Par Rates & Swap Rates

Par Rates & Swap Rates

Par Rate

The par rate for maturity $n$ is the coupon rate that makes a bond price exactly equal to its face value (price = 1):

$1 = p_n \sum_{t=1}^{n} d(t) + d(n)$

Solving for $p_n$:

$p_n = \frac{1 - d(n)}{\sum_{t=1}^{n} d(t)}$

Where $d(t) = \frac{1}{(1 + s_t)^t}$ is the discount factor at time $t$.

Swap Rate

An interest rate swap exchanges fixed payments for floating rate payments. The swap rate is the fixed rate that makes the swap's NPV equal to zero.

The swap rate formula is identical to the par rate formula:

$\text{Swap Rate} = \frac{1 - d(n)}{\sum_{t=1}^{n} d(t)}$

Intuition

Both formulas have the same structure because a fixed-rate bond and a swap have the same cash flow structure at inception:

• Numerator: $1 - d(n)$ = the net difference between the initial notional and the final discounted payment
• Denominator: sum of discount factors = the annuity factor

A rising yield curve means longer swap rates exceed shorter ones.