Spot rates and forward rates

A spot rate is a zero-coupon yield

The n-year spot rate is the annual effective rate at which a single payment n years away is discounted today. There is one spot rate per maturity, and together they are the term structure - a curve rather than a number, which is the whole difference between this topic and everything before it on the syllabus.

Suppose the one-year spot rate is 4.0 percent, the two-year 4.5 percent, the three-year 5.0 percent and the four-year 5.2 percent. A payment of 1000 due in four years is worth 1000 divided by 1.052 to the fourth power, which is 816.46 - not the 822.70 a flat 5 percent would have given.

Nothing here is new machinery. Every cash flow is still discounted from its own date to today; the only change is that the rate used depends on how far away the payment is, so the discounting has to be done flow by flow rather than through an annuity factor.

Spot rates are not observed directly for every maturity, because most traded bonds pay coupons. They are extracted from bond prices by bootstrapping: the one-year spot comes from a one-year instrument, then the two-year spot is whatever makes a two-year bond price correctly given the one-year rate already known, and so on outward. Exam questions usually hand you the curve, but knowing where it came from explains why it is a set of solved values rather than a set of quotes.

Forward rates are implied, not forecast

The forward rate between two dates is the rate the current curve implies for the period between them. It is found by asking what rate makes investing to the later date directly equivalent to investing to the earlier date and reinvesting.

With the curve above, the one-year forward rate starting in one year satisfies 1.04 times one plus that rate, equal to 1.045 squared. That gives 5.0024 percent. The next one-year forward is 6.0072 percent, and the one after it 5.8023 percent. The three-year rate applying from year one to year four, in the same way, is 5.5036 percent.

The word forward is misleading and worth guarding against. A forward rate is not a prediction of where rates will be. It is an arithmetic consequence of today's curve, and it would be exactly the same number if every market participant expected rates to fall.

The no-arbitrage argument

The reason a forward rate is determined rather than chosen is that any other value would allow a riskless profit. If the forward rate on offer exceeded 5.0024 percent, an investor could borrow for two years at 4.5 percent, lend for one year at 4 percent, and lock in the forward - ending with more than the borrowing costs, with no capital and no risk.

That argument is worth reconstructing once rather than memorised, because it explains why the accumulation identity holds exactly. One accumulated at the three-year spot rate gives 1.157625, and one accumulated through the first-year rate and the two implied forwards gives the same 1.157625. Spot rates and forward rates are two descriptions of one curve.

Pricing a coupon bond off the curve

A coupon bond is a bundle of zero-coupon bonds, so it is priced by discounting each coupon at the spot rate for its own date. A three-year bond with face 1000 and annual coupons of 50 gives 50 discounted at 4.0 percent, 50 at 4.5 percent, and 1050 at 5.0 percent.

Those pieces are 48.08, 45.79 and 907.03, so the price is 1000.89. Note that no single rate was used and no annuity factor appeared; the bond's value came from three independent discountings, which is exactly what the term structure requires.

The same method values anything. A four-year level annuity of 100 against this curve is 96.15 plus 91.57 plus 86.38 plus 81.65, which is 355.76 - computed flow by flow because there is no single rate for an annuity factor to use.

Yield is an average, and a misleading one

Having priced the bond at 1000.89, ask what single rate would produce that price. Solving gives 4.9672 percent, which is none of the three spot rates used and is not their average either. It is a payment-weighted compromise dominated by the largest cash flow, which is the redemption at year three.

The four-year annuity behaves the same way: its level-yield equivalent is 4.8593 percent, and the annuity factor at that rate reproduces the 355.76 exactly. The yield is a summary statistic that reproduces the price of that particular set of cash flows and nothing else.

This is why two bonds with the same maturity can have different yields without any arbitrage existing. Their coupon patterns weight the curve differently, so they summarise the same set of spot rates into different single numbers. Comparing them as though the yields were rates is the standard mistake.

Habits that keep this topic simple

Almost every error in this material comes from applying a rate to the wrong interval, and a short discipline eliminates most of them.

• Draw the time line and label each cash flow with its own date before choosing any rate.
• Write the accumulation identity rather than the forward rate formula. One plus the later spot, to the later power, equals one plus the earlier spot to the earlier power, times the forward factor.
• Check that a forward rate lies between neighbouring spot rates in the intuitive direction: a rising curve implies forwards above the spot rates they sit beyond.
• Never use an annuity factor against a non-flat curve. Discount each payment separately, and use the annuity factor only after a single equivalent yield has been found.
• Compute the yield last, and report it as a summary rather than as the rate at which anything was actually discounted.