Zero-Coupon Bond & Spot Rates

Builds onPresent Value of a Single Sum — if this page feels steep, start there.

P = \frac{F}{(1+z_n)^n}

$F$: the face (par) value: the single payment at maturity
$z_n$: the n-year spot rate: THE interest rate for money delivered exactly n years out
$n$: years to maturity
$(1+z_n)^n$: the compound growth the market demands over those n years
$P$: the price — always below face while rates are positive; the discount IS the return

Why it must be true

A zero-coupon bond is the purest promise in finance: one payment, one date, nothing else. Its price is a single present-value calculation, and the discount rate that makes it work is the interest rate for that exact maturity — the spot rate $z_n$ .

This purity is why zeros matter beyond themselves: a coupon bond is just a bundle of zeros (each coupon one mini-zero), so the zero curve — spot rates at every maturity — is the fundamental price list from which every bond, swap and forward is built. Buy a zero at 74.41 against a 100 face in five years, and the deep discount IS the yield: the whole return is baked into the purchase price.

The derivation

One cash flow, so the bond-pricing sum collapses to a single term:

P = \frac{F}{(1+z_n)^n}

Run it backwards and any traded zero reveals its maturity's spot rate:

z_n = \left(\frac{F}{P}\right)^{1/n} - 1

— the same n-th-root maneuver as the implied-return face of compound growth. Coupon bonds then price as portfolios of these:

P_{coupon} = \sum_t \frac{CF_t}{(1+z_t)^t}

which is why "bootstrapping the spot curve" from coupon bonds is the first job of every rates desk's morning.

When to reach for it

Pricing a single future payment off a spot rate, or extracting a maturity's spot rate from a traded zero's price.

Listen for

“zero-coupon bond / pure discount bond”“spot rate for maturity n”“pays no coupons”“STRIPS”

Back-of-the-envelope

Estimate it in your head first — then the calculator only confirms.

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Rule of 72 sizes the discount: at 6%, money halves in ~12 years — so a 12-year zero should cost about half its face. Anchor the magnitude before computing.
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Longer maturity or higher rate → deeper discount, and the effect compounds. A 10-year zero at 7% near 90? Impossible — that's a 1.5-year discount.
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Implied-rate face: (F/P)^(1/n) − 1 is the same n-th-root move as any compound-growth problem — no new machinery.

Traps in applying it

✗Discounting with a coupon bond's YTM instead of the maturity's spot rate — zeros define the spot curve; YTMs average across it.
✗Using semiannual conventions inconsistently (US zeros usually quote semiannual compounding — halve the rate, double the periods).
✗Forgetting that zeros still generate taxable phantom income in many regimes despite paying no cash.

Limits & criticisms

The formula assumes the single payment is certain — sovereign-quality credit; for anything else a spread belongs inside the discounting. Its clean structure also hides the instrument's ferocity: a zero's duration equals its full maturity, making long zeros the most rate-sensitive plain bonds that exist — a 30-year STRIP swings roughly 30% per 100bp, which buyers of "safe Treasuries" rediscover each cycle.

Where it came from

Deep-discount instruments are ancient (bills of exchange traded below face in medieval fairs), but zeros became an asset class in 1982, when Merrill Lynch's TIGRs and then the Treasury's STRIPS program (1985) let dealers split coupon Treasuries into their component zeros. Pension funds devoured them for liability matching — one payment on the exact date a liability falls due, zero reinvestment risk. Every discount factor in every swap and bond system today is a zero-coupon price in disguise.

One identity, 2 questions

The exam can hide any variable. Each face below is the same equation solved for a different unknown — drill them separately.

Price off the spot curve

P = \frac{F}{(1+z_n)^n}

The pricing face: one payment, one discounting — the atomic unit every coupon bond is assembled from.

Drill this face →

Spot rate from a traded zero

z_n = \left(\frac{F}{P}\right)^{1/n} - 1

The curve-building face: each traded zero REVEALS its maturity's true interest rate — the bootstrap's raw material.

Drill this face →

On the BA II Plus

Worked example: Price a 5-year zero (face $1,000.00) when the 5-year spot rate is 2.75%.

1.5 [N] 2.75 [I/Y] 0 [PMT] 1000 [FV]one cash flow, no coupons
2.[CPT] [PV]price (shown negative = cash out)

→ $873.15

Drill this formula Back to the library