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CFA Level I · CFA Level I Exam

The Term Structure of Interest Rates: Spot, Par, and Forward Curves

The term structure of interest rates shows how yields change with maturity. Spot rates discount a single cash flow, forward rates are rates agreed today for future periods, and par rates make a bond price equal par. To solve questions, convert between them using discount factors and no-arbitrage logic.

What this chapter covers

This chapter explains how interest rates vary with maturity and how three related curves describe the same market. The spot curve gives the yield on a zero-coupon bond for each maturity. The forward curve gives rates for loans starting at future dates. The par curve gives the coupon rates at which bonds of each maturity would price at par.

The key skill is moving between these curves. You use discount factors as the common currency: a spot rate gives a discount factor, a discount factor gives a forward rate, and par rates can be bootstrapped into spot rates. You then link this to yield to maturity, yield spreads, and how curves shift or twist.

This chapter sits in Fixed Income and supports much of what follows. Bond valuation with spot rates, duration and convexity, credit spreads, and derivatives pricing such as forward rate agreements and swaps all build on these ideas. It also helps in Economics, where the shape of the yield curve is read as a signal about growth and policy.

Fixed Income carries a weight of 11-14% on the 2027 Level I exam, and this chapter is a foundation for many of its questions. The calculations are short and repeatable, which makes them reliable marks if you practise them. With 180 questions at roughly 90 seconds each, you cannot afford to rederive formulas, so a clear method for spot, forward and par conversions saves time. The concepts also reappear in derivatives and portfolio questions, so the effort pays off across several topics.

The Term Structure of Interest Rates: Spot, Par, and Forward Curves: topics in the order to study them

  1. 1Spot Rates and Discount FactorsStart here because spot rates and discount factors are the base for every other conversion in the chapter.
  2. 2Forward Rates and the Forward CurveForward rates are derived directly from spot rates using no-arbitrage, so learn them once discount factors are comfortable.
  3. 3Par Curve and Bootstrapping Spot RatesBootstrapping uses discount factors and the logic of pricing at par, so it comes after you can handle spot and forward rates.
  4. 4Yield to Maturity, Yield Spreads and Curve MovementsFinish with YTM and spreads because you can now compare it with the spot curve and read curve shifts, twists and spreads properly.

How to prepare The Term Structure of Interest Rates: Spot, Par, and Forward Curves

Treat this chapter as one toolkit built on discount factors. Practise short calculations until the steps are automatic, and keep the concepts tied to the numbers.

  1. Write the core relationships on one page: discount factor = 1 ÷ (1 + spot)^t, and (1 + z_B)^B = (1 + z_A)^A × (1 + f_{A,B-A})^(B-A).
  2. Practise price-from-spot-rates questions by discounting each cash flow at its own spot rate, then compare with discounting at one YTM.
  3. Do forward rate drills in both directions: find a forward rate from two spot rates, and find a spot rate from a spot rate and a forward rate.
  4. Bootstrap spot rates from a par curve by hand. For a par bond, 1 = c × (sum of earlier discount factors) + (1 + c) × the final discount factor, solved for the last factor.
  5. Use the TI BA II Plus power function (y^x) or the HP 12C equivalent for roots and powers, and check you can do a multi-year forward rate in under a minute.
  6. Practise the concept questions: why YTM differs from the spot rate, what a steepening or flattening curve means, and how spreads are defined.
  7. Finish with mixed three-option MCQs, and practise ruling out the two wrong options by checking direction and size, for example whether the forward rate should be above the later spot rate.

Common mistakes in The Term Structure of Interest Rates: Spot, Par, and Forward Curves

  • Using the wrong time periods in a forward rate formula

    Fix: Label the start and length first. For the 2y3y rate, A = 2 and B = 5, and the exponent on the forward factor is 3.

  • Forgetting to take the root at the end

    Fix: Raise the ratio to 1 ÷ (length of forward period), then subtract 1. Write the exponent before pressing keys.

  • Discounting all cash flows at the YTM when the question gives spot rates

    Fix: If spot rates are supplied, discount each cash flow at its own spot rate. Use YTM only when the question asks for it.

  • Treating the par rate as a spot rate

    Fix: Remember the par rate applies to a coupon bond and the spot rate to a single cash flow. Convert by bootstrapping.

  • Mixing compounding conventions

    Fix: Check the stated compounding. Use periods and per-period rates consistently, and convert back only if asked for an annual figure.

  • Misreading curve movement terms

    Fix: Compare the change in the long rate with the change in the short rate. Steepening widens the gap and flattening narrows it.

Last-day revision: The Term Structure of Interest Rates: Spot, Par, and Forward Curves

  • Spot rate is the yield on a zero-coupon bond for a given maturity.
  • Discount factor = 1 ÷ (1 + spot rate)^t, with annual compounding.
  • Bond price = sum of each cash flow times its own discount factor.
  • Forward rate is the rate for a future period implied by today's spot rates through no-arbitrage.
  • Forward rate link: (1 + z_B)^B = (1 + z_A)^A × (1 + f)^(B − A).
  • When the spot curve slopes upward, forward rates lie above spot rates.
  • Par rate is the coupon rate at which a bond prices at par.
  • Bootstrapping solves for spot rates one maturity at a time from par rates.
  • YTM is the single discount rate that equates a bond's price to its cash flows, assuming it is held to maturity and reinvestment at that rate.
  • YTM is a blend of spot rates and generally differs from any single spot rate.
  • Yield spread is the difference between yields of two bonds, often quoted in basis points.
  • A parallel shift moves all rates by the same amount; steepening and flattening change the gap between long and short rates.

The Term Structure of Interest Rates: Spot, Par, and Forward Curves practice questions

The Term Structure of Interest Rates: Spot, Par, and Forward Curves 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 of Interest Rates: Spot, Par, and Forward Curves: frequently asked questions

What is the difference between spot, par and forward rates?

A spot rate is the yield on a zero-coupon bond to a given maturity. A par rate is the coupon rate that makes a bond price at par. A forward rate is the rate for a loan that starts in the future, implied by today's spot curve.

How do I calculate a forward rate on the CFA Level I exam?

Use no-arbitrage: investing for the longer period must give the same result as investing for the shorter period and then rolling at the forward rate. Divide the longer growth factor by the shorter one, then take the root for the forward period length and subtract 1. Do this with the y^x key on your approved calculator.

What is bootstrapping in the term structure?

Bootstrapping derives spot rates from par rates one maturity at a time. You start with the shortest maturity, whose par rate equals its spot rate, and use earlier discount factors to solve for the next. Each new spot rate depends on the ones before it.

Why does YTM differ from the spot rate?

YTM is a single rate applied to all of a bond's cash flows, so it works like an average of the spot rates for those cash flows. A spot rate applies to one cash flow at one maturity. The two are close only when the curve is fairly flat.