Discount Accretion Through The Recurrence

Why a Discount Bond’s Book Value Increases

A bond bought for less than its redemption amount is purchased at a discount. At each coupon date, its book value reflects two opposing effects: interest earned on the amount currently carried, and the coupon paid to the investor. For a discount bond, interest earned exceeds the coupon, so the unpaid difference increases the book value. The book-value recurrence makes that increase explicit.

The calculation depends on matching the yield rate to the time between coupon dates. A yield stated per coupon period can be used directly. If the yield is quoted as a nominal annual rate convertible $m$ times per year and coupons are paid $m$ times per year, the effective rate per coupon period is $i^{(m)}/m$.

Definition: Consider a level-coupon bond with redemption amount $R$, coupon payment $C$ at each of $n$ coupon dates, and effective yield $i$ per coupon period. Let $B_k$ be the book value immediately after the coupon at date $k$, but before redemption of principal. The initial book value $B_0$ is the purchase price at a coupon date. At maturity, $B_n=R$, immediately after the final coupon and before the redemption payment.

The Coupon-Date Recurrence

During one coupon period, the previous book value $B_{k-1}$ earns interest at the yield rate $i$. The amount of interest is $iB_{k-1}$. The accumulated balance is therefore $B_{k-1}(1+i)$. Once the coupon $C$ is paid, the remaining balance is the next book value.

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

Rearranging the recurrence separates the change in book value from the interest and coupon amounts:

B_k-B_{k-1}=iB_{k-1}-C

Thus the increase in book value is not the coupon, nor is it the full interest earned. It is the excess of interest earned over the coupon paid. In the final period, the recurrence gives $B_n=R$; the redemption payment then pays off that balance.

Theorem: If $iB_{k-1}>C$, the book value increases during period $k$ by $iB_{k-1}-C$. If $iB_{k-1}=C$, it is unchanged. If $iB_{k-1}<C$, it decreases.

Why Discount Accretion Is Positive

For a level-coupon bond, the coupon rate per period is $C/R$, while the yield per period is $i$. A bond is at a discount when $C/R<i$, or equivalently $C<iR$. The recurrence then shows why the book value increases at each period, not just overall.

At the purchase date, the bond’s price is the present value of its coupons and redemption amount at yield $i$:

B_0=C a_{\overline{n}|}+Rv^n,\qquad v=\frac{1}{1+i}

Using $a_{\overline{n}|}=(1-v^n)/i$, subtract $C/i$ from both sides. This gives a useful comparison between the starting book value and the amount $C/i$:

B_0-\frac{C}{i} =\frac{C(1-v^n)}{i}+Rv^n-\frac{C}{i} =v^n\left(R-\frac{C}{i}\right)

When $C<iR$, the quantity $R-C/i$ is positive. Since $v^n>0$, it follows that $B_0>C/i$. The recurrence also gives

B_k-\frac{C}{i} =(1+i)\left(B_{k-1}-\frac{C}{i}\right)

Because $1+i>0$, a positive difference $B_{k-1}-C/i$ remains positive at the next date. Therefore $B_{k-1}>C/i$ at every coupon date before maturity, and hence $iB_{k-1}>C$. This proves that the book value increases in every period of a discount bond under the stated level-coupon model.

Accumulate the current balance.
Interest for the period is $iB_{k-1}$, so the balance before the coupon is $B_{k-1}(1+i)$.

Subtract the coupon.
The coupon payment is $C$, leaving $B_k=(1+i)B_{k-1}-C$.

Interpret the change.
The book-value increase is $iB_{k-1}-C$. For a discount bond this amount is positive; at maturity the remaining balance is $R$.

Worked Recurrence Calculations

Worked Example: Annual Coupons on a Discount Bond

A hypothetical bond has redemption amount $R=\$1{,}000$, annual coupon $C=\$40$, and three years to maturity. Its effective annual yield is $i=6\%$. Find the purchase price, the first two coupon-date book values, and the discount accretion in each period. Check the maturity balance.

Method. First value the remaining cash flows at the yield to obtain $B_0$. Then apply the coupon-date recurrence. The coupon rate is $40/1{,}000=4\%$, below the 6% yield, so this is a discount bond.

The discount factor is $v=1/1.06$. The initial price is

B_0=40a_{\overline{3}|}+1{,}000v^3 =40\left(\frac{1-(1.06)^{-3}}{0.06}\right)+1{,}000(1.06)^{-3} =\$946.539761

In the first year, interest earned is $0.06(946.539761)=\$56.792386$. Subtract the $\$40$ coupon:

B_1=946.539761+56.792386-40=\$963.332147

The first-period accretion is $56.792386-40=\$16.792386$. In the second year, interest earned is $0.06(963.332147)=\$57.799929$, so

B_2=963.332147+57.799929-40=\$981.132076

The second-period accretion is $57.799929-40=\$17.799929$. As a maturity check, the third-year interest is $0.06(981.132076)=\$58.867925$, giving $B_3=981.132076+58.867925-40=\$1{,}000$. The recurrence therefore reaches the stated redemption amount. Rounded to cents, $B_1=\$963.33$, $B_2=\$981.13$, and the maturity balance is $\$1{,}000.00$. The accretion grows because each period’s interest is earned on a larger balance.

Common mistake avoided: Do not calculate interest as $iR$. Interest is earned on the current book value $B_{k-1}$, not on the redemption amount. In the first year, using $0.06(1{,}000)-40=\$20$ would incorrectly produce $B_1=\$966.54$, rather than $\$963.33$.

