8.2 Yield Measures, Spot Rates, and Forward Rates

Yield is a measure, not a promise

A bond yield summarizes price and cash flows. It is tempting to treat yield as the investor's future return, but realized return depends on reinvestment rates, sale price, holding period, default experience, taxes, and option exercise. Level I items usually ask which yield measure fits the situation, or test the relationship between yield and price.

Common yield measures

Current yield = annual coupon / flat (clean) price. It ignores principal gain or loss and reinvestment. A 5% coupon bond priced at 95 has a current yield of 5 / 95 = 5.26%, which omits the pull to par.

Yield to maturity (YTM) is the internal rate of return that sets the present value of promised cash flows equal to price. For a conventional bond: priced below par -> YTM above coupon; priced above par -> YTM below coupon; priced at par -> YTM equals coupon. YTM rests on three assumptions: the investor holds to maturity, the issuer makes every payment in full and on time, and all coupons are reinvested at the YTM.

Yield to call (YTC) uses the call date and call price instead of final maturity. The yield to worst is the lowest of YTM and the YTC to each possible call date, a conservative figure for premium callable bonds.

Periodicity and compounding

Yields must be compared on the same compounding basis (periodicity). A semiannual-pay bond quotes a stated annual rate as twice its semiannual rate; this is the bond-equivalent yield and a semiannual-bond basis. To convert across periodicities, equate effective rates:

A stated 6% on a semiannual basis is a 3% periodic rate and an effective annual yield of (1.03)^2 - 1 = 6.09%. Money-market instruments add their own conventions: discount yield (T-bills, on a 360-day, price-of-face basis), add-on yield (CDs), and bond-equivalent yield. The exam point: identify the denominator, day count, and compounding before comparing.

Spot rates

A spot rate is the discount rate for a single payment at one maturity, i.e., a zero-coupon rate. The arbitrage-free value of an option-free bond is the sum of each promised cash flow discounted at its maturity-matched spot rate, not at one YTM. Because the yield curve is rarely flat, the spot-rate method is more precise; YTM is only a summary rate that blends them.

Bootstrapping intuition

Bootstrapping derives spot rates sequentially. If a one-year zero price is known, the one-year spot z1 is known. A two-year coupon bond price then solves for z2 after discounting its first-year coupon at z1. Each new maturity adds exactly one unknown spot rate once earlier ones are solved, producing a zero-coupon spot curve consistent with observed prices.

Forward rates

A forward rate is a rate set today for a loan or investment starting in the future. No-arbitrage requires that investing two years at z2 equals investing one year at z1 and reinvesting at the implied one-year forward rate f(1,1):

(1 + z2)^2 = (1 + z1) x (1 + f(1,1)), so f(1,1) = [(1 + z2)^2 / (1 + z1)] - 1.

Example: z1 = 3.00%, z2 = 4.00%. Then f(1,1) = (1.04^2 / 1.03) - 1 = (1.0816 / 1.03) - 1 = 5.01%. The upward-sloping curve implies a forward rate above both spot rates. With semiannual or other periods, match the compounding basis. The same chaining extends across the curve: the general no-arbitrage link is (1 + z_T)^T = (1 + z_A)^A x (1 + f(A, T-A))^(T-A), where f(A, T-A) is the forward rate for a (T-A)-period loan starting in A periods. When the spot curve is upward sloping, each forward rate sits above the spot rate of the same starting point; when the curve is inverted, forwards lie below spots.

A bond can equivalently be valued by discounting each cash flow at its spot rate or by discounting along the sequence of implied forward rates, and the two methods give the identical price by construction.

Realized return versus quoted yield

The horizon (total return) yield measures what an investor actually earns over a holding period shorter than maturity. It depends on the price at sale (set by the yield curve at that future date) and on the rate at which interim coupons are reinvested. If reinvestment rates fall below the original YTM, the realized return is below YTM; if they rise, the realized return exceeds it. This is the practical reason YTM is an assumption-laden number, not a guarantee.

For a bond held exactly to its Macaulay duration, price risk and reinvestment risk roughly offset for a one-time parallel yield shift, which is the foundation of immunization studied at later levels.

Structured aid: yield selection table

Exam focus

Do not mix yield concepts. A discount bond's YTM exceeds its coupon because par repayment adds return; a premium bond's YTM is below its coupon because price decays to par. For a discount bond, the ordering is coupon rate < current yield < YTM; for a premium bond it reverses to coupon rate > current yield > YTM. Forward rates are break-even reinvestment rates from today's curve, not unbiased forecasts. If a question asks for no-arbitrage value or the future rate consistent with the curve, use the spot/forward relations rather than a narrative outlook.