Question

Suppose initially that two assets, A and B, will each make a single guaranteed payment of \(100 in 1 year. But asset A has a current price of \)80 while asset B has a current price of $90.

1. What are the rates of return of assets A and B at their current prices? Given these rates of return, which asset should investors buy and which asset should they sell?

2. Assume that arbitrage continues until A and B have the same expected rate of return. When arbitrage ends, will A and B have the same price?

Short answer

1. The rates of return of asset A will be 25%, and of asset, B will be 11.11%. The individual will buy asset A and sell asset B.

2. When the arbitrage ends, assets A and B will have the same price.

Step-by-step solution

Step 1. Explanation for part (a)

The rate of return of asset A is calculated below:

$$\begin{gathered} RateofReturn=\frac{ExpectedPayoff-Price}{Price}\times 100 \\ ExpectedPayoff=100 \\ Price=\$80 \\ RateofReturn=\frac{100-80}{80}\times 100 \\ =\frac{20}{80}\times 100 \\ =25\% \end{gathered}$$

The rate of return will be 25%.

The rate of return of asset B is calculated below:

$$\begin{gathered} RateofReturn=\frac{ExpectedPayoff-Price}{Price}\times 100 \\ ExpectedPayoff=\$100 \\ Price=\$90 \\ RateofReturn=\frac{100-90}{90}\times 100 \\ =\frac{10}{90}\times 100 \\ =11.11\% \end{gathered}$$

The rate of return will be 11.11%.

The individual will buy the asset whose return will be more and sell whose return is less. Thus, the individual will buy asset A and sell asset B.

Step 2. Explanation for part (b)

To check whether the prices of the assets are the same or not, when the rate of return is the same; you need to do the following calculations:

$$\begin{gathered} RateofAssetA=ReturnofAssetB \\ \frac{ExpectedPayoff-PriceA}{PriceA}=\frac{ExpectedPayoff-PriceB}{PriceB} \\ \frac{ExpectedPayoff}{PriceA}=\frac{ExpectedPayoff}{PriceB} \\ \frac{1}{PriceA}=\frac{1}{PriceB} \\ PriceA=PriceB \end{gathered}$$

Thus, when the arbitrage ends, the price of both assets is the same.