Portfolio Management Pathway · Yield Curve Strategies

Term Structure Models and Yield Curve Views Explained

Q: What is the Nelson-Siegel model in CFA Level III?

It is an empirical model that fits the yield curve with a few parameters. These represent level, slope and curvature. It is used to describe and track curve shape, not to model equilibrium rates.

Q: What is the difference between Vasicek and CIR?

Both include mean reversion of the short rate. Vasicek has constant volatility, so rates can be negative. CIR makes volatility depend on the square root of the rate, which keeps rates non-negative under standard conditions.

Q: Are forward rates the expected future spot rates?

Only under the pure expectations theory. Other theories add a risk or liquidity premium, so a forward may differ from the expected spot rate. This is why active managers look for gaps between their views and the forwards.

Q: What is an active yield curve view?

It is a view that future spot rates will differ from those implied by forward rates. You position duration or curve exposure to profit if you are right. A passive view assumes the forwards or an unchanged curve.

Updated 9 October 2026 · Fact-checked

A term structure model describes how yields vary by maturity, and how rates may move. You compare the view implied by forward rates with your own view of future spot rates. If you expect rates to differ from the forwards, you position the portfolio to profit from that gap.

Understand Term Structure Models and Yield Curve Views

A yield curve shows yields for different maturities. A forward rate is the rate for a future period that is implied by today's spot curve. It is the break-even rate: if the future spot rate turns out equal to the forward, a long bond and a rolled-over short bond earn the same return.

The expectations view says forward rates are the market's expected future spot rates. Other theories add a risk premium (liquidity preference) or say investors prefer certain maturities (preferred habitat and market segmentation). Forwards may include a term premium, so they need not be unbiased forecasts of future spot rates. Active managers therefore look for gaps between their own views and the forwards.

The active versus passive distinction is about what you assume. A passive view assumes the curve stays unchanged, or that spot rates are realised as the forwards imply. An active view says future spot rates will differ from the forwards. You earn excess return only if your view is right and differs from what is already priced in.

Term structure models give a disciplined way to describe the curve. Nelson-Siegel is a curve-fitting (empirical) model. It describes the whole curve with a few parameters: three components (level, slope and curvature), governed by a decay parameter λ. The Svensson extension adds a second curvature term. It fits observed yields well and lets you track how the components move. It is not built on a no-arbitrage or equilibrium argument.

Vasicek and Cox-Ingersoll-Ross (CIR) are affine equilibrium models of the short rate. Affine means they give a closed-form bond price. Both have mean reversion: the short rate is pulled toward a long-run level. In Vasicek, the random shock does not depend on the rate level, so rates can become negative. In CIR, volatility rises with the square root of the rate, and with the usual parameter conditions rates stay non-negative. Arbitrage-free models (such as Ho-Lee) are instead calibrated to fit today's curve exactly.

Key rules to remember

Forward rate from spot rates: (1 + z_B)^B = (1 + z_A)^A × (1 + f(A,B−A))^(B−A); z is the spot rate, A and B are maturities in years. f(A,B−A) is the forward rate starting at A for B−A years. Solve for f.
Expectations view: Forward rate = expected future spot rate (pure expectations); With a risk premium, forward = expected spot + premium. Do not treat the forward as a pure forecast.
Active view decision rule: Expected future spot rate < forward rate → the longer bond outperforms rolling shorter bonds; expected spot rate > forward → the longer bond underperforms rolling shorter bonds; Compare your expected spot with the forward, not with today's spot. Example: if the one-year spot in one year is 4.5% < 5.01%, rolling one-year bonds earns less than the two-year bond.
Vasicek model: dr = a(b − r)dt + σ dz; a is speed of mean reversion, b the long-run mean, σ constant volatility. Rates can go negative. Affine, with a closed-form bond price.
CIR model: dr = a(b − r)dt + σ√r dz; Volatility rises with the rate level. Rates stay non-negative under standard conditions. Affine, with a closed-form bond price.
Nelson-Siegel components: Yield curve = level + slope + curvature components, with decay parameter λ; Three components (level, slope, curvature), governed by a decay parameter λ, fit the shape. The Svensson extension adds a second curvature term.

How to solve Term Structure Models and Yield Curve Views questions

Use this order for any question that asks you to form, test or justify a yield curve view.

1Identify the task: compute a forward rate, compare a view with forwards, or choose and justify a model.
2Write down the spot rates and maturities given. Keep maturities in years.
3Compute the forward rate for the period in question using the compounding formula. Show the working.
4State the market-implied view: the forward is the break-even spot rate for that period.
5Compare your expected future spot rate with the forward. Decide whether the bond or longer maturity will outperform or underperform.
6Link the decision to the client's objectives and constraints, such as duration limits or risk tolerance.
7For model questions, match the feature to the model: curve fitting means Nelson-Siegel, mean reversion with constant volatility means Vasicek, rate-dependent volatility means CIR.
8Write the conclusion in one sentence that uses the command word, such as justify or recommend.

