Forward Rates And Expected Future Short Rates

Why a Forward Rate Is Not Automatically a Forecast

A forward rate is determined by today’s prices for investments at different maturities. A forecast of a future short rate is a statement about what an analyst expects rates to be later. They may be numerically close, but they answer different questions: one is implied by current market prices, while the other is an expectation under a specified view of future outcomes.

The distinction matters because future rates are uncertain, and investors may require compensation for bearing that uncertainty. Even if the market’s pricing probabilities and an analyst’s real-world probabilities were the same, the calculation that converts state-contingent future rates into a bond price is nonlinear. Therefore, an implied forward rate need not equal the arithmetic expected future rate.

If the annual effective spot rates for years 1 and 2 are \(i_1\) and \(i_2\), respectively, then \(P(0,1)=1/(1+i_1)\) and \(P(0,2)=1/(1+i_2)^2\). Consequently,

This relation is a no-arbitrage pricing identity. It does not, by itself, state what rate anyone expects to prevail in year 2. The earlier material on forward rates establishes how the identity is obtained from prices and how it supports comparisons between locking in a rate and rolling over investments. The remaining question is how that price-based rate relates to an expectation when future rates are uncertain.

A Certain Future Rate

If the future one-year rate is known with certainty, then the forward rate and that known future rate coincide in a market without arbitrage. If the future rate is genuinely uncertain, however, a bond price reflects the values of payments across possible outcomes, not simply the arithmetic average of the possible rates.

How Uncertainty Enters the Price

To make the pricing relationship precise, consider a simple two-period model. Assume the first one-year rate \(i_{0,1}\) is known today, the future one-year rate \(R\) applies from year 1 to year 2, and \(R>-1\) in every possible outcome. Let \(Q\) denote the pricing probability distribution for future outcomes when prices are expressed using the money-market account as numeraire. Under this model, the time-0 price of a payment of 1 at year 2 is the expected value, under \(Q\), of the payment discounted along each possible path:

Dividing the first price by the second gives the forward-rate relation. This is a short derivation of why the implied rate involves an expectation of a discount factor, rather than an expectation of the rate itself:

This model also shows why even the pricing-distribution arithmetic expectation \(E_Q[R]\) is not generally the forward rate. The expectation is taken after transforming each possible rate to its reciprocal accumulation factor, and the reciprocal of that average is then taken to obtain \(1+f_{1,2}\).

For \(R>-1\), the function \(g(R)=1/(1+R)\) is convex because \(g''(R)=2/(1+R)^3>0\). Jensen’s inequality therefore gives \(E_Q[1/(1+R)]\geq 1/(1+E_Q[R])\). Taking reciprocals of positive quantities yields

This inequality applies to the stated one-period pricing model and the pricing distribution \(Q\). It is not a general inequality between a forward rate and an expectation under real-world probabilities. Nor should it be extended without checking assumptions to models with different numeraires, additional pricing factors, or a different definition of the rate being forecast.

Pricing Probabilities, Real-World Expectations, and Risk Premiums

A real-world expectation is usually calculated under the probability distribution an analyst uses to describe actual future outcomes. In contrast, prices are formed using state values: a dollar paid in one state may be worth more or less than a dollar paid in another. The distribution \(Q\) in the simple derivation represents pricing weights after those state values are accounted for. It is not automatically the analyst’s real-world distribution.

When risk is priced, the probability weights that reproduce market prices can differ from real-world probabilities. This is one way of expressing a risk premium. Consequently, the forward rate can differ from the real-world expected future short rate both because the relevant probability weights differ and because the discount-factor transformation is nonlinear. The forward rate is not a pure forecast, and the real-world expected rate is not necessarily a price-consistent rate.

The next example holds the possible future rates fixed but uses different real-world and pricing probabilities. This isolates the importance of specifying which expectation is being calculated.

Interpreting a Forward Rate in Practice

A forward rate is useful because it summarizes the rate that is consistent with current prices under a particular compounding convention. It can serve as a benchmark for comparing a forecast with the term structure, or for identifying the rate embedded in a sequence of investment periods. It should not be described as the market’s guaranteed prediction.

When interpreting a comparison, check these points:

• Match the rate period and convention. A one-year annual effective forward rate should be compared with a forecast for that same one-year period, expressed on the same basis.
• Specify the expectation. An expected rate is incomplete unless the probability distribution or forecasting model is clear.
• Separate pricing from forecasting. Bond prices imply pricing weights and discount factors; an analyst’s forecast may use real-world probabilities.
• Do not infer arbitrage from a forecast difference alone. An arbitrage conclusion requires a price inconsistency and a feasible strategy that locks in a risk-free gain, not merely a disagreement about a future rate.

The key discipline is to label each quantity before calculating: a forward rate comes from current prices, while an expected future short rate comes from a specified probability model. This distinction is aligned with the SOA Exam FM (Financial Mathematics) syllabus.