Question

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 portfolio, portfolio E, is well-diversified with a beta of 0.6 and expected return of 10%. Would an arbitrage opportunity exist? If so, what would the arbitrage strategy be?

I need to solve this for a problem set and I am really confused as to how to go about it. Any explanations and answers would be appreciated.

Answer 1

The first thing we should in this question is that Beta of portfolio F is zero.Therefore F is a risk free portfolio.Now we need to find whether there any arbitrage opportunities to produce higher returns.

To do that we find out RP/B,ratio of risk premium to beta(E(r)-E(rf)/Beta)

A:RP/B Ratio=(12-6)÷1.2

=5

E:RP/B=(10-6)÷.6=6.66

Since RP/B of portfolios are not equal,arbitrage opportunities exist.

Example of this arbitrage opportunity.

We see that Beta of A is 1.2,half of E and also Beta of F is 0

Therefore we can mix A and F in equal weights to produce net beta of .6.Lets call it M portfolio

Expected return from this portfolio

E(rm)=.5×12+.5×6=9%

Beta M=.5×1.2+.5×0=.6=Beta E

Expected return of E=10% and of M=9% with same beta.

Therefore arbitrage opportunity will be buying portfolio E and selling equal amount of M.

For this arbitrage profits will be

rE-rM=10-9=1%