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

Drivers of the Shape of the Term Structure: Expectations, Risk Premium and Convexity

Updated 11 October 2026 · Fact-checked

The shape of the term structure comes from three drivers: expected future short rates, a risk (term) premium, and a convexity effect. Spot rate ≈ average expected short rate + risk premium − convexity. To solve questions, compute each component by maturity, then see which dominates. Convexity grows with maturity and pulls long rates down.

Understand Drivers of the Shape of the Term Structure

A spot rate for maturity T is not one number with one cause. It is the sum of three pieces. The first is the expectations piece: the average of the short rates the market expects over the life of the bond. If rates are expected to rise, this piece slopes upward. If they are expected to fall, it slopes downward.

The second is the risk premium (term premium). Long bonds have more price risk, so investors usually want extra yield to hold them. This piece is typically positive and grows with maturity. It can be small or negative when investors value long bonds as a hedge.

The third is convexity, from Jensen's inequality. Bond prices are convex in yields, so uncertainty about future rates raises the expected bond price. A higher price means a lower yield. This effect lowers the spot rate. In the normal-model formula used below, it grows with the square of maturity times volatility squared, so it is small at short maturities and larger at long ones. That scaling is specific to that model, not a universal law.

The shape comes from how these pieces add up. Upward slope: rising expected rates and a positive premium, with convexity small. Flat: the pieces offset. Inverted: expected rate cuts, or, in a model with high enough volatility, convexity at long maturities large enough to outweigh the premium. Humped: the premium and expectations lift the curve at short to medium maturities, then, in a model like the normal model, convexity grows faster and can pull the long end down.

In the Ho-Lee type normal model with zero drift, the convexity effect on the spot rate is σ²T²/6. The zero-coupon yield is r₀ − σ²T²/6. Always use the formula given in the question.

Key formulas to remember

Spot rate decomposition: Spot rate(T) = Expectations(T) + Risk premium(T) − Convexity(T); Convexity is entered as a positive amount that is subtracted. Check the sign convention in the question.
Expectations component: Expectations(T) = (1 ÷ T) × ∫ E[r(s)] ds, or simple average of expected short rates; With discrete periods, use the simple average of expected one-period rates as an approximation.
Convexity in the zero-drift normal model: Convexity(T) = σ² × T² ÷ 6; This is the spot-rate effect when short-rate volatility is σ (annual, in decimals) and drift is zero under the risk-neutral measure. It grows with T² and σ².
Spot rate in the normal model with constant risk-neutral drift λ: r(T) = r₀ + λ × T ÷ 2 − σ² × T² ÷ 6; Here λ is the constant risk-neutral drift of the short rate per year. It is not a risk premium, and no separate premium term is added in this formula. Use only if the question defines it.
Jensen's inequality for bond prices: E[P(r)] ≥ P(E[r]) for convex P; Because price is convex in yield, uncertainty raises expected price and lowers the yield.

How to solve Drivers of the Shape of the Term Structure questions

Use this approach for any question that asks you to explain or decompose a curve shape.

1Write down the spot rate or forward rate you are asked about and its maturity T.
2Find the expected short rates and average them over 0 to T. This is the expectations piece.
3Find the risk premium piece. Use the stated premium or take the difference between the observed spot rate and the expectations piece, depending on what is given.
4Compute convexity with the formula supplied, usually σ² T² ÷ 6 in the normal model. Keep σ in decimals.
5Combine: spot = expectations + premium − convexity. If one piece is unknown, solve for it.
6Compare maturities. Convexity grows fastest with T, so check whether it overtakes the other pieces at long maturities.
7State the shape (upward, flat, humped, inverted) and name which driver causes it.

Quickest way: Rank the drivers by maturity

When to use it: When the question asks which driver explains a shape and you do not need exact numbers. The ranking below assumes a normal-model setting where convexity scales with T².

At short maturities, expectations dominate. Convexity is almost zero.
At medium maturities, the risk premium adds and lifts the curve.
At long maturities, convexity grows with T² in the normal model and can pull the curve down.
Downward slope with no expected rate cuts points to convexity or a negative premium.
A hump means the premium and expectations win early, then convexity wins late.

