Zero-Coupon Bond

A zero-coupon bond is the simplest instrument in fixed income because it consists of exactly one cash flow. Its price is the face amount divided by one plus the periodic yield, raised to the number of periods remaining. Under the semiannual bond basis convention used for most US market quotes, the stated yield is halved and the number of years is doubled, so a five-year zero yielding 6% on a semiannual basis is priced as par divided by 1.03 raised to the tenth power. Quoting the same bond on an annual basis produces a different number for the same price, which is why the compounding convention must always be specified.

The relationship inverts cleanly: given price and maturity, the yield is the rate that grows the purchase price into par over the term, found by taking the appropriate root of the ratio of par to price and subtracting one. Because there is only one discounting step, no reinvestment assumption is embedded in that yield.

With no interim payments, the weighted average time to receipt of cash equals the maturity itself, so the Macaulay duration of a zero-coupon bond is simply its term to maturity. Modified duration is that figure divided by one plus the periodic yield. Among bonds of identical maturity and yield, the zero therefore has the longest duration and the largest percentage price move for a given change in rates. A 30-year zero can lose a very large share of its value on a modest rise in long yields.

The compensating feature is the absence of coupon reinvestment risk. An investor who holds the bond to maturity earns exactly the yield at which it was purchased, regardless of the path of interest rates, because there is no intermediate cash to reinvest at uncertain future rates. Zeros thus separate the two components of return risk: price risk is maximised, reinvestment risk is eliminated.

Beyond instruments issued as zeros, the concept is the analytical backbone of bond valuation. Every coupon bond can be viewed as a portfolio of zero-coupon claims, one for each scheduled payment, and arbitrage-free valuation requires that each of those claims be discounted at the spot rate matching its own maturity rather than at a single yield to maturity. Discount factors published or derived from the market are nothing more than the prices of unit zeros.

The spot curve is typically extracted from observable coupon bond prices by bootstrapping: the shortest maturity gives the first spot rate directly, and each longer bond is solved for the unknown final-period rate once earlier cash flows are discounted at rates already established. Money market instruments such as Treasury bills, commercial paper and bankers' acceptances are also pure discount instruments, though they are often quoted on a discount-rate or add-on basis rather than as bond-equivalent yields.

Candidates should be able to move fluently between price, yield and maturity for a single cash flow, and to state which compounding convention is in force. Expect to convert an annual-basis zero rate into a semiannual-basis equivalent, to build a bond price from a set of spot rates, and to bootstrap a two- or three-period spot curve from par or coupon bond data. Deriving implied forward rates from adjacent spot rates rests on the same discount arithmetic.

You should also be able to construct an accretion schedule: the carrying value rises each period by the yield applied to the prior carrying amount, and that accrual is the interest expense recognised by the issuer under the effective interest method, even though no cash changes hands. Finally, be ready to compare a zero's duration and price sensitivity with those of a coupon bond of equal maturity.