4.4 Interpretation of the Rate of Return Formula

Although the derivation of the rate of return formula is fairly straightforward, it does not lend itself easily to interpretation or intuition. By applying some algebraic “tricks,” it is possible to rewrite the British rate of return formula in a form that is much more intuitive.

Step 1: Begin with the British rate of return formula derived in Chapter 4 "Foreign Exchange Markets and Rates of Return", Section 4.3 "Calculating Rate of Returns on International Investments":

$$Ro{R}_{\text{£}}=\frac{E_{\text{\$/£}}^{\text{e}}}{E_{\text{\$/£}}}(1+i_{\text{£}})-1.$$

Step 2: Factor out the term in parentheses. Add i_£ and then subtract it as well. Mathematically, a term does not change in value if you add and subtract the same value:

$$Ro{R}_{\text{£}}=\frac{E_{\text{\$/£}}^{\text{e}}}{E_{\text{\$/£}}}+i_{\text{£}}\frac{E_{\text{\$/£}}^{\text{e}}}{E_{\text{\$/£}}}-1+i_{\text{£}}-i_{\text{£}}\text{.}$$

Step 3: Change the (−1) in the expression to its equivalent, $-\frac{E_{\text{\$/£}}}{E_{\text{\$/£}}}$. Also change −i_£ to its equivalent, $-i_{\text{£}}\frac{E_{\text{\$/£}}}{E_{\text{\$/£}}}$. Since $\frac{E_{\text{\$/£}}}{E_{\text{\$/£}}}=1$, these changes do not change the value of the rate of return expression:

$$RoR{}_{\text{£}}=\frac{E_{\text{\$/£}}^{\text{e}}}{E_{\text{\$/£}}}+i_{\text{£}}\frac{E_{\text{\$/£}}^{\text{e}}}{E{}_{\text{\$/£}}}-\frac{E_{\text{\$/£}}^{}}{E{}_{\text{\$/£}}}+i_{\text{£}}-i_{\text{£}}\frac{E_{\text{\$/£}}^{}}{E{}_{\text{\$/£}}}\text{.}$$

Step 4: Rearrange the expression:

$$RoR{}_{\text{£}}=i_{\text{£}}+\frac{E_{\text{\$/£}}^{\text{e}}}{E_{\text{\$/£}}}-\frac{E_{\text{\$/£}}^{}}{E{}_{\text{\$/£}}}+i_{\text{£}}\frac{E_{\text{\$/£}}^{\text{e}}}{E{}_{\text{\$/£}}}-i_{\text{£}}\frac{E_{\text{\$/£}}^{}}{E{}_{\text{\$/£}}}\text{.}$$

Step 5: Simplify by combining terms with common denominators:

$$Ro{R}_{\text{£}}=i_{\text{£}}+\frac{E_{\text{\$/£}}^{\text{e}}-E_{\text{\$/£}}}{E_{\text{\$/£}}}+i_{\text{£}}\frac{E_{\text{\$/£}}^{\text{e}}-E_{\text{\$/£}}}{E_{\text{\$/£}}}\text{.}$$

Step 6: Factor out the percentage change in the exchange rate term:

$$Ro{R}_{\text{£}}=i_{\text{£}}+(1+i_{\text{£}})\frac{E_{\text{\$/£}}^{\text{e}}-E_{\text{\$/£}}}{E_{\text{\$/£}}}\text{.}$$

This formula shows that the expected rate of return on the British asset depends on two things, the British interest rate and the expected percentage change in the value of the pound. Notice that if $\frac{E_{\text{\$/£}}^{\text{e}}-E_{\text{\$/£}}}{E_{\text{\$/£}}}$ is a positive number, then the expected $/£ ER is greater than the current spot ER, which means that one expects a pound appreciation in the future. Furthermore, $\frac{E_{\text{\$/£}}^{\text{e}}-E_{\text{\$/£}}}{E_{\text{\$/£}}}$ represents the expected rate of appreciation of the pound during the following year. Similarly, if $\frac{E_{\text{\$/£}}^{\text{e}}-E_{\text{\$/£}}}{E_{\text{\$/£}}}$ were negative, then it corresponds to the expected rate of depreciation of the pound during the subsequent year.

The expected rate of change in the pound value is multiplied by (1 + i_£), which generally corresponds to a principal and interest component in a rate of return calculation.

To make sense of this expression, it is useful to consider a series of simple numerical examples.

Suppose the following values prevail,

Plugging these into the rate of return formula yields

$$Ro{R}_{\text{£}}=0.05+(1+0.05)\frac{1.10-1.00}{1.00},\infty$$

which simplifies to

RoR_£ = 0.05 + (1 + 0.05) × 0.10 =.155 or 15.5%.

Note that because of the exchange rate change, the rate of return on the British asset is considerably higher than the 5 percent interest rate.

To decompose these effects suppose that the British asset yielded no interest whatsoever.

This would occur if the individual held pound currency for the year rather than purchasing a CD. In this case, the rate of return formula reduces to

RoR_£ = 0.0 + (1 + 0.0) × 0.10 =.10 or 10%.

This means that 10 percent of the rate of return arises solely because of the pound appreciation. Essentially an investor in this case gains because of currency arbitrage over time. Remember that arbitrage means buying something when its price is low, selling it when its price is high, and thus making a profit on the series of transactions. In this case, the investor buys pounds at the start of the year, when their price (in terms of dollars) is low, and then resells them at the end of the year when their price is higher.

Next, suppose that there was no exchange rate change during the year, but there was a 5 percent interest rate on the British asset. In this case, the rate of return becomes

RoR_£ = 0.05 + (1 + 0.05) × 0.0 =.05 or 5%.

Thus with no change in the exchange rate, the rate of return reduces to the interest rate on the asset.

Finally, let’s look back at the rate of return formula:

$$Ro{R}_{\text{£}}=i_{\text{£}}+(1+i_{\text{£}})\frac{E_{\text{\$/£}}^{\text{e}}-E_{\text{\$/£}}}{E_{\text{\$/£}}}\text{.}$$

The first term simply gives the contribution to the total rate of return that derives solely from the interest rate on the foreign asset. The second set of terms has the percentage change in the exchange rate times one plus the interest rate. It corresponds to the contribution to the rate of return that arises solely due to the exchange rate change. The one plus interest rate term means that the exchange rate return can be separated into two components, a principal component and an interest component.

Suppose the exchange rate change is positive. In this case, the principal that is originally deposited will grow in value by the percentage exchange rate change. But the principal also accrues interest and as the £ value rises, the interest value, in dollar terms, also rises.

Thus the second set of terms represents the percentage increase in the value of one’s principal and interest that arises solely from the change in the exchange rate.