Deriving the Par Swap Rate in a Single-Curve Framework
Summary
The document asks how the familiar par swap rate formula follows when the forward and discount curves are identical. The answer values a swap as a fixed-rate bond position offset by a floating-rate bond position. In a single-curve framework, the floating-rate bond with no spread is worth par, so its value can be set to one. Equating the fixed leg’s discounted coupons and principal to that value gives the par rate as the remaining principal value divided by the sum of discount factors for the coupon dates.
This bond-based argument explains why the floating coupons drop out of the final expression and clarifies the role of discounting. The response simplifies notation by assuming annual payments and omitting accrual fractions; with other payment schedules, accrual factors must be included. It also notes that the setup can be adapted to a forward-starting swap. The result relies on the single-curve assumption and should not be carried over unchanged to frameworks with separate projection and discount curves.
Key ideas
- A swap can be represented as a fixed-rate bond position offset by a floating-rate bond position.
- In a single-curve framework, a floating-rate bond with zero spread is valued at par.
- The par fixed rate equals the remaining principal value divided by the sum of discounted coupon payments.
- The simplified derivation omits accrual fractions by assuming annual payments.
- Separate forward and discount curves require a different valuation setup.
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# Derivation of Swap rate formula
# Derivation of Swap rate formula
Assuming usual notation, I derive the floating rate and fixed rate payoffs and set them equal. The par swap rate I get thus is:
$$S_{mn}\mid_{t=0} = {\sum_{i=m}^{N-1} \tau_i L(0, T_{i-1}, T_i)Z_{0i} \over \sum_{n=m}^{N-1} \tau \cdot Z_{0i}}$$
When the $L$ and $Z$ curves are identical, the following can result as per the textbook - but I am not sure how?
$$S_{mn} = {Z_{0m} - Z_{0n} \over \sum_{i=0}^{N-1} \tau \cdot Z_{0i}}$$
## Answer by David Duarte (score 6, accepted)
https://quant.stackexchange.com/a/51305
If $L$ and $Z$ curves are identical, you are in a single curve frame work.
A swap can be seen as a long position in a fixed rate bond and a short position in a floating rate bond. (I'll use yearly payments 30/360 in order to be able to ignore the $\tau$ =1 and simplify the notation)
$$DF_1 \times C^{fixed} + ... + DF_n \times C^{fixed} + DF_n - (DF_1 \times C_1^{float} + ... + DF_n \times C_n^{float} + DF_n ) = 0$$
But because in a single curve framework (forward curve = discount curve) a floating rate bond with zero spread will always be at par, you have:
$$DF_1 \times C_1^{float} + ... + DF_n \times C_n^{float} + DF_n = 1$$
and so you only have to worry about the fixed leg:
$$DF_1 \times C^{fixed} + ... + DF_n \times C^{fixed} + DF_n - 1 = 0 $$
$$DF_1 \times C^{fixed} + ... + DF_n \times C^{fixed} + DF_n = 1 $$
If you rearrange this in terms of the fixed rate ($C^{fixed}$), you get:
$$C^{fixed} = \frac{1 - DF_n}{\sum^n_{i=1}DF_i}$$
Which is the formula you showed, just with a different notation.
The intuition of this formula is that you are determining the fixed rate on a bond where the sum of the present value of the interest payments plus the price of the zero coupon bond will be one.
You can also easily adjust this for a forward starting swap.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.