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Interest rate swaps

Exchange a floating stream for a fixed one; the swap rate is the fixed rate that makes the exchange worth nothing at inception.

The formulas

Swap rate

$$R = \dfrac{1 - P(n)}{\sum_{t=1}^{n} P(t)}, \qquad P(t) = (1+s_t)^{-t}$$

Equivalently

$$R = \dfrac{\sum_{t} P(t) f_{t-1,t}}{\sum_t P(t)}$$

a PV-weighted average of the forward rates

Market value later

$$V = (R_{\text{new}} - R_{\text{old}})\sum_{t} P(t)$$

Where it comes from

1. The floating leg of a swap on notional 1 is worth $1 - P(n)$ at inception - the same as a floating-rate note priced at par minus its redemption.
2. The fixed leg is worth $R\sum_t P(t)$, an annuity of $R$ discounted on the zero curve.
3. Setting them equal gives the swap rate, which is therefore a present-value weighted average of the implied forward rates.

Worked example

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

• Swap rate = (1 − last zero price) / (sum of zero prices). Both pieces come straight off the discount curve.
• A swap is worth zero at inception by construction; it only gains value as rates move.

Traps

• Averaging the spot rates instead of using the discount factors.
• Forgetting that the deferred swap sum starts at the deferral date, not at t = 1.

Related

• Forward rates
• Spot rates and the yield curve

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