Approximating Discrete-Time Forward Rates with Small-Change Expansions

Summary

The document explains an approximation for the forward rate between two dates when spot rates are compounded discretely. It starts from the exact expression, which compares the accumulated growth factors at the two maturities and converts their ratio into a per-period rate. The approximation replaces the logarithms of those growth factors with first-order expansions, then approximates the exponential with its first-order expansion.

This derivation shows why the forward rate can be estimated from the difference in accumulated growth factors divided by the interval between dates. It is a local approximation: the logarithm and exponential expansions are most reliable when their relevant arguments are small. The document offers no numerical example, error bound, or discussion of how accuracy changes with rate levels and maturity, so users should check the approximation against the exact expression when precision matters.

Key ideas

• The exact discrete-time forward rate is obtained from the ratio of compounded growth factors at two maturities.
• First-order expansions of the logarithm and exponential yield a simpler difference-over-interval approximation.
• The approximation relies on small-change expansions and may lose accuracy when those changes are large.
• The document provides an algebraic derivation but no numerical validation or error analysis.

Tags

# Approximation of Forward Rates in discrete time


# Approximation of Forward Rates in discrete time












The forward rate from time $t$ to $T$ ($f_{t,T}$) can be approximated by:

$$ f_{t,T}= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{{T-t}}}-1 \sim \frac{(1+r_T)^T-(1+r_t)^t}{T-t} $$

Why is that the case?

## Answer by Gordon (score 4, accepted)

https://quant.stackexchange.com/a/42243

You may show it as follows: \begin{align*} f_{t,T}&= \left[ \frac{(1+r_T)^T}{(1+r_t)^t} \right]^{\frac{1}{T-t}}-1\\ &=e^{\frac{1}{T-t} \left[\ln (1+r_T)^T - \ln (1+r_t)^t \right]} -1\\ &\approx e^{\frac{1}{T-t} \left[(1+r_T)^T-1 - \big((1+r_t)^t -1\big)\right]} -1\\ &=e^{\frac{1}{T-t} \left[(1+r_T)^T - (1+r_t)^t\right]} -1\\ &\approx 1+ \frac{1}{T-t} \left[(1+r_T)^T - (1+r_t)^t\right] -1\\ &=\frac{1}{T-t} \left[(1+r_T)^T - (1+r_t)^t\right]. \end{align*}

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