# Real Interest Rate

Real interest rate is the interest rate adjusted for the effect of inflation on maturity value of a loan or investment. It approximately equals nominal interest rate minus inflation rate.

Inflation refers to the decrease in purchasing power of money due to increase in general price level. Inflation hurts a debtor because a dollar of principal amount that he receives at maturity has lower purchasing power than the dollar he initially lends. Real interest rate is the interest rate worked out by incorporating the effect of general price level changes between the settlement date and maturity date on loan/investment cash flows. It helps lenders assess whether an investment will result in increase in their real wealth.

## Example

Assume you invest in a certificate of deposit (CD) which has the following terms:

Settlement date (purchase date) 1 January 20X1
Purchase price $100,000 Maturity date 31 December 20X1 Maturity value$110,000
Consumer price index (1 Jan 20X1) 100
Consumer price index (31 Dec 20X1) 105

The interest rate that you will earn on the CD works out to 10%. It is called the nominal interest rate.

$$\text{r} _ \text{n}=\frac{\text{\110,000}\ -\ \text{\100,000}}{\text{\100,000}}=\text{10%}$$

But this calculation ignores that the consumer price index has increased from 100 to 105. The movement in CPI means that the value of a dollar received on 31 December 20X1 is 0.9524 (100/105) times the value of a dollar on 1 January 20X1. It shows that in real terms the maturity value of your investment is only $104,761.90 (=$110,000×100/105). This means that your real return is just 4.76% as calculated below:

$$\text{r} _ \text{r}=\frac{\text{\104,761.90}-\text{\100,000}}{\text{\100,000}}=\text{4.76%}$$

## Finding Real Interest Rate

You do not need to do these long-winded calculations every time to arrive at your real interest rate.

The following formula converts your nominal interest rate (rn) to real interest rate (rr):

$$\text{r} _ \text{r}=\frac{\text{1}+\text{r} _ \text{n}}{\text{1}+\text{i}}-\text{1}$$

Where i is the inflation rate, the percentage change in consumer price index:

$$\text{i}=\frac{{\rm \text{CPI}} _ \text{n}-{\rm \text{CPI}} _ {\text{n}-\text{1}}}{{\rm \text{CPI}} _ {\text{n}-\text{1}}}$$

The approximate real interest equals nominal interest rate (rn) minus inflation rate (i):

$$\text{Approximate r} _ \text{r} = \text{r} _ \text{n}-\text{i}$$

Let us see if these formulas hold. The inflation rate is the example above is 5%:

$$\text{i}=\frac{\text{105}-\text{100}}{\text{100}}=\text{5%}$$

We already know that the nominal interest rate is 10%. Hence, real interest rate works out to 4.76%.

$$\text{r} _ \text{r}=\frac{\text{1}+\text{10%}}{\text{1}+\text{5%}}-\text{1}=\text{4.76%}$$

Using the simple approximation formula, real interest rate comes out to be 5%:

$$\text{Approximate r} _ \text{r}\ = \text{10%}-\text{5%}=\text{5%}$$