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Actuarial Mathematics for Modelling · Term structure of interest rates

Par Yields and Yield to Maturity Explained

Updated 11 October 2026 · Fact-checked

Yield to maturity is the single rate that equates a bond's price to the present value of its cash flows. A spot rate discounts one payment at one term. The par yield is the coupon rate that prices a bond at par. Bootstrap spot rates from the shortest term upward, then par yield = (1 − v_n) ÷ Σv_t.

Understand Par Yields and Yield to Maturity

A spot rate y_t is the annual effective rate for money lent from now until time t, with no payments in between. It is the yield on a zero-coupon bond. Each cash flow at time t is discounted at its own spot rate, using v_t = (1 + y_t)^(−t).

A coupon bond pays many cash flows at different times. Its yield to maturity (YTM) is one rate i that discounts all of them to the bond's price. It is an average of the spot rates, weighted towards the times with the largest cash flows. It is not a spot rate, and it depends on the coupon rate. Two bonds with the same maturity but different coupons usually have different YTMs when the yield curve is not flat.

Bootstrapping turns coupon bond prices into spot rates. Start with the shortest bond, which has one cash flow, and solve for y_1. Then take the 2-year coupon bond. Discount its first coupon at y_1 and solve the remaining unknown for v_2, and so on up the maturities. Each step has one unknown.

The par yield for term n is the coupon rate that makes a bond of term n, redeemed at par, have a price equal to its face value. It is also the YTM of that par bond. You find it from the spot rates: the price of the bond at par is 1, so c × (v_1 + ... + v_n) + v_n = 1.

So the three ideas connect. Prices give spot rates by bootstrapping. Spot rates give par yields by the formula. A bond priced at par has YTM equal to its coupon rate.

Key rules to remember

Discount factor from spot rate
v_t = (1 + y_t)^(−t)
y_t is the annual effective spot rate for term t. Convert the answer back with y_t = v_t^(−1/t) − 1.
Price of a coupon bond using spot rates
P = Σ C_t × v_t, where C_t = Fn × coupon rate for t < n and C_n = Fn × coupon rate + redemption
Each payment is discounted at its own spot rate. Fn is the nominal (face) value.
Yield to maturity
P = Σ C_t ÷ (1 + i)^t
Solve for the single rate i. This usually needs a quadratic for 2 years, or trial and error or interpolation for longer terms.
Par yield (annual coupons, redeemed at par)
c_n = (1 − v_n) ÷ (v_1 + v_2 + ... + v_n)
Per unit of nominal. It comes from c × Σv_t + v_n = 1. It assumes coupons are paid annually.
Bootstrapping step
v_n = (P − C × (v_1 + ... + v_(n−1))) ÷ (C + R)
For a bond with annual coupon C and redemption R at time n. Use the same units for P, C and R.
Par bond check
If the price equals the redemption value and R = Fn, then YTM = coupon rate
For a bond redeemed at par with annual coupons, price at par means YTM equals the coupon rate.

How to solve Par Yields and Yield to Maturity questions

Use this method for any question that links bond prices, YTM, spot rates and par yields. State your assumptions first: annual coupons, redemption at par unless told otherwise, and no default.

  1. 1Write down every bond: nominal, coupon rate, term, redemption and price. Check whether the price is per ₹100 nominal.
  2. 2Identify what is asked: a YTM, a spot rate, a discount factor or a par yield.
  3. 3For spot rates, start with the shortest bond. Solve for v_1 or y_1 from its cash flows.
  4. 4Move to the next maturity. Discount the known earlier coupons with the spot rates you already found. Solve the single remaining unknown v_n, then convert with y_n = v_n^(−1/n) − 1.
  5. 5For a YTM, write the equation of value for the one bond. Solve it using a quadratic in v = 1/(1 + i), or by trial and error with a check on the price.
  6. 6For a par yield, compute v_t from the spot rates. Then apply c_n = (1 − v_n) ÷ Σ v_t.
  7. 7Check reasonableness. The YTM should sit among the spot rates used. A par yield for a rising curve should be slightly below the n-year spot rate.
  8. 8State the answer clearly with units, as a percentage per year, and say which rate you found.

Quickest way: Work in discount factors, not rates

When to use it: Use this when you must bootstrap two or three spot rates or find a par yield under time pressure. Avoid converting to percentages until the last step.

  1. Find each v_t first, as a decimal to at least six places. Rounding early is the main source of lost marks.
  2. Keep a running total of v_1 + ... + v_t. You need it for both bootstrapping and the par yield.
  3. Bootstrap with v_n = (P − C × running total) ÷ (C + R). This avoids solving polynomial equations.
  4. For the par yield, divide (1 − v_n) by the running total. No iteration is needed.
  5. For a 2-year YTM, solve 108x² + 8x − P = 0 style quadratics for x = 1/(1 + i), then invert. For longer terms use trial and error, then interpolate between two guesses.
  6. Convert to y_n only once, at the end.

Common mistakes in Par Yields and Yield to Maturity

  • Treating the YTM of a coupon bond as the spot rate for that term.

    Both are described as the return to maturity, so they look the same.

    Fix: Remember a spot rate prices one payment. YTM is one rate for all payments. They agree only for a zero-coupon bond or a flat yield curve.

  • Discounting every coupon at the YTM when asked to price with spot rates.

    Students default to the single-rate approach they learned first.

    Fix: If the question gives spot rates, use v_t = (1 + y_t)^(−t) for each payment. Use a single rate only when told to use YTM.

  • Forgetting that the final payment includes the redemption value as well as the last coupon.

