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Treat a plain vanilla interest rate swap as a strip of FRAs and use that to value it.

FRM® Part I · Financial Markets and Products · Swaps · FRM1.FM.20.h

The idea

A plain vanilla swap can also be seen as a strip of forward rate agreements, one for each payment date, each exchanging the swap fixed rate for that period's floating rate. Each FRA is valued by assuming that the floating rate turns out equal to today's forward rate. For the fixed payer, the value of the FRA for period i, which settles at the end of that period, is FRAi = N × (Fi − K) × τ × DFi. Here N is the notional, Fi the forward rate for the period, K the fixed rate, τ the accrual fraction and DFi the discount factor to the payment date. The first floating rate was already set at the last reset, so it replaces the first forward rate. Add the FRA values to get the swap value, and flip the sign for the fixed receiver. At initiation the individual FRAs are generally not zero, but their sum is, so the swap rate is a discount-factor weighted average of the forward rates when the accrual fractions are equal. Common mistakes are using zero rates instead of forwards, and not discounting.

Formulas

Value of one FRA in a pay-fixed swap

FRAᵢ = N × (Fᵢ − K) × τ × DFᵢ

N
the notional
Fᵢ
the forward rate for period i, or the rate already set for the first period
K
the swap's fixed rate
τ
the accrual fraction
DFᵢ
the discount factor to the payment date

Sum over all periods for the swap value. Use forward rates, not zero rates, and discount each FRA.

Forward rate from two continuous zero rates

F = (z₂ × T₂ − z₁ × T₁) ÷ (T₂ − T₁)

z₁
the zero rate to T₁
z₂
the zero rate to T₂
T₁
the earlier maturity, in years
T₂
the later maturity, in years

Continuous compounding only. Annually compounded forwards come from the ratio of discount factors instead.

A practice question

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

Treating a plain vanilla interest rate swap as a strip of forward rate agreements, what does each FRA exchange for a fixed payer?

  1. The notional principal at the start and at the end of that period.Why not: Notional is not exchanged in an FRA or in a plain vanilla swap.
  2. The zero rate for that maturity paid against the swap's fixed rate received.Why not: Zero rates are not exchanged; the FRA exchanges the forward rate.
  3. The swap's fixed rate paid against the forward rate received for that period. - correct
  4. The swap's fixed rate received against the forward rate paid for that period.Why not: Reverses the direction: the fixed payer pays the fixed rate in each FRA.

A pay-fixed swap is a series of FRAs, each paying the swap fixed rate and receiving the forward rate for its period. The swap value is the sum of the FRA values.

A harder one

The 1-, 2- and 3-year discount factors are 0.9800, 0.9350 and 0.8700. What is the 3-year annual-pay swap rate when the swap is seen as a strip of FRAs, closest to?

  1. 4.77%Why not: Takes the simple average of the three forward rates; the weights must be the discount factors.
  2. 4.88%Why not: Weights the forwards with the discount factors in reverse order.
  3. 7.47%Why not: Takes the last forward rate rather than a weighted average of all three.
  4. 4.67% - correct

Forward rates: 2.04%, 4.81%, 7.47%. Swap rate = sum(F_i x DF_i)/sum(DF_i) = (1 - 0.8700)/2.7850 = 4.67%.

The hardest tier

A 3-year annual-pay swap has notional USD 50 million. You receive 4.4% fixed. Continuously compounded zero rates are 3.8%, 4.2% and 4.6% for 1, 2 and 3 years. The year-1 floating rate is already set at the 1-year annual rate implied by the 3.8% zero, and annually compounded forward rates apply to years 2 and 3. Value the swap to you as FRAs (USD million), closest to?

  1. -0.39 - correct
  2. -0.46Why not: Does not discount the FRA values.
  3. 0.29Why not: Uses the zero rates in place of the forward rates when valuing each FRA.
  4. 0.39Why not: Reverses the sign, valuing the swap for the fixed payer.

Forwards (annual): 3.87%, 4.71%, 5.55%. Fixed receiver value = sum of 50 x (4.4% - F_i) x exp(-z_i x i) = -0.39.

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