Exam FMBondsFree to read

Premium, discount and amortisation of premium

The gap between price and redemption value is written off over the bond's life; the write-down each period is a geometric series like a loan's principal repayments.

The formulas

Premium

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

Write-down in coupon t

$$(Fr-Ci)v^{n-t+1}$$

Interest earned in coupon t

$$I_t=i{B}_{t-1}$$

Book value recursion

$$B_t=B_{t-1}(1+i)-Fr$$

Where it comes from

1. Each coupon $Fr$ splits into interest actually earned, $i{B}_{t-1}$, and a return of capital.
2. For a premium bond the coupon exceeds the interest earned, so the excess $(Fr-Ci)v^{n-t+1}$ is written OFF the book value.
3. For a discount bond the same expression is negative and the book value accumulates UP towards $C$.

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

• Write-downs grow by (1 + i) each period, exactly like loan principal repayments — same algebra, different name.
• Premium bond: book value falls to C. Discount bond: book value rises to C. Both end at C.

Traps

• Signing the write-down wrongly for a discount bond.
• Using n − t rather than n − t + 1 in the exponent.

Related

• Bond pricing
• Book value and Makeham's formula
• Loan amortisation

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