Converting Nominal Interest Rates to Real Interest Rates

Once in a while, not often, it is useful to do calculations in real dollars (or any other currency). For example, when planning for retirement it may be easier to think in real terms, because you are used to thinking in terms of the money that you currently earn.

Let’s start with some definitions:

• Nominal Interest Rate ($R_N$): the nominal interest rate is the stated rate of interest. It has an expected inflation rate already built into it. Interest rates that are quoted by banks or for investment products are nominal interest rates.

• Inflation Rate ($R_I$): the rate of inflation is the rate at which your money will lose purchasing power.

• Real Interest Rate ($R_R$): the real rate of interest is the increase in purchasing power that you (or the institution that you are paying it to) can expect to receive.

So, nominal interest rates are what we usually see, but real interest rates are what we are ultimately interested in because we care about how much our purchasing power changes. This is particularly true over long periods of time. Fortunately, it is quite simple to convert nominal rates to real rates, or vice versa, as long as you know the expected inflation rate.

Very often you will see a slightly simplified expression:

$$R_R=R_N−R_I$$

where $R_R$ is the real interest rate, $R_N$ is the nominal interest rate, and $R_I$ is the expected rate of inflation. For example, if you expect to earn a rate of 8% on your investment and you think that inflation will average about 3% per year, then you would expect a real return of about 5% per year. This approximation is fine as long as expected inflation is low and the time frame is relatively short.

The exact relationship between nominal and real interest rates is only slightly more complex. The key is to realize that inflation rates compound, just like interest rates. Therefore, the relationship between real rates and inflation rates is multiplicative, not additive as in the approximation above. So, the correct relationship is:

$$R_R=\frac{(1+R_N)}{(1+R_I)}\,−1$$

We can, of course, rearrange that equation to see that the nominal rate is determined by combining the real rate and the inflation rate:

$$R_N = (1+R_R)(1+R_I)\,-1$$

This is known as the Fisher relation or Fisher equation, after the famous economist Irving Fisher. Note that if we expand that equation, there is a cross product term that shows the interaction between the real rate and the inflation rate:

$$R_N = R_R + R_I + R_R \times R_I$$

The interaction term is ignored in the simplified expression above. It is typically small and safely ignored, until inflation is high. If you wish to be precise, as you should be over long periods of time, then you should not ignore that interaction term.

An Example Using Retirement Calculations

Suppose that you are currently 25 years old and that you plan to retire at age 65 (40 years from now). You wish to have accumulated $2,000,000 in today's dollars at the time that you retire. If you can earn a nominal return of 8% per year on your investments and you expect inflation to average 3% annually, how much must you invest each year to meet your goal?

The problem here is that your retirement nest egg (\$2 million) is stated in today’s dollars, while the rate of return is in nominal terms. We need to adjust one or the other so that they agree before we can solve the problem. It really doesn’t matter which one we adjust, but it tends to be easier to think in current dollars so we will convert the nominal interest rate into a real rate:

$$R_R=\frac{(1+0.08)}{(1+0.03)}\,−1=0.04854=4.85\% \text{ per year}$$

Now, we can solve for the annual payment amount using the future value of an annuity formula:

$$\rm{Annual\ Investment}=\frac{\text{2,000,000}}{\left[\frac{(1+0.04854)^{40} \,−1}{0.04854}\right]}=\text{17,153.85}$$

So, if you were to invest 17,153.85 (in today’s dollars) each year for 40 years, you would end up with \$2 million (again, in today’s dollars).

Now, let’s work this problem the other way around. We first need to find out what $2 million in today’s dollars will be in 40 years if inflation averages 3% per year. This is a simple future value problem:

$$FV=\text{2,000,000}(1.03)^{40}=\text{6,524,075.58}$$

This means that you will need to have accumulated over \$6.5 million (nominal) at the time that you retire to achieve your goal. What annual investment will result in that amount after 40 years if you earn a nominal rate of 8% per year? Again, we solve for the payment using the future value of an annuity equation:

$$\rm{Annual\ Investment}=\frac{\text{6,524,075.58}}{\left[\frac{(1+0.08)^{40} \,−1}{0.08} \right]}=\text{25,183.98}$$

So, if you were to invest $25,183.98 each year then you would meet your goal.

Notice that these two solutions are equivalent, however there is a subtle difference. The real solution ($17,153.85 in today’s dollars) requires that you invest the inflation-adjusted amount each year in order to reach the goal of having \$6,524,075.58. So, we need to grow each payment at the rate of inflation. The table below shows the first five actual payments:

First Five Investments
Payment #	Payment Amount
1	17,668.47
2	18,198.52
3	18,744.48
4	19,306.81
5	19,886.01
continue for all 40 payments
40	55,956.51

The nominal solution requires that you invest \$25,183.98 each and every year. Again, though, after adjusting for inflation you end up with the same amount of money after 40 years.

The important thing to remember is that inflation compounds. Therefore, to extract a real rate from a nominal rate, you must divide the nominal rate by the inflation rate (after adding 1 to each). This technique will work for any time value of money problem, though I have only shown a future value of an annuity example.

I hope that has been helpful.