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CFA Level II Exam · The Term Structure and Interest Rate Dynamics

Yield to Maturity and Par, Spot and Forward Curves

Updated 7 October 2026 · Fact-checked

Yield to maturity is the single rate that discounts a bond's cash flows to its price. A spot rate discounts one cash flow from a zero-coupon bond. A par rate is the coupon that prices a bond at par. Bootstrap spot rates from par rates one maturity at a time using discount factors, then get forwards from spot ratios.

Understand Yield to Maturity and Par, Spot and Forward Curves

A yield to maturity (YTM) is one discount rate applied to every cash flow of a bond. It makes the present value of the coupons and principal equal to the price. It is a summary number. It assumes you hold to maturity, the issuer does not default, and you reinvest every coupon at the YTM itself.

A spot rate is different. It is the yield on a zero-coupon bond for one maturity. The 3-year spot rate discounts only a cash flow received in year 3. The spot curve (or zero curve) plots spot rates against maturity. A bond is correctly valued by discounting each cash flow at its own spot rate. That is why two bonds with the same maturity but different coupons can have different YTMs.

A par rate is the coupon rate at which a bond with that maturity prices exactly at par (100). Its YTM equals its coupon. The par curve plots these rates by maturity. Par rates are what you usually see quoted, for example for government benchmark bonds. Spot rates are what you need for valuation, so you convert par rates to spot rates by bootstrapping.

A forward rate is a rate agreed today for a loan that starts at a future date. The notation f(A, B) means the rate for a loan of B years that starts A years from now. Forward rates come from spot rates by a no-arbitrage argument: investing for A+B years at the spot rate must give the same result as investing for A years and then rolling into the forward contract. So the spot curve and the forward curve contain the same information.

When the spot curve slopes upward, the forward curve lies above it, and the par curve lies below it at the same maturity. When the curve slopes downward, the order reverses. Use this as a sense check on your answers, not as a replacement for calculation.

Key formulas to remember

Discount factor from spot rate
DF(n) = 1 ÷ (1 + z(n))^n
z(n) is the n-year spot rate, annual compounding. Work in discount factors; they make every other step linear.
Bond price using spot rates
P = Σ [CF(t) ÷ (1 + z(t))^t]
Each cash flow is discounted at the spot rate for its own maturity.
Bootstrapping a spot rate from par rates
DF(n) = [1 − c(n) × Σ DF(t) for t = 1 to n−1] ÷ [1 + c(n)]; then z(n) = DF(n)^(−1/n) − 1
c(n) is the n-year par rate (annual coupons, price 100). Needs all earlier discount factors first.
Par rate from spot rates
c(n) = [1 − DF(n)] ÷ Σ DF(t) for t = 1 to n
The reverse conversion. The sum runs through year n, including the final year.
Forward rate from spot rates
(1 + z(A+B))^(A+B) = (1 + z(A))^A × (1 + f(A, B))^B
Rearrange for f(A, B). For one-period forwards: 1 + f = DF(A) ÷ DF(A+1).
Spot rate from forward rates
(1 + z(n))^n = (1 + z(1)) × (1 + f(1,1)) × (1 + f(2,1)) × ... × (1 + f(n−1,1))
The spot rate is a geometric average of the one-year forward rates.

How to solve Yield to Maturity and Par, Spot and Forward Curves questions

Almost every question on this topic is a conversion between par, spot and forward rates. Convert everything to discount factors first, then convert out at the end.

