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Value a currency swap as two bond positions held simultaneously.

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

The idea

A fixed-for-fixed currency swap equals a bond in each currency held in opposite positions. BD is the domestic bond, valued at domestic zero rates in domestic currency, and BF is the foreign bond, valued at foreign zero rates in foreign currency. S0 is the spot rate, in domestic units per foreign unit. For the party receiving domestic currency and paying foreign currency, the swap value is V = BD − S0 × BF. For the other party the sign flips. Each bond holds its coupons and its principal, and the principals are exchanged at the end, so they must be included. At initiation the value is zero, because BD equals S0 × BF. Other things equal, when the foreign currency appreciates, a party paying the foreign currency loses. The usual errors are failing to convert BF, using the inverse quote, and discounting both bonds at the same curve.

Formulas

Value of a currency swap as two bonds

V = BD − S0 × BF

BD
the domestic bond's value, at domestic zero rates
S0
the spot rate, domestic units per foreign unit
BF
the foreign bond's value in foreign currency, at foreign zero rates

For the party receiving domestic and paying foreign; flip the sign for the other. Both bonds include their principal.

A practice question

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

A party receives USD and pays EUR in a currency swap. With B_D the USD bond value, B_F the EUR bond value in EUR, and S0 the spot in USD per EUR, what is the swap's value in USD?

  1. B_D - B_FWhy not: Subtracts the EUR bond without converting it to USD.
  2. S0 x B_D - B_FWhy not: Converts the USD bond with the spot rate and subtracts a EUR amount; the units are mixed.
  3. B_D - S0 x B_F - correct
  4. S0 x B_F - B_DWhy not: Reverses the sign, giving the value for the party that pays USD and receives EUR.

The party receives the USD bond and pays the EUR bond. The EUR bond is converted to USD at the spot rate, so V = B_D - S0 x B_F.

A harder one

A currency swap has 2 years left, with annual payments. You receive USD 5.0% on USD 60 million and pay EUR 3.2% on EUR 50 million, principals exchanged at maturity. USD zero rates are 4.0% and EUR zero rates 3.0% (continuous, flat). The spot rate is 1.20 USD per EUR. What is the swap's value to you (USD million), closest to?

  1. 10.89Why not: Leaves the EUR bond unconverted when subtracting it from the USD bond.
  2. 0.86 - correct
  3. -0.86Why not: Reverses the sign, valuing the swap for the party that pays USD.
  4. 19.25Why not: Divides the EUR bond by the spot rate, inverting the quote.

B_D = 3.0 e^(-0.04) + 63.0 e^(-0.08) = 61.04. B_F = 1.6 e^(-0.03) + 51.6 e^(-0.06) = 50.15 EUR million. V = 61.04 - 1.20 x 50.15 = 0.86.

The hardest tier

You pay USD 4.0% on USD 40 million and receive EUR 2.5% on EUR 36 million, annually for 3 years, principals exchanged at maturity. Continuously compounded zero rates: USD 3.5%, 3.8%, 4.0% and EUR 2.0%, 2.3%, 2.6% for years 1, 2, 3. The spot rate is 1.12 USD per EUR. What is the swap's value to you (USD million), closest to?

  1. -7.89Why not: Divides the EUR bond by the spot rate instead of multiplying.
  2. -0.25Why not: Reverses the sign, valuing the swap for the party that receives USD.
  3. 0.25 - correct
  4. -1.36Why not: Discounts the EUR bond at USD zero rates instead of EUR rates.

B_D = 39.92; B_F = 35.87 EUR million. You are long the EUR bond and short the USD bond: V = 1.12 x 35.87 - 39.92 = 0.25.

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