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.
V_{swap} = B_{fix} – B_{fl}
Where V_{swap} = Value of the swap
B_{fix} = The value of the fixed rate bond
B_{fl} = 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. V_{swap} = B_{fl} – B_{fix}
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.
- The principal amounts are exchanged at the beginning and at the end.
- The interest payments are made once a year
Just like a swap, the currency swap can be used to transform liabilities and assets.
Comparative advantage
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.
V_{swap} = B_{D} – S_{0}B_{F}
Most of the calculations for valuing a currency is same as that of a swap.