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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

$$P=Fr{a}_{\overline{n}|i}+C{v}^n$$

Premium/discount formula

$$P=C+(Fr-Ci)a_{\overline{n}|i}$$

Makeham

$$P=K+\frac{g}{i}(C-K),K=C{v}^n,g=\frac{Fr}{C}$$

Symbols

$$F\text{ face},r\text{ coupon rate},C\text{ redemption},i\text{ yield},n\text{ periods}$$

Where it comes from

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

Worked example

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

• Premium, discount and amortisation of premium
• Book value and Makeham's formula
• Bond yield and callable bonds

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