Quickest way: Forward versus view shortcut

When to use it: Use it when you are given spot rates and an expected future spot rate and must pick a position.

Compute the forward with (1+z_B)^B ÷ (1+z_A)^A, then take the root and subtract 1.
Compare your expected spot with the forward.
Expect rates lower than forward: lengthen duration or buy the longer bond.
Expect rates higher than forward: shorten duration or favour the shorter bond.
Match model questions by one keyword: fits curve, negative rates possible, or volatility depends on the rate.

Common mistakes in Term Structure Models and Yield Curve Views

Comparing the expected spot rate with today's spot rate instead of the forward rate.
Students think a rise from today's rate means a loss.
Fix: The forward is already priced in. You gain or lose only against the forward.
Treating forward rates as unbiased forecasts.
The pure expectations theory is taught first.
Fix: Forwards can include a risk premium. Say so when asked about forecasting ability.
Forgetting to raise to the power of the maturity when computing forwards.
Rushing and mixing simple and compounded rates.
Fix: Always write (1+z)^n for each leg, then take the root of the forward period.
Saying Vasicek keeps rates non-negative.
Mixing up Vasicek and CIR.
Fix: Vasicek has constant volatility and can give negative rates. CIR volatility scales with √r and avoids this.
Calling Nelson-Siegel an equilibrium or arbitrage-free model.
All three names are grouped as term structure models.
Fix: Nelson-Siegel is an empirical curve-fitting model with level, slope and curvature factors.

Worked examples

Example 1

The one-year spot rate is 3.0% and the two-year spot rate is 4.0%, both annual compounding. (a) Calculate the one-year forward rate starting in one year. (b) You expect the one-year spot rate in one year to be 4.5%. Should you favour the two-year bond or rolling one-year bonds?

Show the solution

(a) (1.04)^2 = 1.0816.
Divide by (1.03)^1 = 1.03: 1.0816 ÷ 1.03 = 1.05010.
Forward rate = 1.05010 − 1 = 5.01%.
(b) Your expected one-year spot rate in one year is 4.5%, which is below the forward of 5.01%.
Two-year bond growth over two years: (1.04)^2 = 1.0816.
Rolling growth: buy a one-year bond at 3.0%, then reinvest at the expected 4.5%: 1.03 × 1.045 = 1.07635, about 3.75% a year.
1.0816 is greater than 1.07635, so the two-year bond is expected to outperform rolling one-year bonds.

Answer: (a) Forward rate ≈ 5.01%. (b) Favour the two-year bond, because the expected spot rate (4.5%) is below the forward (5.01%). Two-year growth of 1.0816 beats rolling growth of 1.07635.

Example 2

A portfolio manager needs a model to describe the observed yield curve using level, slope and curvature, and a separate model with mean reversion in which volatility rises with the rate level. Name the model for each need and give one reason for each.

Show the solution

Level, slope and curvature factors describe the shape of the observed curve: this is Nelson-Siegel.
Nelson-Siegel is an empirical curve-fitting model with a small number of parameters.
Mean reversion with volatility that rises with the rate level is the Cox-Ingersoll-Ross model.
CIR volatility is proportional to √r, so volatility rises with the rate level and rates stay non-negative under standard conditions.

Answer: Nelson-Siegel fits the curve shape with level, slope and curvature factors. CIR has mean reversion and rate-dependent volatility (σ√r), unlike Vasicek, which has constant volatility.

Exam tips

Show the forward rate calculation fully. A correct number on its own earns credit, but showing the working protects you if you slip.
Read the command word. 'Calculate' needs a number, 'justify' needs a reason tied to the client.
Always compare the view to the forward, not to today's spot rate.
Memorise one contrast line for each model: Nelson-Siegel fits, Vasicek constant volatility, CIR √r volatility.
Item set choices often test whether you know forwards may include a risk premium.

Term Structure Models and Yield Curve Views in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Term Structure Models and Yield Curve Views: frequently asked questions

What is the Nelson-Siegel model in CFA Level III?

It is an empirical model that fits the yield curve with a few parameters. These represent level, slope and curvature. It is used to describe and track curve shape, not to model equilibrium rates.

What is the difference between Vasicek and CIR?

Both include mean reversion of the short rate. Vasicek has constant volatility, so rates can be negative. CIR makes volatility depend on the square root of the rate, which keeps rates non-negative under standard conditions.

Are forward rates the expected future spot rates?

Only under the pure expectations theory. Other theories add a risk or liquidity premium, so a forward may differ from the expected spot rate. This is why active managers look for gaps between their views and the forwards.

What is an active yield curve view?

It is a view that future spot rates will differ from those implied by forward rates. You position duration or curve exposure to profit if you are right. A passive view assumes the forwards or an unchanged curve.