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Calculate forward rates from spot rates and interpret what a forward rate tells you.

FRM® Part I · Valuation and Risk Models · Interest Rates · FRM1.VR.10.c

The idea

A forward rate is the rate, fixed today, for borrowing or lending over a future period. It is implied by spot rates through no-arbitrage. Lending to time T2 must give the same result as lending to T1 and rolling at the forward rate. Let s1 and s2 be the spot rates to T1 and T2, and f the forward rate between them. So with annual compounding, the two routes give Growth = (1 + s2)^T2 = (1 + s1)^T1 × (1 + f)^(T2 − T1). Solving that equation for f gives the forward rate as Forward rate = [(1 + s2)^T2 ÷ (1 + s1)^T1]^(1 ÷ (T2 − T1)) − 1. With continuous compounding it simplifies to Forward rate = (s2 × T2 − s1 × T1) ÷ (T2 − T1). The forward rate is the break-even rate. If the realised future spot rate ends up above it, rolling short loans beats the long bond, and below it the long bond wins. It is not a forecast. A common error is annualizing incorrectly by omitting the root.

Formulas

Forward rate from annually compounded spot rates

f = [(1 + s2)^(T2) ÷ (1 + s1)^(T1)]^(1 ÷ (T2 − T1)) − 1

s1
the spot rate to the earlier date T1
s2
the spot rate to the later date T2
T1
the start of the forward period, in years
T2
the end of the forward period, in years

No-arbitrage between lending to T2 and lending to T1 then rolling. Omitting the root leaves a cumulative, not annual, rate.

Forward rate from continuously compounded spot rates

f = (s2 × T2 − s1 × T1) ÷ (T2 − T1)

s1
the continuously compounded spot rate to T1
s2
the continuously compounded spot rate to T2
T1
the start of the forward period, in years
T2
the end of the forward period, in years

Exact only with continuous compounding. With annual compounding it is merely an approximation.

A practice question

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

A 1-year-forward 1-year rate (the rate for borrowing from year 1 to year 2, fixed today) is best interpreted as:

  1. the market's unbiased forecast of next year's 1-year spot rate.Why not: Treats the forward rate as an expectation; it is a no-arbitrage break-even and may include risk premia.
  2. today's 1-year spot rate, because both are 1-year rates.Why not: Ignores that the forward rate applies to a future period and generally differs from the spot rate.
  3. the 2-year spot rate, because the borrowing ends at year 2.Why not: Confuses the rate for the single second year with the annualized rate over the whole two years.
  4. the rate implied by today's spot curve that makes lending for 2 years equal to lending for 1 year and rolling over. - correct

No-arbitrage requires (1 + s2)² = (1 + s1)(1 + f). The forward rate is the break-even rate for the second year; it is not necessarily the expected future spot rate.

A harder one

The annually compounded spot rates are 4% for 2 years and 5% for 5 years. The implied 3-year rate beginning in 2 years (the 2y3y forward) is closest to:

  1. 5.00%Why not: Reports the 5-year spot rate rather than the rate for the forward period.
  2. 18.00%Why not: Does not annualize: this is the total 3-year growth, not a per-year rate.
  3. 5.67% - correct
  4. 4.50%Why not: Averages the 2-year and 5-year spot rates, ignoring the weights of the different periods.

(1.05^5 / 1.04²)^(1/3) - 1 = (1.2763/1.0816)^(1/3) - 1 = 5.67%. The ratio of growth factors is the 3-year growth and must be annualized by the third root.

The hardest tier

Semi-annually compounded spot rates are 2.0% for 6 months and 4.0% for 1 year. A bank agrees today to lend USD 10 million for the six months that start in 6 months, at the forward rate implied by these spot rates. The interest paid at the end of that period is closest to:

  1. USD 200,000Why not: Uses the 1-year spot rate of 4.0% for the half-year; it is an average over a full year.
  2. USD 100,000Why not: Uses the 6-month spot rate of 2.0% for the half-year, which applies to the first period only.
  3. USD 300,990 - correct
  4. USD 601,980Why not: Uses the annualized forward rate for a half-year, without halving it.

The half-year forward rate is (1.02)² / (1.01) - 1 = 0.03010 (annualized 6.02%). The interest on USD 10 million is 10m x 0.03010 = USD 300,990.

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