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FRM® bond pricing: use the spot rate for each cash flow

A coupon bond has payments at several dates, so each payment needs the spot rate for its own maturity. Follow a small worked example and avoid the final-rate trap.

ConceptOmni Curriculum Team2 min read

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To find a coupon bond’s theoretical price from spot rates, discount each promised cash flow at the spot rate for its payment date, then add the present values. An early coupon and the final principal do not share a discount rate unless the spot curve happens to be flat.

Why the payment date matters

A spot rate prices a payment at a particular maturity. A coupon bond can therefore be viewed as a collection of separate dated payments. The CFA Institute explanation of spot curves uses that same maturity-by-maturity pricing logic; GARP’s FRM study materials place interest-rate work in the Part I financial markets curriculum.

P=∑t=1TCFt(1+st)tP = \sum_{t=1}^{T} \frac{CF_t}{(1+s_t)^t}
Bond price P is the sum of each annual cash flow CF at year t discounted by its annual-compounded spot rate s for that maturity.

The formula shown assumes annual payments and annually compounded spot rates. If the question gives another payment or compounding frequency, make the cash-flow periods and rate periods consistent before calculating.

A small illustrative calculation

Assume an invented bond with £100 face value, an annual 4% coupon, and maturity in two years. The annual-compounded one-year spot rate is 3% and the two-year spot rate is 5%. These are illustrative inputs, not observed market rates.

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Article data
End of year one£43%£3.88
End of year two£1045%£94.33

Discount the first £4 by 1.03. Discount the final £104, which includes the coupon and principal, by 1.05 squared. The unrounded values are £3.8835 and £94.3311, giving a theoretical price of £98.2146, or £98.21 to the nearest penny. All figures in this calculation are illustrative.

The common wrong turn

Using the two-year 5% spot rate for both payments treats the first coupon as if it faced the same time price as the later payment. Equally, putting only £100 into the final cash flow drops the last coupon. Write a cash-flow timeline first, pair every payment with its own maturity, and check whether each quoted spot rate is annual or per period.

For adjacent rate concepts, the forward-rate explanation covers a rate implied between dates. The FRM program page provides the broader programme context.

Sources

  1. FRM Study Materials | GARP · garp.org · Retrieved
  2. The Term Structure of Interest Rates: Spot, Par, and Forward Curves | CFA Institute · cfainstitute.org · Retrieved

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Educational content, not investment advice. This article is written to help candidates prepare for professional exams. It is not a recommendation to buy, sell or hold any security, and it does not take your personal circumstances into account.

Omni Finance Academy is an independent exam-preparation provider and is not affiliated with or endorsed by CFA Institute, GARP, CAIA Association or CFP Board. CFA®, FRM®, CAIA® and CFP® are trademarks of their respective owners. Full disclaimer.

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