# Calculating the real interest rate, why would my thesis supervisor want me to subtract inflation instead of dividing by inflation?

I'm writing my thesis and for the data I'm using I had calculated the real interest rate by dividing the nominal interest rate by the inflation rate. My supervisor recommended me to change it to instead just subtracting the inflation but I didn't want to ask him why since I've already asked him too many stupid questions.

I always thought dividing is the "real" way and IIRC one of my professors called subtracting the inflation rate a mere "approximation". So why would my supervisor want me to subtract it?

Two things to note here. First, subtracting inflation from the nominal interest rate is an approximation to the real interest rate, but only in discrete time. Furthermore, the "true" relationship it's approximating isn't division of one rate by the other--you have to add 1 to all three of your quantities (inflation, real interest rate, and nominal interest rate) first to get the true relationship.

Here's a brief overview. Consider the Fisher equation of

$r = i - \pi$

where $r$ is the real interest rate, $i$ is the nominal interest rate, and $\pi$ is the inflation rate.

This equation is often introduced as a linear approximation to the true real rate of interest, given by the equation

$\frac{1 + i}{1 + \pi} = 1 + r$

Let's see how this holds in a discrete time model. Denote your nominal income as $Y$ and the price level as $P$. Your real income is $Y/P$. If all this income is invested in some interest-bearing asset in a discrete time model, your real income in the next time period becomes

$\frac{Y (1 + i)}{P (1 + \pi)}$

and if we want to find a real interest rate that summarizes this change in real income, we would need to write your real income in the next time period as

$\frac{Y}{P} (1 + r)$

which gives us the identity $\frac{1 + i}{1 + \pi} = 1 + r$ that the Fisher equation approximates. However, if we work in continuous time, this breaks down. First, the units don't work out--inflation rates and interest rates are measured in percentage change per year (or some other unit of time), so they cannot be added to the dimension-less number $1$. Second, it turns out that Fisher's approximation is actually completely correct in continuous time. Using derivatives, we define our quantities as follows:

$i = \frac{dY}{dt} \frac{1}{Y}$

$\pi = \frac{dP}{dt} \frac{1}{P}$

$r = \frac{d(Y/P)}{dt} \frac{1}{(Y/P)}$.

Using the quotient rule, we can rewrite $r$ as

$r = \frac{\frac{dY}{dt}P - \frac{P}{dt}Y}{P^{2}} \frac{1}{(Y/P)}$

which simplifies to

$r = (\frac{dY}{dt} \frac{1}{P} - \frac{dP}{dt} \frac{Y}{P^{2}}) \frac{P}{Y} = \frac{dY}{dt} \frac{1}{Y} - \frac{dP}{dt} \frac{1}{P} = i - \pi$

which gives us the Fisher equation, no approximations about it!

Note that this assumes continuous compounding from your nominal interest rate. If you instead have a compounding rate of $\tau$, we would define $i$ differently, and our equation becomes

$r = \tau \ln(1 + \frac{i}{\tau}) - \pi$.

However, for the purposes of the data you're working with, I am 90% sure that this modification is completely superfluous. If you want to use something more precise than the Fisher equation, you need to know exactly how your data was computed. What price index was used to calculate the inflation rate? Under what assumptions was the nominal interest rate calculated?

In short, while I can't speak to the reasons for your supervisor's recommendation, they're definitely correct that you should subtract inflation rather than divide by it. There's a reason why we use the Fisher equation.

• Can you explain how the assumption of continuous compounding plays into your definition of i in continuous time? – tjnel Dec 30 '17 at 2:45
• @tjnel Sure! Without continuous compounding, an initial nominal investment of Y becomes Y(1+i/n)^tn at time t, with n as the compounding rate. If we treat the price level and the real investment as growing continuously (with instant compounding), we have (Y(1+i/n)^tn)/(Pe^t(pi)) = (Y/P)e^rt. Dividing both sides by Y/P and taking the natural logarithm of both sides, we have (n)ln(1+i/n) - pi = r. If this is the case, we can't define the nominal interest rate as (dy/dt)/Y, because i is not equal to (n)ln(1+i/n). – SilasLock Dec 30 '17 at 9:36

Rules to keep in mind when dealing with nominal variables.

• To convert a nominal quantity (i.e. nominal GDP) to real value (i.e. real GDP, you divide the nominal value by a measure of price level (i.e. GDP deflator or CPI).
• To convert a nominal interest rate to real interest rate, you subtract inflation rate from nominal interest rate. Read the post about fisher equation @Herr K. suggested above.

Your supervisor wants you to do this because its based on the fisher equation.

$$i =r+\pi$$

where $i$ is nominal interest rate, $r$ is real interest rate and $\pi$ is inflation rate.

rearranging this equation we get the basis for your supervisors recommendation.

$$r=i-\pi$$

Hope this helps.