Deriving the Par Fixed Rate for a Plain Vanilla Interest Rate Swap

Summary

The document explains how to derive the fair fixed coupon, or par rate, for a plain vanilla interest rate swap that begins at zero value. It separates the swap into fixed and floating legs, discounts each scheduled cash flow using discount factors, and equates the two present values. Solving that equality gives the fixed rate as the discounted sum of forecast floating payments divided by the discounted fixed-leg accrual factors.

The calculation uses two market curves: discount factors for valuing cash flows and forward rates for projecting floating payments. It allows the fixed and floating schedules to differ, which matters when their payment dates or accrual periods are not aligned. The explanation is a compact valuation formula rather than a numerical example. It does not discuss curve construction, collateral conventions, resets, or other market details that may affect an applied swap valuation.

Key ideas

• The swap’s fair fixed rate makes the present values of its fixed and floating legs equal.
• Discount factors value payments at their respective dates.
• Forward rates estimate the floating coupons over future accrual periods.
• The fixed and floating schedules can differ, so their cash flows may use separate indices.

Tags

# analytical formula for FV of fixed rate of a IRS


# analytical formula for FV of fixed rate of a IRS












IRS plain vanilla - expiry in 5 years - principal is 1$ - semianual payment

How could the analytical formula be derived for the fair value of the fixed rate (initially no value of the swap)?

## Answer by Attack68 (score 6)

https://quant.stackexchange.com/a/46389

The key inputs to this calculation are two yield curves obtained from market data: $\{v_i\}$ the discounting factors (value today of \$1 received at time i) and $\{r_i\}$ the forecasting curve (forward semiannual rates for period i to i+1).

The calculation itself proceeds as follows. There are two legs to a fixed/floating interest rate swap.

The fixed leg, which has a present value (PV) equal to the sum of its cashflows discounted based on their payment date:

$PV = N R \sum_i d_i v_i $

for $N$ the notiional, and $d$ the day count fraction and $v$ the discount period for period $i$.

The floating leg has a present value:

$PV = N \sum_j d_j v_j r_j $

for $r$ the floating rate forecast for the period $j$. $i$ and $j$ differ if the schedules are not aligned.

The to derive the mid-maket rate you set these equal to each other:

$ \implies R = \frac{\sum_j v_j d_j r_j}{\sum_i v_i d_i} $

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.