Core Path · L1.4.03

Pricing a bond

12 minFixed incomeValuation

Know these first

  • L1.4.01 What is a bond?
  • L1.3.03 Present value and discounting
  • L1.3.04 Annuities and perpetuities

Math Atoms

  • ∑ Exponents
  • ∑ Geometric series

By the end you will be able to

  1. 1Price a fixed-coupon bond as the present value of its cash flows at a single yield.
  2. 2Explain when a bond trades at par, at a premium or at a discount.
  3. 3Price a bond that pays semi-annual coupons.
  4. 4Explain in words why each cash flow could use its own discount rate.

Why it matters

“Safe” government bonds lost value in 2022 when interest rates rose. How can a bond that promises fixed payments lose value? Pricing explains why.

Core explanation

A promise of future cash, valued today

A bond is a promise of cash in the future. Its price is what that promise is worth today.

Would you rather have £100 today or £100 in a year? Most people say today: you could put today's money in a savings account and earn interest. So a future payment is worth less than the same amount now. Shrinking a future amount to a value today is called discounting, and the value today is its present value (PV).

Say you can earn 5% a year. Then £105 in a year is worth £105 ÷ 1.05 = £100 today. The 5% is the discount rate.

To price a bond

  1. List every cash flow the bond will pay (see L1.4.01).
  2. Discount each one back to today.
  3. Add them up. Price = PV of the coupons + PV of the face value.

Which discount rate? Here we use one rate for every cash flow, called the yield: the annual return investors currently demand from this bond. Prices are quoted per 100 of face value, so a price of 95.67 means £95.67 for each £100 of face value.

Par

Price = 100

Coupon rate equals the yield.

Discount

Price < 100

Coupon rate is below the yield. The bond pays less than investors now demand.

Premium

Price > 100

Coupon rate is above the yield. The bond pays more than investors now demand.

Why prices fall when rates rise. A higher yield shrinks each future payment more, so the price falls. That is what hit bond investors in 2022: the coupons did not change, but the yield investors demanded went up, and prices had to come down.

Semi-annual bonds. Many bonds pay coupons twice a year. Halve the coupon, halve the yield, and double the number of periods.

One rate or many? Using one yield for every payment is a simplification. The market may ask a different rate for a payment in one year than for one in ten, so each cash flow could be discounted at its own rate. Those rates are spot rates, and together they form a spot curve, which later lessons build on.

The formal version

Now let's write the same idea as a formula, one piece at a time. We use four letters: CC is the coupon paid each year, FF the face value, yy the yield and TT the number of years to maturity.

Adding up TT separate terms gets slow for a 30-year bond. Because every coupon is the same, there is a shortcut using the annuity factor.

The two formulas, for reference

P=∑t=1TC(1+y)t+F(1+y)T=C×1−(1+y)−Ty+F(1+y)TP = \sum_{t=1}^{T} \frac{C}{(1+y)^t} + \frac{F}{(1+y)^T} = C \times \frac{1 - (1+y)^{-T}}{y} + \frac{F}{(1+y)^T}

With two coupons a year, use half the coupon (C/2), half the yield (y/2) and twice as many periods (2T):

P=C2×1−(1+y/2)−2Ty/2+F(1+y/2)2TP = \frac{C}{2} \times \frac{1 - (1+y/2)^{-2T}}{y/2} + \frac{F}{(1+y/2)^{2T}}

Try it

Before the worked example, get a feel for it. Start at a 4% coupon and a 5% yield, then slide the yield down to 4% and then 3%. Watch the price cross 100.

Try itPrice explorer

Open the full bond pricer →

Coupons paid

Price per 100Discount
95.67
PV of coupons17.32
PV of face value78.35
Each cash flow, and its value today paid worth today
0521041y2y3y4y5y
Price against yieldThe dot is your bond. It crosses 100 where yield = coupon.
0%2%4%6%8%10%100

Worked example

A 5-year, 4% bond at three yields

Face value 100, annual coupon 4% (so C=4C = 4), 5 years to maturity. Try each step yourself before revealing it.

Price at a 5% yield

Step 1 of 6
  1. 1

    Discount factor for year 5

    1.055=1.2762821.05^5 = 1.276282, so 1.05−5=0.7835261.05^{-5} = 0.783526.

The same steps at the other two yields give:

YieldAnnuity factorPV couponsPV facePriceTrades at
3%4.579718.31986.261104.58Premium
4%4.451817.80782.193100.00Par
5%4.329517.31878.35395.67Discount

From 4% to 5% the price falls by 4.33; from 4% to 3% it rises by 4.58. Prices and yields move in opposite directions, and not quite symmetrically. Lesson L1.4.06 explains why.

Semi-annual version at a 5% yield

Step 1 of 5
  1. 1

    Per-period inputs

    Coupon 4 ÷ 2 = 2, rate 5% ÷ 2 = 2.5%, periods 2 × 5 = 10.

Regulatory lens · looking ahead

No rules apply yet. Banks and insurers that hold bonds at fair value price them in just this way for their accounts; see L4.7 Accounting for risk.

Sparks

SparkPrices in 32nds
US Treasuries are quoted in 32nds of a point, not decimals. A quote of “99-16” means 99 and 16/32, which is 99.50. Always check how a price is quoted before you put it into a formula: 99.16 and 99-16 are very different prices.
SparkA price fall is not a default
When a bond's price falls after rates rise, the issuer has done nothing wrong. The coupon and face value are unchanged; only the rate used to discount them has moved. That is why even a bond certain to be repaid in full can show a loss if you sell before maturity.

Key takeaways

  • A bond's price is the present value of all its cash flows: the coupons plus the face value, discounted at the yield.
  • The annuity shortcut turns the sum of equal coupons into one factor.
  • Coupon rate equals yield: par. Below the yield: discount. Above the yield: premium.
  • For semi-annual coupons, halve the coupon and the yield, and double the number of periods.
  • Each cash flow could be discounted at its own rate (a spot curve); one yield is the standard starting point.

Reaction

Check what stuck

Question 1 · objective 1

A 2-year bond has face value 100 and an annual coupon of 4%. The yield is 5%. What is its price, to two decimal places?

Question 2 · objective 2

A bond has a 4% coupon rate and the market yield is 6%. The bond trades:

Question 3 · objective 3

A 1-year bond has face value 100 and a 6% coupon paid twice a year. The yield is 4% a year (2% per half-year). What is its price, to two decimal places?

Question 4 · objective 4

Why might each cash flow of a bond use its own discount rate?

Glossary

Present value
What a future cash flow is worth today after discounting.
Discount rate
The annual rate used to shrink a future cash flow to its value today.
Yield
The annual return investors demand from a bond, used here as one discount rate for all of its cash flows.
Par
A bond price equal to its face value, quoted as 100.
Premium
A bond price above par.
Discount
A bond price below par.
Annuity factor
The value today of receiving 1 each period for a set number of periods.

Deep Dives & Origins

Deep Dives

Tuckman and Serrat, Fixed Income Securities, chapters 1 to 3. Intermediate.

Origins

  • Fabozzi, Bond Markets, Analysis and Strategies.
  • Tuckman and Serrat, Fixed Income Securities.

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Previous · L1.4.02

Types of bonds

Next · L1.4.04

Yield to maturity