Deriving the Single-Curve Interest Rate Swap Par Rate
Summary
The document derives the fixed rate that makes the present values of a swap’s fixed and floating legs equal under a single-curve framework. It first expresses each leg as a discounted sum of accrual-period cash flows, then solves for the fixed rate as the ratio of the floating-leg value to the fixed-leg annuity.
The answer explains the further simplification of the floating-leg sum. Under single-curve discounting, each projected forward rate is obtained from the ratio of adjacent discount factors over its accrual period. Substituting that relation into the sum makes the intermediate terms cancel, leaving the initial discount factor less the final one. The result depends on the single-curve assumption; the document does not extend the derivation to multi-curve valuation or discuss market conventions beyond its stated notation.
Key ideas
- The par swap rate is found by equating fixed-leg and floating-leg present values.
- Under single-curve assumptions, forward rates are implied by adjacent discount factors and accrual fractions.
- Substituting those forward rates into the floating-leg sum causes intermediate discount-factor terms to cancel.
- The simplified floating-leg value is the difference between the initial and final discount factors.
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Full text
# Par rate of Interest Rate Swap
# Par rate of Interest Rate Swap
I'm interested in deriving the par rate of an interest rate swap priced under the single curve framework. Let's follow the corresponding Wikipedia article for the sake of notation simplicity.
The present value of the fixed leg can be calculated as $$PV_{fixed} = NR\sum_{i=1}^{n_1}d_i x_i,$$ where $N$ is the notional, $R$ is the fixed rate, $n_1$ is the number of payments of the fixed leg, $d_1$ is the decimalised day count fraction of the accrual in the $i$'th period and $x_i$ is the corresponding discounting factor.
Similarly the present value of the floating leg is given by $$PV_{float} = N\sum_{j=1}^{n_2}r_j d_j x_j,$$ where $n_2$ is the number of payments of the floating leg and $r_j$ are the forecasting (forward) rates.
In order to find the par rate we set $PV_{fixed} - PV_{float} = 0$ and solve for $R$, the resulting expression is $$R = \frac{\sum_{j=1}^{n_2} r_j d_j x_j}{\sum_{i=1}^{n_1} d_i x_i}.$$
However, Wikipedia claims that under the single curve framework this expression can be simplified further to $$R = \frac{x_0 - x_{n_2}}{\sum_{i=1}^{n_1} d_i x_i}$$
The above expression isn't obvious to me. How do we conclude that $\sum_{j=1}^{n_2} r_j d_j x_j = x_0 - x_{n_2}$?
## Answer by yoggi-yalla (score 4, accepted)
https://quant.stackexchange.com/a/73384
It follows from the fact that, under the single-curve framework, the projected rate $r_j$ is found via:
$$r_j = \frac{\frac{x_{j-1}}{x_j} - 1}{d_j}$$
If you plug in this expression for $r_j$ in the summation then all terms will cancel out except for $x_0$ and $x_{n_2}$.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.