Pricing a Libor-Plus Swap with a Separate Risk-Free Discount Curve
Summary
The document explains how to find the par fixed rate for an interest rate swap whose floating leg pays Libor plus a constant spread when discounting uses a separate risk-free curve. It sets the present value of fixed payments equal to the discounted value of floating payments, using Libor forward rates, accrual fractions, and risk-free discount factors. Once the spread is specified, the equation can be rearranged to solve for the par fixed rate; if both the rate and spread are unknown, the information given does not uniquely determine both.
The responses distinguish swap valuation from estimating a swap spread relative to government yields. One notes that such a spread may reflect supply and demand and cannot be inferred from the listed rates alone. The discussion gives a valuation framework rather than market data or a worked numerical example, and it does not address conventions, curve construction, or how the spread itself is agreed.
Key ideas
- The par fixed rate equates the present values of the fixed and floating legs.
- Floating payments use Libor forward rates plus the agreed spread.
- Discount factors can come from a risk-free curve distinct from the Libor projection curve.
- The listed information alone does not determine both the par rate and an unknown spread.
- A government yield does not by itself determine the term structure of swap spreads.
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Full text
# Swap rate calculation if reference rate differs from risk free rates
# Swap rate calculation if reference rate differs from risk free rates
I want to find a swap rate, for an IRS where the floating is Libor+x bp where x is a constant. I have a risk free curve which is not the libor curve. I also have the libor rates.
How can I calculate the swap rate and x ?
## Answer by Helin (score 2)
https://quant.stackexchange.com/a/24445
This is actually no different from pricing a "standard" swap. The par swap rate is the "c" solved from $$ \sum_{i=1}^n c \cdot \delta_i \cdot d(t_i) = \sum_{j=1}^N \Delta_j \cdot (l_j + x) \cdot d(t_j). $$
The left hand side is the present value of the fixed payments, where $n$ is the number of fixed leg payments, $\delta_i$ is the day count fraction for each period, $d(t_i)$ is the discount factor (taken from the risk free discount curve, usually the OIS curve).
The right hand side is the present value of the floating payments. Here $\Delta_j$ is the day count fraction corresponding to each payment period, $l_j$ is the LIBOR forward rate for each period, $x$ is the agreed-upon spread, and $d(t_j)$ is once again the risk-free discount factor.
The risk-free curve not being LIBOR is not an issue. In fact, LIBOR is NOT risk-free and shouldn't be used as the discount curve even for a standard swap.
## Answer by dm63 (score 1)
https://quant.stackexchange.com/a/24443
You can't calculate the term swap rate from that information. The problem is that the swap spread (ie difference between swap rate and government bond yield) has a term structure determined by supply and demand, that cannot be calculated.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.