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CFA Level II · CFA Level II Exam

The Term Structure and Interest Rate Dynamics for CFA Level II

The term structure describes how yields change with maturity. You learn to move between spot rates, forward rates and par rates, explain curve shapes with theories and models, measure curve risk with factor models, and value bonds with a binomial interest rate tree. In the exam, you find the rates in the vignette and apply the model step by step.

What this chapter covers

This chapter is about the yield curve and how it moves. You start with spot rates (the yield on a zero-coupon bond for one maturity) and forward rates (the rate agreed today for a loan starting later). You learn to convert between them, and to build spot curves from par yields. You then see what forward rates imply for returns, including riding the yield curve, and how the swap rate curve and spreads act as a benchmark.

The middle of the chapter asks why curves have the shapes they do. Traditional theories (pure expectations, liquidity preference, segmented markets, preferred habitat) give the intuition. Modern models (Cox-Ingersoll-Ross, Vasicek, and the Ho-Lee and Kalotay-Williams-Fabozzi type arbitrage-free models) give the mathematics. Factor models, such as level, steepness and curvature, and key rate duration, show how portfolios react when the curve moves in different ways.

The chapter ends with arbitrage-free valuation using a binomial interest rate tree. This links directly to the Fixed Income readings on bonds with embedded options and credit analysis. It also supports Derivatives (forward pricing and swaps) and Portfolio Construction (liability-driven investing and duration matching). Every question comes from an item set vignette, so you must pick the right rates and dates from the exhibit, then apply the model.

Fixed Income carries 10-15% of the exam, and this chapter is its foundation. Spot and forward rate calculations, tree valuation and curve-risk ideas are reused in later Fixed Income chapters and in Derivatives. The questions are often calculation-led and follow a repeatable method, so careful practice converts directly into points. Weak understanding here also costs you in related item sets, since the same rate logic runs through them.

The Term Structure and Interest Rate Dynamics: topics in the order to study them

  1. 1Spot Rates and Forward RatesEverything else uses these two rates, so master discounting with spot rates and the forward rate formulas first.
  2. 2Yield to Maturity and Par, Spot and Forward CurvesIt builds on spot and forward rates and shows how the three curves relate and how you bootstrap spot rates from par yields.
  3. 3Return of Forward Contracts and Riding the Yield CurveIt applies forward rates to returns and needs the curve concepts you just learned.
  4. 4Swap Rate Curve and SpreadsIt introduces the swap curve as a benchmark and measures spreads against it, so it comes once curve basics are solid.
  5. 5Traditional Theories of the Term StructureThese are the conceptual explanations of curve shape, and they are easier once you know forward rates.
  6. 6Modern Term Structure ModelsThese add equilibrium and arbitrage-free models to the theories, so study them after the intuition is clear.
  7. 7Yield Curve Factor Models and DurationIt uses the idea of curve movements to measure risk, and it makes sense after you know how curves are modelled.
  8. 8Arbitrage-Free Valuation and Binomial Interest Rate TreeIt is the most computational topic and pulls together spot rates, forward rates and modelling, so finish with it.

How to prepare The Term Structure and Interest Rate Dynamics

Treat this chapter as a method chapter. Most marks come from a few calculations done accurately, plus clear concept answers.

  1. Learn the core formulas first: discounting with spot rates, (1 + z_B)^B = (1 + z_A)^A × (1 + IFR_A,B−A)^(B−A), and the link between par rate and discount factors.
  2. Practise bootstrapping spot rates from par yields until you can do it without notes. Always keep the time periods and compounding frequency visible.
  3. Work vignette-style questions. Underline the maturity, the rate type (spot, forward, par) and the compounding before you calculate.
  4. Build a one-page comparison of the theories and models: what each says, what drives the curve shape, and whether it is arbitrage-free.
  5. For curve risk, learn what level, steepness and curvature mean, and practise reading a key rate duration table to find where a portfolio is exposed.
  6. Practise the binomial tree by working backwards from maturity. At each node, use V = [0.5 × (V_up + C) + 0.5 × (V_down + C)] ÷ (1 + f). Here V_up and V_down are the values at the next date (ex-coupon), C is the coupon paid at that next date, and f is the forward rate at the node. At maturity the ex-coupon value is par, so the cash flow at the final date is par plus the final coupon.
  7. Finish with mixed item sets under timed conditions, then review every error by cause: wrong rate, wrong period or arithmetic.

