Zero-Coupon Bond

A zero-coupon bond skips the periodic interest payments entirely. You buy it below face value — say 70 for a bond that pays 100 in ten years — and your whole return is that built-in climb from 70 to 100.

Nothing to reinvest, nothing to track: one payment, one date. That makes zeros perfect for funding a known future expense — a tuition bill in 2035, a pension payment in 2040 — with no uncertainty about reinvesting coupons along the way.

The flip side: with all the money arriving at the very end, zeros are the most rate-sensitive bonds of their maturity. Long zeros can swing like stocks when yields move.

Where zeros come from

• Issued directly: T-bills are natural short-term zeros; some governments and corporates issue longer ones.
• Stripping: dealers separate a coupon bond into individually tradable pieces — each coupon and the principal become standalone zeros (US "STRIPS").

Yield and price

The relationship is a single compound-interest equation. At an annually compounded yield \(y\) and maturity \(T\): price = 100/(1+y)^T. A 10-year zero at 4% costs 67.56; the same bond at 5% costs 61.39 — a 9% price drop for one point of yield.

Duration = maturity

Because there is only one cash flow, a zero's (Macaulay) duration equals its maturity exactly — the cleanest possible rate exposure. Pension funds discharging a 2045 liability buy 2045 zeros and simply stop worrying about rates ("immunisation" in its purest form).

Tax wrinkle

Many jurisdictions tax the accrued discount annually ("phantom income") even though no cash arrives — one reason zeros often live in tax-sheltered accounts.

The discount function

Zero prices are the discount factors of the curve: \(P(0,T) = e^{-z(T)\,T}\) in continuous compounding. Every fixed-income valuation reduces to a weighted sum of these:

Bootstrapping the discount function from coupon-bond or swap quotes is the first step of every pricing library.

Forward rates from zeros

The instantaneous forward curve \(f(t) = -\partial_T \ln P(0,T)\) is the input to HJM-class term-structure models, where the entire no-arbitrage drift is determined by volatility structure alone.

Convexity of zeros

Modified duration \(T/(1+y)\) and convexity ≈ \(T(T+1)/(1+y)^2\): convexity grows with maturity squared. Long zeros are therefore the cheapest convexity in bond markets — barbelling zeros against selling coupon bonds is a classic convexity trade.

Zeros in models

In short-rate models, the zero price has closed form, e.g. Vasicek/Hull–White: \(P(t,T) = A(t,T)e^{-B(t,T) r_t}\) — making zeros the analytic building block for calibrating swaptions, caps and everything above them.