The value of an interest rate swap
1. The idea
A swap can be valued as two bonds. Receiving fixed and paying floating is like owning a bond that pays the fixed rate and owing one that pays the floating rate, both repaying the notional at the end. Just after a payment date the floating bond is worth exactly its notional, because its next payment is set at the market’s rate. This holds for a plain swap whose floating rate has no added margin, is reset on each payment date and is discounted with the same zero rates. The notionals in the two bonds are never paid: they cancel out. The fixed bond is priced with the zero rates, as in the lesson on bonds. Per $100 of notional:
\[ \begin{aligned} \text{receive fixed: value} &= B_{\text{fix}} - 100,\\ \text{pay fixed: value} &= 100 - B_{\text{fix}}, \end{aligned} \]
where \(B_{\text{fix}}\) is the present value of the fixed payments and the notional at the end, each discounted with its zero rate, and $100 the value of the floating bond just after a payment date.
When the floating payments are set by one curve of rates and discounted with another, we value each payment on its own. Section 4 shows how, next to its exercise.
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
In the question we receive 5% a year on a notional of $100 for two more years, and pay the floating rate.
The floating bond is worth $100 just after a payment date. The fixed side is priced with the zero rates: \(5 \times e^{-0.04 \times 1} + 105 \times e^{-0.045 \times 2} = 100.7667\). So the swap in the question is worth \(\$100.7667 - \$100 = \$0.77\) to us. Now give every date the same zero rate and move it: the swap is worth $0 to both sides at one rate, and whatever one side gains, the other loses.
4. One worked example, then your turn
The worked example is the side that pays fixed.
Now three years left, with hints, one step at a time.
With two curves. The floating payments may be set by one curve of rates and discounted with another. The floating bond is then not necessarily worth its notional just after a payment date, so each payment is valued on its own, like a forward rate agreement. With payments every six months on a notional \(L\):
\[ \begin{aligned} V_{\text{fix}} &= \sum_t L\, R_{\text{fix}} \times 0.5 \times P_t,\\ V_{\text{fl}} &= \sum_t L \left(e^{0.5 f_t} - 1\right) P_t, \end{aligned} \]
where \(V_{\text{fix}}\) and \(V_{\text{fl}}\) are the present values of the fixed and the floating payments, \(R_{\text{fix}}\) is the fixed rate a year, \(f_t\) the floating rate for the six months that end at date \(t\) (already set for the first period, a forward rate for the later ones) with continuous compounding, both as decimals, and \(P_t = e^{-z_t t}\) the discount factor from the zero rate \(z_t\) for date \(t\). Receiving fixed, the swap is worth \(V_{\text{fix}} - V_{\text{fl}}\); paying fixed, \(V_{\text{fl}} - V_{\text{fix}}\).
Then a swap valued with two curves, 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 receiving fixed and one paying fixed, each with one curve, and one for a swap valued with two curves. 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: