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FRM® Part I: find a forward rate from spot rates

Today’s spot curve implies a rate for a future lending period. Work through annual compounding, verify two matching investment paths, and avoid the forecast trap.

ConceptOmni Curriculum Team2 min read

Two blank paper strips showing separate investment paths beside a financial calculator on a white desk.
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For annually compounded spot rates, the one-year forward rate beginning after the first year is the rate that makes a two-year investment from today grow by the same amount as a one-year investment followed by a second year at a rate agreed today. Divide the two-year growth factor by the first-year growth factor, then subtract one. The result is an implied break-even rate, not a forecast of the rate that will actually prevail.

Match the two paths to the same end date

A spot rate begins now; a forward rate applies to a future interval. To infer a forward, compare two strategies with the same starting amount and maturity. One holds an investment through the full horizon at the longer spot rate. The other locks in the shorter spot rate and the forward for the remaining interval. With matching cash-flow terms and no arbitrage, their terminal values must agree.

The European Central Bank’s discussion of forward rates describes implied forwards as rates calculated from observed spot rates through replication. It also cautions that a forward can contain a risk premium. GARP’s FRM study materials identify learning objectives for the programme; in revision, practise both the rate calculation and what the answer means.

(1+s2)2=(1+s1)(1+f1,2)(1+s_2)^2=(1+s_1)(1+f_{1,2})
With annual compounding, s1 and s2 are spot rates to the end of years one and two; f1,2 is the one-year forward rate between those dates.
f1,2=(1+s2)21+s1−1f_{1,2}=\frac{(1+s_2)^2}{1+s_1}-1
Solve the equal-growth condition for the forward rate between years one and two.

A worked example with invented rates

Assume, purely for illustration, that the annually compounded one-year spot rate is 2% and the annually compounded two-year spot rate is 4%. These are invented rates, not a market curve. Insert them as decimals in the annual-compounding equation.

f1,2=1.0421.02−1=0.060392…≈6.04%f_{1,2}=\frac{1.04^2}{1.02}-1=0.060392\ldots\approx 6.04\%
The illustrative implied forward rate for the second year, rounded to two decimal places.

Check with an illustrative £100 invested today. The two-year route ends at £108.16, since £100 grows by the factor 1.0816. The rolling route first reaches £102; multiplying that amount by the unrounded forward growth factor of approximately 1.060392 also gives £108.16. Rounding the forward rate before the check can create a small discrepancy.

What candidates misread

  • Use the specified convention. Annual compounding requires growth factors; a continuously compounded formula will give a different answer.
  • Read the interval. The rate from year one to year two uses the one-year and two-year spot rates, while a rate starting after year two requires a later maturity.
  • Keep implication separate from prediction. Today’s curve fixes a break-even rate under the stated assumptions; future realised rates may differ.

A quick reasonableness check helps: with this upward-sloping illustrative spot curve, the second-year forward exceeds the two-year spot rate because the first year earns less than that two-year rate. The FRM programme page provides the wider exam context, and the study methodology is a reference when organising practice around calculations and interpretation.

Sources

  1. FRM® Study Material, Guide, Books, Practice Exams | GARP · garp.org · Retrieved
  2. The ECB survey of Monetary Analysts: an introduction · ecb.europa.eu · Retrieved

Explore FRM study resources

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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