What a Zero-Coupon Bond Pays
A bond’s price reflects the value today of the cash flows its holder expects to receive. A zero-coupon bond makes no periodic coupon payments. Instead, under the basic valuation model, it promises one payment of face value at maturity. Pricing it therefore requires discounting one future cash flow rather than a series of coupon payments.
This tutorial considers valuation at time 0, a known maturity payment, and a required yield stated as an effective annual rate. The calculation is aligned with the SOA VEE Accounting and Finance syllabus. The model treats the promised payment as certain; it does not adjust for default risk, taxes, transaction costs, or other features that could affect an actual market price.
For the model in this tutorial, \(n\) is a whole number of years and \(i\) is an effective annual rate. A positive \(i\) means a dollar invested today grows by a factor of \(1+i\) over one year. The discount factor is \(v=1/(1+i)\). Since there is only one payment, the price is:
The face value is not the price paid today. When the required yield is positive, the price is below face value because money paid today could grow to the larger maturity payment. If the required yield is zero, the price equals face value. A negative yield would make the model price exceed face value, so the common statement that a zero-coupon bond sells at a discount assumes a positive yield.
Why the Discounting Formula Holds
The formula follows from comparing the bond with an investment that earns the required yield. If an investor pays \(P_0\) now and can earn the effective annual rate \(i\), that amount grows to \(P_0(1+i)^n\) after \(n\) years. To match the bond’s promised payment at maturity, the accumulated amount must equal \(F\). Solving that equality for the amount invested today gives the price:
This argument assumes that the required yield is appropriate for the bond and is constant over the full term in the calculation. It also assumes annual effective compounding. If a rate is quoted using a different compounding convention or the time to maturity is expressed in a different unit, it must be made consistent with the period count before applying the formula. Do not insert a number of months as though it were a number of years.
Calculating the Price
A reliable approach is to identify the payment and its date before calculating. Then use the required yield for the full number of years, discount the payment, and interpret the resulting price in context. Keep intermediate calculations unrounded where possible; round the final dollar price to cents.
For the basic zero-coupon model, the single payment at maturity is the face value \(F\).
Confirm that \(i\) is an effective annual yield and \(n\) counts years.
Calculate \(F/(1+i)^n\) without rounding intermediate values unnecessarily.
For positive \(i\), verify that the price is below \(F\). Also confirm that accumulating the price at \(i\) for \(n\) years returns the promised payment.
Worked Example: Pricing a Three-Year Bond
A hypothetical zero-coupon bond promises a face value of $1,000 in exactly 3 years. The required yield is 5.00% effective annually, compounded annually. Assume the maturity payment is certain. Find the price today.
Method. Discount the single $1,000 payment by the effective annual yield for three years. This method avoids treating the face value as though it were already the present price.
The accumulation factor over three years is:
Therefore, the price is:
Rounded to the nearest cent, the price is $863.84. The difference between face value and price is $1,000.00 − $863.84 = $136.16, but it is not a coupon paid during the term; the bond makes no interim payments.
Check. Accumulating the unrounded price for three years at 5% gives approximately $863.8376 × 1.157625 = $1,000. The computed price is below face value, as expected for a positive yield.
How Face Value, Time, and Yield Affect Price
Each input has a distinct interpretation. Face value determines the size of the maturity payment: holding the yield and term fixed, a larger payment has a proportionally larger present value. Time to maturity determines how many annual growth periods separate the price date from the payment date. The required yield determines how strongly that future payment is discounted.
| Input change | Effect on price, other inputs fixed | Reason |
|---|---|---|
| Higher face value \(F\) | Price increases in the same proportion | The single payment being discounted is larger |
| Longer term \(n\), with positive \(i\) | Price decreases | The payment is discounted for more years |
| Higher required yield \(i\) | Price decreases | Each year’s discounting factor is larger |
| Yield of zero | Price equals face value | No discount is applied |
These comparisons assume that only the named input changes. They are useful for interpreting a calculation, but they do not determine an actual market price when the bond’s promised payment is uncertain or other market features matter.