Common mistakes in Drivers of the Shape of the Term Structure

Adding convexity instead of subtracting it
Convexity sounds like a benefit, so students treat it as extra yield.
Fix: Convexity raises bond prices, so it lowers yields. Subtract it from the spot rate.
Assuming convexity is large at short maturities
Students forget that it scales with T² in the normal-model formula.
Fix: Compute it for each T. In the σ²T²÷6 formula, doubling maturity quadruples the effect.
Using σ in percent in the formula
Volatility is quoted as, say, 1%, and students enter 1.
Fix: Convert to decimals first: 1% = 0.01, so σ² = 0.0001.
Treating an inverted curve as always signalling expected rate cuts
The pure expectations story is the first one taught.
Fix: Check whether a large convexity term or a low premium could explain the slope before concluding.
Confusing the risk premium with the forward-minus-spot gap
Both involve a difference between rates.
Fix: The forward rate minus the expected future spot rate equals the risk premium minus the convexity effect. The forward-minus-spot gap is a different quantity: it mainly reflects the slope of the curve. Do not read it as the premium.

Worked examples

Example 1

In a zero-drift normal model, the current short rate is 4.00% and annual short-rate volatility is 1.20%. The investor requires no risk premium. Find the 10-year spot rate and the convexity effect.

Show the solution

Convert: σ = 0.012, so σ² = 0.000144.
Convexity = σ² × T² ÷ 6 = 0.000144 × 100 ÷ 6 = 0.0144 ÷ 6 = 0.0024, or 0.24%.
Expectations = 4.00% (zero drift, no premium).
Spot = 4.00% + 0 − 0.24% = 3.76%.

Answer: The 10-year spot rate is 3.76%, and convexity lowers it by 0.24%.

Example 2

Expected average short rates are 3.00% for a 5-year bond and 3.50% for a 20-year bond. Risk premiums are 0.40% and 1.10%. Short-rate volatility is 1.00% a year. Using convexity = σ² T² ÷ 6, compute both spot rates and describe the shape between 5 and 20 years.

Show the solution

σ² = 0.01² = 0.0001.
5-year convexity = 0.0001 × 25 ÷ 6 = 0.0025 ÷ 6 = 0.000417, about 0.042%.
5-year spot = 3.00% + 0.40% − 0.042% = 3.358%, about 3.36%.
20-year convexity = 0.0001 × 400 ÷ 6 = 0.04 ÷ 6 = 0.00667, about 0.667%.
20-year spot = 3.50% + 1.10% − 0.667% = 3.933%, about 3.93%.
The curve rises from 3.36% to 3.93%, so it is upward sloping over this range. Expectations plus premium rise by 1.20% (from 3.40% to 4.60%), while convexity rises by only about 0.63%, so convexity damps the rise but does not overtake the other pieces. There is no hump in this example.

Answer: 5-year spot is about 3.36% and 20-year spot is about 3.93%. The curve is upward sloping. Convexity damps the long end but does not create a hump here.

Exam tips

Write the three components in a column for each maturity before you look at the options.
Check units. Volatility given in percent or basis points must become decimals in σ².
Remember convexity scales with T², so a long-end fall is a convexity clue.
If a question asks what happens when volatility doubles, convexity rises four times.
Read the model definition in the stem. Some give the formula with a different divisor or sign.

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

Drivers of the Shape of the Term Structure: frequently asked questions

Why can the yield curve be humped?

Expectations and the risk premium lift yields at short to medium maturities. Convexity grows with T², so at long maturities it pulls yields back down. The net effect is a peak in the middle.

Does an inverted curve always mean the market expects rate cuts?

No. A low or negative risk premium or a large convexity effect can also lower long rates. Decompose the curve before you read it as a forecast.

Is convexity a risk premium?

No. The risk premium is compensation for bearing interest rate risk. Convexity is a mathematical effect from Jensen's inequality and lowers yields when volatility is positive.

How do I decompose a spot rate in the exam?

Average the expected short rates, add the stated risk premium, and subtract the convexity term from the given formula. If one part is missing, solve for it from the observed spot rate.