FRM Exam Part II · The Art of Term Structure Models: Drift

Model 2: Constant Drift and Risk-Neutral Drift Explained

Updated 11 October 2026 · Fact-checked

Model 2 adds a constant drift λ to the normal rate process: dr = λdt + σdw. Under risk-neutral pricing, λ is the drift after adjusting for the risk premium. A positive λ lifts spot rates, but the volatility term subtracts a convexity effect that grows with maturity, so the spot curve first rises, then bends down.

Understand Model 2: Constant Drift and Risk-Neutral Drift

Model 1 assumes rates move randomly with zero drift: dr = σdw. Model 2 adds a constant drift: dr = λdt + σdw. Here r is the short rate, λ is the drift per year, σ is the annual volatility in rate units, and dw is a normal shock with mean zero.

The drift you use for pricing is the risk-neutral drift. It is not the same as the real-world drift. Investors in long bonds want compensation for rate risk. So the risk-neutral drift equals the real-world expected drift plus a risk premium term. In the text's framing, λ = real-world drift + risk premium component. Pricing uses the risk-neutral one, because it makes the model match market prices of bonds.

The drift moves the whole term structure. The spot rate for maturity T in Model 2 is approximately the starting short rate plus the drift effect, less a volatility effect: Ẑ(T) = r0 + λT/2 − σ²T²/6. The drift term grows linearly in maturity. The volatility term is a convexity effect that grows with the square of maturity and pulls long rates down.

The result is a curve shaped by two forces. For short maturities the drift dominates, so the curve slopes up when λ is positive. For long maturities the −σ²T²/6 term dominates and the curve bends down. A positive drift alone cannot keep the curve rising forever.

Keep two ideas apart. Drift is an expectation of where rates go. The risk premium is the extra return investors demand for bearing rate risk. The model bundles both into one risk-neutral number, so you cannot separate them from prices alone.

Key formulas to remember

Model 2 rate process: dr = λdt + σdw; λ is the constant risk-neutral drift per year; σ is annual volatility in rate units; dw is a normal shock.
Expected short rate: E[r(t)] = r0 + λt; Rates drift linearly. The rate distribution is normal with standard deviation σ√t.
Spot rate (continuous compounding): Ẑ(T) = r0 + λT/2 − σ²T²/6; Drift term is linear in T, convexity term is quadratic. Use this to describe curve shape.
Zero-coupon bond price: Z(T) = exp(−r0T − λT²/2 + σ²T³/6); Ẑ(T) = −ln Z(T) ÷ T gives the spot rate formula above.
Risk-neutral drift: λ = real-world drift + risk premium component; Pricing uses the risk-neutral drift. Real-world drift is for forecasting.
Spot rate with zero drift (Model 1): Ẑ(T) = r0 − σ²T²/6; Setting λ = 0 gives Model 1, a downward-bending curve.

How to solve Model 2: Constant Drift and Risk-Neutral Drift questions

Use this order for any Model 2 question, whether it asks for a rate, a price or the curve shape.

1Write down the inputs: r0, λ, σ and maturity T, all in annual units. Convert basis points to decimals (80 bp = 0.0080).
2Identify what is asked: expected rate, spot rate, bond price or interpretation of drift versus risk premium.
3For expected short rate, use r0 + λt. For rate dispersion use σ√t.
4For a spot rate, compute the drift term λT/2 and the convexity term σ²T²/6 separately.
5Combine: Ẑ(T) = r0 + drift term − convexity term. Check the sign of each term.
6For a bond price, compute the exponent r0T + λT²/2 − σ²T³/6 and take exp of its negative. Or discount at the spot rate.
7Interpret: say whether drift or convexity dominates at this maturity, and note that λ here is risk-neutral, including the risk premium.

Quickest way: Drift minus convexity shortcut

When to use it: Multiple-choice questions asking for a spot rate or the curve direction at a given maturity.

Compute λT/2 first. This is the upward push.
Compute σ²T²/6. This is the downward pull.
Spot rate = r0 + push − pull. Stop there.
To compare answer options, check the sign and rough size of the net adjustment before doing exact arithmetic.
If a question asks which force dominates, compare λ/2 with σ²T/6. Drift wins when T < 3λ/σ².

