Why Floating-Rate Legs Need the Final Notional Exchange

# Why do floating legs have zero delta risk?


# Why do floating legs have zero delta risk?












I am puzzled as to why floating coupon legs do not have a risk to the interest rate to which they are indexed. Consider a quarterly SOFR floating leg (imagine it is part of a float vs. float cross-currency swap, which only has a risk to the cross-currency basis, not either of the floating legs).

Consider for simplicity only two coupons: the first one $r_1$ in 30 days and the second one $r_2$ in 60 days. According to the New York Fed formula for SOFR compounding:

$$r_1=\left[\prod_{i=1}^{i=30}\left(1+\frac{SOFR_i*n_i}{360}\right)-1\right]*\frac{360}{d_c}$$

And analogously for $r_2$.

For simplicity, assume $n_i$ is always one and $d_c$ is always 30. Assume $SOFR_i$ is initially flat at 3%. Then we have:

$$r_1=\left[\prod_{i=1}^{i=30}\left(1+\frac{0.03}{360}\right)-1\right]/12\approx0.030036$$

And same for $r_2$.

The disount factors would be:

$$DF_1=\frac{1}{1+r_1}\approx0.97084, DF_2=\frac{1}{1+r_1}*\frac{1}{1+r_2}\approx0.94253$$

So the initial PV would be:

$$PV=DF_1*r_1+DF_2*r_2\approx0.05747$$

It is already obvious from the PV formula that an increase (decrease) in the $SOFR_i$ rates and a resultant increase (decrease) in $r_1$ and / or $r_2$ and the corresponding effect on the discount factors do not exactly offset each other.

As an example, assume a hike of 1% is projected just after the end of the first compounding period (i.e. the hike happens at $i=31$ and affects all $SOFR_i$ for $i\geq30$), then:

$$r_1\approx0.030036, r_2\approx040065, DF_1\approx0.97084, DF_2\approx0.933442$$

And the resultant $PV\approx0.067682$.

So from the formulas, as well as from the numbers, it should be clear that a floating leg is sensitive to an increase (or decrease) of the indexed rates. Then why does a float/float cross-currency swap only have sensitivity to the basis, and not the floating coupons on each side?

## Answer by Jan Stuller (score 5, accepted)

https://quant.stackexchange.com/a/81363

## Short Story:

The OP forgot to include the final notional exchange: this component is crucial for the floating leg to be insensitive to rate changes.

## Long story:

Let's do an example: a floating-rate note (i.e. FRN, or we can call it a floating rate bond: this is identical in cash-flows to a floating leg in a Xccy swap).

Let's assume two coupons and a final notional repayment (say notional of 100 for argument's sake).

Below, we can think of $r_1$ and $r_2$ as the compounded rates as per OPs question, but without loss of generality, these can be any rates that drive the coupon payments. The only assumption below is that the discount factors are using the same rates.

$$PV=\frac{100*r_1}{(1+r_1)}+\frac{100*r_2}{(1+r_1)(1+r_2)}+\frac{100}{(1+r_1)(1+r_2)}$$

We claim that the PV is always equal to par, i.e. by definition that would mean that the PV is not sensitive to changes in interest rates.

Proof:

$$100\stackrel{?}{=}\frac{100*r_1}{(1+r_1)}+\frac{100*r_2}{(1+r_1)(1+r_2)}+\frac{100}{(1+r_1)(1+r_2)}$$

Divide by notional:

$$1\stackrel{?}{=}\frac{r_1}{(1+r_1)}+\frac{r_2}{(1+r_1)(1+r_2)}+\frac{1}{(1+r_1)(1+r_2)}$$

Multiply by $(1+r_1)(1+r_2)$:

$$(1+r_1)(1+r_2)\stackrel{?}{=}r_1(1+r_2)+r_2+1$$

Multiply out the brackets:

$$1+r_2+r_1+r_1r_2\stackrel{?}{=}r_1+r_1r_2+r_2+1$$

Matching terms, we see they are equal. So the LHS is always equal to RHS: in other words, we showed that the PV is always equal to par, irrespective of the magnitude of $r_1$ and / or $r_2$, therefore by definition, the PV is insensitive to movements in interest rates.

If the OP included the final notional payment, he'd also get the same answer. So the final notional payment is crucial, otherwise indeed, if we consider just the floating coupons, these are sensitive to interest rate changes, even when discounted by the same curve.

In conclusion, the notional exchange at the beginning is not relevant to the sensitivity of the leg. But it is relevant to making the PV of the leg zero (otherwise the PV of the leg would equal the notional).