Pricing and Valuation of Interest Rate and Other Swaps

A plain vanilla swap looks like a strip of forwards, but it charges ONE fixed rate across every settlement date. That single tension is the whole reading.

A pay-fixed, receive-floating interest rate swap with eight quarterly settlements exchanges (floating − fixed) × notional × the accrual fraction on each payment date. The accrual fraction is one quarter here. Each settlement works like an interest rate forward that pays at the end of its period.

A standard forward rate agreement (FRA) settles at the start of its period instead, so it pays the same interest gap discounted one period. After that timing adjustment, the swap is economically a strip of eight FRAs. You describe how a swap differs from that strip by looking at the fixed rate. Each separately priced, at-market FRA uses its own implied forward rate and has zero value at initiation.

Implied forward rates describe how borrowing costs vary across swap periods. A swap charges ONE fixed rate across all settlements. A lower implied forward means negative value to the fixed-payer.

Common mistakes

• Treating a swap as a strip of at-market forwards. When the single swap rate is applied, the implied forwards are generally OFF-market. Their values sum to zero at inception, but individually they are positive or negative unless a forward rate happens to equal the swap rate.
• Forgetting that the floating leg prices to par at reset. This shortcut is what makes the pricing formula work. Trap: trying to forecast every floating coupon when a reset just occurred.
• Confusing price with value. Price is the contractual fixed rate. Value is the dollar mark-to-market. Trap: answering "value at initiation" with the swap rate (e.g., 3.94%) when the correct answer is zero.

Bottom line

• A par swap is a portfolio of forwards sharing one fixed rate. The components are generally off-market and their values sum to zero at initiation. With a flat forward curve, all components can be at-market.
• Discount factors come from spot rates: $D(t) = 1/(1 + s_t)^t$; implied forwards are $D(t-1)/D(t) - 1$.
• Swap price (the fixed rate) is set at initiation so PV(fixed leg) = PV(floating leg), making value zero on day one.
• The swap rate per period solves $s^* = (1 - D(t_n)) / \sum D(t_i)$; divide by the year fraction to annualize a semiannual or quarterly rate.

Exam shortcut

For pricing, memorize the words: "one minus last discount factor, over sum of discount factors." If the question gives spot rates, convert each to a discount factor first, compounding over the full maturity. With semiannual or quarterly payments, divide the result by the year fraction.

The full lesson (about 3,192 words, 21 min read) adds 6 worked examples, all 7 common mistakes, a self-check, free in the app.

Learning objectives

• pricing and valuation of swaps

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