Use a set of spot rates to obtain forward interest rates.

FRM® Part I · Financial Markets and Products · Properties of Interest Rates · FRM1.FM.16.h

The idea

A forward rate is the rate for a future period that is implied by today's spot curve through no-arbitrage. Investing for T2 years at the spot rate R2 must give the same result as investing for T1 years at the spot rate R1 and then rolling at the forward rate F. With annual compounding, F is the ratio of the two growth factors raised to the power 1 ÷ (T2 − T1), minus one: F = [(1 + R2)^(T2) ÷ (1 + R1)^(T1)]^(1 ÷ (T2 − T1)) − 1. With continuous compounding, the rate-times-time terms simply subtract: F = (R2 × T2 − R1 × T1) ÷ (T2 − T1). The reverse is also true: with annual compounding, one plus a long spot rate is the geometric average of one plus each forward rate, not the arithmetic mean, while with continuous compounding the long spot rate is the time-weighted average of the forward rates. The common errors are averaging spot rates, taking differences of rates, and forgetting to annualise over the length of the forward period.

Formulas

Forward rate from two annually compounded spot rates

F = [(1 + R₂)^(T₂) ÷ (1 + R₁)^(T₁)]^(1 ÷ (T₂ − T₁)) − 1

R₁: the spot rate to the earlier date
R₂: the spot rate to the later date
T₁: the earlier date, in years
T₂: the later date, in years

No-arbitrage, not a forecast. Forgetting to annualise over T₂ − T₁ is the common error when the forward period exceeds one year.

Forward rate from two continuously compounded spot rates

F = (R₂ × T₂ − R₁ × T₁) ÷ (T₂ − T₁)

R₁: the spot rate to the earlier date
R₂: the spot rate to the later date
T₁: the earlier date, in years
T₂: the later date, in years

Continuous rates only; with annual compounding the growth factors must be divided, not the rates subtracted.

A practice question

From the question bank, with the reason each wrong option is wrong.

The 1-year and 2-year continuously compounded spot rates are 4.0% and 4.5%. The implied 1-year continuously compounded forward rate starting in one year is:

4.5%Why not: Reports the 2-year spot rate as the forward rate.
4.25%Why not: Averages the two spot rates.
0.5%Why not: Subtracts one spot rate from the other without weighting by the maturities.
5.0% - correct

With continuous compounding F = (R2 x T2 - R1 x T1)/(T2 - T1) = (4.5 x 2 - 4.0 x 1)/1 = 5.0%.

A harder one

The 3-year and 5-year continuously compounded spot rates are 5.0% and 5.6%. The implied continuously compounded forward rate for the period from year 3 to year 5 is closest to:

6.5% - correct
5.30%Why not: Averages the two spot rates.
0.60%Why not: Subtracts the 3-year spot rate from the 5-year spot rate.
13.0%Why not: Subtracts the rate-time products but forgets to divide by the 2-year length of the forward period.

F = (5.6 x 5 - 5.0 x 3)/(5 - 3) = (28.0 - 15.0)/2 = 6.5%.

The hardest tier

A bootstrapped curve gives one-year forward rates, annually compounded, of 3.0% for year 1, 4.5% for year 2 and 6.5% for year 3. The implied 3-year spot rate is closest to:

4.66% - correct
4.50%Why not: Takes the middle forward rate (year 2) as the spot rate, ignoring the other two years.
14.63%Why not: Compounds the three years but does not take the cube root to annualize.
6.50%Why not: Takes the last forward rate as the 3-year spot rate.

(1 + S3)³ = 1.03 x 1.045 x 1.065 = 1.14631, so S3 = 4.66%.

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