The swap price: solving for the par swap rate

How is the fixed rate in a new swap set?

The fixed rate of a new swap is chosen so that the swap is worth nothing to either side on day one. That rate is the par swap rate , and it is the swap’s price. Think of the price as the single fixed rate the market quotes, and of the value as what the contract is worth to you at any moment. The price is set once, at inception. The value starts at zero.

The swap price is found by setting two present values equal.

What must the fixed rate equal?

The present value of the fixed and floating legs must match. The floating leg is valued at today’s implied forward rate for each period, because the forward rate is the market’s expectation of each future reset. The fixed leg pays the swap rate every period. Both legs are discounted with the same discount factor for each date.

$$DF_i = \frac{1}{(1 + z_i)^i}$$

$$s_N \sum_{i=1}^{N} DF_i = \sum_{i=1}^{N} IFR_i \, DF_i = 1 - DF_N$$

The last equality is the shortcut. A floating-rate note that pays the reference rate is worth par at each reset, so its coupons are worth par minus the present value of the final principal, which is 1 − DF_N. Solving for the swap rate gives the exam formula.

$$s_N = \frac{1 - DF_N}{\sum_{i=1}^{N} DF_i}$$

Par swap rate, s_Nthe price of the swap: set at inception, fixed for its life

• Fixed rate that equates the two legs Memorize PV(fixed payments) = PV(floating payments at forward rates)
• Par yield Use it the coupon of a bond priced at par on the same curve
• Weighted average of the forward rates Use it weights DF_i ÷ ΣDF, so it lies between the lowest and highest forward

Drag the spot rates up and down. The flat swap-rate line always sits among the forward steps, at the level where the shaded areas above and below it have the same present value.

Exam tip Numerator: one minus the last discount factor only. Denominator: the sum of all of them. Never average the spot rates.

What if the swap pays more often than once a year?

Use one discount factor per payment date and the same formula. The answer is a rate per period, so multiply it by the number of periods in a year to annualize it.

Why is the swap rate the same as a par yield?

A bond with coupon c that is priced at par satisfies c × ΣDF + DF_N = 1, which gives exactly the formula above. So the swap rate is the par yield : the coupon a bond would need to sell at par on the same curve. That is why the swap curve works as a benchmark yield curve, and it links this module to the term-structure lessons in Fixed Income.

Go deeper The swap rate is a weighted average of the forwards

Rewrite the equality as s_N = Σ (DF_i ÷ ΣDF) × IFR_i. The weights are the discount factors divided by their sum, so they are positive and add to 1. In the example they are 34.57%, 33.37% and 32.06%, and 0.3457 × 2.2000% + 0.3337 × 3.6048% + 0.3206 × 4.1047% = 3.279%. A weighted average must lie between the lowest and highest forward rate, which is a fast reasonableness check on any answer.

Go deeper Why earlier forwards carry more weight

Discount factors shrink with time, so earlier periods get slightly bigger weights. On a steeply upward-sloping curve this pulls the swap rate a little toward the early, lower forwards, which is why the three-year swap rate (3.279%) is below the simple average of the forwards (3.30%).