Bond Pricing
A bond pays a level coupon of Fr every period plus a redemption value C at maturity. Its fair price at yield rate i is just the present value of those two cash flows: \(P = Fr \cdot a_n + C \cdot v^n\).
By the end you'll be able to price a bond as the sum of a coupon annuity's present value and a discounted redemption payment, and predict whether price rises or falls when yield changes.
Predict: when the required yield rises, does the bond's price rise or fall? Drag the yield slider and check which of the two bar segments moves the most.
This bond has face and redemption value F = C = 1000. Drag the yield i, coupon rate r, and term n to reshape the price. The bar stacks two contributions: the coupon annuity's present value, Fr·aₙ, and the discounted redemption, C·vⁿ. Their sum, marked by the dashed line, is the price. Hover or focus either segment for the exact numbers.
P = 1147.20 = Coupon PV 588.81 + Redemption PV 558.39
A bond is just a set of dated cash flows, so its fair price is nothing more than the present value of those cash flows at the market's required yield.
Think of buying a bond as buying two separate promises: a stream of level coupon payments while you hold it, and a single lump-sum payback when it matures. You'd price each promise on its own — a coupon stream is priced like any level annuity, and a single future payment is priced by discounting it back to today — then just add the two prices together. That sum is the bond's price.
With face value F, coupon rate r per period (so each coupon is Fr), redemption value C (C = F when redeemed at par), n periods to maturity, and yield rate i with discount factor v = 1/(1+i):
\(P = Fr \cdot a_n + C \cdot v^n\), where \(a_n = \dfrac{1 - v^n}{i}\)
The first term prices the coupon stream as a level annuity-immediate; the second discounts the single redemption payment n periods back to today. Every other bond result — premium/discount, amortization, pricing between coupon dates — is an algebraic rearrangement of this one formula.
A pension fund shopping for a 20-year corporate bond doesn't care about the coupon rate printed on the certificate in isolation — it cares what that coupon is worth at today's market yield. If the market suddenly demands a higher yield on similar bonds, the fund reprices every bond it's evaluating downward, because both the coupon annuity and the redemption payment get discounted harder. That's the whole reason bond prices move day to day even though the coupon and redemption amounts printed on the certificate never change.
A 10-year bond has face and redemption value F = C = 1000, pays annual coupons at r = 8% (Fr = 80), and is bought to yield i = 6%. Then v\(^{10}\) = 1.06\(^{-10}\) = 0.558395, a\(_{10}\) = (1 − 0.558395)/0.06 = 7.360087, so
\(P = 80 \times 7.360087 + 1000 \times 0.558395 = 588.81 + 558.39 =\) 1147.20
The price exceeds the redemption value because the 8% coupon beats the 6% the market demands — set r = 8% and i = 6% on the sliders above and check the bar matches.
Same formula, different numbers: a 5-year bond, F = C = 1000, coupon rate r = 5% (Fr = 50), bought to yield i = 7%. Then v\(^5\) = 1.07\(^{-5}\) = 0.712986, a\(_5\) = (1 − 0.712986)/0.07 = 4.100200, so the coupon PV is \(50 \times 4.100200 = 205.01\). Now finish it: the redemption PV is \(1000 \times 0.712986 = \) ____, and the price \(P = 205.01 + \) ____ \(=\) ____
Reveal the answer
Redemption PV = \(1000 \times 0.712986 = 712.99\). Price \(P = 205.01 + 712.99 = \) 918.00. Set i = 7%, r = 5%, n = 5 on the sliders above and check the readout matches.
More info — two other ways to write the same price
Substituting \(C \cdot v^n = C - C \cdot i \cdot a_n\) into the basic formula gives the premium/discount form: \(P = C + (Fr - Ci) \cdot a_n\). The sign of \(Fr - Ci\) — coupon cash flow minus the interest the redemption alone would earn — decides whether the bond trades above or below its redemption value; the next lesson builds on exactly this rearrangement. There's also Makeham's formula: with \(K = C \cdot v^n\) (the discounted redemption you just computed) and modified coupon rate \(g = Fr/C\), \(P = K + (g/i)(C - K)\) — handy whenever \(v^n\) is easy to get but \(a_n\) is awkward. Both are the same price, just regrouped.
Check your understanding
An 8-year bond, face and redemption value 1000, annual coupon rate 6% (so Fr = 60), is priced to yield 5% annually. Which is closest to the price?
If the required yield i rises while the coupon rate r and term n stay fixed, what happens to the bond's price P = Fr·aₙ + C·vⁿ?
Suppose C = F and the coupon rate r exactly equals the yield rate i. What does the price formula P = Fr·aₙ + C·vⁿ simplify to?
In P = Fr·aₙ + C·vⁿ, how should you read the two terms?
Recap
- A bond's price is the present value of its cash flows at the yield rate i: \(P = Fr \cdot a_n + C \cdot v^n\).
- \(Fr \cdot a_n\) prices the level coupon annuity-immediate; \(C \cdot v^n\) discounts the single redemption payment back n periods.
- Raising the yield i shrinks both \(a_n\) and \(v^n\), so the price falls.
- When \(r = i\) and \(C = F\), the two terms combine to exactly \(P = F\) — the knife-edge between premium and discount pricing.
Dive deeper
- Khan Academy — Introduction to bonds Build an intuitive picture of what a bond is before pricing it.
- thismatter.com — Bond Pricing, Accrued Interest, Illustrated with Examples Review present-value pricing of coupons plus redemption with worked examples.
- Corporate Finance Institute — Bond Pricing See the price-as-present-value-of-cash-flows formula explained.
- Marcel Finan — A Basic Course in the Theory of Interest (Exam FM study guide, PDF) Study the exam-oriented derivation of the basic bond price formula.
Sources
- Bond Pricing — The Basic Price Formula