Valuing swaps between payment dates
1. The idea
In the value of an interest rate swap, we valued a plain swap just after a payment date, when its floating side is worth exactly its notional. Between payment dates one more thing is known: the next floating payment, set at the last payment date. Just after that payment the floating side is worth its notional again, so its value today is the present value of the notional plus that payment:
\[ B_{\text{float}} = (L + k^*)\,e^{-r_1 t_1}, \]
where \(L\) is the notional, \(k^*\) the next floating payment, and \(r_1\) the zero rate for its date \(t_1\). The fixed side is a bond priced with the zero rates, as before, and the swap is worth \(B_{\text{fix}} - B_{\text{float}}\) to the side that receives fixed.
An example. On $10 million, the next floating payment, in 3 months, was set at 5% a year for six months: \(k^* = 10 \times 0.05 \times 0.5 = \$0.25\) million. With a zero rate of 4% for 3 months, the floating side is worth \((10 + 0.25)\,e^{-0.04 \times 0.25} = \$10.148\) million.
We value a currency swap the same way, each side as a bond, but in two currencies. Section 4 shows how, next to its exercises.
2. Try it
If an answer is wrong, we will see what went wrong and how to fix it.
3. The value on a chart
First, the floating payment that is already known.
Now move the floating rate set at the last payment date. The higher it is, the more the next floating payment is worth, and the less the swap is worth to the side that pays it.
4. One worked example, then your turn
The worked example is a swap in which the company pays fixed.
Currency swaps exchange interest in two currencies, and the principals at the end. Each side is a bond in its own currency, discounted with that currency’s interest rate. The foreign bond is changed into dollars at the spot exchange rate \(S_0\), in dollars per unit of the foreign currency, and the swap is worth
\[ S_0 B_F - B_D \]
to the side that receives the foreign currency, where \(B_F\) is the foreign bond’s value in the foreign currency and \(B_D\) the dollar bond’s.
Now a currency swap, with hints, one step at a time.
The same value comes out when each exchange of payments is valued as a forward contract on the currency. At a payment date \(t\), the foreign payment is worth, in dollars, the payment times the forward exchange rate for that date, \(F_t = S_0 e^{(r - r_f)t}\), where \(r\) and \(r_f\) are the dollar and the foreign interest rate. Less the dollar payment, that is the exchange’s net payment in dollars, and its present value, that amount times \(e^{-rt}\), is the exchange’s value today. The swap is worth the sum of these values over all the payment dates, the last one with the principals.
Then a currency swap valued as forward contracts, with hints.
And one on your own.
5. Practice
Each question has new numbers. You master the skill when you get three right in a row, with at least one right answer for an interest-rate swap, a currency swap valued as two bonds, and one valued as forward contracts. You can always open the solution, but then that question does not count.
6. Coming back to it
When you have mastered a skill, it comes back on the start page for a short review after 1 day. After each right answer the next review comes later: after 3, 7, 14 and then 30 days. After a wrong one, the reviews start again at 1 day.
This skill: