Par Swap Rates from Discount Factors and Forward Rate Curves
Summary
The document derives the par fixed rate for a forward-starting swap by equating the present values of its fixed and floating legs. Under the stated setup, the floating leg can be expressed as a telescoping sum, giving a par rate determined by discount factors at the swap’s start and end dates divided by the discounted fixed-leg accrual periods.
It then raises a curve-construction question: when discounting with an OIS curve, how are future floating-rate fixings obtained, including for long-dated swaps? The example distinguishes a fixing already known today from a later fixing that must be estimated. The document poses this practical issue but provides no answer about market instruments, forward curves, or how to construct those projections. Its par-rate expression also reflects the notation and conventions assumed in the question.
Key ideas
- A par swap rate is found by equating the present values of the fixed and floating legs.
- The floating leg can telescope under the discounting assumptions in the setup.
- The resulting par rate uses discount factors at the swap dates and across fixed-leg payment dates.
- Future floating fixings require projections when they are not yet known.
- The document asks how to construct those projections but does not provide a method.
Tags
Full text
# How to get forward rate fixing when valuating a swap
# How to get forward rate fixing when valuating a swap
Suppose I want to value a (fwd) starting swap, that means I would like to calculate the fixed rate $S_{\alpha, \beta}(t)$. Note, I'm using Brigo's Notation here. We know that the discounted payoff of the cash flows are of the form (payer swap)
$$ D(t,T_i) N \tau_i(L(T_{i-1},T_i)-S_{\alpha, \beta}(t))$$
where $N$ is the notional, $\tau_i$ the daycount between $T_{i-1}$ and $T_i$ and $D(t,T_i)$ the discount factor for maturity $T_i$.
By observing that the floating rate side can be written as a telescoping sum we can infer $S_{\alpha,\beta}$ from the NA condition that both legs need the same present value, that is
$$S_{\alpha,\beta}(t)=\frac{P(t,T_{\alpha})-P(t,T_\beta)}{\sum_{i=\alpha+1}^\beta P(t,T_i)}$$
Now, I have a very practical question. How are the forward fixing in reality constructed to evaluate such a swap? It seem to me that we have multiple unknows, the swap rate and the forward fixings. However, to evaluate all the present values of the cash flows, I need to know the $L(T_{i-1},T_i)$ at time $t$. Are there other instruments out there with very long maturity to get this forward fixings?
Example
Suppose I have a 1y spot starting swap which pays floating and fixed semi-annually. That means I have the cash flows for the fixed rate
$$ K\frac{1}{2}N (P(0,6M)+P(0,1Y))$$
where we get the discount values for a discount curve, say OIS. The $\frac{1}{2}$ is the perfect half year between payments. On the other hand, for the floating I need
$$\frac{1}{2}N(L(0,6M)P(0,6M) + L(6M,1Y)P(0,1Y)) $$
As said, if we use a different curve (OIS) for the discounting, i.e. $P(0,\cdot)$ I need to know the $L(0,6M)$ and $L(6M, 1Y)$. The former is the current fixing and known today. However, how do I get the next one? This gets more complicated for swaps with maturity say 30Y+.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.