# Arbitrage Opportunities

## Arbitrage Opportunity

Arbitrage is risk-free trading that does not require an initial investment of money but earns an expected positive net return. An arbitrage opportunity exists if an investor can make a deal that would give an immediate profit, with zero initial cost, no risk of future loss, and a non-zero probability of future profit.

### Characteristics of Arbitrage Opportunity

• It neither costs anything today nor in any state in the future.
• It has a positive payoff either today or at a future date (or both).

#### Example: Exploiting an Arbitrage Opportunity

Assume you are an investor in an investment firm that uses a two-factor model to evaluate assets. The following data for portfolios A, B, and C is provided:

$$\begin{array}{c|c|c|c} \textbf{Portfolio} & \textbf{Expected Return} & \bf{{\beta}_{1}} & \bf{{\beta}_{2}} \\ \hline A & 12\% & 1 & 0.4 \\ \hline B & 16\% & 0.8 & 0.8 \\ \hline C & 12\% & 0.9 & 0.6 \end{array}$$

The arbitrage opportunity is closest to:

#### Solution

If we allocate 50% of funds to portfolio A and the remaining 50% to B, we can obtain a portfolio (D) with $$\beta_1$$ and $$\beta_2$$ equal to the Portfolio C betas.

$$\beta_1$$ for portfolio D = 0.5(1.0) + 0.5(0.8) = 0.9

$$\beta_2$$ for portfolio D = 0.5(0.4) + 0.5(0.8) = 0.6

The expected return for portfolio D is 0.5(12%) + 0.5(16%) = 14% while that of C is 12%. Therefore, portfolios D and C have the same risk, but D has a higher expected return.

Logically, by buying portfolio D and short-selling portfolio C, we expect to earn a 14% − 12% = 2% return. Investors will exploit the arbitrage opportunity, which will lead to the depreciation of prices of assets in portfolio C, and the rise of the expected return for portfolio C to its equilibrium value.

The APT assumes that there are no market irregularities that prevent investors from exploiting arbitrage opportunities. Therefore, investors can hold extremely long and short positions, making mispricing vanish immediately. This makes arbitrage opportunities to be exploited and eliminated immediately.

## Question

An investment firm uses a single-factor model to evaluate assets. Consider the following data for portfolios ABC, XYZ, and MNO:

$$\begin{array}{c|c|c|c} \textbf{Portfolio} & \textbf{Expected Return} & \bf{{\beta}_{1}} & \bf{{\beta}_{2}} \\ \hline ABC & 20\% & 1.00 & 0.50 \\ \hline XYZ & 15\% & 0.80 & 0.40 \\ \hline MNO & 17\% & 0.92 & 0.46 \end{array}$$

Which of the following statements is least likely true about portfolio MNO?

1. It is overvalued.
2. It is undervalued.
3. It is fairly valued.

#### Solution

We construct a portfolio with ABC and XYZ portfolios having the same risks as the MNO portfolio. To do so, we solve the following equations simultaneously:

\begin{align*} x+0.8y & = 0.92 \\ 0.5x+0.4y & = 0.46 \\ x = 0.6 & \text{and} y = 0.4. \end{align*}

Therefore a new portfolio, NEW, can be created by allocating $$\frac{0.6}{0.6+0.4}\times100=60\%$$ of funds to ABC and the remaining 40% to XYZ. This portfolio will have the same risks are MNO. However, the expected return for portfolio NEW is $$0.6(20\%) + 0.4(15\%) = 18\%$$.

This implies that the MNO portfolio is overvalued, and thus, arbitrageurs will short sell MNO and buy the NEW portfolio.

LOS 41 (b) Define arbitrage opportunity and determine whether an arbitrage opportunity exists.

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