# 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 ?

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.

• I totally agree with the answer from @haginile. Only one point: The OP asked for the calculation of both, swap rate and the spread $x$. This is not possible, i.e. one of the two determines the other. So I would fix the spread and calculate the swap rate with the formula and approach given by haginile. – BerndH Feb 22 '16 at 7:33
• @BerndH I'm a bit confused. My excercise asks: "What is the fixed semiannual swap rate calculated from the risk-free rates? Let that be the the fixed leg payment. Now what is the constant spread 's' that sets the PV of the swap to zero?" – user010010001 Feb 22 '16 at 8:38
• From my understanding it's the definition of the swap rate that it is exactly the rate for the fixed leg which ensures the swap has an initial PV of zero. So the spread would be zero for that rate. Things are a little more complicated in the so called multi curve environment where the discount curve is OIS (or similar) and different from the forward rate. So you have to be more precise in which context you determine the swap rate as it depends e.g. on the collateralization of the swap. – BerndH Feb 22 '16 at 10:16

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.