Accreted Value of a Zero-Coupon Bond Before Maturity

Why an Intermediate Value Matters

A zero-coupon bond makes no periodic interest payments. Its value therefore grows from its purchase price to a single redemption amount at maturity. Between those dates, an investor or analyst may need the bond’s accreted value: the value implied at that date by the original purchase price and yield, assuming the bond is carried forward under the original yield.

This value is useful for tracking the bond’s book value and for checking how much of the original discount has been accreted. It is not automatically the price at which the bond could be sold on that date. A sale price depends on the yield used to value the remaining cash flow at that time. The distinction is important in calculations aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Definition: Consider a zero-coupon bond bought at time 0 for price $P$, with redemption value $R$ payable at time $n$. If its original effective annual yield is $i$, its accreted value at time $t$, where $0 \leq t \leq n$, is the value obtained by accumulating $P$ to time $t$ at that original yield. Under annual effective accumulation, this value is $P(1+i)^t$. Equivalently, it is the value at time $t$ of the redemption payment discounted at the same yield over the remaining term.

The model assumes a single redemption payment, no interim cash flows, and a constant original effective annual yield used for the accumulation or discount calculation. For fractional times, use fractional exponents under the annual effective accumulation model. If the yield changes and the question asks for a current market price, use the yield at the valuation date for the remaining term instead of treating the original-yield accreted value as a sale price.

Two Equivalent Ways to Calculate the Value

Let $B_t$ denote the accreted value at time $t$. Starting from the purchase price, accumulate for $t$ years. Alternatively, discount the redemption value back for the $n-t$ years left until maturity. Since the original price and redemption amount satisfy the original yield equation, both approaches give the same result.

P = R(1+i)^{-n}

B_t=P(1+i)^t=R(1+i)^{-(n-t)}

In actuarial notation, $v=1/(1+i)$, so the equivalent discount expression is $B_t=Rv^{n-t}$. The exponent in the discount expression is the remaining term, not the elapsed term.

Key result: For a zero-coupon bond with original effective annual yield $i$, purchase price $P$, redemption value $R$, and maturity at time $n$, the accreted value at time $t$ is $P(1+i)^t$, or $R/(1+i)^{n-t}$. The equality holds because $P=R/(1+i)^n$.

Why the Two Expressions Agree

The reconciliation follows directly from the original pricing equation. The purchase price is the redemption value discounted over the full term. Accumulating that price for the elapsed time cancels the corresponding part of the full-term discount.

Write the original price equation.
At time 0, the only future payment is $R$ at time $n$, so $P=R(1+i)^{-n}$.

Accumulate to the valuation date.
Multiply the original price by the accumulation factor for the elapsed time $t$: $B_t=P(1+i)^t$.

Substitute the original price.
$B_t=R(1+i)^{-n}(1+i)^t=R(1+i)^{-(n-t)}$, which is discounting over the remaining term.

This is an algebraic identity under the stated model; it does not require the bond to be sold or the market yield to stay unchanged. It does require using the same original yield in both calculations. If one side uses the original yield and the other uses a new yield, the expressions need not agree.

Worked Calculations

Worked Example: Accreting a Bond and Reconciling to Redemption

A hypothetical zero-coupon bond is purchased for $10,000 and pays $13,310 at the end of three years. Its original effective annual yield is 10%. Find its accreted value at the end of year 1, then verify the result by discounting the redemption value over the remaining term.

Method. Accumulate the purchase price for one year at the original yield, then use the redemption-value expression as an independent check. The common mistake to avoid is discounting for the elapsed year instead of the two years remaining.

Step 1: Accumulate the purchase price. The elapsed time is one year, so

B_1=10{,}000(1.10)^1=10{,}000(1.10)=\$11{,}000

Step 2: Discount from redemption. There are $3-1=2$ years remaining. Thus

B_1=\frac{13{,}310}{(1.10)^2} =\frac{13{,}310}{1.21} =\$11{,}000

Check. The two independently calculated values agree. Also, the original pricing equation holds: $10{,}000(1.10)^3=10{,}000(1.331)=13{,}310$. The accreted value is therefore $11,000 at the end of year 1.

Worked Example: Finding the Value Near Maturity

A hypothetical bond has a redemption value of $8,640 payable in three years. Its purchase price is $5,000, and its original effective annual yield is 20%. Calculate the accreted value at the end of year 2 by both methods.

Method. Accumulate for the two elapsed years, and then check by discounting for the one remaining year. This example emphasizes that a bond can have a large accreted value close to maturity even though no payment has yet been made.

