Interest rate swaps and forward rate agreements

Definition Fix dates $0 \le T_1 < T_2$ in years, the accrual fraction $\delta > 0$ of $[T_1, T_2]$ under the contract’s day count, a notional $N$ and a fixed rate $K$ quoted as a simple annual rate. The reference rate $L(T_1,T_2)$ is a simple rate for $[T_1, T_2]$ that is fixed at $T_1$. An end-settled forward rate agreement pays, at $T_2$, the party that receives floating and pays fixed the amount

where a negative amount flows the other way, and the party that pays floating has the opposite position. In the single-curve model, default-free zero-coupon bonds trade without frictions at prices $P(t,T)$ for one currency unit paid at $T$, and the reference rate is the simple lending rate of the same curve, $1 + \delta L(T_1,T_2) = 1/P(T_1,T_2)$. Define the simple forward rate $F = F(0;T_1,T_2)$ by

Then the value at time $0$ to the party that receives floating is

so the fair fixed rate is $F$. The floating payment is replicated by buying $N$ bonds maturing at $T_1$ and selling $N$ bonds maturing at $T_2$. At $T_1$ the $N$ received is lent until $T_2$ at the rate $L(T_1,T_2)$ and returns $N(1 + \delta L(T_1,T_2))$, of which $N$ repays the bonds sold, which leaves $N\delta L(T_1,T_2)$. The replicating bonds cost $N(P(0,T_1) - P(0,T_2)) = N\delta F P(0,T_2)$, and the fixed payment is worth $N\delta K P(0,T_2)$. A start-settled agreement instead pays $N\delta(L - K)/(1 + \delta L)$ at $T_1$, with $L = L(T_1,T_2)$. In the single-curve model this has the same value, and in general its amount and date have to be specified separately. Once the rate has been fixed, the payment is a known amount and is simply discounted.

Definition An interest rate swap exchanges fixed for floating interest on a schedule $T_0 < T_1 < \dots < T_n$, with accrual fractions $\delta_k > 0$ for the periods $[T_{k-1}, T_k]$, a notional $N$ and a fixed rate $K$. At each payment date $T_k$ one party pays the fixed amount $N\delta_k K$ and receives the floating amount $N\delta_k L(T_{k-1}, T_k)$, whose rate is set at the start of the period. In an ordinary same-currency swap the notional only scales the payments and is never exchanged. A payer swap pays fixed and receives floating, and a receiver swap does the opposite. The value to the payer is $\mathrm{PV}_{\text{float}} - \mathrm{PV}_{\text{fixed}}$, the present value of the floating leg minus that of the fixed leg, and the receiver’s value is its negative. When both legs use the same schedule, the swap is a portfolio of forward rate agreements, one for each period, all with the fixed rate $K$. Traded swaps often use different schedules and day counts on the two legs, and swaps on an overnight rate compound the daily rate over each period under their own observation conventions. Those cases need their exact schedules and are not modeled here.

Definition The fixed leg pays $N\delta_k K$ at each $T_k$, so its present value is

The swap annuity $A_0$ is the value, per unit of notional, of receiving $\delta_k$ at every payment date, that is, of the payments of a fixed rate equal to one. It is positive because every $\delta_k$ and every discount factor is positive. Keeping it per unit of notional makes the notional appear exactly once in each leg. Each fixed payment is discounted from its own payment date, not from the start of its period. A change of one basis point, $0.0001$, in the fixed rate changes the value of the fixed leg by $0.0001\,N A_0$. Valuing the fixed leg needs only the discount curve.

Definition When the first floating rate is fixed today or later, so that $T_0 \ge 0$, the floating leg has the present value

where $F_k$ is the forward rate of the reference rate for $[T_{k-1}, T_k]$. In the single-curve model $F_k = F(0;T_{k-1},T_k)$ comes from the discount factors as in the forward rate agreement card, and each term is the value of one floating payment. In a multi-curve setup the reference rate is projected from its own forwarding curve, and the payments are discounted with the curve that matches the collateral of the trade. The floating leg is still this sum, now with the projected forward rates, which is stated here without a stochastic model. If the valuation date lies inside the first period, the first rate is already fixed, and that coupon enters as the known amount $N\delta_1 L(T_0,T_1)$ discounted with $P(0,T_1)$. An amortizing notional $N_k$ or a delay between the end of a period and its payment date changes the corresponding terms.

Definition Suppose that one default-free curve both projects the reference rates and discounts the payments, that each rate is fixed at the start $T_{k-1} \ge 0$ of its period and paid at its end $T_k$, that the rate accrues over the same fraction $\delta_k$ that defines the payment, and that each period starts where the previous one ends. Then $\delta_k F_k P(0,T_k) = P(0,T_{k-1}) - P(0,T_k)$ for every $k$, and the sum telescopes,

For a swap that starts today, $T_0 = 0$ and $\mathrm{PV}_{\text{float}} = N(1 - P(0,T_n))$. Equivalently, the floating payments together with a final payment of the notional are worth exactly the notional at $T_0$, like a floating-rate note on a reset date. The identity is a consequence of these assumptions, not a general formula. With a separate forwarding curve, a coupon already fixed, a payment delay or mismatched accrual conventions, the floating leg has to be computed from its full sum.

Definition The par swap rate $K_{\text{par}}$ is the fixed rate that makes the two legs equally valuable, so that a new swap at that rate is worth zero to both parties. Dividing the floating leg by the positive quantity $N A_0$ gives

with the weights

The weights are positive and add up to one, so the par rate is a weighted average of the forward rates and lies between the smallest and the largest of them. It is not a simple average of spot rates. In the single-curve model the numerator telescopes, and

In a multi-curve setup the definition by equal leg values is unchanged, but the numerator is the full projected sum. A swap made earlier with the fixed rate $K$ is worth $N A_0 (K_{\text{par}} - K)$ to the payer and $N A_0 (K - K_{\text{par}})$ to the receiver, where $A_0$ and $K_{\text{par}}$ are computed today for the remaining payments.