**THE BOOTSTRAP ALGORITHM**

The analysis of off-market (i.e. non-par) swaps requires a set of discount factors with the effects of regular swap coupon payments removed. A commonly used algorithm to create such is the bootstrap algorithm. Data inputs consist of cash deposit rates and futures or forward rate agreements, followed by a set of level principal, par rate swap rates.

Bootstrapping involves solving for the unknown last discount factor in an equation for which the earlier discount factors are known. Suppose we wish to solve for the nth discount factor given the n-period par swap rate. The equation which defines the present value of the fixed payments on this swap per USD1 notional principal is:

We observe the par swap rate C_{n}
as a market input rate and know the accrual fractions by the terms of the swap. If the discount factors DF_{i}
for periods i = 1... n-1 are known, then equation [1] is a single equation in one unknown and we can solve it for the unknown discount factor DF_{n}:

Equation [2] is applied sequentially out the yield curve. At least one known discount factor is required to start the process, hence the name bootstrap. Remember that equation [2] takes the par swap coupon, C_{n}, as given.

Given a set of discount factors we can backwards and solve for the par rate on an n-period swap. Simply rearrange equation [1] to obtain:

Note that we have said nothing yet about the behavior of the floating side of the input swaps, using only fixed swap coupon rates, payment dates and day counts.

**THE FRN METHOD**

A par swap has zero net present value (NPV)--that is, the fixed rate must be set such that there are equal present values of the floating and fixed sides. The floating rate note (FRN) method provides a quick way to solve for the present value of the floating side.

Given that the fixed side of a swap is much like an annuity, solving for the fixed side present value per USD1 notional is straightforward:

The present value of the expected floating side cash flows is given by:

where E[ ] is the expectations operator and CF_{i}
is the floating side cash flow at the end of period i.

The expected floating side cash flow per USD1 notional in any period is the forward rate for the period, F_{i}, times the accrual factor for the period

The definition of the forward rate is

Substituting equations [6] and [7] into equation [5]^{1}:

Expanding the summation in [8] and canceling terms ends in a remarkably simple result:

which states that the present value of the floating side per USD1 notional is equal to the discount factor at the start of the swap less the same at maturity. Note that the
present value of the floating side is independent of the reset frequency^{2}.)

Letting the swap have a spot starting date (i.e. DF_{start}
= 1), then

We now have expressions for the present values of both sides of the swap. Equating equations [4] and [10] and solving for the fixed swap coupon rate produces:

Equation [11] should look familiar because it is identical to equation [3], the expression for the fixed rate on a par swap using the bootstrap method.

**CONCLUSION**

We have shown that the commonly used bootstrap algorithm implicitly assumes that the floating side of the par swaps used in the bootstrap are valued using the FRN method. Input par fixed side swap rates can then only be reproduced identically using FRN-based present value algorithms^{3}. A fixed swap rate solved using a true forecast method in which forward rates for each floating side period are forecast may result in a different rate than the input par swap rate (i.e. a non-zero NPV).

This can occur because of small differences between the accrual fractions of swap cash flow periods (which are determined by the terms of the specific swap in question) and the accrual fractions used to forecast future floating interest rates (which are determined by the spot market conventions of the reference floating rate). So it is entirely possible that what we usually think of as a par swap is in fact not a par swap, but rather something very close to a par swap. On large notional underlyings, the difference can be noticeable.

^{1}
An assumption made in the FRN method is that the AF's used to determine the forward rate are identical to the AF's used to compute the cash flow amount (i.e. the floating side rate setting period and the cash flow period are assumed to be identical).

^{2}
This statement is true for all periods which have not yet started (i.e. forward starting).

^{3}
The fair (i.e. zero net present value) fixed swap rate for a given term is defined as the present value of the floating side divided by the relevant cumulative discount factor.

*This week's Learning Curve was written by David J. Novak, principal at
Financial Science and Technology Inc.
in Toronto.*