Exact and Approximate Links Between Real and Nominal Rates

Summary

The document compares an exact relationship between real, nominal, and inflation rates with the familiar approximation that the real rate equals the nominal rate minus inflation. It asks how these formulas relate to a bond valuation example involving a real coupon and inflation-adjusted cash flows. The author is uncertain about when real coupon accrual applies to the original principal versus an inflation-adjusted principal, and how to handle inflation over the bond’s final period.

The document supplies the equations and an illustrative valuation setup, but it contains no answer resolving the questions. It therefore serves mainly as a statement of the distinction and a prompt to clarify timing and compounding assumptions. In general, the exact relation accounts for multiplicative compounding, while subtracting inflation is an approximation; applying either in a valuation requires consistent definitions of rates, cash-flow dates, and the inflation index used. Those explanatory conclusions are context for the question, not a worked solution contained in the source. The example does not provide enough information to settle the accrual convention or resolve the timing issue.

Key ideas

• The document contrasts an exact multiplicative relation with an approximate subtraction of inflation from nominal rates.
• It frames the distinction through a real coupon bond valuation example.
• The questions turn on whether coupon accrual uses initial or inflation-adjusted principal.
• Consistent rate definitions, compounding, and cash-flow timing are needed for valuation.
• No response or worked resolution is included in the document.

Tags

# What are the differences between the two equations relating inflation, real interest rates and nominal interest rates?


# What are the differences between the two equations relating inflation, real interest rates and nominal interest rates?












My lecturer showed two equations to describe the relationship between inflation, nominal and real interest rates, denoted as $I, RN, RR$, respectively.

Then we have the following equations $$ 1+ RR_{1-0} = \frac{1+NR_{1-0}}{1+I_{1-0}} \tag{1} $$

$$ RR_{1-0} \approx NR_{1-0} - I_{1-0} \tag{2} $$

My understanding of the two equations is as follows.

$(2)$ becomes an equality once we apply $RR$ in advance on $V_{0}$, where $V_i$ denotes the principal at time $i$. My reasoning for this is:

\begin{equation}\tag{3} V_1 =V_0 \cdot (1+NR_{1-0}) = V_0 \cdot (1+I_{1-0}) + V_0\cdot RR_{1-0}\\ \implies NR_{1-0} = I_{1-0} + RR_{1-0} \end{equation}

My troubles comes with when to use $(1)$.

Consider a bond with $PV=1000$ with $RC_{2-0} = 2.5%$. Then the $NPV$ calculation becomes:

$$ 1000 = \frac{25}{1+RY_{2-0}} + \frac{1025}{(1+RY_{2-0})^2} $$

So when we calculate this in terms of $NY$ we get: $$ 1000=\frac{25}{1+NY_{2-0}} + \frac{25(1+I_{2-0}) + 1000(1+I_{2-0})^2}{(1+NY_{2-0})^2} $$

We get $25$ as we apply the $RC$ on the principal value of 1000 at $t=0$, we get $25(1+I_{2-0})$ as we apply $RC$ on principal at $t=1$, which has undergone inflation.

My questions are:

- My lecturer wrote "If $RC$ accrues ahead on $V_1$ not $V_0$ then we get $(1)$." How is this the case? Since for the $NPV$ calculation at $t=1$ we're applying $RC$ on $V_0$.

- Why do we get $1000(1+I_{2-0})^2$? Since we get principal back at $t=2$ and we only know inflation rate of $t=2$ at $t=2+\delta$ where $\delta > 0.$

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.