its price is expected to go up to $65 or down to $45 in one year. A one year forward contract on AXE stock is offered at $55.

The risk free interest rate is 4% is there any arbitrage profit making opportunity to profit from the situation without taking any risk and without using out of pocket funds based on the information? If so,

1.

Show the steps to take an arbitrage profit.

2.

What is the impact of arbitrage on the AXE stock price?

3.

What if the forward price is $51?

4.

What should be the equilibrium forward price?

Answer 1

We have following information -

AXE current trading price (P) = $ 50

risk free interest rate(r) = 4 % p.a

One year Forward rate of AXE (F) = $ 55

1. Steps to take an arbitrage profit

Step-1 - Calculate the fair value of Forward Contract.

Fair value of Forward = P(1+r)^t - D

Where,

P = current trading price of stock

r= risk free interest rate

t = forward maturity time

D = dividend

In our case dividend is 0.

Thus,

Fair value of Forward = 50(1+0.04)¹

= $ 52

It means, Forward price is overvalued as Forward Price ($ 55) is greater than its fair value, Hence arbitrage opportunity exist. here.

Step-2 - Borrow money at risk free interest rate and buy AXE stock today and sell forward contract of AXE.

Step-3- At end of year, sell the AXE stock at $ 55 as per forward contract and repay the borrowed money.

Profit from Arbitrage = Sale value of Stock - Repayment of Laon

Sale value of Stock = $ 55

Repayment of Loan = $ 50 (1+0.04)¹

= $ 52

Thus, Profit = $ 55 - $ 52

= $ 3

By following above steps, arbitrager can earn $ 3 profit without making any investment from his pocket.

2. Impact of arbitrage on AXE stock price

As we discussed above, forward price is overvalued hence arbitrage opportunity exist.

To take advantage of this opportunity, arbitrager buy more stocks of AXE and sell forward contracts.

Buying more stocks of AXE force AXE stock price to go up till arbitrage benefits nullify.

Thus, Impact of arbitrage on AXE stock price is increase in AXE Stock price.

3. If Forward price is $ 51

If forward price is $ 51 and other things remains constant then forward price is under valued.

In this situation also, arbitrage opportunity exist. Arbitrager may follow below strategy to earn arbitrage profit.

Today - Sell the AXE stocks and invest the proceeds to Bank at risk free interest rate. Buy forward contract at $ 51

At the end of year - Receive money from bank and buy the AXE stock at $ 51

Profit = Proceeds from Bank - $ 51

= 50(1+1.04)¹ - 51

= 52 -51

= $ 1

Thus, Arbitrage Profit would be $ 1

4. Equilibrium Forward Price

As calculated above fair value of Forward is equilibrium forward price -

Fair value of Forward = P(1+r)^t - D

Where,

P = current trading price of stock

r= risk free interest rate

t = forward maturity time

D = dividend

In our case dividend is 0.

Thus,

Fair value of Forward = 50(1+0.04)¹

= $ 52

Thus, in this case, equilibrium price of Forward is $ 52.

Please note , equilibrium forward price can also be calculated using continuous compound interest.

equilibrium forward price = P*e^rt