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Describe a swap transaction and how par rates come to be defined in the swap market.

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

The idea

An interest rate swap is an agreement to exchange interest payments on a notional principal that is not itself exchanged. In a plain vanilla swap one party pays a fixed rate and receives a floating rate, such as an overnight or term benchmark rate reset each period, and the other does the opposite. Only the net difference is paid on each date. Valuing it, the floating leg is worth par immediately after each reset, so at inception the swap has zero value only if the fixed leg is also worth par. The par swap rate is therefore the fixed rate that makes the fixed leg worth par, given by Par rate = (1 − DFn) ÷ (DF1 + DF2 + … + DFn), the same formula as for a par bond. Later the swap gains or loses value as market swap rates move away from the contract rate.

Formulas

Par rate from discount factors

c = (1 − DFn) ÷ (DF1 + … + DFn)

DF1
the discount factor for the first coupon date
DFn
the discount factor for the final date n

In a swap this is the fixed rate that makes the fixed leg worth par, matching the floating leg at inception. It is not the final spot rate.

Value of an existing swap to the fixed payer

V = N × (cm − ck) × (DF1 + … + DFn)

N
the notional principal
cm
the current market par swap rate for the remaining term
ck
the fixed rate in the contract
DF1
the discount factor for the first remaining payment date
DFn
the discount factor for the final payment date

For annual payments. The fixed payer gains when the market rate rises above the contract rate; the receiver's value is the negative.

A practice question

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

In a plain vanilla interest rate swap:

  1. one party pays a fixed rate and receives a floating rate on a notional amount, and the notional itself is not exchanged. - correct
  2. the parties swap the notional principal at the start and return it at the end, with interest paid in between.Why not: Describes a currency swap with principal exchange; in an interest rate swap the notional is only a reference.
  3. both parties pay fixed rates on different notional amounts.Why not: A vanilla swap exchanges fixed for floating, not fixed for fixed.
  4. the fixed-rate payer is paid a premium upfront in exchange for taking default risk.Why not: Confuses swaps with options or credit protection; a vanilla swap has no upfront premium at par.

A vanilla swap exchanges fixed-rate payments for floating-rate payments on a notional principal; only the net amount is paid on each date and the notional is never exchanged.

A harder one

A company pays fixed at 4.0% on a USD 10 million 3-year annual-pay swap. The current 3-year par swap rate is 5.0%, and the discount factors for years 1, 2 and 3 are 0.9709, 0.9426 and 0.9151. The value of the swap to the fixed payer is closest to:

  1. a loss of USD 282,860.Why not: Reverses the sign: the payer locks in 4%, below the market's 5%, so the position has positive value.
  2. a gain of USD 282,860. - correct
  3. a gain of USD 100,000.Why not: Takes one year of the 1% rate gap and ignores the annuity of three discounted payments.
  4. a gain of USD 300,000.Why not: Sums three years of the 1% gap without discounting.

The payer pays 4% instead of the market 5%: value = notional x (5% - 4%) x sum of DFs = 10m x 0.01 x 2.8286 = USD 282,860.

The hardest tier

A fixed payer has a swap with USD 100 million notional, 1.5 years remaining and semi-annual payments at a fixed rate of 5.0%. The next floating payment is already set at 4.6% annualized, so USD 2.3 million will be paid in 6 months. Discount factors for 0.5, 1.0 and 1.5 years are 0.985, 0.969 and 0.951. The value of the swap to the fixed payer, in USD million, is closest to:

  1. -2.36Why not: Values the floating leg at par, ignoring that the coming payment is already fixed at 2.3 million.
  2. -1.60 - correct
  3. 1.60Why not: Reverses the sign, valuing the swap from the fixed receiver's side.
  4. -3.86Why not: Values the floating leg as 100 x DF(0.5), leaving out the coupon of 2.3 million due then.

Floating leg = (100 + 2.3) x 0.985 = 100.77 (par plus the known coupon at the next reset). Fixed leg = 2.5 x (0.985 + 0.969 + 0.951) + 100 x 0.951 = 102.36. The fixed payer's value is floating minus fixed = -1.60.

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