Common mistakes in The Term Structure and Interest Rate Dynamics

  • Mixing up spot, forward and par rates when reading an exhibit

    Fix: Before calculating, label each number with its type, maturity and compounding. Write the label next to the figure.

  • Using the wrong time periods in the forward rate formula

    Fix: Draw a short timeline. Mark the start and end of the forward period and use its length as the exponent for the forward rate.

  • Forgetting to add the coupon at each node in the binomial tree

    Fix: At every node, use V = [0.5 × (V_up + C) + 0.5 × (V_down + C)] ÷ (1 + f), where V_up and V_down are the next-date values (ex-coupon) and C is the coupon paid at that date. At the final date the ex-coupon value is par, so the cash flow is par plus the final coupon.

  • Claiming riding the yield curve always adds return

    Fix: State the condition: it adds return when the curve is upward-sloping and does not shift enough to offset the roll-down. A rise in yields can wipe out the gain.

  • Confusing equilibrium and arbitrage-free models

    Fix: Remember that equilibrium models (Vasicek, CIR) derive rates from assumed economic factors and may not match market prices. Arbitrage-free models start from observed prices and fit them.

  • Treating all curve moves as parallel shifts

    Fix: Use key rate duration or the level, steepness and curvature factors whenever the question describes a twist or a change in shape.

Last-day revision: The Term Structure and Interest Rate Dynamics

  • A spot rate is the yield on a zero-coupon bond for a given maturity; a forward rate is a rate for a loan starting at a future date.
  • Forward rate link: (1 + z_B)^B = (1 + z_A)^A × (1 + IFR_A,B−A)^(B−A).
  • Par rates come from a curve where each bond is priced at par; bootstrap spot rates from them one maturity at a time.
  • When the forward rate covering the period from A to B exceeds the A-period spot rate, the B-period spot rate is higher than the A-period spot rate, so the spot curve is upward-sloping between A and B.
  • Riding the yield curve earns extra return if the curve is upward-sloping and stays unchanged, as the bond rolls down to a lower yield.
  • Swap spread = swap fixed rate minus the government bond yield of the same maturity. I-spread = the bond's yield minus the interpolated swap rate for the bond's maturity.
  • Pure expectations: forward rates are unbiased predictors of future spot rates. Liquidity preference adds a premium for longer maturities.
  • Segmented markets and preferred habitat explain curve shape through supply and demand across maturities.
  • Vasicek and CIR are equilibrium models with mean reversion; Ho-Lee and similar models are arbitrage-free and fit the observed curve.
  • Level, steepness and curvature are the main curve factors; level explains most of the movement in yields.
  • Key rate duration shows sensitivity to a shift at one maturity, with the rest of the curve unchanged.
  • In a binomial tree, value backwards from maturity: V = [0.5 × (V_up + C) + 0.5 × (V_down + C)] ÷ (1 + f), where V_up and V_down are the next-date values (ex-coupon), C is the coupon paid at that date and f is the node's forward rate. At maturity the ex-coupon value is par, so the cash flow is par plus the final coupon.

The Term Structure and Interest Rate Dynamics in other exams

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

The Term Structure and Interest Rate Dynamics: frequently asked questions

How do I get a spot rate from par rates?

You bootstrap. The first spot rate equals the first par rate. For each later maturity, you set the bond's price at par equal to the coupons discounted at the known spot rates plus the final cash flow discounted at the unknown spot rate, and solve for that rate.

What is the difference between an equilibrium and an arbitrage-free term structure model?

An equilibrium model, such as Vasicek or CIR, starts from assumptions about the economy and the behaviour of interest rates, and it may not match observed bond prices exactly. An arbitrage-free model is calibrated to the current market curve so that it reproduces observed prices.

Do I need to memorise the Vasicek and CIR equations?

Focus on what the models say: both include mean reversion. Vasicek has constant volatility, so it can produce negative rates. CIR scales volatility with the rate level, which keeps rates non-negative. Be able to interpret the parameters and compare the models, and learn any formula the vignette is likely to require you to apply.

How should I practise the binomial interest rate tree?

Start with a two-period tree and work backwards by hand. Then try three-period trees with coupons. Check that the tree values a benchmark bond at its market price, which tells you whether it is calibrated properly.