Common mistakes in Model 2: Constant Drift and Risk-Neutral Drift

Treating λ as the real-world expected drift in pricing questions
The word drift sounds like a forecast.
Fix: Say it out loud: pricing uses risk-neutral drift, which includes the risk premium. Real-world drift is a different number.
Using σ in basis points without converting
Volatility is quoted as, say, 100 bp per year.
Fix: Convert to decimals first: 100 bp = 0.01. Then σ² = 0.0001.
Dividing by the wrong number in the spot formula
Drift term is T/2 and convexity term is T²/6, which are easy to mix up.
Fix: Memorize Ẑ(T) = r0 + λT/2 − σ²T²/6. Check with T small: the curve should start at r0.
Assuming positive drift always gives an upward-sloping curve
Students ignore the convexity term.
Fix: Because the convexity term grows with T², long maturities eventually bend down for any σ > 0.
Confusing the rate's expected value with the spot rate
Both rise with drift.
Fix: E[r(t)] = r0 + λt is the expected short rate at time t. The spot rate is an average over the life, so its drift term is λT/2, and it also has the convexity adjustment.
Thinking the risk premium can be observed separately from the drift
The text splits λ into two parts.
Fix: Market prices only reveal the sum. Without a real-world forecast, you cannot isolate the premium.

Worked examples

Example 1

In Model 2, r0 = 4.00%, λ = 0.60% per year and σ = 1.00% per year. Compute the 5-year spot rate, continuously compounded.

Show the solution

Drift term: λT/2 = 0.0060 × 5 ÷ 2 = 0.0150.
Convexity term: σ²T²/6 = 0.0001 × 25 ÷ 6 = 0.000417.
Spot rate = 0.0400 + 0.0150 − 0.000417 = 0.054583.

Answer: About 5.46%. Drift lifts the spot rate by 1.50%, and convexity lowers it by about 0.04%.

Example 2

A risk manager has r0 = 3.00%, σ = 1.20% per year and λ = 0.30% per year. Using Model 2, find the expected short rate in 4 years and the standard deviation of that rate. Then say whether the risk-neutral drift is the right input for forecasting.

Show the solution

Expected rate: r0 + λt = 0.0300 + 0.0030 × 4 = 0.0420.
Standard deviation: σ√t = 0.0120 × √4 = 0.0240.
The drift λ is risk-neutral, so it includes a risk premium component.

Answer: Expected short rate is 4.20% with a standard deviation of 2.40%. The risk-neutral drift is for pricing; it overstates or misstates the real-world forecast by the risk premium component, so it is not a clean forecast.

Exam tips

Expect questions that give λ and σ and ask for a spot rate or curve shape at a stated maturity. Do the two terms separately.
Watch the label: risk-neutral drift versus real-world drift. Many wrong options swap them.
Know that Model 2 reduces to Model 1 when λ = 0, and that the curve then only bends down.
If the question mentions a risk premium, expect the answer to say the drift used in pricing exceeds the real-world drift when investors require compensation.
Check units. Basis points, percent and decimals are the most common source of wrong options.

Practice questions from The Art of Term Structure Models: Drift

Model 2: Constant Drift and Risk-Neutral Drift: frequently asked questions

What is the difference between risk-neutral drift and real-world drift?

Real-world drift is the expected change in rates under actual probabilities. Risk-neutral drift is the drift used for pricing, and it includes a risk premium component. They differ because investors demand compensation for rate risk.

How does drift affect the spot rate curve in Model 2?

A positive drift raises spot rates by λT/2, so the curve slopes up at short maturities. The convexity term σ²T²/6 reduces spot rates more at long maturities. The net shape rises, then bends down.

Why does Model 2 still give a downward bend at long maturities?

Convexity grows with the square of maturity, while the drift term grows only linearly. Eventually convexity wins, as long as σ is positive.

Is Model 2 a no-arbitrage model?

Its single constant drift cannot fit an arbitrary observed curve. It is a stepping stone toward models with time-dependent drift that can match market prices exactly.