  1. 1Read the vignette and list what is given: par rates, spot rates or forward rates, the coupon frequency, and the compounding convention. Note what is asked.
  2. 2Check that the bond is priced at par if the data are par rates. Par rates assume price 100 and coupon equal to the rate.
  3. 3Convert the given rates to discount factors. For spot rates use DF(n) = 1 ÷ (1 + z)^n. For par rates, bootstrap one maturity at a time, starting with year 1.
  4. 4For the bootstrap, at each maturity n compute DF(n) = [1 − c(n) × sum of earlier DFs] ÷ [1 + c(n)]. Keep at least six decimals in the DFs.
  5. 5Convert to the required output. Spot rate: z(n) = DF(n)^(−1/n) − 1. Par rate: (1 − DF(n)) ÷ sum of DFs. Forward rate: ratio of the discount factors or of the compounded spot growth factors.
  6. 6Check the exponents. For f(A, B) the root is B, the forward period length, not A+B.
  7. 7Sense check against the curve shape. In an upward-sloping curve, forward rate > spot rate > par rate at similar maturities. If your answer breaks the pattern, recheck.

Quickest way: Discount-factor shortcut

When to use it: Use it whenever you must go from par rates to spot rates, or compute one-year forwards, under time pressure.

  1. Compute DF(1) = 1 ÷ (1 + c1). Store it in the calculator memory.
  2. For each next year, accumulate a running sum of DFs and use DF(n) = (1 − c(n) × running sum) ÷ (1 + c(n)).
  3. For a one-year forward between years n and n+1, divide: f = DF(n) ÷ DF(n+1) − 1. No roots needed.
  4. Only take roots at the very end, if the question asks for a spot rate.
  5. Eliminate options early: a forward rate for an upward-sloping curve must exceed the later spot rate, and the par rate must sit below the spot rate.

Common mistakes in Yield to Maturity and Par, Spot and Forward Curves

  • Treating YTM as the spot rate for that maturity

    Both are quoted as yields for a maturity, so they look alike.

    Fix: Only a zero-coupon bond has YTM equal to its spot rate. A coupon bond's YTM is a blend of several spot rates.

  • Leaving out the final coupon when bootstrapping

    Students discount the principal at year n but forget that the year-n cash flow is 100 plus the coupon.

    Fix: Use the formula with (1 + c(n)) in the denominator. The last cash flow is 1 + c(n) per 1 of par.

  • Bootstrapping in the wrong order

    Students try to find the 3-year spot rate before they have the 2-year spot rate.

    Fix: Always start at year 1. Each new DF needs all the earlier DFs.

  • Using the wrong exponent for a forward rate

    The notation f(A, B) mixes the start date and the loan length.

    Fix: Write (1 + z(A+B))^(A+B) ÷ (1 + z(A))^A first. Then take the root equal to B, the length of the forward loan.

  • Averaging spot rates arithmetically to get a forward or a spot rate

    It looks simpler than compounding.

    Fix: Rates compound. Use geometric relationships on growth factors, such as (1 + z) raised to the maturity.

  • Applying the par-rate bootstrap to a bond not priced at par

    The vignette gives a coupon and a price, and students reuse the par formula.

    Fix: If price is not 100, use the general price equation and solve for the last discount factor: DF(n) = (Price − coupon × sum of earlier DFs) ÷ (100 + coupon).

Worked examples

Example 1

An analyst is given annual-coupon government par rates: 1-year 2.00%, 2-year 3.00%, 3-year 4.00%. All bonds price at par. Questions: (1) What is the 2-year spot rate? A. 3.00% B. 3.02% C. 3.50%. (2) What is the 3-year spot rate? A. 4.00% B. 4.05% C. 4.10%. (3) What is the forward rate f(2,1), the one-year rate two years from now? A. 5.00% B. 6.17% C. 7.10%.

