Adjusting interest for inflation

What's better: a 5% EUR bond subject to 3% inflation, or a 10% RON bond subject to 8% inflation? Let's calculate.

Checking our intuition

Intuitively, we'd assume that 5% nominal interest with 3% inflation is the same as 10% nominal interest with 8% inflation. After all, \(5\%-3\%=2\%=10\%-8\%\), so both seem to work out to 2% real interest. But is that actually true?

Let's start by plugging in some real numbers into the rates. Suppose, we buy a one year bond of 100. How much money do we have at the end of the year?

\[ \begin{align} \text{EUR}:\quad& 100 \times 1.05 = 105 \\ \text{RON}:\quad& 100 \times 1.10 = 110 \end{align} \]

We discount these with the interest rates to get the present values:

\[ \begin{align} \text{EUR}:\quad& \frac{105}{1.03} = 101.94 \\[0.2cm] \text{RON}:\quad& \frac{110}{1.08} = 101.85 \end{align} \]

We immediately see that the present values aren't the same. It seems \(5\%-3\%=1.94\%\) is better than \(10\%-8\%=1.85\%\).

The Fisher equation

It's the same story as with growth rates (see Averaging growth rates). The correct thing to do is to multiply and divide rather than add and subtract.

We can derive the correct formula for real interest rates by doing the math in the section above backwards:

\[ \begin{align} 1.94 &= \frac{101.94 - 100}{100} \\[0.2cm] 1.94 &= \frac{\frac{105}{1.03} - 100}{100} \\[0.2cm] 1.94 &= \frac{100 \times \frac{1.05}{1.03} - 100}{100} \\[0.2cm] r &= \frac{100 \times \frac{1 + i}{1 + \pi} - 100}{100} \\[0.2cm] r &= \frac{1 + i}{1 + \pi} - 1 \\[0.4cm] 1+r &= \frac{1+i}{1 + \pi}\quad\quad(\textit{Fisher equation}) \end{align} \]

The last formula is known as the Fisher equation and is the correct way to turn a nominal interest rate into a real one.

Funnily enough, the Wikipedia article goes on to say that "the approximation of \(r=i−\pi\) is often used instead since the nominal interest rate, real interest rate, and inflation rate are usually close to zero". This is attributed to Fisher who must have written it in the 1930s. That must've been a simpler and more innocent time when tens of basis points of interest could just be brushed off as insignificant.

Continuous compounding

We started by talking about a one year bond which pays its interest and principal at the end of the year. But what if we put our 100 in a term deposit with continuous compounding? Mind you, this not how term deposits usually work, but it's easier to talk about than some derivative, so bear with me. How much money will we have at the end of the year?

\[ \begin{align} \text{EUR}:\quad& 100 * e^{0.05} = 105.13 \\ \text{RON}:\quad& 100 * e^{0.10} = 110.52 \end{align} \]

And if we discount back to the present value, we get:

\[ \begin{align} \text{EUR}:\quad& \frac{105.13}{e^{0.03}} = 102.02 \\[0.2cm] \text{RON}:\quad& \frac{110.52}{e^{0.08}} = 102.02 \end{align} \]

So, the real interest rates do work out to be the same 2.02% in this case. In other words, the approximation \(r=i-\pi\) is actually correct when continuous compounding is used.

This shouldn't be particularly surprising since multiplications and divisions turn into additions and subtractions under a logarithm (or the other way around when raising to the power).

RON is weird

This is a bit offtopic, but the exchange rate for RON is pretty weird in that it's been essentially 0.2€ since 2021, but there's no official peg in effect.

The RON inflation and interest rates have been fairly consistently some 3% higher than those for EUR. For example, in March 2025, you could get a 2.5% deposit rate from the ECB, but a 5.5% deposit rate from the BNR.

Conceptually, this shouldn't happen—RON should be losing value against the EUR because of the discrepancy in interest rates, but in practice, this hasn't been happening for the last few years. This means that you could've taken EUR, converted to RON, got the high RON interest rate, then converted back to EUR and gotten the EUR inflation rate. So, the logic used in this post to turn a nominal inflation rate to a real one wouldn't have worked in practice because of the weird exchange dynamics.

If you believed that this state of affairs will continue, there's a trade here (not one that I would engage in, mind you). This is basically the ¥ carry trade situation, but localized to Europe.

Conclusion

The TL;DR is use the Fisher equation of \(1+r = \frac{1+i}{1 + \pi}\) to convert a nominal interest rate to a real one when discrete compounding is in effect, or the approximation of \(r=i-\pi\) when continuous compounding is used. In practice, this means use the first formula for discounting real-world cashflows, but use second formula when dealing with derivatives and fancier interest rates.

I think this an interesting topic because it highlights the big pitfall with finance calculations—it is very easy to use a wrong formula somewhere, get a number that is almost but not quite right, then magnify the problem with a long time horizon or a large amount of money.