# Relationship between nominal and real interest rates and inflation in a term structure model

I am studying the basics of the term structure of interest rates from Hillier et al. "Fundamentals of Corporate Finance" (3rd ed., 2017) (here is a link to a slightly different edition). An illustration in the book suggests the nominal rates are a sum of three underlying components: the real rate, the inflation premium and the interest rate risk premium.

On the other hand, the real rate is previously defined as the increase in the purchasing power, with the following relationship holding between the nominal rate, the real rate and the inflation rate: $$1+r=\frac{1+R}{1+h}$$ where $$r$$ is the real rate, $$R$$ is the nominal rate and $$h$$ is the inflation rate. (I think this relationship defines the real rate.)

I think there is a discrepancy between the model of the term structure and the definition of the real rate; the former includes the interest rate risk premium while the latter does not. How do I reconcile the two?

I think the textbook you are using is really confusing.

Normally the real rate of interest includes risk premium. In case you are using formula:

$$1+r = \frac{1+R}{1+h} \text{ or } r \approx R -h$$

the real interest rate itself includes risk premium as it is a compensation for all real factors (impatience and risk).

However, you could always decompose $$r$$ into risk free rate $$r_f$$ and risk premium $$r_r$$.

For some reason your textbook calls $$r_f$$ the real rate, even though it should be more correctly called risk free rate. Together risk free rate and risk premium give you real rate and then adding inflation on top of it gives you nominal rate:

$$r \approx R-h \implies r+h \approx R$$

• Hm. The bonds underlying the term structure are risk free in the sense of having zero credit risk. If I understand it right, the risk premium on them is due to the uncertainty around the future discount rate on such bonds. Meanwhile, the $r$ vs. $r_f$ from your penultimate paragraph reminds me more of bonds that have nonzero credit risk vs. bonds with zero credit risk. Also, the "real rate" in the picture is meant to reflect impatience or something like it but not other things like discount rate risk, so I guess the name "real rate" in the picture is misleading and should be changed. Sep 22, 2022 at 17:29
• @RichardHardy generally speaking the real rate in economics reflects the real factors, credit risk or uncertainty about future is all part of real interest rate, since they are real in a sense that they all are caused by some underlying economic reality. Nominal interest rate is just compensation for real factors + inflation (inflation is the nominal/monetary factor).
– 1muflon1
Sep 22, 2022 at 20:18
• That makes sense. Thanks! Sep 23, 2022 at 6:57