Pricing Zero-Coupon Bonds with Discount Factors

Why a Zero-Coupon Bond Has One Price-Deriving Cash Flow

A zero-coupon bond pays no periodic coupons. Under the standard model used here, an investor pays a price at time 0 and receives one specified redemption amount at the bond’s maturity. Pricing the bond therefore requires identifying that payment, its date, and the discount factor from that date back to time 0.

The central calculation is short, but its timing matters: multiply the redemption amount by the discount factor for the bond’s maturity. The discount factor must match the time to the redemption payment and the interest basis specified in the problem. A discount factor for another date, or a factor applied in the wrong direction, gives the wrong price.

Definition: A zero-coupon bond is a bond that makes no coupon payments before maturity and pays a specified redemption amount $R$ at maturity. If its maturity is $n$ years after the purchase date and $v^n$ is the discount factor from year $n$ to year 0 on the stated interest basis, its price at time 0 is $P=Rv^n$. More generally, if the applicable discount factor to maturity is denoted by $D(0,n)$, its price is $P=R D(0,n)$.

Here, $P$ is the purchase price and $R$ is the amount paid at maturity. Neither the definition nor the formula assumes that $P$ equals the bond’s face amount. Under positive interest, the price is typically less than the future redemption amount; the calculation determines the price from the stated discount factor.

Deriving the Price Formula

Suppose the interest model gives an accumulation factor $1/v^n$ from time 0 to time $n$. An amount invested as $P$ at time 0 then accumulates to $P/v^n$ at maturity. To fund the bond’s redemption payment, that accumulated amount must equal $R$. Thus $P/v^n=R$, and multiplying both sides by $v^n$ gives the price formula.

\frac{P}{v^n}=R \quad\Longrightarrow\quad P=Rv^n

For a constant effective annual interest rate $i$, with annual compounding and an integer maturity of $n$ years, $v=1/(1+i)$. Substitution gives $P=R/(1+i)^n$. This version requires that the stated annual effective rate applies throughout the term and that the maturity is measured in whole years. If the problem instead supplies a maturity discount factor directly, use that factor; do not replace it with a factor based on an unstated rate.

v=\frac{1}{1+i}, \qquad v^n=\frac{1}{(1+i)^n}, \qquad P=R v^n=\frac{R}{(1+i)^n}

The formula applies to the bond’s value at the date from which the stated discount factor is measured. It assumes there are no payments between that valuation date and maturity. If a security makes interim payments, those payments are additional cash flows and cannot be omitted from its value. This tutorial focuses on a single-payment bond.

Worked Applications

Worked Example: Pricing a Bond at a Stated Annual Effective Rate

A hypothetical zero-coupon bond will redeem for $1,000 exactly four years from today. The valuation basis is a constant effective annual interest rate of 5%, compounded annually. Find the price today and the dollar difference between redemption value and price. Give the price to the nearest cent.

Method. There is one payment, $1,000, at time 4. Use the discount factor for four years, not the one-year factor by itself.

First calculate the annual discount factor:

v=\frac{1}{1.05}=0.95238095\ldots

The four-year discount factor is:

v^4=\frac{1}{(1.05)^4} =\frac{1}{1.21550625} =0.8227024748\ldots

Multiply the redemption value by this factor:

P=1{,}000(0.8227024748\ldots) =822.7024748\ldots

Rounded to the nearest cent, the price is $822.70. The difference between the redemption amount and price is $1{,}000-822.70=$ $177.30. As a check, accumulating the unrounded price for four years gives $822.7024748\ldots(1.05)^4=1{,}000$, as required. A common mistake is to discount for only one year because the annual rate is quoted; the bond’s payment is four years away, so the exponent must be 4.

Worked Example: Scaling a Discounted Price to the Bond’s Face Amount

A hypothetical bond pays $5,000 at maturity in three years and makes no earlier payments. The applicable constant effective annual interest rate is 4%, compounded annually. Determine the time-0 price to the nearest cent, then find the price per $100 of redemption value.

Method. Find the price of the whole bond first. The per-$100 amount is the same price calculation scaled to $100 of redemption value.

Calculate the three-year accumulation factor and discount factor:

(1.04)^3=1.124864, \qquad v^3=\frac{1}{1.124864}=0.88899636\ldots

Apply the maturity discount factor to the $5,000 redemption payment:

P=5{,}000(0.88899636\ldots) =4{,}444.9818\ldots

The price is $4,444.98. For $100 of redemption value, the corresponding price is:

\frac{100}{5{,}000}(4{,}444.9818\ldots) =88.899636\ldots

Thus the price per $100 of redemption value is $88.90, rounded to the nearest cent. As a check, the price per $100 is $100(0.88899636\ldots)$, consistent with applying the same discount factor to $100. The common mistake is to report $88.90 as the price of the entire $5,000 bond; that is the scaled price per $100, not the total price.

