Zero-Coupon Pricing and Yield Problem Workshop

One Cash Flow, Several Possible Yield Conventions

A zero-coupon bond makes no periodic coupon payments. Its value is determined by one future redemption payment and the yield used to discount that payment. The calculations become more demanding when a problem combines different yield conventions, a fractional term, and a valuation date before maturity. A timeline helps keep the cash flow, elapsed time, and remaining time distinct.

The key is to state the accumulation model before calculating. An effective annual yield compounds once per year. A nominal annual rate convertible $m$-thly compounds $m$ times per year, at a rate of $i^{(m)}/m$ per conversion period. If a fractional term is given, the question must also specify, or the solution must state, how fractional periods are treated. In the examples below, nominal-rate accumulation is extended to fractional numbers of conversion periods using the factor $(1+i^{(m)}/m)^{mt}$.

Definition: A zero-coupon bond with redemption value $F$, payable at time $T$, has no other cash flows. Under an effective annual yield $i$, its price at time 0 is $P=F(1+i)^{-T}$. Under a nominal annual yield $i^{(m)}$, convertible $m$-thly, and the stated fractional-period convention, its price is $P=F(1+i^{(m)}/m)^{-mT}$. The yield must be positive for these formulas to describe discounting to a price below redemption.

These formulas assume a fixed yield and a single certain redemption payment. They do not account for default, embedded call rights, taxes, or transaction costs. If the yield is quoted using a different convention, first translate it into the accumulation factor that applies over the relevant term.

Pricing and Yield Equations

The valuation equation comes directly from equating the price at time 0, accumulated at the specified yield until redemption, to the redemption payment. In the effective annual model, the accumulated value of $P$ after $T$ years is $P(1+i)^T$. Setting that equal to $F$ and solving for $P$ gives the price formula. Solving the same equation for the yield gives

P(1+i)^T=F \qquad\Longrightarrow\qquad P=\frac{F}{(1+i)^T}, \qquad i=\left(\frac{F}{P}\right)^{1/T}-1.

For a nominal annual rate convertible $m$-thly, the accumulation over $T$ years is $(1+i^{(m)}/m)^{mT}$. Thus the corresponding price and rate equations are

P=\frac{F}{(1+i^{(m)}/m)^{mT}}, \qquad i^{(m)} = m\left[\left(\frac{F}{P}\right)^{1/(mT)}-1\right].

The effective annual rate equivalent to a nominal rate convertible $m$-thly satisfies

1+i=\left(1+\frac{i^{(m)}}{m}\right)^m.

At an intermediate time $t$, the value under the same yield convention is the present value, at time $t$, of the remaining redemption payment. For an effective annual yield, this is $V_t=F/(1+i)^{T-t}$. The elapsed time $t$ determines where the valuation date sits on the timeline; the exponent uses the remaining term, $T-t$.

A Reliable Timeline and Valuation Process

Mark the cash flow.
Write the redemption value $F$ at maturity $T$. A zero-coupon bond has no intervening coupon amounts.

Identify the yield convention.
Record whether the quote is effective annual, nominal convertible $m$-thly, or another specified convention.

Express time in the model's units.
For nominal $m$-thly compounding, the number of periods over $T$ years is $mT$, including any fractional periods allowed by the stated model.

Write the valuation equation before calculating.
Set the price accumulated to maturity equal to the redemption value, or discount the redemption value back to the valuation date.

Check the result.
For a positive yield and $t<T$, an intermediate value should be below $F$ and above the original price, assuming the yield remains unchanged.

Worked Example: Pricing with a Fractional Term and Semiannual Yield

A hypothetical zero-coupon bond will redeem for $1,000 in 3.5 years. Its yield is a nominal annual rate of 6% convertible semiannually. Find the price, the equivalent effective annual yield, and the value 1.5 years after purchase. Assume the nominal rate is applied over fractional semiannual periods as specified by $(1+i^{(2)}/2)^{2t}$.

Method and timeline. There is one cash flow: $1,000 at $t=3.5$. The semiannual rate is $0.06/2=0.03$, and maturity is $2(3.5)=7$ semiannual periods away.

0 \quad \xrightarrow{\text{price }P} \quad 1.5 \quad \longrightarrow \quad 3.5:\ \$1{,}000

The time-0 valuation equation is $P(1.03)^7=1{,}000$. The accumulation factor is $1.03^7=1.2298738654$, so

P=\frac{1{,}000}{1.03^7} =\frac{1{,}000}{1.2298738654} =\$813.09.

The equivalent effective annual yield is $(1.03)^2-1=0.0609$, or 6.09%. At $t=1.5$, two years remain, equal to four semiannual periods. The intermediate value is therefore

V_{1.5}=\frac{1{,}000}{1.03^4} =\frac{1{,}000}{1.12550881} =\$888.49.

As a check, accumulating the original price for three semiannual periods gives $813.09(1.03)^3$, approximately $888.49; using the unrounded price gives the same value. Both calculated values are below $1,000, and the intermediate value exceeds the purchase price. Common mistake avoided: using 3.5 periods instead of seven. The yield is semiannual, so the model counts two periods per year.

Worked Example: Recovering a Nominal Yield and Valuing at an Intermediate Date

A hypothetical zero-coupon bond has a redemption value of $1,500 in 2.25 years and a price of $1,053.88. Infer the nominal annual yield convertible quarterly, then find the value after one year. Assume the quarterly compounding model applies for all nine periods to maturity. Use unrounded values in the calculations and report money to the nearest cent.

