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 Irving Fisher's Theory of Interest Rates With and Without Adjustment for Tax Rates and Risk Premiums

## The Original Fisher Model

Irving Fisher's theory of interest rates relates the nominal interest rate i to the rate of inflation π and the "real" interest rate r. The real interest rate r is the interest rate after adjustment for inflation. It is the interest rate that lenders have to have to be willing to loan out their funds. The relation Fisher postulated between these three rates is:

(1+i) = (1+r) (1+π) = 1 + r + π + r π

This is equivalent to:

i = r + π(1 + r)

Thus, according to this equation, if π increases by 1 percent the nominal interest rate increases by more than 1 percent.

This means that if r and π are known then i can be determined. On the other hand, if i and π are known then r can be determined and the relationship is:

1+r = (1+i)/ (1+π)

or

r = (i - π)/ (1+π)

When π is small then r is approximately equal to i-π, but in situation involving a high rate of inflation the more accurate relationship must be taken into account.

## Adjustment for Variation in Tax Rates

The next step in the analysis is to take into account the effect of taxes on the real rate of return. Let iC be the nominal risk-free interest rate in the country with currency C and rC and πC be the corresponding real interest rate and expected rate of inflation, respectively. Let tC be the corresponding tax rate on interest income and r*C be the after-tax real rate of return. The rate of return after-taxes is iC(1-tC). Then

r*C = [iC(1-tC)- πC] /(1+πC).

If we know r*C,tC and πC and want to determine iC the formula is:

iC = [r*C(1+πC)+ πC]/ (1-tC)
= r*C/(1-tC)
+ (1 + r*CC/(1-tC).

This means that when the rate of inflation increases the nominal interest rate increase by some multiple of the increase in the rate of inflation; i.e.,

∂iC/∂πC = (1+r*C)/(1-tC).

William Crowder and Dennis Hoffman in their article, "The Long- Run Relationship between Nominal Interest Rates and Inflation: the Fisher Effect Revisted," Journal of Money, Credit and Banking (Feb. 1996) report that a 1.0 percent increase in the inflation rate yields a 1.34 percent increase in the nominal interest rate. This is consistent with a marginal tax rate of about 25 percent.

## Adjustment for Variation in Risk

The preceding analysis presumes that the level of risk is the same in all countries. If countries differ in risk, lenders and investors will need a risk premium, an increment in the interest rate, to compensate them for accepting higher levels of risk. Let sC be the risk premium required for country C. If the international capital market is in equilibrium the real, after-tax rates of return in the different countries must be equal. Then rC-sC=r* for all countries and hence

(iC(1-tC)- πC)/ (1+πC) = r* + sC.

Thus,

iC = [(r*+sC)(1+πC) +πC)]/(1-tC)

Suppose tC = 0.4 so 1-tC=0.6 and r* + sC = 0.05. Then

iC = [0.05(1+πC) + πC]/0.6 = 0.0833 + 1.75πC)

so that each 1 percent increase in the expected rate of inflation gets translated in to a 1.75 percent increase in the nominal interest rate.

An alternate approach to incorporating country risk premiums into the analysis is to reformulate Fisher's original equation to include a factor of (1+ρ) where ρ is the risk premium for the country. This means that the nomimal interest rate is given by:

(1+i) = (1+r)(1+ρ) (1+π)

Thus when inflation increases by 1 percent the nomimal rate will increase by (1+r)(1+ρ) percent, which could be significantly greater than 1.0.

To take into account the tax rate on interest, the term on the left should be 1 plus the after-tax nominal interest rate; i.e.,

(1+i(1-t)) = (1+r)(1+ρ) (1+π)

Thus the before-tax nominal interest rate is given by:

i = [(1+r)(1+ρ) (1+π) - 1]/(1-t)
and hence

∂i/∂π = (1+r*)(1+ρ)/(1-t).

## Finding the Expected Rate of Inflation from the Nominal Interest Rate

In order to use the PPP principle for forecasting future exchange rates we need the expected rate of inflation. The way this would be determined for a country would be.

(1+π) = (1+i(1-t))/(1+r*)(1+ρ)

For two countries in financial equilibrium the values of r* would be the same. Thus the factor required for forecasting exchange rates by the PPP principle is given by:

(1+πF)/(1+π\$) =
(1+iF(1-tF))/(1+i\$(1-t\$))/
[(1+ρF)/(1+ρ\$)]

Suppose the nominal risk-free interest rates in the U.S. and France are 8% and 11%, respectively and the tax rates are 0.3 and 0.4, also respectively. Furthermore, suppose the country risk premiums for the U.S. and France are 0% and 0.5 of 1%, respectively. Then the after-tax nominal rates are 5.6% and 6.6%. The ratio of 1 plus the expected rates of inflation is given by:

(1+πF)/(1+π\$) = (1.066/1.056)/[1.005/1.0] = 1.00445.

Thus the French franc should depreciate 0.445 of 1% per year with respect to the dollar.