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Bond pricing

A bond is an annuity of coupons plus a lump at redemption; price it with the basic formula and every other bond formula follows.

The formulas

Basic formula
P=Fr an‾∣i+CvnP = Fr\,a_{\overline{n}|i} + C v^{n}
Premium/discount formula
P=C+(Fr−Ci) an‾∣iP = C + (Fr - Ci)\,a_{\overline{n}|i}
Makeham
P=K+gi(C−K),K=Cvn,  g=FrCP = K + \dfrac{g}{i}(C-K), \quad K = Cv^{n},\; g = \dfrac{Fr}{C}
Symbols
F face,  r coupon rate,  C redemption,  i yield,  n periodsF\text{ face},\; r\text{ coupon rate},\; C\text{ redemption},\; i\text{ yield},\; n\text{ periods}

Where it comes from

  1. The bond pays FrFr each period for nn periods and CC at the end, so its price is the present value of exactly those cash flows.
  2. Substituting an‾∣=1−vnia_{\overline n|} = \frac{1-v^n}{i} into the basic formula and rearranging gives the premium/discount form, which isolates the excess coupon Fr−CiFr - Ci.
  3. Makeham's form groups the redemption KK and expresses the coupon stream as a fraction g/ig/i of the capital not yet repaid.

Worked example

A 1,000 par-value 10-year bond with 8% annual coupons is bought to yield 6% effective. Find the price.

  1. Fr=1,000×0.08=80Fr = 1{,}000 \times 0.08 = 80 per year, C=F=1,000C = F = 1{,}000, n=10n = 10, i=0.06i = 0.06.
  2. a10‾∣0.06=7.360087a_{\overline{10}|0.06} = 7.360087 and v10=0.558395v^{10} = 0.558395.
  3. P=80(7.360087)+1,000(0.558395)=588.807+558.395P = 80(7.360087) + 1{,}000(0.558395) = 588.807 + 558.395.
  4. P=1,147.20P = 1{,}147.20 - a premium bond, because the coupon rate exceeds the yield.

Answer: 1,147.20

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

  • Coupon rate is applied to FACE; the yield discounts to REDEMPTION. They are different bases and the exam checks that you know it.
  • r > i means premium (price above C); r < i means discount.

Traps

  • Applying the coupon rate to the redemption value when C ≠ F.
  • Using an annual yield with semiannual coupons - halve the coupon rate AND the yield, and double n.

Related

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