Introduction
A zero-coupon bond makes no scheduled payments before maturity. Its value today is therefore determined by one future cash flow: the amount payable at maturity. The key pricing decision is which rate to use. A spot rate applies to a particular maturity, so a bond’s promised payment must be discounted using the spot rate that matches its payment date, with the quoted compounding convention handled correctly.
This tutorial develops that calculation for annual effective, nominally convertible, and continuously compounded spot rates. The examples assume deterministic payments and no default or other uncertainty in the promised amount. The pricing method is aligned with the SOA Exam FM (Financial Mathematics) syllabus.
The Maturity-Specific Discount Factor
Let \(F\) be the bond’s face value and let \(t\) be its maturity in years, measured from the valuation date. Write \(D(t)\) for the price today of one dollar payable at time \(t\). Thus \(D(t)\) is the discount factor for that dated payment, and the zero-coupon bond price \(P\) is
When the spot rate to maturity \(t\) is an annual effective rate \(i_t\), a dollar invested today accumulates for \(t\) years by the factor \((1+i_t)^t\). Its present value is the amount that accumulates to one dollar, so \(D(t)=(1+i_t)^{-t}\). This assumes the annual effective rate applies over the stated term, including any fractional year under that rate convention.
For a nominal annual spot rate \(i_t^{(m)}\), convertible \(m\) times per year, the periodic rate is \(i_t^{(m)}/m\). If the maturity contains \(mt\) complete conversion periods, the accumulation factor is \(\left(1+i_t^{(m)}/m\right)^{mt}\). For a continuously compounded spot rate \(\delta_t\), the accumulation factor is \(e^{\delta_t t}\). In each case, the bond price is the face value divided by the corresponding accumulation factor.
| Spot-rate quotation | Accumulation factor to maturity | Zero-coupon price |
|---|---|---|
| Annual effective \(i_t\) | \((1+i_t)^t\) | \(F/(1+i_t)^t\) |
| Nominal \(i_t^{(m)}\), convertible \(m\) times yearly | \((1+i_t^{(m)}/m)^{mt}\), for \(mt\) complete periods | \(F/(1+i_t^{(m)}/m)^{mt}\) |
| Continuous force \(\delta_t\) | \(e^{\delta_t t}\) | \(F e^{-\delta_t t}\) |
The nominal formula as written requires the maturity to line up with a whole number of conversion periods. If it does not, the problem must specify how to handle the fractional period; do not silently round the number of periods. Likewise, do not substitute one quotation convention for another without a valid conversion.
Why the Pricing Formula Holds
The formula follows directly from the meaning of a discount factor. Suppose an investment of \(P\) today grows at the matching spot rate until maturity. To produce the promised face value \(F\), its accumulated value must equal \(F\). Solving this equation for \(P\) gives the present price.
For an annual effective spot rate \(i_t\), an amount \(P\) today accumulates to \(P(1+i_t)^t\). A zero-coupon bond pays \(F\) at maturity, so \(P(1+i_t)^t=F\).
Divide both sides by the accumulation factor: \(P=F/(1+i_t)^t\). Equivalently, the present value of one dollar at maturity is \(D(t)=1/(1+i_t)^t\), so \(P=F D(t)\).
If the rate is nominally convertible, use its periodic rate and number of periods. If it is continuous, use \(e^{\delta_t t}\) as the accumulation factor. The same maturity-equation reasoning applies.
This reasoning assumes the stated spot rate is the applicable rate for the bond’s maturity and that the promised payment is certain under the pricing model. It does not require a probability distribution or mortality assumption: there is one deterministic payment. If there are interim coupons, the security is not a zero-coupon bond, and its price must account for those additional dated cash flows.
Pricing with an Annual Effective Spot Rate
Worked Example: A Two-Year Bond with an Annual Effective Spot Rate
A hypothetical zero-coupon bond pays \(F=\$12{,}000\) exactly two years from today. The two-year spot rate is 5% effective annually. Find the price. Assume the payment is deterministic, there is no default, and the annual effective accumulation convention applies for both years. No mortality model or probability distribution is involved.
Method. Use the annual effective spot rate for the bond’s two-year maturity, and discount the single face-value payment for two years:
Rounding only the final price to the nearest cent gives \(\$10{,}884.35\). As a check, \(\$10{,}884.353741\ldots \times 1.1025=\$12{,}000\), as required at maturity.
