Consider the following data for a one-factor economy. All portfolios are well
diversi�ed.
Table 1: One-factor Economy
Portfolio E(r) Beta
A 10% 1.0
F 4 0
Suppose another portfolio E is well diversi�ed with a beta of 2/3 and expected return of
9%. Would an arbitrage opportunity exist? If so, what would the arbitrage strategy be?
To determine if there is an arbitrage opportunity in a one-factor economy, we need to compare the expected return of the portfolio in question with its systematic risk (beta) and the risk-free rate.
1. Calculate the risk-free rate:
Since the data doesn't provide the risk-free rate, we assume it to be 0%.
2. Calculate the expected excess return:
The expected excess return is the difference between the expected return of the portfolio and the risk-free rate.
Expected Excess Return of Portfolio E = 9% - 0% = 9%
3. Compare the expected excess return with the systematic risk:
Arbitrage opportunities arise when a portfolio has a higher expected excess return than its systematic risk multiplied by the market risk premium.
Market Risk Premium = Expected Return of Portfolio A - Risk-Free Rate = 10% - 0% = 10%
Maximum Acceptable Systematic Risk = Beta of Portfolio A x Market Risk Premium = 1.0 x 10% = 10%
Since the expected excess return of Portfolio E (9%) is less than the maximum acceptable systematic risk (10%), there is no arbitrage opportunity.
In this scenario, no arbitrage strategy would be necessary as the market is already efficiently pricing the assets.
To determine if an arbitrage opportunity exists, we need to compare the expected return of the portfolio with its beta.
In a one-factor economy, the expected return of a portfolio can be calculated using the following equation:
E(r) = Rf + Beta * (E(rm) - Rf)
Where:
- E(r) is the expected return of the portfolio
- Rf is the risk-free rate
- Beta is the beta of the portfolio
- E(rm) is the expected return of the market
Given the following data:
Portfolio E:
- E(r) = 9%
- Beta = 2/3
Let's assume the risk-free rate (Rf) is 3% and the expected return of the market (E(rm)) is 8%.
Using the equation above, we can calculate the expected return of Portfolio E:
E(r) = 3% + (2/3) * (8% - 3%)
= 3% + (2/3) * 5%
= 3% + 10/3%
≈ 6.33%
Since the expected return of Portfolio E is lower than the actual expected return of 9%, there is an arbitrage opportunity.
The arbitrage strategy would involve buying Portfolio E and short selling Portfolio F. By doing this, you will earn the expected return of Portfolio E (9%) and pay the borrowing cost associated with short-selling Portfolio F (expected return of 0%). This difference will result in a riskless profit.