Swap valuation: Inferring prevailing rates through forward-rate calcs

Apologies, I won't go into swap theory much as I'm assuming that if you're reading this, you already know the in's and out's of a swap. The below focuses on floating legs and how we can infer our rates if you find yourself in a bit of a bundle in retrieving your rates for any reason -either weird tenor dates or anything of the like in your valuation.

A generic swap simply transforms one cash flow to another, as multi-period extensions of forward contracts (FRAs), they usually involve the exchange of a fixed interest rate for a floating rate or visa-versa.

The fixed leg is calculated as:

While the floating leg is calculated as:

The focus in this section will be on the calculation of the floating leg by use of forward rates.

Note, a 365 day count convention for ZAR payments.

Floating Leg

For swap transactions consisting of a floating leg, the reference rate is often used for the interest calculations; the JIBAR in the South African context.

We can infer prevailing rates by calculating the forward rate as per the tenor of the swap:

In the valuation we count the days since valuation for discounting purposes and the days between reset dates for our forward rate calculation.

The forward rate is the expected value of future JIBAR, under the risk-neutral measure, thus, in instances where we don't have a curve for us to determine the interest payments at a certain date/ tenor, we calculate the forward rate from the ZAR Swap-Zero and use it to derive our floating rates.

We start by determining the Discount Factors for each period, derive the rates from said DF’s by use of the LN function in Excel, then we key our values to the forward-rate formula above, interpolating appropriately.

The Discount Factors were retrieved from the ZAR Swap-Zero curve which are the exponent of r times t (e^r*t) from ZAR Swap-Zero yields to find the day values.

Thus,

The swap zero curve is the curve we used to:

1. Imply JIBAR forward rates, and

2. Discount future cash flows.

This curve is the building block curve for all FRA and swap pricing.

Note that we do not know today what the future JIBAR cash-flows on the floating leg are going to be, yet we use the forward rates from the swap zero curve as a proxy. The reason is not really because it is the market’s prediction of future JIBAR, but because every future JIBAR rate can be hedged at the forward rate today. If the fair forward rates are not used in the valuation of the floating leg, then arbitrage is possible.

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