# Expected inflation in the real interest rate equation

Real interest rate = Nominal rate - Expected inflation

In the above equation, in a quarterly data-set, which expected inflation shall be used? next quarter (q+1) or the same quarter of next year (q+4)? and why?

• Use next quarter. Since you are modeling (based on dsge tag), you have quarterly data so you are observing changes in real interest rate quarterly. So every quarter you have a new inflation expectation based on results of previous quarter. Dec 4 '20 at 5:33

Your formula isn't quite right. If $$r$$ is the nominal interest rate, and $$\pi$$ is the inflation rate, the real interest rate is $$\frac {1+r}{1+\pi}-1$$. The formula $$r-\pi$$ is approximation for small rates. (Note: for all of this, I'm using "rate" to mean the multiplicative factor minus one. So if the balance is multiplied by 1.2, the rate is 0.2.)

The inflation should be over the period over which the interest rate is being calculated. It's like to like: the real interest rate at a particular time is given by the nominal interest rate at that time compared to the inflation at that time.

The fact that you think you are wondering whether you should use the future inflation suggests that you may be confusing the spot inflation rate with the Consumer Price Index. You can get the real interest rate by taking the total nominal interest times the ratio between the CPI at the end of the period to the CPI at the beginning. That is, if $$r$$ is the total nominal interest, $$CPI_i$$ is the starting CPI, and $$CPI_f$$ is the final CPI, then the total real interest is $$(1+r) \frac {CPI_f}{CPI_i}-1$$.

It is the expected inflation rate over the life of the instrument. So if it is a 10-year bond, it is expected inflation over the next 10 years.

If you use future values of the price index to determine “expected” inflation, you are assuming bond investors can predict the future perfectly. Based on my experience, that assumption is implausible. It could be justified, but one would really need to be careful about how the data are interpreted.

Note: I missed a clarifying comment. The original poster was only interested in whether to use the next quarter, or +4 quarters for “inflation expectations.”

For determining the real rate, it should still be matched to the term of the bond. E.g., subtracting 3 months inflation from a 10-year bond makes little sense.

For a model, 1 quarter ahead is used in some contexts (linearisations of models), otherwise, it is the entire model horizon. E.g., a model that projects a solution out to infinity, it’s the entire inflation trajectory.

• Thank you Brian for your answer. I am asking in the context of a macroeconomic model, not the bond market. In this model, different economic agents (i.e. households, firms, etc.) set their expectations to be forward-looking. Accordingly, the question remains which expected inflation shall be used? next quarter (q+1) or the same quarter of next year (q+4)? and why? Jun 6 '20 at 12:40
• Whether or not they are bond investors, the logic is the same. If you want to discuss a real rate, you need to match the terms. If you are not interested in a real rate, and just inflation expectations, you need to ask why the agent is interested in inflation? I am unconcerned about inflation over the next three months if I am thinking about my retirement plans. Jun 6 '20 at 16:13
• Yes, this is what i am asking. It is a specific question, rather than a general problem. In a macro-model (QPM or DSGE), what is the difference between using next quarter (q+1) or the same quarter of next year (q+4) to reflect inflation expectations? Jun 8 '20 at 10:07
• I believe @Brian Romanchuk is saying that the term of the inflation rate must match the term of the real and nominal interest rates being quoted. So if you have 3month real and nominal rates, they reflect expected inflation over the next 3months. Not a forward period
– dm63
Nov 3 '20 at 11:15