Common mistake avoided. The rate is a two-year spot rate, not a rate to be applied to a different maturity. Also, the exponent is two because the bond has two years to maturity; the face value is not discounted for just one year.
Pricing with a Nominally Convertible Spot Rate
Worked Example: A Three-Year Bond with Quarterly Conversion
A hypothetical zero-coupon bond pays \(\$5{,}000\) in three years. Its matching spot rate is 8% nominal per year, convertible quarterly. Find the price. Assume the quarterly nominal-rate convention applies for all 12 quarters, the payment is deterministic, and there is no default. No mortality assumption or probability distribution is involved.
Method. First convert the nominal annual quote into a quarterly rate. Then count the conversion periods over the full maturity:
The three-year accumulation factor is
Therefore,
Rounding only the final price to the nearest cent gives \(\$3{,}942.47\). Accumulating the unrounded price by the factor \(1.268241794562545\ldots\) returns \(\$5{,}000\), subject to displayed rounding.
Common mistake avoided. Do not treat 8% nominal convertible quarterly as an 8% effective annual rate. The incorrect calculation \(5{,}000/(1.08)^3\) uses a different convention. The correct quarterly periodic rate is 2%, and there are 12 quarterly periods—not three.
Pricing with a Continuous Spot Rate
Worked Example: A Two-Year Bond with Continuous Compounding
A hypothetical zero-coupon bond pays \(\$20{,}000\) in two years. The matching continuously compounded spot rate is 4.5% per year. Find the price. Assume a constant force of interest of 0.045 per year over the two-year term, a deterministic payment, and no default. There is no mortality assumption or probability distribution.
Method. Under continuous compounding, the accumulation factor is the exponential of the force multiplied by time. Here the exponent is \(0.045 \times 2=0.09\):
Using \(e^{-0.09}=0.913931185271228\ldots\),
Rounding only the final price to the nearest cent gives \(\$18{,}278.62\). The check is that \(18{,}278.623705\ldots \times e^{0.09}=\$20{,}000\), subject to displayed rounding.
Common mistake avoided. Do not use \((1+0.045)^2\) as the accumulation factor. That expression applies annual effective compounding, whereas this problem states a continuous force of interest.
A Reliable Exam Calculation
Most errors in zero-coupon bond pricing come from choosing the wrong rate, not from the final division. Before calculating, separate three pieces of information: the payment amount, the time to payment, and the spot-rate quotation convention. Then identify the matching maturity-specific rate and translate its quotation into an accumulation factor.
A useful final check is to reverse the calculation. Multiply the unrounded price by the accumulation factor used in the denominator. The result should equal the face value. If it does not, inspect the maturity, rate conversion, number of periods, and sign of the exponent before rounding.
A second check is qualitative. With a positive spot rate, the present price of a positive maturity payment should be below its face value. A zero rate gives a price equal to face value. A negative rate can produce a price above face value, so “price below face value” is not a universal rule; it depends on the rate’s sign.
These checks are especially useful when a question gives several rates for different maturities. A rate that is numerically convenient is not automatically the right one: only the spot rate matching the bond’s payment date belongs in this single-payment calculation. Keep the conventions explicit rather than relying on labels such as “annual rate” alone.
Check Your Understanding
For each problem, use the spot rate and compounding convention stated. Round only the final dollar price to the nearest cent.
- A hypothetical zero-coupon bond pays \(\$10{,}000\) in two years. Its matching spot rate is 4% effective annually. What is its price? (Answer: \(\$9{,}245.56\))
- A hypothetical bond pays \(\$7{,}500\) in four years. The four-year spot rate is 6% nominal, convertible semiannually. What is its price? (Answer: \(\$5{,}920.57\))
- A hypothetical zero-coupon bond pays \(\$15{,}000\) in five years. Its matching spot rate is 3.2% continuously compounded. What is its price? (Answer: \(\$12{,}782.16\))
- A hypothetical bond pays \(\$20{,}000\) in 18 months. Its spot rate is 4% nominal, convertible quarterly. Assume the maturity is exactly six quarters. What is its price? (Answer: \(\$18{,}840.90\))
- A hypothetical zero-coupon bond pays \(\$6{,}000\) in two years. The matching spot rate is 3% continuously compounded. What is its price, and what accumulation factor should be used in the check? (Answer: \(\$5{,}650.59\); accumulation factor \(e^{0.06}\))