Question

Next, consider another pair of assets, C and D. Asset C will make a single payment of \(150 in one year while D will make a single payment of \)200 in one year. Assume that the current price of C is \(120 and that the current price of D is \)180.

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

d. Assume that arbitrage continues until C and D have the same expected rate of return. When arbitrage ends, will C and D have the same price?

Compare your answers to questions a through d before answering question e.

e. We know that arbitrage will equalize rates of return. Does it also guarantee to equalize prices? In what situations will it equalize prices?

Short answer

c. The return on Asset C will be 25%, and on Asset D will be 11.11%. The individual will buy Asset C and sell Asset D.

d. The price of Asset C and D will not have the same price once the arbitrage ends.

e. It will only equalize the price when the payoff is the same; thus, not every time it equalizes the price.

Step-by-step solution

Step 1. Explanation for (c)

The rate of return of Asset C is calculated below:

$$\begin{gathered} RateofReturn=\frac{ExpectedPayoff-Price}{Price}\times 100 \\ ExpectedPayoff=\$150 \\ Price=\$120 \\ RateofReturn=\frac{150-120}{120}\times 100 \\ =\frac{30}{120}\times 100 \\ =25\% \end{gathered}$$

The rate of return will be 25%.

The rate of return of Asset D is calculated below:

$$\begin{gathered} RateofReturn=\frac{ExpectedRate-Price}{Price}\times 100 \\ ExpectedPayoff=\$200 \\ Price=\$180 \\ Rateofreturn=\frac{200-180}{180}\times 100 \\ =\frac{20}{180}\times 100 \\ =11.11\% \end{gathered}$$

The rate of return will be 11.11%.

The individual will buy the asset with a high return, and the asset with a low return will be sold.

Thus, Asset C will be bought, and Asset D will be sold.

Step 2. Explanation for part (d)

When the expected payoff is the same then,

$$\begin{gathered} ReturnofAssetC=ReturnofAssetD \\ \frac{ExpectedPayoff-PriceC}{PriceC}=\frac{ExpectedPayoff-PriceD}{PriceD} \\ \frac{ExpectedPayoff}{PriceC}-1=\frac{ExpectedPayoff}{PriceD}-1 \\ \frac{ExpectedPayoff}{PriceC}=\frac{ExpectedPayoff}{PriceD} \end{gathered}$$

Expected Payoff C ≠ Expected Payoff D

Price C≠ Price D

Asset C and D's expected payoff are not the same; thus, the price cannot be the same.

Step 3. Explanation for part (e)

It is given that the rate of return is equal to two assets, but the payoff is not equal. Thus, the price cannot be equal for both assets. For the price of the assets to be equal, it requires the payoff to be equal.