Zero-Coupon Bonds

Start with the absence of interim cash

Investor.gov describes zero-coupon bonds by their lack of periodic interest payments. The essential modeling property is a single promised maturity receipt. There is no sequence of spendable coupons along the way. A statement that the bond earns interest over time therefore should not be confused with a statement that it distributes interest in cash before the final payment date.

TreasuryDirect's STRIPS explanation provides an important structural example: eligible interest and principal payments can become separately traded claims. Each separated payment has its own maturity. This guide uses the single-payment idea, not any specific STRIP or transaction quote. It also does not assume that every zero-coupon instrument shares the same issuer, liquidity, tax treatment or operational arrangements as Treasury STRIPS.

Price one payment with a stated clock

Let F denote the maturity payment and T the whole years until it arrives. With annual nominal yield y compounded m times, the model price is F/(1+y/m)^(m*T). Even without coupons, the yield still needs a compounding convention. Here m identifies the rate clock; it does not imply that the issuer makes m payments per year when the coupon is zero.

For a hypothetical USD 1,000 payment three years from now and a 5% annually compounded yield, price is 1000/(1.05)^3 = USD 863.837599. Equivalently, the three-year discount factor is approximately 0.863837599 per dollar payable at maturity. Multiplying that factor by face gives the price. No separate coupon annuity or assumed reinvestment income enters the valuation.

Check the yield by reversing the calculation

For annual compounding, the reverse calculation is y = (F/P)^(1/T)-1. Applying it to the unrounded hypothetical price returns 5%. This is a useful independent consistency test because it begins with the price and maturity payment rather than reusing the original discount factor. In the general nominal convention, multiply the periodic root result by the stated frequency m.

If the payment arrives in full and the instrument is held to maturity, this single-receipt structure has no intermediate coupon-reinvestment requirement. The quoted nominal yield still differs from an effective annual rate when m exceeds one. Taxes, costs and any external funding arrangement can affect the holder's outcome, so the clean reverse calculation should not be described as a universal after-cost return.

Observe price sensitivity without a coupon distraction

Raise the assumed annual yield in the three-year example from 5% to 6%, leaving the USD 1,000 maturity payment unchanged. Price becomes 1000/(1.06)^3 = USD 839.619283. That is USD 24.218315 below the unrounded starting value. The scenario is instantaneous: no time passes, no payment occurs, and only the discount rate changes. It is not a forecast or an observed market move.

All the present-value weight sits at maturity, so Macaulay duration equals the remaining maturity. By contrast, a positive-coupon bond of the same maturity distributes some weight to earlier dates. This timing difference helps explain why zeros are useful for teaching sensitivity. It does not establish a universal ranking across instruments with different maturities, yields, currencies or contractual payment contingencies.

Do not make discount status part of the definition

Introductory explanations often describe zeros as sold below face because they assume positive yields. The defining feature is instead the missing interim coupon. With zero yield, the formula gives price equal to face. With a negative yield and a positive compounding base, price exceeds face. These cases are mathematically consistent and should not be rejected simply because an instrument has no coupon.

A price above face means the holder pays more now than the single nominal amount promised later. It does not imply that the contractual maturity payment has become negative or that an extra coupon must exist. Keeping cash-flow identity separate from valuation makes the negative-yield case much easier to understand and provides a clean check on code that incorrectly assumes all zero-coupon prices must be discounts.

Separate payment matching from complete risk elimination

A single payment date can be compared directly with a dated obligation, but matching dates alone does not prove that the obligation is funded under every circumstance. Currency, amount, contractual payment certainty and any need to sell beforehand remain relevant. A nominal payment also does not automatically match an obligation whose amount changes with inflation or other conditions after the comparison is made.

The lack of interim cash can itself matter operationally. An obligation arising before maturity cannot be met from coupons that do not exist. Selling part of the claim would introduce a transaction price and liquidity assumptions. Possible tax recognition before cash receipt is another separate issue noted in official educational material; this guide neither calculates it nor supplies jurisdiction-specific tax treatment.

Use the zero-coupon case as a model boundary check

Within regular bullet mathematics, setting the coupon rate to zero should remove the annuity component completely while retaining principal discounting. The parent model's other restrictions remain: whole years from one through thirty and the selected annual, semiannual or quarterly compounding clock. A zero coupon does not add support for Treasury-bill discount quotations, arbitrary settlement dates or fractional-year money-market conventions.

Review the calculation by asking whether exactly one positive receipt remains, whether its time is unchanged and whether the inverse yield formula reproduces the input. Then interpret the result narrowly. The example demonstrates the value of a stipulated future payment under an assumed discount rate; it does not certify a real issuer, establish an executable price or determine an allocation to zero-coupon securities.

Primary references

1. TreasuryDirect: Separate Trading of Registered Interest and Principal of Securities
2. Investor.gov: Zero Coupon Bond