Bond Amortization

Bond amortization is the division of each coupon of a bond into interest earned at the yield rate and a principal adjustment that moves the book value from the price $P$ to the redemption value $C$ . The bond amortization schedule sets this out coupon by coupon, the way a loan's amortization schedule does for its payments.

$I_t = j\,B_{t-1}$

$PR_t = Fr - I_t = (Fr - Cj)\,v^{\,n-t+1}$

$B_t = B_{t-1} - PR_t$

$Fr$ is the coupon, $j$ the yield rate per coupon period at which the bond was bought, $n$ the number of coupons and $v = 1/(1+j)$ . $B_t$ is the book value just after the $t$ -th coupon, starting at $B_0 = P$ and ending at $B_n = C$ .
$PR_t > 0$ when the bond was bought at a premium: the book value is written down each period (amortization of premium). $PR_t < 0$ when it was bought at a discount: the book value is written up (accumulation of discount).
The principal adjustments form a geometric progression with ratio $1 + j$ and add up to the premium or discount, $\sum_{t=1}^{n} PR_t = P - C$ . The interest column adds up to the coupons less that amount, $\sum_{t=1}^{n} I_t = nFr - (P - C)$ .
One row can be found without the rest of the table: value $B_{t-1}$ at the original yield from the remaining coupons and redemption value, then take $I_t$ and $PR_t$ from it.

A Full Schedule for a Bond Redeemable Above Par A 3-year bond with face value $1{,}000$ pays 8% annual coupons and is redeemable at $1{,}050$ . It is bought to yield 7%. Construct the bond amortization schedule.
Answer The coupon is $Fr = 80$ and $Cj = 1{,}050(0.07) = 73.50$ , so the bond is bought at a premium:
 $\begin{align*} P &= C + (Fr - Cj)\,a_{\overline{3}|7\%} \\ &= 1{,}050 + 6.50(2.624316) \\ &= 1{,}067.058 \end{align*}$ 
$t$ Coupon Interest $I_t$ Principal adjustment $PR_t$ Book value $B_t$

0 $1{,}067.058$

1 $80$ $74.694$ $5.306$ $1{,}061.752$

2 $80$ $74.323$ $5.677$ $1{,}056.075$

3 $80$ $73.925$ $6.075$ $1{,}050.000$

Total $240$ $222.942$ $17.058$

Each $I_t$ is $7\%$ of the book value above it (e.g. $0.07 \times 1{,}067.058 = 74.694$ ). The adjustments grow by the factor $1.07$ and add up to the premium $P - C = 17.058$ , so the book value ends at the redemption value $1{,}050$ , not at the $1{,}000$ face value.

$t$	Coupon	Interest $I_t$	Principal adjustment $PR_t$	Book value $B_t$
0				$1{,}067.058$
1	$80$	$74.694$	$5.306$	$1{,}061.752$
2	$80$	$74.323$	$5.677$	$1{,}056.075$
3	$80$	$73.925$	$6.075$	$1{,}050.000$
Total	$240$	$222.942$	$17.058$

One Row of a Long Schedule A bond with face amount $1{,}000$ , redeemable at par, pays a coupon of 5% per period and has 20 coupons until maturity. It is bought to yield 6% per period. Find the interest earned and the principal adjustment in the 5th coupon, and the book value just after it.
Answer The book value just after the 4th coupon is the value of the 16 coupons and the redemption value still to come:
 $\begin{align*} B_4 &= 50\,a_{\overline{16}|6\%} + 1{,}000\,v^{16} \\ &= 50(10.105895) + 1{,}000(0.393646) \\ &= 898.94 \\ I_5 &= 0.06(898.94) \\ &= 53.94 \\ PR_5 &= 50 - 53.94 \\ &= -3.94 \\ B_5 &= 898.94 + 3.94 \\ &= 902.88 \end{align*}$ 
The bond was bought at a discount (price $885.30$ ), so the 5th coupon writes the book value up by $3.94$ . As a check, $(Fr - Cj)\,v^{16} = (50 - 60)(0.393646) = -3.94$ .