Question

9..Consider the following data for a one-factor economy. All portfolios are well diversified.

Portfolio                 E(r)              Beta

A                           12%               1.2

F                             6%                0.0

Suppose that another protfolio E is well diversified with a beta of 0.6 and expected return of 7%.

Construct an arbitrage strategy by investing $1 in the long position and $1 in the short position. What is the profit for the arbitrage strategy?

Your answer should be in dollars and accurate to the hundredth.

Answer 1

In this Question we have 3 Portfolio's -

Return and Beta

Portfolio	Return	Beta
A	12%	1.2
F	6%	0
E	7%	0.6

We have create an Arbitrage Strategy. An Arbitrage Strategy means a situation where we have a riskless profit and 0 investment. Which also means that we Buy and Sell a portfolio/stocks of similar amount and we have a guaranteed profit.

In the above Question we have to Buy and Sell a portfolio/combination of portfolio which would give us a higher return for the same risk.

Now Assume we create a portfolio being a mix of Portfolio A and Portfolio F with each having a weight of 50%.

Then we would have a return and risk of:

Return = Expected Return of A * 50% + Expected Return of F * 50%

Return of Combined Portfolio = 12% * 50% + 6% * 50% = 6% + 3% = 9%

Similarly Beta of the combined Portfolio = 1.2 * 50% + 0 * 50% = 0.6 + 0 = 0.6

So the Return and Risk of the Combined Portfolio A & F is 9% and 0.6 respectively.

Now comparing the Combined Portfolio to Portfolio E we see that the combined portfolio is giving us a higher return (9% v/s 7%) for the same Beta (0.6 v/s 0.6).

So in order to have an Arbitrage Opportunity we would short sell Portfolio E and Long the Combined Portfolio of A & F.

There are 3 steps in an Arbitrage opportunity:

Sell the Expensive Portfolio

Buy the Cheap portfolio

Settle in the future by selling the Cheap portfolio at a Higher rate and buying the expensive portfolio at a lower rate and earning the profit.

Step 1-

Sell the Expensive portfolio which in this case is Portfolio E

So we receive = $1

Amount that we have to Purchase this portfolio for in 1 year if expected return is 7%= $1+ $1*7% = $1+ $.07

Price after 1 year = $1.07

Step 2-

Purchase the Cheap portfolio which in this case is the combined portfolio-

Doing the same Calculation as above

We pay $1 now to receive = $1+ $1*9% = $1+$0.09 = $1.09 after 1 year.

At the end of 1 year we will now sell the Cheap portfolio that is Combined portfolio for $1.09 and purchase the Expensive portfolio that is Portfolio E for $1.07 and square off the transaction by making a profit of=

$1.09 - $1.07 = $0.02

Hence we made a profit of $0.02 without investing any money of our own and without taking any additional risk.