Bond Amortization Schedule
A coupon is never pure income. Part of it is the interest the yield rate earns on the money currently tied up in the bond, and whatever's left over nudges the book value toward the redemption value. Splitting each coupon that way, period after period, builds the amortization schedule.
By the end you'll be able to split any coupon into its interest and write-down pieces, recompute the book value one period at a time, and explain why premium and discount bonds glide toward C in opposite directions.
Predict: for a premium bond, does the book value rise or fall over time — and what does that do to each period's write-down? Step through k and check.
This is a 1000 face/redemption, 10-period bond. Each period, interest earned equals i times the prior book value, and coupon minus interest is the write-down of premium — or, when it comes out negative, the accumulation of discount. Drag k to pin one period and read its exact split; drag r and i to flip between a premium and a discount bond.
Period 1 of 10: coupon Fr = 80.00 · interest \(i\cdot B_0\) = 68.83 · write-down = 11.17 · new book value \(B_1\) = 1136.03
Let \(B_t\) be the book value after period t, with \(B_0 = P\) and \(B_n = C\). For period t: interest earned \(= i\cdot B_{t-1}\), and the principal adjustment \(= Fr - i\cdot B_{t-1}\), giving the recursion \(B_t = B_{t-1}(1+i) - Fr\) — accumulate the old book value one period, then remove the coupon. When \(Fr > i\cdot B_{t-1}\) (a premium bond) the adjustment is positive and is called the write-down of premium; when \(Fr < i\cdot B_{t-1}\) (a discount bond) it's negative and is called the accumulation of discount. Either way, the recursion always lands on \(B_n = C\) at maturity.
An accountant carrying a bond on the books needs to know how much of each coupon is "real" interest income versus a return of the premium originally paid. The amortization schedule answers that question period by period, which is exactly why bond issuers and holders both keep one: it reconciles the fixed cash coupon against the shrinking (or growing) carrying value on the balance sheet.
1000 par, 8% annual coupon, bought to yield 6%: \(B_0 = P = 1147.20\). Interest in period 1: \(0.06 \times 1147.20 = \) 68.83. Write-down: \(80 - 68.83 = \) 11.17. New book value: \(B_1 = 1147.20 - 11.17 = \) 1136.03 — matching \(80\cdot a_9 + 1000\cdot v^9\) at 6%.
Same idea, but now a discount bond: 1000 par, 5% annual coupon, 10 periods, yield 7%. Pricing gives \(B_0 = P = 859.53\). Interest in period 1: \(0.07 \times 859.53 = 60.17\). Since the coupon (50.00) is smaller than the interest, the adjustment is negative: \(50.00 - 60.17 = \) ____ (this is the period's accumulation of discount, not a write-down).
Reveal the answer
\(50.00 - 60.17 = \) −10.17. The negative sign means the book value grows: \(B_1 = 859.53 - (-10.17) = \) 869.70, a step closer to \(C = 1000\) from below. Set r = 5% and i = 7% on the sliders above and step k to 1 to see the curve climbing instead of falling.
More info — checking a book value two ways
The recursion \(B_t = B_{t-1}(1+i) - Fr\) is fast to iterate, but you can always check any single \(B_t\) from scratch with the prospective formula \(B_t = Fr\cdot a_{n-t} + C\cdot v^{n-t}\) — the present value of whatever coupons and redemption are still left after period t. For the worked example above, \(B_1 = 80\cdot a_9 + 1000\cdot v^9\) at 6% comes out to 1136.03, matching the recursion exactly, because both are just the price of the bond as seen from a different starting date. See the UT Austin handout in Dive deeper below for a fully tabulated schedule.
Check your understanding
A 1000 par, 8% annual-coupon bond is bought to yield 6% (B₀ = 1147.20). What is period 1's write-down of premium?
In period 7 of a bond's amortization schedule, what should the interest column equal?
A bond's coupon rate is below its yield rate, so period 1's coupon-minus-interest adjustment comes out negative. What does that mean for the book value?
A bond is priced at a premium (P > C), so the coupon rate exceeds the yield. As the book value amortizes toward C over the life of the bond, what happens to each period's write-down?
Recap
- Each period's interest earned \(= i\cdot B_{t-1}\) — the yield rate times the PRIOR book value, never the face value and never the coupon rate.
- Principal adjustment \(= Fr - i\cdot B_{t-1}\): positive is a write-down of premium (book value falls), negative is an accumulation of discount (book value rises).
- Recursion: \(B_t = B_{t-1}(1+i) - Fr\), and always \(B_n = C\) at maturity.
- Any \(B_t\) can also be checked from scratch as \(Fr\cdot a_{n-t} + C\cdot v^{n-t}\) — the present value of the remaining cash flows.
Dive deeper
- UT Austin — Bond Amortization Schedule (actuarial course handout) Work through the interest and write-down columns of a bond amortization schedule.
- Marcel Finan — A Basic Course in the Theory of Interest (Exam FM study guide, PDF) Study amortization of premium and accumulation of discount via book-value recursions.
Sources
- The Bond Amortization Schedule