Method and timeline. The maturity term is $2.25(4)=9$ quarters. Let $q$ be the yield per quarter. The price equation is $1{,}053.88(1+q)^9=1{,}500$, so the nominal annual rate is $i^{(4)}=4q$.

0:\ \$1{,}053.88 \quad \longrightarrow \quad 1.00 \quad \longrightarrow \quad 2.25:\ \$1{,}500

Solving for the quarterly rate using the displayed price gives

q=\left(\frac{1{,}500}{1{,}053.88}\right)^{1/9}-1 \approx 0.040000.

Thus $i^{(4)}=4(0.04)=0.16$, or 16.00% nominal annual convertible quarterly. Because the price is stated to cents, the inferred rate is approximate; the underlying unrounded price at a quarterly rate of 4% is $1{,}500/(1.04)^9$, which rounds to $1,053.88. The equivalent effective annual yield is

$$ (1.04)^4-1=0.16985856, $$

or 16.985856% (approximately 16.99%). At $t=1$, the remaining term is 1.25 years, or five quarters. Therefore,

V_1=\frac{1{,}500}{(1.04)^5} =\frac{1{,}500}{1.2166529024} =\$1{,}232.91.

The check is consistent with the timeline: after one year, five of the original nine quarters have elapsed and four years? No: nine quarters equal 2.25 years, so four years is not the remaining term; the remaining term is five quarters, or 1.25 years. The value is below redemption and above the initial price. Common mistake avoided: treating a 2.25-year maturity as 2.25 quarterly periods. It is nine quarters.

Worked Example: Converting an Effective Yield Before a Fractional-Term Valuation

A hypothetical zero-coupon bond pays $2,000 in 2.5 years. Its effective annual yield is 4%. Find its price, the equivalent nominal annual yield convertible quarterly, and its value at $t=1.5$. Use fractional-year accumulation under the effective annual model.

Method and timeline. Apply the effective annual rate directly to the 2.5-year term. At the intermediate date, only one year remains.

0:\ P \quad \longrightarrow \quad 1.5:\ V_{1.5} \quad \longrightarrow \quad 2.5:\ \$2{,}000

At time 0, the equation is $P(1.04)^{2.5}=2{,}000$. Since $1.04^{2.5}=1.04^2\sqrt{1.04}\approx1.1030199$,

P=\frac{2{,}000}{1.1030199} \approx \$1{,}813.20.

For the equivalent nominal quarterly yield, let $q$ be the quarterly rate. Equivalence over one year requires $(1+q)^4=1.04$, so

i^{(4)}=4q =4\left((1.04)^{1/4}-1\right) \approx 0.039414,

or approximately 3.9414% nominal annual convertible quarterly. At $t=1.5$, one year remains, so

V_{1.5}=\frac{2{,}000}{1.04} =\$1{,}923.08.

Using the converted quarterly rate for the remaining four quarters gives the same result before rounding: $2{,}000/(1+q)^4=2{,}000/1.04$. The intermediate value is above the time-0 price and below the redemption value. Common mistake avoided: applying the nominal quarterly rate as though 3.9414% were the rate for each quarter. The per-quarter rate is $i^{(4)}/4$.

Checks That Catch Timing Errors

The arithmetic in a zero-coupon problem is usually short; most errors come from using the wrong exponent or a mismatched rate. A price is measured at the start of the full term. An intermediate value is measured at the intermediate date and therefore uses only the remaining term. The interest rate and time unit must agree: an annual effective yield uses years, while a nominal rate convertible quarterly uses quarters.

For a positive yield under a fixed accumulation model, these inequalities provide a useful reasonableness check:

P_0<V_t<F \qquad (0<t<T).

This check assumes the same positive yield is used from purchase through the intermediate valuation date. It is not a claim that an actual bond's market price must rise over time: market yields can change. It also does not apply as stated when the yield is zero or negative.

A frequent wrong calculation is to value the bond at an intermediate date using the original maturity term. For example, in the first worked example, the incorrect calculation at $t=1.5$ would be $1{,}000/(1.03)^7=\$813.09$. That is the time-0 price, not the value at $t=1.5$. The correction is to use the four periods remaining: $1{,}000/(1.03)^4=\$888.49$. A second trap is confusing a nominal annual rate with its periodic rate; for 6% nominal convertible semiannually, each half-year rate is 3%, not 6%.

When a problem gives a price and asks for a yield, write the price equation first and isolate the periodic or annual rate only afterward. When the problem gives a yield and asks for an intermediate-date value, draw the timeline and count forward from the valuation date to redemption. These habits make fractional-term calculations easier to audit and help distinguish a rate-conversion error from a time-counting error.

Check Your Understanding

For each problem, state the yield convention and use the specified time model before calculating.

A hypothetical zero-coupon bond pays $1,200 in two years. Its effective annual yield is 5%. Find its price. (Answer: $1,088.44)
A hypothetical zero-coupon bond pays $800 in 2.5 years at a nominal annual yield of 8% convertible semiannually. Find its price. (Answer: $657.54)
A hypothetical zero-coupon bond with a $1,000 redemption value due in two years sells for $907.03. Find the implied effective annual yield, to two decimal places. (Answer: 5.00%)
A hypothetical zero-coupon bond pays $2,500 in 1.75 years at a nominal annual yield of 6% convertible quarterly. Find its value at $t=0.75$, using the quarterly model. (Answer: $2,355.47)
A hypothetical zero-coupon bond pays $1,000 in 2.5 years. Its nominal annual yield is 12% convertible quarterly, with fractional periods valued using the stated compound-power model. Find its time-0 price and its value at $t=0.5$. (Answer: price $744.09; intermediate value $789.41)

What You'll Learn