Topic 6: International Constraints on Domestic Real Interest Rates

The last topic developed the following relationship between the domestic and foreign nominal interest rates, the combined foreign exchange and country-specific risk premiums, and the expected rate of change (depreciation) of the domestic exchange rate:

   1.    i_d = i_f + E_π + ρ

where  i_d and  i_f  are the domestic and foreign interest rates,  ρ  is the combined country-specific and foreign exchange risk premium and  E_π  is the expected rate of change in the domestic currency price of foreign currency.

We now turn to the implications of this relationship for the determination of domestic real interest rates. In the Lesson entitled Interest Rates and Asset Values it was noted that real interest rates equal nominal rates minus the expected rate of inflation as follows:

   2.    r = i - E_p

where  E_p  is the rate of inflation expected during the term of the loan, and  r  is the real interest rate on which people base their decisions. The basis for this relationship is the fact that inflation reduces the amount of real goods and services the principal and interest on bonds and other assets that are fixed in nominal terms will buy when they come due. The real interest rate that the contracting parties expect to pay and receive is therefore the nominal rate minus an allowance for the expected erosion of the real value of the principal and interest due to inflation.

Using Equation 2 to characterize the real interest rates in both the domestic economy and the rest of the world, we can express the domestic and foreign real interest rates as

   2a.    r_d = i_d - E_pd
   2b.    r_f = i_f - E_pf

The domestic/foreign real interest rate differential can be obtained by subtracting Equation 2b from Equation 2a to yield

   3.    r_d - r_f = i_d - i_f - E_pd + E_pf

If we obtain the nominal interest rate differential by subtracting  i_f  from both sides of Equation 1 and substitute the resulting equation into Equation 3 we obtain

   4.    r_d = r_f + ρ - E_pd + E_π + E_pf

Two conditions must hold for domestic and foreign real interest rates to be equal. The combined risk premium, ρ , must be zero and so must be the term E_pd - E_π - E_pf . Common sense tells us that the risk premium must be zero for domestic and foreign interest rates to be the same. The condition that

    E_pd - E_π - E_pf = 0

also has a straight-forward interpretation. The first term on the left  E_pd  is the expected rate of increase in the level of prices of domestic output in domestic currency. The second term  E_π  is the expected rate of increase in the price of foreign currency in terms of domestic currency and the third term  E_pf  is the expected rate of increase in the level of prices of foreign output in foreign currency. The sum of these latter two terms  E_π + E_pf  is the expected rate of increase in the domestic currency price of foreign output. Suppose, for example, that the price of foreign currency in units of domestic currency goes up by 3% and the price of foreign output in units of foreign currency goes up by 2%. Then it follows that the price of foreign output in units of domestic currency will go up by approximately 5%. The term

    E_pd - E_π - E_pf = 0

thus equals the expected rate of increase in the price level of domestic output minus the expected rate of increase in the price level of foreign output when both price levels are measured in domestic currency.

The ratio of the price level of domestic output to the domestic currency price level of foreign output,

   5.   P_d / (Π P_f )

is called the real exchange rate. The real exchange rate is the relative price of domestic output in terms of foreign output or, in other words, the rate at which domestic output can be exchanged for foreign output. The expression

    E_pd - E_π - E_pf

can thus be interpreted as the expected percentage change in the domestic real exchange rate. This interpretation and others in the preceding screens follow from two elementary mathematical relationships:

1) that the percentage change in a ratio equals the percentage change in the numerator minus the percentage change in the denominator, and

2) that the percentage change in the product of two numbers equals the sum of their percentage changes.

Equation 4 can thus be rewritten as

   6.    r_d = r_f + ρ - E_q

where  E_q  is the expected rate of change in the domestic real exchange rate as defined above. The domestic real interest rate equals the foreign real interest rate plus a risk premium minus the expected rate of increase in the domestic real exchange rate. The role of the risk premium in determining the domestic real interest rate is obvious. But why does the domestic real interest rate fall in response to an expected increase in the relative price of domestic output in terms of foreign output?

An increase in the relative price of domestic in terms of foreign goods represents an increase in the value of the income flow from capital employed in the domestic economy relative to the value of the income flow from capital employed abroad. A positive value of  E_q  thus represents an expected future capital gain on domestic relative to foreign capital which makes it more profitable to hold domestic capital at equal risk-adjusted interest rates. The prices of the sources of real income from capital employed in the domestic economy will thus be bid up, and the interest rate on that capital bid down, until net yield on capital, taking everything into account, is the same in the domestic economy as abroad.

Equation 6 imposes a constraint on the domestic real interest rate similar to the constraint on the domestic nominal interest rate imposed by Equation 1. It says that, regardless of the mechanics by which government policy affects the economy, the authorities can only bring about a change in the domestic real interest rate by either inducing a change in the risk of holding domestic assets or inducing a change in the expected rate of change of the domestic real exchange rate. This, of course, assumes that the domestic government cannot influence the foreign real interest rate. To affect the real interest rate the government must change either the risk premium or the expected rate of change in the country's real exchange rate---to affect the nominal interest rate, it must change either the risk premium or the expected rate of change in the country's nominal exchange rate.

Finally, from the relationships between the nominal and real interest rates in the domestic and foreign economies, given by the two country's Fisher Equations

   7a.    i_d = r_d + E_pd
   7b.    i_f = r_f + E_pf

which are simply reorganizations of equations 2a and 2b, it is clear that the domestic/foreign nominal interest rate differential equals the domestic/foreign real interest rate differential plus the difference between the domestic and foreign expected inflation rates:

   8.    i_d - i_f = r_d - r_f + E_pd - E_pf

A change in the domestic relative to the foreign expected inflation rate will lead to an equal change in the domestic nominal interest rate relative to the nominal interest rate abroad---assuming, of course, that real interest rates are unaffected. If real interest rates are approximately the same in different countries---that is, if risk differences are not too great---nominal interest rates will differ across countries by the differences in their expected inflation rates.

It's time for a test. Be sure to think up your own answers before looking at the ones provided.

Question 1
Question 2
Question 3

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