Yield to Maturity

There is no closed-form algebraic solution for yield to maturity on a multi-period bond. The price equation sets the market price equal to the sum of each coupon discounted at one over one plus the periodic rate raised to the period number, plus the redemption amount discounted the same way over the full number of periods. Because the unknown rate appears in every denominator with a different exponent, the equation is a polynomial that must be solved numerically.

In practice a financial calculator does the iteration: enter the number of periods, the periodic coupon payment, the redemption value and the price as a negative present value, then compute the interest rate per period. For a semiannual-pay bond the output is a semiannual rate. Doubling it gives the yield stated on a semiannual bond basis, the market convention; compounding it, one plus the semiannual rate squared minus one, gives the effective annual yield. Candidates must be explicit about which of the two they are quoting.

Yield to maturity is a promised yield, and it delivers the promised return only under three conditions holding simultaneously. First, the issuer makes every coupon and the principal payment in full and on time, so no credit event intervenes. Second, the investor holds the position to the maturity date rather than selling into a changed market. Third, every coupon received is reinvested at the yield to maturity itself until maturity.

The third condition is the one that breaks most often. Coupons arriving over a twenty-year horizon are reinvested at whatever short rates then prevail, and the realised horizon return will exceed or fall short of the original yield depending on whether rates rose or fell. A rise in rates depresses the price but lifts reinvestment income; the two effects offset exactly at a horizon equal to the Macaulay duration, which is the foundation of duration matching for liability-driven portfolios.

Yield to maturity is a single blended rate that summarises an entire term structure of spot rates. A coupon bond can equivalently be valued by discounting each cash flow at its own zero-coupon rate; the yield to maturity is the internal rate of return that reproduces that same price, effectively a cash-flow-weighted average of the relevant spot rates. This is why two bonds of identical maturity but different coupons can carry slightly different yields to maturity when the curve is sloped, an effect known as the coupon effect.

Around it sit related measures candidates must distinguish. Current yield ignores principal repayment and time value entirely. Yield to call applies the same internal rate of return logic to a call date and call price; for a callable bond the lowest of the yields to each call date and to maturity is the yield to worst, the conservative quote. Floating-rate notes are quoted on a discount margin rather than a yield, and money market instruments use add-on or discount-rate conventions that require conversion before comparison.

At Level I the yield to maturity appears in both directions. Given a yield, discount the promised cash flows to obtain price, and recognise that price and yield move inversely along a convex curve, so equal yield changes up and down do not produce equal price changes. Given a price, solve for the yield and compare it to the coupon rate to classify the bond as trading at a discount, at par or at a premium, then describe the pull to par as the price converges on the redemption amount over the remaining life.

Candidates should also be able to convert between quotation bases, to state a yield on an annual basis when comparing a semiannual-pay corporate bond with an annual-pay European issue, and to decompose a realised horizon return into price change and reinvestment income. Knowing when the promised yield overstates expected return, notably for credit-risky or callable issues, is as important as the arithmetic itself.