    The coupon and redemption are written separately in the question.

    Fix: At time n the cash flow is C + R. Write the cash flow line out in full before you solve.

  • Using the wrong denominator in the par yield, such as (v_1 + ... + v_(n−1)) or v_n alone.

    Students mix up the annuity sum with the bootstrapping formula.

    Fix: Derive it each time: c × (v_1 + ... + v_n) + v_n = 1, so c = (1 − v_n) ÷ Σ v_t, with the sum running to n.

  • Rounding discount factors to three or four decimals during bootstrapping.

    Students want tidy numbers in the working.

    Fix: Carry at least six decimals through. Errors compound across maturities, because every later spot rate depends on the earlier ones.

  • Mixing price units, for example price per ₹100 with coupons on a ₹1,000 nominal.

    Questions switch between nominal amounts and per-₹100 quotes.

    Fix: Convert everything to per ₹100 nominal, or everything to the actual nominal, before you write the equation.

Worked examples

Example 1

The annual effective spot rates are y_1 = 4%, y_2 = 5% and y_3 = 6%. Calculate the 3-year par yield, assuming annual coupons and redemption at par.

Show the solution
  1. Find the discount factors. v_1 = 1 ÷ 1.04 = 0.961538.
  2. v_2 = 1 ÷ 1.05² = 1 ÷ 1.1025 = 0.907029.
  3. v_3 = 1 ÷ 1.06³ = 1 ÷ 1.191016 = 0.839619.
  4. Sum of discount factors: 0.961538 + 0.907029 + 0.839619 = 2.708186.
  5. Apply c_3 = (1 − v_3) ÷ Σ v_t = (1 − 0.839619) ÷ 2.708186 = 0.160381 ÷ 2.708186.
  6. This gives c_3 = 0.05922, or 5.92% per year.
  7. Check: 0.05922 × 2.708186 + 0.839619 = 0.160381 + 0.839619 = 1, as required.

Answer: The 3-year par yield is about 5.92% per year. It is below the 3-year spot rate of 6%, as expected for a rising curve.

Example 2

A 1-year zero-coupon bond has a spot rate of 5% per year. A 2-year bond pays an annual coupon of 8% per ₹100 nominal and is redeemed at par at the end of year 2. Its price is ₹103.50 per ₹100 nominal. (a) Find the 2-year spot rate. (b) Find the yield to maturity of the 2-year bond. (c) Comment on the two rates.

Show the solution
  1. (a) Cash flows of the 2-year bond are ₹8 at time 1 and ₹108 at time 2. Discount ₹8 at the 1-year spot rate: 8 ÷ 1.05 = 7.619048.
  2. Equation of value: 103.50 = 7.619048 + 108 × v_2. So v_2 = (103.50 − 7.619048) ÷ 108 = 95.880952 ÷ 108 = 0.887786.
  3. y_2 = v_2^(−1/2) − 1 = √(1 ÷ 0.887786) − 1 = √1.126395 − 1 = 1.061318 − 1 = 0.0613, so 6.13%.
  4. (b) For the YTM, let x = 1 ÷ (1 + i). Then 103.50 = 8x + 108x², so 108x² + 8x − 103.50 = 0.
  5. x = [−8 + √(8² + 4 × 108 × 103.50)] ÷ (2 × 108) = [−8 + √44,776] ÷ 216 = (−8 + 211.6034) ÷ 216 = 0.942608.
  6. i = 1 ÷ 0.942608 − 1 = 0.06089, so 6.09%.
  7. (c) The YTM of 6.09% lies between the 1-year spot rate of 5% and the 2-year spot rate of 6.13%. It is close to y_2 because most of the cash flow arrives at time 2.

Answer: (a) The 2-year spot rate is about 6.13%. (b) The YTM is about 6.09%. (c) The YTM is a weighted average of the spot rates, so it differs from y_2 because the coupon at time 1 is discounted at the lower 5% rate.

Exam tips

  • Show the equation of value before any numbers. Marks are given for method, even if the arithmetic slips.
  • State your assumptions: annual coupons, redemption at par, no default and no tax. Examiners reward explicit assumptions.
  • In written parts, explain why the YTM differs from the spot rate. Say that it depends on the coupon and is a single rate for several payments.
  • Keep six decimal places in discount factors and round only the final percentage. This matters most in bootstrapping questions.
  • In computer-based Paper B questions, build the bootstrapping as a column of discount factors and use a running sum. Label each column and state the formula you used.

Practice questions from Term structure of interest rates

Par Yields and Yield to Maturity in other exams

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

Par Yields and Yield to Maturity: frequently asked questions

What is the difference between yield to maturity and spot rate?

A spot rate is the yield on a single payment at a single future time, such as a zero-coupon bond. Yield to maturity is one rate that discounts all the cash flows of a coupon bond to its price. For coupon bonds the YTM depends on the coupon, so it can differ from the spot rate for the same term.

How do I bootstrap spot rates from coupon bond prices?

Start with the shortest bond and solve for its spot rate. For each longer bond, discount the coupons that fall before maturity using the spot rates you already know. Solve the remaining equation for v_n and then convert to y_n.

What is a par yield?

It is the coupon rate at which a bond of a given term is priced at par. It is found from the spot rates using c_n = (1 − v_n) ÷ Σ v_t. It also equals the YTM of that par bond.

Is the par yield always below the spot rate?

No. It lies below the n-year spot rate when the spot curve is rising, and above it when the curve is falling. If the curve is flat, the par yield equals the spot rate.