# Fisher Effect

Fisher effect is the concept that the real interest rate equals nominal interest rate minus expected inflation rate. It is based on the premise that the real interest rate in an economy is constant and any changes in nominal interest rates stem from changes in expected inflation rate.

The assumption that the real interest rate in an economy should stay constant in the long-run is based on the notion that changes in money supply affects only nominal values such as prices, exchange rates and have no bearing on real indicators such as employment, real GDP, etc.

## Formula

The fisher effect postulates the following relationship between nominal interest rate (n), real interest rate (r) and expected inflation rate (i):

$$\text{n}=\text{r}+\text{i}$$

## Example

The nominal interest rate is the interest rate quoted on most loans and investments. For example, Treasury bonds and most of corporate bonds quote nominal coupon rates and yields. The real interest rate is the effective interest rate that is earned on an investment or paid on a loan after considering the deterioration of purchasing power due to inflation. The interest rate on treasury inflation protected securities (TIPS) or any inflation-indexed bonds is the real interest rate.

Let’s say you invested $20 million at the start of 2001 when the consumer price index was 1,000. It is end of 2010 now and the CPI is 1,100. Your investment has accumulated to$30 million. Your holding period return over the 10-year period is 50%.

$$\text{HPR}=\frac{\text{\30 million}-\text{\20 million}}{\text{\20 million}}=\text{50%}$$

HPR of 50% translates to nominal annual interest rate of 4.14%.

$$\text{n}=\left(\text{1}+\text{HPR}\right)^\frac{\text{1}}{\text{n}}-\text{1}=\left(\text{1}+\text{50%}\right)^\frac{\text{1}}{\text{10}}-\text{1}=\text{4.14%}$$

However, if we factor in the deterioration in purchasing power of dollar, we find out that $30 million received at the end of 2010 are equal to$27.27 million in term of 2001 dollars:

$$\text{\30 million in 2001 dollars}=\text{\30 million}\times\frac{\text{1,000}}{\text{1,100}}=\text{\27.27 million}$$

This translates to holding period return of 36.36% and annual holding period yield i.e. real interest rate of 3.15%.

$$\text{r}=\left(\text{1}+\text{HPR}\right)^\frac{\text{1}}{\text{n}}-\text{1}=\left(\text{1}+\text{36.36%}\right)^\frac{\text{1}}{\text{10}}-\text{1}=\text{3.15%}$$

The difference between the nominal interest rate of 4.14% and the real interest rate of 3.15% is attributable to inflation.

$$\text{i}=\text{n}-\text{r}=\text{4.14%}-\text{3.15%}=\text{0.99%}$$

This roughly equals the inflation rate determined using CPI:

$$\text{i}=\frac{\text{1,100}-\text{1,000}}{\text{1,000}}=\text{1%}$$