A ten-year zero-coupon bond and the cost of waiting

Specify one payment and one convention

This constructed security pays no interim coupon and returns $1,000 at the end of year ten. Its baseline yield is 4% with annual compounding. There is no default, tax charge, sale cost or optional redemption in the example. The absence of coupons does not mean the security has no cash flow: the single terminal principal payment is its entire value.

The annual compounding choice matters even though the security makes only one payment. A quoted yield with semiannual compounding would imply a different discount factor if the same numerical yield were entered. The coupon-frequency selector also determines the yield-compounding convention in this calculator. For this example select annual frequency, zero coupon, ten years and the stated face amount.

Discount the terminal amount

The present value is 1,000 divided by 1.04 to the tenth power, or $675.5641688258. Its price per 100 face is 67.5564168826. Multiplying the present value by 1.04 to the tenth power returns the terminal amount, providing a direct independent identity check without a multi-payment schedule.

That mathematical identity is not an assurance of a particular investor's realized outcome. It relies on receiving the assumed payment and holding the instrument under the stated conditions. Selling earlier introduces a sale price, and changing purchasing power affects what the nominal amount can buy. The worksheet does not convert nominal payments into a real consumption measure.

Separate duration definitions

Because all present value is attached to the payment at year ten, the present-value-weighted payment time is exactly ten years. That is Macaulay duration for this example. Modified duration divides it by 1.04, producing 9.6153846154 years. These two values should not be substituted for one another in the percentage price-change approximation.

The first describes payment timing under this discounting convention; the second measures the local normalized price derivative with respect to the annual yield. The baseline DV01 magnitude is $0.6495809316. Expressing sensitivity in dollars requires both the normalized sensitivity and the actual present value, so the face amount alone is insufficient.

Reprice rather than extrapolate indefinitely

With the yield increased from 4% to 5%, exact present value becomes 1,000 divided by 1.05 to the tenth power, or $613.9132535408. The change is -$61.6509152850, equal to -9.1258415011% of the original value. The scenario has no elapsed time: it is still a payment ten years away.

Duration alone estimates -9.6153846154%. The convexity coefficient is 101.7011834320 years squared, and the quadratic estimate is -9.1068786982%. Its error, approximately +0.018963 percentage points, is larger than the default five-year coupon example's error. That comparison concerns these fixtures and shock sizes, not a universal accuracy ranking across all securities.

Understand what removing coupons changes

With no interim coupons, this particular instrument has no interim coupon reinvestment decision. That does not eliminate the need to decide what happens to principal when it matures. A liability that falls earlier than year ten would also require a separate funding solution or an assumed early sale. The terminal amount is not spendable cash today.

TreasuryDirect describes STRIPS as separately traded interest and principal components of eligible securities. That is useful instrument context, but this case does not price a specific STRIP, identify a CUSIP or import a Treasury quotation. Its selected yield is an educational assumption. Actual instruments require their own documentation, tax treatment and market conventions.

Document the reconciliation

A compact review file can retain the terminal cash flow, discount factor, compounding convention, initial price and shocked price. Then add a separate row for each approximation. This prevents an analyst from comparing an exact dollar loss with a percentage approximation or from mixing a ten-year horizon with a ten-period semiannual horizon.

Use the downloadable reference JSON to inspect full precision and reproduce the case through the calculator. Change face value as a second check: doubling it should double both prices and DV01 while leaving the durations and relative price change unchanged. If it does not, investigate the quantity units before drawing any portfolio conclusion.

Do not treat a discount as a bargain signal

A price below face is expected for this positive-yield, zero-coupon construction. It is not by itself evidence that the asset is cheap relative to alternatives. Comparing value would require a relevant curve, issuer risk, liquidity and actual purchase terms. The case deliberately stops at a transparent calculation rather than inferring a trading recommendation from the discount.

Primary references

1. TreasuryDirect: STRIPS
2. FINRA: Bonds, interest-rate changes and duration
3. Financial library: net present value convention