Premium Amortization Through The Recurrence

How the Recurrence Amortizes a Bond Premium

A premium bond is purchased for more than its redemption amount. That difference does not disappear in one step: as coupons are paid, the book value is reduced by the part of each coupon that exceeds the effective interest earned for the period. The coupon-date recurrence makes this accounting relationship explicit and gives a reliable way to follow the carrying value from one period to the next.

The key distinction is between the coupon, which is the cash paid to the bondholder, and effective interest, which is calculated using the yield and the book value at the start of the period. Their difference is the period’s premium amortization. The method applies when book value is carried at the original yield and coupon periods and yield periods are aligned.

Definition: Let $B_k$ be the book value immediately after the $k$th coupon payment, $C$ the level coupon paid at the end of each period, and $i$ the effective yield per coupon period. The effective interest for period $k$ is $iB_{k-1}$. The premium amortization for that period is $C-iB_{k-1}$, and the book value recurrence is $B_k=B_{k-1}(1+i)-C$.

Why the Recurrence Holds

Between two coupon dates, the opening book value earns interest at the effective yield. Immediately before the coupon is paid, the value is therefore $B_{k-1}(1+i)$. The coupon $C$ is then paid out, leaving the closing book value. Subtracting the coupon gives the recurrence.

B_k=B_{k-1}(1+i)-C

Rearrange the recurrence by separating the interest earned from the opening balance:

\begin{aligned} B_k &=B_{k-1}+iB_{k-1}-C\\ &=B_{k-1}-\bigl(C-iB_{k-1}\bigr). \end{aligned}

Thus the book value falls by $C-iB_{k-1}$ when the coupon exceeds effective interest. For a premium bond valued at its original yield, that excess is the period’s premium amortization. The recurrence uses the opening book value as the base for effective interest, not the redemption amount or the closing book value.

Key takeaway: For a premium bond, the period’s premium amortization is coupon paid minus effective interest earned. The closing book value equals the opening book value minus that amortization.

Worked Coupon-Date Calculations

For a standard level-coupon bond, the coupon-date book value can be obtained by discounting the remaining coupons and redemption payment at the yield. The recurrence then carries that value forward one period at a time. In the examples below, each yield is an effective rate per annual coupon period, and book values are measured immediately after coupons unless stated otherwise.

Worked Example: First-Year Amortization on a Two-Year Bond

A hypothetical bond has a redemption amount of $1,000, annual coupons of $120, and an effective annual yield of 10%. It has two years remaining at purchase. Find its purchase price, first-year premium amortization, and book value immediately after the first coupon. Round final dollar amounts to the nearest cent.

Method. First value the two remaining cash flows at the yield; then use the recurrence. This avoids treating the whole coupon as interest or calculating interest on the redemption amount.

The purchase-date book value is the present value of the $120 coupon at the end of year 1 and the $1,120 coupon-plus-redemption payment at the end of year 2:

\begin{aligned} B_0 &=\frac{120}{1.10}+\frac{1{,}120}{1.10^2}\\ &=109.090909\ldots+925.619835\ldots\\ &=1{,}034.710744\ldots \end{aligned}

Since $1,034.710744 is greater than the $1,000 redemption amount, this is a premium bond. First-year effective interest is $0.10B_0$:

\begin{aligned} \text{Interest}_1&=0.10(1{,}034.710744\ldots)=103.471074\ldots,\\ \text{Amortization}_1&=120-103.471074\ldots=16.528926\ldots,\\ B_1&=1{,}034.710744\ldots(1.10)-120\\ &=1{,}018.181818\ldots. \end{aligned}

The first-year premium amortization is $16.53, and the book value after the first coupon is $1,018.18. As a check, $B_0-16.528926\ldots=1{,}018.181818\ldots$, matching the recurrence. The calculation avoids the mistake of using the $1,000 redemption amount as the interest base.

Worked Example: Following Amortization Through Maturity

A hypothetical three-year bond pays an annual coupon of $150 and redeems for $1,000. Its effective annual yield is 10%. Find the initial book value and the premium amortization and closing book value in each year. Round final dollar amounts to the nearest cent.

Method. Discount each remaining payment to find $B_0$, then apply $B_k=B_{k-1}(1.10)-150$ without rounding intermediate values. This avoids compounding rounding errors and makes the maturity check exact before rounding.

\begin{aligned} B_0 &=\frac{150}{1.10}+\frac{150}{1.10^2}+\frac{1{,}150}{1.10^3}\\ &=136.363636\ldots+123.966942\ldots+864.012021\ldots\\ &=1{,}124.342600\ldots \end{aligned}

Year 1 effective interest is $0.10(1{,}124.342600\ldots)=112.434260\ldots$. The coupon exceeds that interest by $150-112.434260\ldots=37.565740\ldots$, so:

\begin{aligned} B_1 &=1{,}124.342600\ldots(1.10)-150\\ &=1{,}086.776860\ldots. \end{aligned}

In year 2, effective interest is $0.10(1{,}086.776860\ldots)=108.677686\ldots$. Amortization is $150-108.677686\ldots=41.322314\ldots$, giving:

B_2=1{,}086.776860\ldots(1.10)-150=1{,}045.454545\ldots.