Show the solution
  1. Year 1: DF(1) = 1 ÷ 1.02 = 0.980392. The 1-year spot rate is 2.00%.
  2. Year 2: DF(2) = (1 − 0.03 × 0.980392) ÷ 1.03 = 0.970588 ÷ 1.03 = 0.942318.
  3. 2-year spot: z(2) = (1 ÷ 0.942318)^(1/2) − 1 = 1.061209^0.5 − 1 = 1.030150 − 1 = 3.015%, about 3.02%.
  4. Year 3: sum of earlier DFs = 0.980392 + 0.942318 = 1.922710. DF(3) = (1 − 0.04 × 1.922710) ÷ 1.04 = 0.923092 ÷ 1.04 = 0.887588.
  5. 3-year spot: z(3) = (1 ÷ 0.887588)^(1/3) − 1 = 1.126649^(1/3) − 1 = 1.04055 − 1, about 4.05%.
  6. Forward f(2,1) = DF(2) ÷ DF(3) − 1 = 0.942318 ÷ 0.887588 − 1 = 1.061662 − 1 = 6.17%.
  7. Sense check: the curve slopes upward, so spot (4.05%) is above par (4.00%), and the forward (6.17%) is above both.

Answer: (1) B, about 3.02%. (2) B, about 4.05%. (3) B, about 6.17%.

Example 2

The annual spot curve is: 1-year 3.00%, 2-year 4.00%, 3-year 5.00%. Questions: (1) What is the 2-year par rate (annual coupon)? A. 3.50% B. 3.98% C. 4.00%. (2) What is the forward rate f(1,2), the 2-year rate starting one year from now? A. 5.50% B. 6.01% C. 6.50%.

Show the solution
  1. DF(1) = 1 ÷ 1.03 = 0.970874. DF(2) = 1 ÷ 1.04² = 1 ÷ 1.0816 = 0.924556.
  2. Par rate: c(2) = (1 − DF(2)) ÷ (DF(1) + DF(2)) = (1 − 0.924556) ÷ (0.970874 + 0.924556) = 0.075444 ÷ 1.895430 = 3.98%.
  3. Check the shape: the par rate (3.98%) is below the 2-year spot rate (4.00%), as expected on an upward-sloping curve.
  4. Forward f(1,2): (1 + f)² = 1.05³ ÷ 1.03 = 1.157625 ÷ 1.03 = 1.123907.
  5. Take the root equal to the forward period, 2: 1 + f = 1.123907^(1/2) = 1.06015. So f = 6.01%.
  6. Sense check: the forward rate exceeds the 3-year spot rate of 5.00%, as expected when the curve slopes upward.

Answer: (1) B, 3.98%. (2) B, about 6.01%.

Exam tips

  • Expect a vignette that gives a par curve and asks for a spot rate or a bond price. Do the bootstrap in a clean column of discount factors and reuse it for every question in the item set.
  • Check whether the vignette gives par rates, spot rates or YTMs. The data label decides which formula you use.
  • Use the curve-shape rule to remove one or two options quickly: in an upward-sloping curve, forward > spot > par at similar maturities.
  • Round only at the end. Small rounding in discount factors can move the answer to a neighbouring option.
  • Watch for the wording on forward notation, such as 2y1y or f(2,1). Write out the start date and length before calculating.

Yield to Maturity and Par, Spot 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.

Yield to Maturity and Par, Spot and Forward Curves: frequently asked questions

What is the difference between YTM and the spot rate?

YTM is one rate applied to all cash flows of a coupon bond, while a spot rate applies to a single cash flow at one maturity. A coupon bond's YTM is a blend of spot rates. The two are equal only for a zero-coupon bond.

How do you bootstrap spot rates from par rates?

Start with the 1-year par rate, which equals the 1-year spot rate. For each later maturity, solve the par bond equation for the final discount factor, using the earlier discount factors. Then convert the discount factor to a spot rate with z = DF^(−1/n) − 1.

What is the relationship between the spot curve and the forward curve?

Forward rates are implied by spot rates through no-arbitrage. The growth factor to a later date equals the growth factor to an earlier date times the forward growth factor. If the spot curve slopes upward, the forward curve lies above it.

Why is the par curve below the spot curve when the curve is upward sloping?

A par bond pays coupons earlier, and those early cash flows are discounted at lower short-maturity spot rates. So the single YTM that prices the bond at par is a weighted average of spot rates and falls below the long-maturity spot rate.