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Tax-Exempt (Municipal) Swap Curve

## BMA Swaps and BMA Swap Curve

A BMA swap is an interest rate swap in which the payments of one leg are variable and are based upon fixings of the US SIFMA Municipal Swap Index (formerly the BMA Municipal Swap Index or "BMA Index"). This index is produced weekly, reflecting the average rate of issues of tax-exempt variable-rate debt, and serves as a benchmark floating rate in municipal swap transactions. The BMA index is usually 65%-70% of its taxable equivalent 1-month LIBOR. This ratio is subject to tax-risk, i.e., the risk that marginal tax rates will change or that there will be revisions to the US Tax Code.

The BMA Swap Curve represents the expected future values of the BMA index, where expectations are taken in the corresponding forward probability measure; the forward rates that are encoded in the curve can be used to calculate expected future cash flows for the purpose of valuing the BMA leg. Similar to other curve generation processes, the BMA Swap Curve is generated using a set of quoted cash rates and par rates for BMA fixed/floating swaps. Another important input is the risk-free discount factor curve (usually the LIBOR curve), which is used to calculate the present value of expected future cash flows. The par rate for a BMA fixed/floating swap of a particular maturity (e.g., 10 years) can be derived from the BMA Basis factor for that maturity (e.g., 75%) and the corresponding LIBOR swap rate. A BMA Basis factor of 75% means that a BMA/LIBOR basis swap, in which one leg pays 75% of LIBOR, and the other leg pays BMA, is a par swap. Thus, if the 10Y LIBOR swap rate (for a 10Y fixed/floating LIBOR swap) is, say, 4%, then the par rate for a BMA fixed/floating swap is 75% x 4% = 3%.

## Technical Details

Bootstrapping starts with the shortest term swap and steps through them all in ascending order of maturity. At every step, forward rates inferred from the preceding swaps are considered as known, and subsequent forward rates are constrained to recover the price of the current swap. When valuing the BMA leg, the BMA Swap Curve is intended to be used as the "accruing curve" (used to calculate expected cash flows from its implied forward rates). The LIBOR curve is typically used as the "discounting curve".

The bootstrapping process iterates through the given par rates for BMA fixed/floating swaps and outputs discount factors and optionally forward rates. Note that the discount factor curve for discounting (i.e., the LIBOR curve) is already known, and therefore it is only necessary to build a curve that encodes implied forward rates.

## Analysis Supported

The FINCAD average swap curve function can be used to calculate an accruing curve that implies forward rates for the Municipal Swap Index, given a discounting (LIBOR) curve and par rates (or basis factors) of tax-exempt municipal swaps whose floating leg payments are based upon the average index rate.

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