Estimating Forward Starting Swap Rates from Spot Swap Rates

Summary

The document gives a simple way to estimate the fixed rate on a forward starting interest rate swap from spot starting swap rates. It treats the longer term as a combination of an initial period, or head, and a later period, or tail. Under a chosen fixed-leg payment frequency, the compounded rates for the full term and initial period are related to the rate for the remaining tail; rearranging this relation gives the implied forward rate.

An example applies the calculation to a swap beginning in five years and ending in eight, using annual payments and illustrative euro swap rates. The method is presented as a spreadsheet-friendly approximation and relies on the quoted spot swap rates and stated compounding convention. It does not discuss curve bootstrapping, discount factors, floating-leg details, collateral conventions, or how market conventions may affect precision, so it is best understood as a back-of-the-envelope estimate.

Key ideas

• A forward starting swap rate can be inferred from spot swap rates for the full term and the initial period.
• The calculation treats the full maturity as an initial segment followed by a forward segment.
• Payment frequency determines how the rates are compounded in the relationship.
• The example estimates a forward rate for a swap starting in five years and ending in eight.
• The approach is approximate and depends on consistent rate and payment conventions.

Tags

# Back of the enveloppe forward irs pricing


# Back of the enveloppe forward irs pricing












trying to have a back of the enveloppe way of working out generic forward starting swap rates like 2y2y or 5y3y to put in a spreadsheet without too much loss of accuracy. Whats a good way to look at it / work it out for exple 3y2y? Thanks all!

## Answer by oronimbus (score 1)

https://quant.stackexchange.com/a/53142

Swap rates are essentially weighted averages of forwards. If you have a swap curve ready at hand then you can calculate the forwards as follows.

$$term = head \cdot tail \Longrightarrow (1+\frac{r_1}{f})^{f t_1} = (1+\frac{r_{t_2}}{f})^{ft_2} \cdot (1+\frac{\color{red}{r_{t_3}}}{f})^{ft_3} $$

where $f$ is the fixed leg payment frequency, $t_i$ the time to maturity for each component and $r_i$ the fixed interest rate.

So if we take your 5y3y example, we would have $t_1 = 8$ (term), $t_2 = 5$ (head) and $t_3 = 3$ (tail, i.e. our forward rate). Since you have the spot starting swaps you only have one unknown variables, $r_3$ which is the forward rate starting in 5y and maturing in 8y from today. Approximately, for EUR we'd have an 8y rate of -3.2bp, 5y at -1.6bp and payment frequency is annual. The rest is just simple algebra.

Here's an example in Excel:

`=100*(((1-0.00032)^8/(1-0.0016)^5)^(1/3)-1)`

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.