Step 1: Accumulate the price. The two-year accumulation factor is

$$ (1.20)^2=1.44 $$

Therefore,

B_2=5{,}000(1.44)=\$7{,}200

Step 2: Check using the remaining term. One year remains, so discount the redemption value for one year:

B_2=\frac{8{,}640}{1.20}=\$7{,}200

Step 3: Verify the original yield equation. Over the full three years, the accumulation factor is $1.20^3=1.728$. Hence $5{,}000(1.728)=\$8{,}640$, which matches the stated redemption value.

Check. The two methods give $7,200, and one more year of accumulation at 20% gives $7{,}200(1.20)=\$8{,}640$. The result is consistent with redemption at maturity.

Worked Example: Accreted Value Versus Current Market Price

A hypothetical zero-coupon bond is bought for $10,000 at an original effective annual yield of 10%. It pays $13,310 in three years. At the end of year 1, find (a) its accreted value under the original yield and (b) its market price if the yield for the remaining two years is then 5%.

Method. Keep the two valuations separate. The accreted value uses the original yield, while the market price uses the yield applicable to the remaining cash flow at the valuation date. A common mistake is to assume that a change in market yield resets the bond’s original-yield book value.

Step 1: Calculate the original-yield accreted value. One year has elapsed:

B_1=10{,}000(1.10)=\$11{,}000

Step 2: Calculate the market price at the new yield. The bond still pays $13,310 in two years. At a 5% effective annual yield for those remaining two years, its price is

\text{Market price at time 1} =\frac{13{,}310}{(1.05)^2} =\frac{13{,}310}{1.1025} =\$12{,}072.56

The last value is rounded to the nearest cent. The original-yield accreted value is $11,000, while the price based on the changed yield is $12,072.56.

Check. Accumulating $10,000 for one year at the original 10% yield gives $11,000. Discounting the remaining payment at 5% gives a different value because it answers a different question. The difference is not an inconsistency: one is an original-yield carrying value, and the other is a market valuation using a different yield.

Fractional Valuation Dates

The same relationship applies when the valuation date falls between integer anniversaries, provided the problem uses annual effective accumulation for fractional periods. If $t$ is fractional, use $(1+i)^t$ for elapsed time and $(1+i)^{n-t}$ for the remaining term. Do not round the elapsed or remaining time to a whole number of years.

For example, if a bond is valued at time $2.5$ in a six-year term, the elapsed period is $2.5$ years and the remaining period is $6-2.5=3.5$ years. Under the same yield,

B_{2.5}=P(1+i)^{2.5} =\frac{R}{(1+i)^{3.5}}

This fractional-exponent formula follows from the specified annual effective accumulation model. If a problem instead specifies a different convention for fractional periods, follow that convention rather than imposing this formula.

Common Timing and Interpretation Traps

Using the wrong term in the discount expression. At time $t$, the remaining term is $n-t$, so the redemption-value expression is $R/(1+i)^{n-t}$, not $R/(1+i)^t$.
Mixing elapsed and remaining time. Accumulation from purchase uses $t$; discounting from redemption uses $n-t$. Both must refer to the same valuation date.
Changing the yield without changing the question. Original-yield accreted value uses the original yield. A market price after a yield change discounts the remaining payment at the market yield then applicable.
Treating accretion as a cash payment. A zero-coupon bond has no interim coupon. The accreted value records the value implied by the yield calculation; it is not cash received by the holder at that date.
Rounding too early. Retain unrounded values through the calculation, especially when comparing the accumulation and discount methods. Round the final monetary amount as requested.

A useful final check is to move one period forward or backward. Under the original yield, a value at time $t$ must accumulate to the next date by the relevant accumulation factor. At maturity, the value must equal the redemption amount. These checks often reveal an incorrect time exponent or an inconsistent yield.

Check Your Understanding

For each problem, use the stated effective annual yield and retain full precision until the final answer.

A hypothetical zero-coupon bond costs $6,400, earns an original effective annual yield of 8%, and is valued three years after purchase. What is its accreted value? (Answer: $8,062.16)
A bond pays $15,000 at the end of year 5 and has an original effective annual yield of 10%. What is its accreted value at the end of year 2? Verify by using the remaining term. (Answer: $11,269.72)
A zero-coupon bond is purchased for $9,000 at an effective annual yield of 5%. Under annual effective accumulation for fractional periods, what is its value after 2.5 years? (Answer: $10,167.54)
A hypothetical bond costs $7,000, earns an original effective annual yield of 10%, and matures in four years. Find its accreted value at the end of year 3, and state the redemption value implied by the original yield. (Answer: Accreted value $9,317.00; redemption value $10,248.70)
A zero-coupon bond costs $10,000 and pays $13,310 in three years. Its original effective annual yield is 10%, but at the end of year 1 the market yield for the remaining two years is 5%. Find both the original-yield accreted value and the market price at that date. (Answer: Accreted value $11,000.00; market price $12,072.56)

What You'll Learn