Interest Rate Swap Valuation and Applications › Swap Valuation as a Portfolio of FRAs

An alternative and equally valid approach views a swap as a series of forward rate agreements. Rather than thinking of the swap as two bonds, we can decompose it into individual exchanges of fixed for floating payments at each payment date. Each payment exchange in the swap can be treated as a separate FRA. This perspective connects swap pricing directly to forward rates, which we discussed in the context of the term structure in Part II, Chapter 3.

The FRA approach offers different insights than the bond approach. While the bond view emphasizes the swap's relationship to fixed income securities, the FRA view highlights how each payment period contributes to the swap's total value. This decomposition is especially useful for analyzing where swap value comes from along the yield curve.

Before applying this to swaps, let's review forward rate agreements and their valuation. A forward rate agreement is a contract to exchange a fixed interest payment for a floating payment at a future date, based on a rate determined at an earlier fixing date. The contract specifies a notional amount, a forward period, and a contracted rate against which the realized floating rate will be compared.

The payoff of a receive-fixed FRA settling at time $T+\delta$ for a period of length $\delta$ starting at $T$ is:

$$\text{FRA Payoff} = \delta \cdot N \cdot (K - L_T)$$

where:

• $\delta$: accrual period length
• $N$: notional principal
• $K$: contracted forward rate
• $L_T$: realized floating rate at time $T$

The payoff is positive when the contracted rate exceeds the realized floating rate, meaning you receive the difference. Conversely, the payoff is negative when the floating rate exceeds the contracted rate.

To determine the present value before settlement, we use the simple-accrual forward rate implied by today's curve. In this single-curve model, $F(0,T,T+\delta)=[Z(0,T)/Z(0,T+\delta)-1]/\delta$. Its expectation interpretation uses the payment-date forward measure, associated with the bond maturing at $T+\delta$, rather than an unqualified physical forecast. For the stated end-of-period payoff, the present value is:

$$V_{\text{FRA}} = \delta \cdot N \cdot (K - F(0, T, T+\delta)) \cdot Z(0, T+\delta)$$

where:

• $V_{\text{FRA}}$: present value of the FRA
• $F(0, T, T+\delta)$: simple-accrual forward rate observed today for the period starting at $T$, with the same accrual fraction as the payoff
• $Z(0, T+\delta)$: discount factor to the payment date
• $T$: start date of the forward period
• $\delta$: accrual period length
• $N$: notional principal
• $K$: contracted forward rate

This formula tells us that the FRA has positive value when the contracted rate $K$ exceeds the market forward rate $F$, and negative value when the market forward rate exceeds the contracted rate.

With the FRA valuation framework established, we can now view the entire swap as a portfolio of these individual contracts. A swap with $n$ payment dates can be decomposed into $n$ forward rate agreements, each corresponding to one payment exchange. The value of the entire swap is the sum of these FRA values:

$$V_{\text{swap}} = \sum_{i=1}^{n} \delta_i \cdot N \cdot (K - F_i) \cdot Z(0, t_i)$$

where:

• $V_{\text{swap}}$: total value of the swap
• $n$: total number of payment dates
• $\delta_i$: day count fraction for period $i$
• $N$: notional principal
• $K$: fixed rate
• $F_i$: forward rate for period $i$ ($F(0, t_{i-1}, t_i)$)
• $Z(0, t_i)$: discount factor for payment at time $t_i$

For a par swap entered without an upfront premium, choose K so that the initial swap value is zero; an off-market coupon can instead require an upfront value adjustment. By setting the swap value to zero and solving for the fixed rate, we can derive the par swap rate. The derivation proceeds as follows:

$$\begin{aligned} 0 &= \sum_{i=1}^{n} \delta_i \cdot N \cdot (K - F_i) \cdot Z(0, t_i) && \text{(set swap value to zero)} \\ 0 &= N \left( K \sum_{i=1}^{n} \delta_i Z(0, t_i) - \sum_{i=1}^{n} \delta_i F_i Z(0, t_i) \right) && \text{(factor out } N \text{ and separate terms)} \\ K \sum_{i=1}^{n} \delta_i Z(0, t_i) &= \sum_{i=1}^{n} \delta_i F_i Z(0, t_i) && \text{(divide by } N \text{ and rearrange)} \end{aligned}$$

Solving for $K$:

$$K = \frac{\sum_{i=1}^{n} \delta_i \cdot F_i \cdot Z(0, t_i)}{\sum_{i=1}^{n} \delta_i \cdot Z(0, t_i)}$$

where:

• $K$: par swap rate
• $n$: number of payment periods
• $F_i$: forward rate for period $i$
• $\delta_i$: day count fraction
• $Z(0, t_i)$: discount factor
• $\sum_{i=1}^{n} \delta_i \cdot F_i \cdot Z(0, t_i)$: present value of the expected floating leg payments per unit of notional
• $\sum_{i=1}^{n} \delta_i \cdot Z(0, t_i)$: annuity factor for payments of $\delta_i$ per unit notional, corresponding to a fixed annual rate of 1

The par swap rate is a weighted average of simple forward rates, with normalized weights proportional to $\delta_i Z(0,t_i)$. For equal accrual fractions and declining discount factors, later periods receive lower weights. Neither that ordering nor a forecast of physical expected short-term rates follows without additional assumptions. In this single-curve model, the floating payments telescope because $\delta_i F_i Z(0,t_i)=Z(0,t_{i-1})-Z(0,t_i)$, with $Z(0,0)=1$.

The example below takes each accrual fraction to equal the difference between its maturity times in years, so the semiannual schedule uses $\delta_i=0.5$. For a contractual day-count convention, use that convention's accrual fraction in the simple-forward conversion and payment calculation; it need not equal the zero curve's time difference.