Spot Rate

A bond's arbitrage-free value is the sum of its cash flows, each discounted at the spot rate matching the date on which that cash flow arrives. For a bond paying annual coupons, the first coupon is divided by one plus the one-year spot rate, the second coupon by one plus the two-year spot rate squared, the third by one plus the three-year spot rate cubed, and so on, with the final payment of coupon plus principal discounted at the rate for its own maturity. Nothing about this calculation requires a single yield.

This is why the spot curve is the foundation of fixed-income valuation rather than a curiosity. Any security can be decomposed into a portfolio of dated single payments, and the law of one price requires that identical payments on the same date carry the same discount rate regardless of which bond they come from. If a coupon bond's market price differs from the value implied by the spot curve, the mispricing can in principle be exploited by stripping or reconstituting cash flows.

When zero-coupon instruments are unavailable across every maturity, spot rates are extracted recursively from coupon bonds. The procedure begins at the short end, where a single remaining payment makes the yield to maturity itself a spot rate. That rate is then used to discount the near-dated coupons of the next longest bond; the price of that bond, less the present value of those earlier coupons, must equal the final payment discounted at the unknown spot rate, which can then be solved directly.

A compact illustration: suppose the one-year spot rate is three percent and a two-year bond with a four percent annual coupon trades at par. Its price of one hundred must equal four discounted one year at three percent plus one hundred and four discounted two years at the unknown two-year spot rate. Solving gives a two-year spot rate slightly above four percent. Repeating the step with a three-year bond yields the three-year rate, and the curve is built maturity by maturity.

The spot curve, the par curve and the forward curve are three representations of one set of information, and each can be derived from the others. A par rate is the coupon rate that makes a bond price at par given the prevailing spot rates, so it behaves like a weighted average of spot rates across the bond's life. Forward rates are implied by ratios of compounded spot rates: one plus the two-year spot rate, squared, equals one plus the one-year spot rate multiplied by one plus the one-year rate expected to prevail one year forward.

Curve shape drives the ordering. When the spot curve slopes upward, the par rate and the yield to maturity of a coupon bond of a given maturity both lie below the spot rate for that maturity, because earlier cash flows are discounted at lower rates. Forward rates lie above spot rates. All three relationships reverse when the curve is inverted, and collapse into equality when the curve is flat.

Exam tasks are concrete. Given a short set of spot rates, price a coupon bond by discounting each payment at its own rate, then compute the single yield to maturity that reproduces that price and observe that it differs from every spot rate used. Given a par curve or a sequence of bond prices, bootstrap the missing spot rates one maturity at a time. Given two spot rates, extract the implied forward rate between them, and given a forward sequence, compound it back into a spot rate.

Also expect qualitative judgments: identifying whether a stated relationship between spot, par and forward rates is consistent with an upward-sloping or inverted curve, and recognising that discounting with spot rates, rather than with one yield, is the arbitrage-free benchmark against which relative-value measures such as the zero-volatility spread are defined.