Worked Example: Comparing a Calculated Price with an Asking Price
A hypothetical investor is considering a zero-coupon bond that pays $25,000 in 4 years. The investor’s required yield is 4.50% effective annually. A dealer offers the bond for $21,100. Assume the payment is certain and annual compounding applies. Calculate the model price and compare it with the offer.
Method. First calculate the present value at the investor’s required yield. Then compare that value with the asking price. A quoted price is not automatically a fair price for every investor; the required yield matters.
The four-year accumulation factor is:
The model price is:
Rounded to cents, the model price is $20,964.03. The asking price exceeds it by $21,100.00 − $20,964.03 = $135.97. At the stated required yield and under the stated assumptions, the offer is above the investor’s model value by that amount.
Check. The calculated price is below the $25,000 maturity payment because the required yield is positive. Comparing the asking price to face value alone would not establish whether the offer meets the investor’s required return.
Worked Example: Interpreting a Lower Required Yield
A hypothetical zero-coupon bond has a face value of $10,000 and matures in 5 years. Calculate its price at a required yield of 3.20% effective annually. A market quote is $8,600. Assume the promised payment is certain and the quote is for the same valuation date.
Method. Discount the maturity payment using the stated annual yield, then compare the resulting model price with the quote. This avoids assuming that a bond’s face value is the amount an investor must pay.
The accumulation factor is built year by year:
Thus:
Rounded to cents, the model price is $8,542.82. The quote is $8,600.00 − $8,542.82 = $57.18 above the model value. That comparison is conditional on the investor’s 3.20% required yield and the certainty assumption; a different required yield would produce a different model price.
Check. Because the required yield is positive, the price is below the $10,000 face value. Accumulating the unrounded price for five years at 3.20% gives approximately $10,000.
Common Mistakes and Exam Traps
The arithmetic is short, so errors often come from interpreting the inputs rather than manipulating the formula. In particular, the face value is the maturity payment, while the price is the amount today. The required yield is the rate used to discount that payment; it is not an additional cash flow.
- Using face value as today’s price. This ignores the time value of money. Discount the maturity payment unless the yield is zero.
- Multiplying by the accumulation factor. Multiplying \(F\) by \((1+i)^n\) moves a value forward in time. Pricing today requires dividing by that factor.
- Using the wrong number of periods. If maturity is stated in years and the rate is effective annually, use years as \(n\). Do not treat months as years.
- Changing the rate without changing its convention. The formula here uses an effective annual yield. A rate stated with a different compounding convention must be converted consistently before use.
- Assuming every zero-coupon bond must be below face value. That conclusion follows from a positive required yield. At a zero yield price equals face value; with a negative yield, this model gives a price above face value.
- Rounding too early. Rounding the accumulation factor or intermediate price can change the final cents. Keep full calculator precision and round only the final dollar amount.
A zero-coupon bond is a useful setting for separating three ideas: the contractual payment, the time remaining until that payment, and the return an investor requires. The price calculation combines them in one present-value expression. When reading a problem, first identify the promised maturity amount, then check the rate’s basis and the time unit, and only then discount.
Check Your Understanding
For each problem, assume the stated maturity payment is certain and the required yield is an effective annual rate with annual compounding. Round final dollar prices to cents.
- A hypothetical bond pays $2,000 in 2 years. Its required yield is 4.00%. Calculate its price today. (Answer: $1,849.11)
- A zero-coupon bond pays $6,000 in 3 years, and the required yield is 5.00%. Calculate the price and state whether it is below or above face value. (Answer: $5,183.03; below face value)
- A hypothetical bond pays $10,000 in 2 years. At a required yield of 10.00%, calculate its price. If its quoted price is $8,300, by how much is the quote above or below the model price? (Answer: $8,264.46; the quote is $35.54 above the model price)
- A zero-coupon bond pays $5,000 in 2 years. An investor requires 5.00% annually, but the bond is offered for $4,500. Calculate the model price and determine whether the offer is above or below it. (Answer: $4,535.15; the offer is $35.15 below the model price)
- A bond with face value $10,000 is due in 3 years. The required yield is 10.00% effective annually. Calculate its price and the difference between face value and price. (Answer: $7,513.15; the price is $2,486.85 below face value)