Worked Example: Pricing a Portfolio With Two Redemption Dates

A hypothetical portfolio consists of one zero-coupon bond that redeems for $2,000 in three years and one that redeems for $3,000 in five years. The applicable discount factors to time 0 are $D(0,3)=0.88$ and $D(0,5)=0.76$, respectively. Find the portfolio’s time-0 price. Assume both discount factors apply to the stated redemption dates and that neither bond makes interim payments.

Method. Price each bond using its own maturity discount factor, then add the two prices. Do not apply one bond’s factor to both payments.

The first bond’s price is:

P_1=2{,}000(0.88)=1{,}760

The second bond’s price is:

P_2=3{,}000(0.76)=2{,}280

The portfolio price is the sum:

P_{\text{portfolio}}=P_1+P_2 =1{,}760+2{,}280 =4{,}040

The portfolio is worth $4,040 at time 0. As a check, its two separate discounted payments contribute $1,760 and $2,280, which sum to the stated total. The common mistake is to discount the combined $5,000 redemption amount with either 0.88 or 0.76. Because the payments occur at different dates, each requires its own applicable discount factor.

Applying the Formula Carefully

Before calculating, write down the bond’s cash-flow timeline. For a zero-coupon bond, the timeline has a purchase price at time 0 and one redemption payment at maturity. Then identify the discount factor from that maturity date to the valuation date. These steps make the formula’s inputs explicit and help distinguish the purchase price from the future payment.

Identify the redemption amount.
Use the amount actually payable at maturity, $R$, as specified in the problem. Do not substitute the purchase price or assume the two amounts are equal.

Identify the time to redemption.
Measure the period from the valuation date to the one maturity payment. The discount factor must correspond to this period.

Select the applicable discount factor.
Use the factor supplied in the question or, under a stated constant effective annual rate and whole-year maturity, calculate $v^n=1/(1+i)^n$.

Multiply and check the result.
Calculate $P=R D(0,n)$. Check that the price and redemption amount have the same currency units and that the factor belongs to the correct maturity.

The formula is linear in the redemption amount: if the redemption amount is multiplied by a constant, the price is multiplied by the same constant, provided the maturity and discount factor stay unchanged. It is also the basis for valuing a collection of zero-coupon payments: value each payment at its own date and add the resulting present values. That addition works because the portfolio’s total value is the sum of the values of its separate cash flows.

Common Mistakes and Exam Traps

A frequent error is to treat discounting as subtracting interest from the redemption amount. Compound-interest discounting instead divides by the accumulation factor, or equivalently multiplies by the discount factor. For example, at a 5% effective annual rate over four years, subtracting $4(5\%)$ of $1,000 would give $800. That is not the correct compound-interest price; the correct calculation is $1{,}000/(1.05)^4=$$822.70.

Using the wrong time period. A four-year payment needs a four-year discount factor. The annual rate alone is not the discount factor for the entire term.
Reversing accumulation and discounting. Multiplying a future redemption amount by $(1+i)^n$ moves it further forward in time. To find its value today, divide by $(1+i)^n$.
Discounting to the wrong date. The price formula gives a value at the date from which the discount factor is measured. If that date is not the purchase date in the problem, the result is not yet the requested price.
Confusing a quoted unit price with a total price. A price per $100 of redemption value must be scaled to the full redemption amount when the question asks for the whole bond’s cost.
Using one factor for payments at different maturities. In a portfolio, match every redemption amount to its own maturity discount factor before adding values.

Key takeaway: A zero-coupon bond has one payment to discount. At time 0, its price equals the maturity redemption amount multiplied by the discount factor from maturity to time 0: $P=R D(0,n)$. Correct pricing depends on matching that factor to the payment date and the stated interest basis.

Check Your Understanding

For each problem, identify the applicable maturity discount factor before calculating.

A hypothetical zero-coupon bond redeems for $2,000 in two years. Its applicable discount factor is 0.94. Find its time-0 price. (Answer: $1,880)
A bond with no interim payments redeems for $10,000 in five years. The applicable discount factor is 0.72. Find its price. (Answer: $7,200)
A zero-coupon bond pays $4,200 in two years. The valuation basis is a constant effective annual interest rate of 5%, compounded annually. Find its price to the nearest cent. (Answer: $3,809.52)
A portfolio contains one zero-coupon bond redeeming for $3,000 in two years and another redeeming for $5,000 in four years. Their applicable discount factors are 0.90 and 0.70, respectively. Find the portfolio price. (Answer: $6,200)
A hypothetical zero-coupon bond has a redemption value of $2,500 and a maturity discount factor of 0.84. Find its price per $100 of redemption value and its total price. (Answer: $84.00 per $100; total price $2,100)

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