FRM Exam Part II · Expectations, Risk Premium, Convexity and the Shape of the Term Structure

Expectations Hypothesis and Forward Rates Explained

Updated 11 October 2026 · Fact-checked

Under the pure expectations hypothesis, the forward rate for a future period equals the expected future spot rate for that period. You get the forward rate from today's spot rates by compounding: (1 + z₂)² = (1 + z₁)(1 + f). Any gap between forwards and expected spots is a risk premium, which this hypothesis sets to zero.

Understand Expectations Hypothesis and Forward Rates

A spot rate is the yield today on a zero-coupon bond for a given maturity. A forward rate is the rate for a future period that you can lock in today using bonds of different maturities. It comes from no-arbitrage, not from any view on the economy.

Suppose the 1-year spot rate is 3% and the 2-year spot rate is 4%. You can buy a 2-year zero, or buy a 1-year zero and roll it over. The forward rate is the 1-year rate one year from now that makes both choices give the same terminal wealth.

The pure expectations hypothesis adds a behavioural claim: the forward rate equals the expected future spot rate. Investors are risk-neutral and need no extra reward for holding longer bonds. So an upward-sloping curve means the market expects short rates to rise. A downward-sloping curve means it expects them to fall.

This is a benchmark, not a proven fact. Real investors dislike risk, and long bonds have price risk. Liquidity preference theory says forwards are higher than expected spots by a term premium. Then a rising curve can reflect compensation for risk, not expected rate rises. Empirically, forward rates tend to overshoot later realised short rates on average, which fits a positive term premium. Treat this as a general tendency, not a rule for every period.

In FRM, the hypothesis is the zero-risk-premium reference case. You measure everything else, such as risk premium and convexity, as a deviation from it.

Key formulas to remember

Forward rate from spot rates (annual compounding): (1 + z_T)^T = (1 + z_t)^t × (1 + f(t,T))^(T − t); z are spot rates, f(t,T) is the annualised forward rate from year t to year T.
One-year forward, one year ahead: f(1,2) = (1 + z₂)² ÷ (1 + z₁) − 1; The most common exam case.
Continuous compounding forward: f(t,T) = (z_T × T − z_t × t) ÷ (T − t); Use only when the question gives continuously compounded rates.
Pure expectations hypothesis: f(t,T) = E[future spot rate for t to T]; Risk premium is zero. Forward is an unbiased predictor of the expected spot.
Forward with a term premium: f = E[future spot] + term premium; A positive premium means forwards sit above expected spots (liquidity preference view).

How to solve Expectations Hypothesis and Forward Rates questions

Use this order for any question on forwards and the expectations hypothesis.

1Identify the rates given: spot, par or forward. Convert par yields to spots first if needed.
2Note the compounding convention: annual, semi-annual or continuous.
3Define the forward period: start t and end T, so T − t is its length.
4Write the no-arbitrage equation: growth to T equals growth to t times growth over the forward period.
5Solve for the forward rate. Take the root of length T − t if the period is longer than one year.
6If the question asks for an expected spot rate, apply the stated theory: under the pure hypothesis expected spot = forward; otherwise subtract the term premium.
7Interpret: compare the forward to today's spot and say whether the curve implies rising or falling rates, and whether a premium explains it.

Quickest way: Ratio of growth factors

When to use it: Annual compounding, one-year forward, and answer choices that are spread apart.

Compute the 2-year growth factor (1 + z₂)² and the 1-year factor (1 + z₁).
Divide the longer by the shorter and subtract 1.
Sanity check: a 1-year forward is roughly 2 × z₂ − z₁. Use it to eliminate options.
If rates are continuous, skip compounding and use the difference of z × T.

Common mistakes in Expectations Hypothesis and Forward Rates

Taking the forward rate as z₂ − z₁.
The difference looks natural and is close to right.
Fix: Use the growth-factor ratio. The difference is only an approximation.
Forgetting to raise to the power T − t for multi-year forwards.
Students solve for the total growth and stop.
Fix: Take the (T − t)th root to annualise before subtracting 1.
Mixing compounding conventions.
Questions switch between annual, semi-annual and continuous rates.
Fix: Write the convention next to each rate and use one formula consistently.
Saying forwards are always accurate forecasts.
The pure hypothesis is treated as a fact.
Fix: It is a benchmark. With risk-averse investors, forward = expected spot + term premium.
Reading an upward-sloping curve only as expected rate rises.
The pure hypothesis is the only story remembered.
Fix: A positive slope can come from expected rises, a term premium or both. Only the pure hypothesis ties it to expectations alone.

Worked examples

Example 1

The 1-year spot rate is 3.00% and the 2-year spot rate is 4.00%, both annually compounded. (a) What is the 1-year forward rate one year from now? (b) Under the pure expectations hypothesis, what is the expected 1-year spot rate in one year?

Show the solution

Growth to year 2: (1.04)² = 1.0816.
Growth to year 1: 1.03.
Forward: 1.0816 ÷ 1.03 = 1.05010, so f(1,2) ≈ 5.01%.
Pure expectations: expected spot = forward.

Answer: (a) About 5.01%. (b) About 5.01%.

Example 2

Using the same curve, the forward rate is 5.01%. A risk manager believes the market's expected 1-year spot rate in one year is 4.60%. What term premium is implied, and what does it say about the theory?

Show the solution

Forward = expected spot + term premium.
Term premium = 5.01% − 4.60% = 0.41%.
The premium is positive and non-zero, so the pure expectations hypothesis does not hold for this view.

Answer: The implied term premium is about 0.41% (41 basis points). This fits the liquidity preference view, where forwards exceed expected spots.

Exam tips

Check the compounding convention first. It decides which formula applies.
Distinguish the pure expectations hypothesis (zero premium) from liquidity preference (positive term premium) in wording-based questions.
Use f ≈ 2z₂ − z₁ to eliminate options fast, then compute exactly if the options are close.
When asked about interpretation, mention both expected rates and risk premium.
Link the topic to risk premium and convexity, which are the deviations from this benchmark.

Practice questions from Expectations, Risk Premium, Convexity and the Shape of the Term Structure

Expectations Hypothesis and Forward Rates: frequently asked questions

What is the expectations hypothesis in the term structure?

It says the forward rate equals the expected future spot rate. Long rates are then an average of expected short rates. The pure version assumes no risk premium.

How do I calculate a forward rate from spot rates?

Equate the growth of the long bond to the growth of the short bond times the forward period. For a 1-year forward one year ahead, f = (1 + z₂)² ÷ (1 + z₁) − 1. For longer periods, take the root of the forward length.

What is the difference between the expectations hypothesis and liquidity preference theory?

The expectations hypothesis sets the forward equal to the expected spot rate. Liquidity preference adds a positive term premium for holding longer bonds. So forwards sit above expected spots under liquidity preference.

Is a forward rate a good forecast of future rates?

It is the benchmark forecast under the pure hypothesis. In practice forwards tend to include a risk premium, so they can overstate later realised short rates on average. Do not treat them as exact predictions.