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Value a plain vanilla interest rate swap as two bond positions held simultaneously.

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

The idea

A plain vanilla swap equals a fixed-rate bond and a floating-rate bond held in opposite positions, with the same notional. The fixed payer is long the floating bond, worth Bfloat, and short the fixed bond, worth Bfixed, so its value is V = Bfloat − Bfixed. The fixed receiver holds the reverse position, so its value is Bfixed − Bfloat. The fixed bond is worth the present value of its coupons and principal at the current zero rates. A floating bond is worth par just after a reset date, because its next coupon is set at the market rate. Between reset dates its next payment is already known, so it is worth the notional plus that payment, discounted from the next payment date only. The swap is worth zero at initiation, when its fixed rate is the par swap rate. The common errors are reversing the sign, valuing the floating bond at par between reset dates, and discounting at the wrong compounding convention.

Formulas

Value of a swap to the fixed payer, as two bonds

V = Bfloat − Bfixed

Bfloat
the value of the floating-rate bond
Bfixed
the value of the fixed-rate bond, coupons and principal at current zero rates

Reverse the sign for the fixed receiver. Bfloat is par just after a reset, but between resets it is the notional plus the known next payment, discounted from that date.

A practice question

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

Treating a plain vanilla interest rate swap as two bonds, what is the value of the swap to the fixed-rate payer?

  1. The sum of the values of the fixed-rate bond and the floating-rate bond.Why not: Adds the bonds; the swap is a long position in one and a short position in the other.
  2. The value of the fixed-rate bond minus the notional principal.Why not: Treats the floating bond as par at all times, which holds only on reset dates.
  3. The value of the fixed-rate bond minus the value of the floating-rate bond.Why not: Reverses the sign: this is the value to the fixed receiver.
  4. The value of the floating-rate bond minus the value of the fixed-rate bond. - correct

The fixed payer is short the fixed bond and long the floating bond, so V = B_float - B_fixed. The fixed receiver's value is B_fixed - B_float.

A harder one

A swap with notional USD 100 million has just reset. It pays semi-annually for 1.5 more years, and you receive 6% fixed and pay floating. Continuously compounded zero rates are 4.0%, 4.3% and 4.6% for 0.5, 1.0 and 1.5 years. What is the swap's value to you (USD million)?

  1. 10.56Why not: Uses the annual 6% as the semi-annual coupon, doubling the fixed payments.
  2. 1.95 - correct
  3. -1.95Why not: Reverses the sign, valuing the swap for the fixed payer.
  4. 101.95Why not: Values only the fixed bond and forgets to subtract the floating bond, which is worth par.

B_fixed = 3 x e^(-0.02) + 3 x e^(-0.043) + 103 x e^(-0.069) = 101.95. B_float = 100 just after a reset. Fixed receiver's value = 101.95 - 100 = 1.95.

The hardest tier

A swap on USD 40 million has just reset and pays semi-annually for 1.5 years. You pay 5.0% fixed and receive floating. Zero rates with semi-annual compounding are 5.4%, 5.8% and 6.2% for 0.5, 1.0 and 1.5 years. What is the swap's value to you (USD million), closest to?

  1. 0.67 - correct
  2. -0.67Why not: Reverses the sign, valuing the swap for the fixed receiver.
  3. 3.93Why not: Divides the rates by one instead of two, treating the semi-annual rates as annual compounding.
  4. -2.16Why not: Doubles the fixed coupon, using the annual 5.0% for each semi-annual payment.

Semi-annual coupon = 40 x 2.5% = 1.00. B_fixed = 1/1.027 + 1/1.029² + 41/1.031³ = 39.33. As the fixed payer, V = B_float - B_fixed = 40 - 39.33 = 0.67.

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