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Book value and Makeham's formula

Book value is the price of the same bond with fewer coupons left; Makeham's formula prices the whole thing from the redemption value alone.

The formulas

Book value after coupon t: $B_{t} = F r a_{\overline{n - t} ∣ i} + C v^{n - t}$
Makeham: $P = K + \frac{g}{i} (C - K)$
Modified coupon rate: $g = \frac{F r}{C}$
Price at par: $P = C ⟺ g = i$

Where it comes from

Nothing about the remaining cash flows depends on history, so the book value is just the price of the shorter bond.
Makeham writes the coupon stream as $F r a_{\overline{n} ∣} = \frac{F r}{i} (1 - v^{n}) = \frac{g}{i} (C - C v^{n}) = \frac{g}{i} (C - K)$ .
Adding $K = C v^{n}$ gives the price. When $g = i$ the whole bracket collapses and $P = C$ .

Worked example

Find the book value of the 1,000 par 10-year 8% bond yielding 6% immediately after the 4th coupon.

Six coupons remain, so value a 6-period bond: $B_{4} = 80 a_{\overline{6} ∣0.06} + 1, 000 v^{6}$ .
$a_{\overline{6} ∣0.06} = 4.917324$ and $v^{6} = 0.704961$ .
$B_{4} = 393.386 + 704.961 = 1, 098.35$ .
The book value has fallen from 1,147.20 towards the redemption value of 1,000, as a premium bond's must.

Answer: 1,098.35

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

Book value = price of the bond that is left. Forget the past.
Makeham needs only K, g and i — useful when the redemption value is given but the coupon amount is not.

Traps

Using the ORIGINAL n in the book-value formula.
Confusing g (Fr/C, the modified coupon rate) with r (the coupon rate on face).

Drill this: the Exam FM question bank has original questions on this topic, and today’s free round is open to everyone.

Book value and Makeham's formula

The formulas

Where it comes from

Worked example

Memory hooks

Traps

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