Worked Example: Quarterly Coupons and a Nominal Yield

A hypothetical bond pays $\$15$ at the end of each quarter and $\$1{,}000$ redemption at the end of the fourth quarter. Its nominal annual yield is $i^{(4)}=8\%$, convertible quarterly. Calculate its price, the first two book values, and the quarterly accretion.

Method. Convert the nominal annual yield to the effective rate per coupon period before pricing or applying the recurrence. Here $i=0.08/4=0.02$ per quarter. The coupon rate per quarter is $15/1{,}000=1.5\%$, below the 2% quarterly yield.

The price is

B_0=15a_{\overline{4}|}+1{,}000(1.02)^{-4} =15\left(\frac{1-(1.02)^{-4}}{0.02}\right)+1{,}000(1.02)^{-4} =\$980.961356

For the first quarter, interest earned is $0.02(980.961356)=\$19.619227$. After the $\$15$ coupon, $B_1=980.961356+19.619227-15=\$985.580584$. The accretion is $\$4.619227$.

For the second quarter, interest earned is $0.02(985.580584)=\$19.711612$. Therefore $B_2=985.580584+19.711612-15=\$990.292196$, and the second-quarter accretion is $\$4.711612$. Rounded to cents, $B_0=\$980.96$, $B_1=\$985.58$, and $B_2=\$990.29$. The positive increases are consistent with the bond’s discount.

Common mistake avoided: Applying 8% in each quarter would use the nominal annual rate as though it were a quarterly effective rate. The correct recurrence uses 2% per quarter.

Worked Example: Inferring the Yield From Consecutive Book Values

A bond’s book value immediately after one coupon is $\$913.223140$. The coupon paid at the next date is $\$50$, and the book value immediately after that coupon is $\$954.545455$. Use the recurrence to infer the effective yield per period, calculate the interest earned and accretion, and verify the next balance if the following coupon is also $\$50$ and the redemption amount is $\$1{,}000$.

Method. Rearrange the recurrence to solve for the yield. The interest accumulated before the coupon must equal the next book value plus the coupon:

1+i=\frac{B_1+C}{B_0} =\frac{954.545455+50}{913.223140} \approx 1.10

Thus the effective yield is $i=10\%$ per period. Interest earned in the first period is $0.10(913.223140)=\$91.322314$. After paying $\$50$, the balance is $913.223140+91.322314-50=\$954.545454$, agreeing with the given next book value up to displayed precision. The first-period accretion is $\$41.322314$.

In the next period, interest earned is $0.10(954.545455)=\$95.454546$. After the $\$50$ coupon, the balance is $954.545455+95.454546-50=\$1{,}000.000001$, which is $\$1{,}000.00$ to the nearest cent. This verifies the maturity balance; the tiny displayed difference is due to rounding the supplied book values. The inferred 10% yield also exceeds the coupon rate $50/1{,}000=5\%$, consistent with a discount.

Common mistake avoided: The coupon is added when solving for the accumulated balance before payment: $B_0(1+i)=B_1+C$. Omitting $C$ would infer the wrong yield.

Reading the Recurrence Correctly

The recurrence is an accounting identity at the bond’s yield. It describes how the original book value is carried forward when the contractual coupon is removed at each date. It does not say that the bond’s market price must change by the same amount. If market yields change, a market valuation can change even though the original-yield book-value recurrence remains based on the original yield.

Keep the following quantities distinct:

Interest earned: $iB_{k-1}$, calculated on the opening book value.
Coupon paid: $C$, determined by the bond’s contract.
Discount accretion: $iB_{k-1}-C$, the increase in book value for that period.
Redemption: $R$, paid after the final coupon when the maturity book value is $B_n=R$.

A common exam trap is to identify the total interest earned with the discount accretion. They are not equal when a coupon is paid: the coupon is the portion of the accumulated balance distributed to the investor, while only the excess interest remains in the bond’s book value. Another trap is to compare the coupon with yield without matching their time units. Compare $C/R$ with the effective yield per coupon period, not automatically with a quoted annual rate.

Key takeaway: For a discount bond, the coupon per period is less than the yield earned on the redemption amount. The coupon-date recurrence $B_k=(1+i)B_{k-1}-C$ shows the period’s accretion precisely: $B_k-B_{k-1}=iB_{k-1}-C$. Use the effective yield per coupon period and calculate interest on the opening book value.

Check Your Understanding

For each problem, use the stated effective yield per period and apply the recurrence where needed.

A bond has $R=\$1{,}200$, annual coupon $C=\$36$, two years remaining, and effective annual yield $i=5\%$. Find its price and its book value immediately after the first coupon. (Answer: price $\$1{,}155.37$; first book value $\$1{,}177.14$.)
At a coupon date, a bond’s book value is $\$4{,}950$. The next coupon is $\$60$, and the effective yield per period is $1.5\%$. Find the interest earned, accretion, and next book value. (Answer: interest $\$74.25$; accretion $\$14.25$; next book value $\$4{,}964.25$.)
A bond has opening book value $\$980$, coupon $\$30$, and effective yield $4\%$ per period. Calculate the next book value and the period’s accretion. (Answer: next book value $\$989.20$; accretion $\$9.20$.)
A quarterly bond has opening book value $\$980$, quarterly coupon $\$15$, and nominal annual yield $i^{(4)}=8\%$, convertible quarterly. Find the quarterly interest earned, accretion, and next book value. (Answer: interest $\$19.60$; accretion $\$4.60$; next book value $\$984.60$.)
A bond has opening book value $\$900$, next coupon $\$35$, and next book value $\$910$. Infer the effective yield per period, then calculate the interest earned and accretion. (Answer: yield $5\%$; interest $\$45$; accretion $\$10$.)

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