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

Risk Premium and Term Premium in Bond Returns

Updated 11 October 2026 · Fact-checked

The risk premium is the extra expected return investors demand for holding a risky bond instead of earning the short rate. The term premium is that compensation in long maturities. Yield = average expected short rate + term premium (+ convexity effect). To solve questions, split a yield into these parts.

Understand Risk Premium and Term Premium in Bond Returns

Start with a simple choice. You can hold a 10-year bond or roll a 1-day deposit for 10 years. The deposit has no price risk. The bond's price falls if rates rise. Most investors dislike that risk, so they ask for extra expected return to hold the bond. That extra expected return is the risk premium.

The term premium is the risk premium built into a long-term yield. Think of a long yield as two pieces. The first is the average of the short rates you expect over the bond's life. The second is the term premium. A third, usually small piece is the convexity effect. It pulls long yields down because convexity helps the bond holder and rises with maturity and volatility.

The key distinction: the expected short rate is a forecast of policy and money-market rates. The term premium is payment for bearing rate risk. Two yield curves can look identical but mean different things. One may reflect expected rate rises. The other may reflect a high term premium with flat expected rates.

In a model with a risk premium, the price of risk is often written as λ. Expected excess return on a bond is roughly λ × (the bond's volatility). Longer bonds have higher volatility, so they earn higher expected excess returns. This is why the curve usually slopes up on average. In the usual setup, the risk-neutral drift of the short rate equals the real-world drift plus λ × σ. With a positive λ, risk-neutral rates are higher, bond prices are lower and the term premium is positive. Sign conventions for λ differ across texts, so follow the convention stated in the question. Pricing uses the risk-neutral drift. Forecasting uses the real-world drift.

The term premium is not directly observed. It is estimated from models, survey forecasts of short rates, or regressions. It can be negative, for example when long bonds hedge equity risk or when demand for safe assets is strong. Treat any estimate as model-dependent.

Key formulas to remember

Yield decomposition: Long yield = average expected short rate over the bond's life + term premium (− convexity effect); The convexity effect is small at short maturities and grows with maturity and volatility.
Term premium: Term premium = observed long yield − average expected short rate (ignoring convexity); Not observable directly. It depends on the expected-rate forecast you use.
Expected excess return: Expected excess return = expected bond return − short rate ≈ λ × σ (per unit of duration-scaled volatility); λ is the price of risk. Higher volatility or higher λ means a higher premium.
Real-world vs risk-neutral drift: Risk-neutral drift = real-world drift + λ × σ (usual setup); With a positive λ, the risk-neutral drift is higher. Some texts define λ with the opposite sign, so the formula holds only under the convention stated. Pricing uses the risk-neutral drift.
One-period forward rate link: Forward rate = expected future short rate + forward (term) premium; If the premium is zero, the expectations hypothesis holds. A positive premium means forwards overstate expected rates.

How to solve Risk Premium and Term Premium in Bond Returns questions

Use this method for any question on risk premium, term premium or curve shape.

1Identify what is given: observed yields or forward rates, forecasts of short rates, or model parameters such as λ and σ.
2Decide what is asked: the term premium, the expected short rate, the excess return, or the curve shape.
3Write the decomposition: long yield = average expected short rate + term premium (− convexity if it is given).
4Compute the average expected short rate from the path of forecasts. Use a simple average unless the question says otherwise.
5Solve for the missing piece. Subtract to find the premium, or subtract the premium to find the expected rate.
6For model questions, apply risk-neutral drift = real-world drift + λ × σ in the usual setup, and keep the sign convention stated in the question.
7Interpret: a positive premium means forwards sit above expected rates. Say what this implies for the curve slope and for holding-period returns.

Quickest way: Subtract and compare

When to use it: Use when the question gives a yield and a rate forecast and asks for the premium or the implied expectation.

Average the expected short rates over the bond's life.
Subtract this average from the observed yield. The gap is the term premium (plus any convexity adjustment).
Check the sign. Yield above expectations means a positive premium.
Scan the options. Discard any with the wrong sign or an obviously wrong size.

Common mistakes in Risk Premium and Term Premium in Bond Returns

Treating the forward rate as the market's expected future short rate.
The expectations hypothesis is taught first and feels like the default.
Fix: Forward = expected rate + premium. Equate them only if the question assumes a zero premium.
Calling any upward-sloping curve a forecast of rising rates.
Slope looks like a signal about the future.
Fix: A slope can come from expected rises, a positive term premium, or both. Separate them before concluding.
Adding the convexity effect instead of subtracting it.
Students mix up the effect on yield and the effect on price.
Fix: Convexity raises bond prices, which lowers yields. It reduces the long yield relative to the expected rate.
Getting the sign of λ wrong in the drift adjustment.
Textbooks differ in conventions.
Fix: Use the convention in the question. In the usual setup with a positive price of risk, risk-neutral drift is above real-world drift.
Assuming the term premium is always positive and directly observed.
Averages over history are mistaken for rules.
Fix: State that premia are model estimates and can be negative at times.

Worked examples

Example 1

The 3-year zero-coupon yield is 4.60%. A survey implies expected one-year rates of 4.0%, 4.2% and 4.4% for years 1, 2 and 3. Ignoring convexity, what is the 3-year term premium?

Show the solution

Average expected short rate = (4.0 + 4.2 + 4.4) ÷ 3 = 12.6 ÷ 3 = 4.20%.
Term premium = 4.60% − 4.20% = 0.40%.

Answer: The term premium is 0.40%, or 40 basis points.

Example 2

In a model, the real-world drift of the short rate is 0.50% per year, volatility σ is 1.00% per year and the price of risk λ is 0.30. Using risk-neutral drift = real-world drift + λ × σ, what is the risk-neutral drift, and how large is the risk premium adjustment?

Show the solution

λ × σ = 0.30 × 1.00% = 0.30%.
Risk-neutral drift = 0.50% + 0.30% = 0.80% per year.
The risk-neutral drift is higher than the real-world drift, so the risk-neutral rate path is higher than the expected path.
The sign of the adjustment depends on the convention used for λ. The difference λ × σ = 0.30% measures the risk premium adjustment.

Answer: The risk-neutral drift is 0.80% per year. It sits 0.30% above the real-world drift, which is the risk premium adjustment.

Exam tips

Always ask whether the question wants real-world or risk-neutral values. The two differ by the risk premium.
Look for the phrase 'expectations hypothesis'. It means the term premium is zero and forwards equal expected rates.
In decomposition questions, do the subtraction first, then check the sign against the story in the stem.
When asked about curve shape, name all drivers: expected rates, term premium and convexity.
Treat estimated premia as model-dependent. Options that call them certain or fixed are usually wrong.

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

Risk Premium and Term Premium in Bond Returns: frequently asked questions

What is the difference between term premium and expected short rate?

The expected short rate is the forecast of future short-term rates averaged over the bond's life. The term premium is the extra yield investors require for holding a long bond instead of rolling short bonds. A long yield is the sum of the two, less a small convexity effect.

How does the risk premium affect the yield curve shape?

A positive premium that grows with maturity pushes long yields above expected short rates. This tends to make the curve slope upward even when rates are expected to stay flat. A falling or negative premium can flatten or invert the curve.

Can the term premium be negative?

Yes. If investors value long bonds as a hedge or demand safe assets strongly, they may accept a lower expected return on long bonds. Premium estimates are model-based, so the sign can change over time.

Is the term premium observable?

No. You cannot observe it directly. Analysts estimate it from term structure models, survey forecasts of short rates, or regressions, so estimates differ across methods.