In year 3, effective interest is $0.10(1{,}045.454545\ldots)=104.545455\ldots$. Amortization is $150-104.545455\ldots=45.454545\ldots$, and:

B_3=1{,}045.454545\ldots(1.10)-150=1{,}000.

The annual amortizations are $37.57, $41.32, and $45.45; the corresponding closing book values are $1,086.78, $1,045.45, and $1,000.00. The maturity value checks: after the final coupon and redemption, no book value remains. Before rounding, the amortizations sum to $37.565740\ldots+41.322314\ldots+45.454545\ldots=124.342600\ldots$, equal to $B_0-1{,}000$.

Worked Example: Two Periods from a Known Carrying Value

At a coupon date, a hypothetical bond has a book value of $12,400. It pays annual coupons of $700, and its carrying value is calculated at an effective annual yield of 4.5%. Find the effective interest, premium amortization, and closing book value for each of the next two periods. Do not round until the final amounts.

Method. Apply the coupon-date recurrence separately in each period, using the latest closing book value as the next opening value. This avoids incorrectly calculating both years’ interest on the original $12,400.

For the first period, effective interest is $0.045(12{,}400)=558$. The coupon exceeds interest by $700-558=142$, so the book value falls to $12{,}400-142=12{,}258$. Equivalently, $12{,}400(1.045)-700=12{,}258$.

For the second period, use the new opening value. Effective interest is $0.045(12{,}258)=551.61$. Amortization is $700-551.61=148.39$, so the closing book value is $12{,}258-148.39=12{,}109.61$. Equivalently, $12{,}258(1.045)-700=12{,}109.61$.

The two periods’ effective interest amounts are $558.00 and $551.61; their amortizations are $142.00 and $148.39; and the book value after the second coupon is $12,109.61. As a check, the total decrease of $290.39 equals $142+148.39$.

Premium Direction and Common Exam Traps

When the coupon rate exceeds the yield per coupon period, a standard level-coupon bond is priced above its redemption amount. Its effective interest is calculated on a book value that is above redemption, while the coupon is set by the larger coupon rate; the coupon therefore exceeds effective interest and the premium is amortized. As the book value declines, the interest amount generally declines too, so the amount of each coupon allocated to amortization generally increases.

A frequent error is to identify the coupon minus the interest amount but calculate that interest on redemption value. In the three-year example, that shortcut gives $150-0.10(1{,}000)=50$ in year 1. The correct interest base is the opening book value, $1,124.342600..., so the correct amortization is $150-112.434260\ldots=37.565740\ldots$, or $37.57. The shortcut overstates the first-year amortization by $12.43.

Use the opening book value. Effective interest for period $k$ is $iB_{k-1}$, not $iB_k$ and not $iR$, where $R$ is redemption.
Keep the cash-flow sign straight. The coupon leaves the bond account, so it is subtracted after interest is added.
Do not round each period prematurely. Carry sufficient precision through the recurrence and round the requested dollar results at the end.
Keep the rate and period aligned. The rate in the recurrence must be the effective yield for one coupon period. If the coupon frequency and quoted yield convention differ, convert consistently before using the recurrence.
Do not reuse a market yield after a yield change. This recurrence carries book value at the yield used to establish the carrying basis; a changed market yield does not by itself reset that basis.

Practical check: After a coupon payment, compare the new book value with the old one. When the coupon exceeds effective interest, the difference $C-iB_{k-1}$ must be positive and the book value must decrease by exactly that amount.

Check Your Understanding

Use the coupon-date recurrence, keeping unrounded values until the final answers.

A two-year bond has redemption of $1,000, annual coupons of $125, and an effective annual yield of 10%. Find its purchase price and first-year premium amortization. (Answer: price $1,043.39; amortization $20.66.)
A two-year bond has redemption of $2,000, annual coupons of $260, and an effective annual yield of 10%. Find the purchase price and the book value after the first coupon. (Answer: price $2,104.13; book value $2,054.55.)
A two-year bond has redemption of $5,000, annual coupons of $350, and an effective annual yield of 6%. Find the purchase price, first-year effective interest, and first-year amortization. (Answer: price $5,091.67; effective interest $305.50; amortization $44.50.)
At a coupon date, a bond’s carrying value is $2,500. The next coupon is $175 and the effective yield per period is 6%. Find effective interest, amortization, and the new book value. (Answer: effective interest $150.00; amortization $25.00; new book value $2,475.00.)
A bond has opening book value $8,400, coupon $600, and effective yield per coupon period 5%. Find the first period’s amortization and closing book value, then the second period’s amortization and closing book value if the coupon and yield remain unchanged. (Answer: first amortization $180.00 and closing value $8,220.00; second amortization $189.00 and closing value $8,031.00.)

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