# Valuation of Swaps

Valuation of Interest Rate Swaps

An interest rate swap is worth zero when it is entered into.

The valuation of a swap can be seen as the difference between two bonds.

As mentioned earlier, a swap can be considered as a fixed income security with long position in a fixed-rate bond and a short position in a floating-rate bond.

Vswap = Bfix – Bfl

Where Vswap = Value of the swap

Bfix = The value of the fixed rate bond

Bfl = The value of the floating rate bond.

The expression can be inversed when we consider the opposite (long position in a floating-rate bond and a short position in a fixed-rate bond)

i.e. Vswap = Bfl – Bfix

Example 7.2 from ‘Options, Futures and other Derivatives’ by John C. Hull

Suppose that a financial institution has agreed to pay 6-month LIBOR and receive 8% p.a. (with semi-annual compounding) on a notional principal of \$100 million. The swap has a remaining life of 1.25 years. LIBOR rates with continuously compounding for 3-months, 9-month and 15-month maturities are 10%, 10.5% and 11%, respectively. The 6-month LIBOR rate at the last payment date was 10.2% (semi-annual compounding). Find the value of the swap.

Let’s find the value of the fixed cash flow:

 timeline Fixed Cash flow Libor rates Discount factor Present value of fixed cash flow 0.25 4.0 10.0% 0.9753 3.9012 0.75 4.0 10.5% 0.9243 3.6971 1.25 104.0 11.0% 0.8715 90.6396 98.2379

The value of the floating cash flow:

 time Floating rate Floating cash flow Discount factor Present value 0.25 10.20% 105.1* 0.9753 102.5051 0.75 0.9243 1.25 0.8715 102.5051

*The floating-rate bond can be regarded as a bond paying single cash flow because immediately after the payment is made, the bond is equal to the current price of the bond.

Hence the value of the swap is 98.2379 – 102.5051 = ─4.2672

The valuation of a swap can be seen as a portfolio of FRAs.

Consider the same example as above.

Value of the fixed cash flow:

 time Fixed Cash flow Libor rates Discount factor Present value of fixed cash flow 0.25 4.0 10.0% 0.9753 3.9012 0.75 4.0 10.5% 0.9243 3.6971 1.25 4.0 11.0% 0.8715 3.4861 11.0845

Value of the floating cash flow:

 time Forward rate Floating rate semi-annual Floating cash flow Discount factor Present value 0.25 10.20% 5.1 0.9753 4.9741 0.75 10.75% 11.0442% 5.52208 0.9243 5.1039 1.25 11.75% 12.1020% 6.05101 0.8715 5.2737 15.3516

The forward rates are calculated from the LIBOR rates.

Present value of cash flow is 11.0845 – 15.3516 = ─4.2672

Currency Swap

A currency swap is the exchange of principal and interest payments in one currency for principal and interest payments in another currency.

Example

Consider a 3-year currency swap agreement between 2 companies A and B who need money in GBP and USD respectively. A pays a fixed rate of 4% on GBP while B pays a fixed rate of 5% on dollars. The principal amounts are adjusted such that they are almost equivalent. In this case, let’s say the principal amounts are GBP 100 million and \$126 million

The above is a fixed-for-fixed currency swap.

1. The principal amounts are exchanged at the beginning and at the end.
2. The interest payments are made once a year

Just like a swap, the currency swap can be used to transform liabilities and assets.

 USD GBP Company A 4.5% 4% Company B 5% 5.6%

Consider the borrowing rates for the companies A and B. A can borrow USD at 0.5% lower than B, however, A can borrow GBP at 1.6% lower than B. So, A has a comparative advantage in the GBP market. From the table, we can see that A is better off borrowing GBP and B is better off borrowing USD.

In the above case the total gain for all the parties is 1.6% ─ 0.5% = 1.1% p.a.

Valuation of currency swaps

Like swaps, currency swaps can be considered the difference between two bonds or a portfolio of forward contracts.

The value of the bond is defined as the difference between the value of the bond defined by the domestic market and the value of the bond defined in foreign currency.

Vswap = BD – S0BF

Most of the calculations for valuing a currency is same as that of a swap.