Why Forward Rates Are Not Generally Expected Short Rates

Summary

The document poses a question about the relationship between instantaneous forward rates and the short rate in a zero-coupon bond framework. It writes the bond price in two forms: as the exponential of the negative integral of forward rates, and as the conditional expectation of discounting at the future short rate. From these expressions, it asks whether a forward rate at a future date equals the current conditional expectation of the short rate at that date, and whether this identity needs special assumptions.

No answer or derivation is included, so the document does not resolve the question or provide empirical evidence. The issue concerns the distinction between rates implied by bond prices and expectations of future short rates, which can depend on the pricing measure and risk premia. A reader should treat the proposed equality as an open question here, rather than as a result established by the text. Further analysis would need to state the stochastic assumptions and measure under which the bond pricing relation is being used.

Key ideas

• The document presents two expressions for a zero-coupon bond price using forward rates and discounted future short rates.
• It asks whether a forward rate equals the conditional expectation of the future short rate.
• The document supplies no answer, derivation, or special assumptions that settle the proposed equality.
• Any assessment would need to specify the pricing measure and relevant risk premia.

Tags

# Relation between short term rate and forward rates


# Relation between short term rate and forward rates












I'm trying to understand relationship between short rates and forward rates

Let $f(t,T)$ is forward rates compounding at $T$ as seen from $t$, and $r(t)$ just a short rate

For a Zero Coupon Bond paying \$1 at maturity $T$ following relations holds:

$$P(t,T) = e^{-\int_t^Tf(t,s)ds}$$ $$P(t,T) = E[e^{-\int_t^Tr(s)ds}|\mathcal{F}_t]$$

Can we say that forward rates is conditional expectations of short rates? $$f(t,T) = E[r(T)|\mathcal{F}_t]$$

Does it holds under special assumption if not?

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