Twenty-Four Per Cent Minus Twice the Floating Rate Is Three Ordinary Swaps

Two banks agree the following. A pays B the three-month floating rate, call it $L$. B pays A twenty-four per cent minus twice $L$. Same notional, same payment dates, same basis on both legs. What is this deal?

It looks like it needs a model. It needs one line of algebra. The deal is three ordinary swaps at eight per cent, and the twenty-four printed on the term sheet is not its fixed rate.

The deal, netted

Because both legs settle on the same dates on the same notional, the cash flows can be added period by period rather than valued separately. On each date A receives $24 - 2L$ and pays $L$:

That is the whole deal, in one expression, in percentage points of notional per period. Nothing has been assumed about how $L$ behaves.

One factorisation

Now look at equation (1) and notice that 24 is 3 times 8:

And $8 - L$ is the net cash flow of the most ordinary instrument in the market: a vanilla swap in which A receives eight per cent fixed and pays floating. So the structure is three of those, side by side, on the original notional. Equivalently it is a single vanilla swap at eight per cent on three times the notional.

The fixed rate was never twenty-four

The fixed rate of a swap is the floating level at which its net cash flow vanishes. Solve equation (1):

so the deal changes hands at eight per cent, not twenty-four. Reading the twenty-four as the fixed rate is a factor-of-three error and it is easy to check how expensive it is. A genuine twenty-four per cent swap breaks even when the floating rate reaches twenty-four. This one, at that same rate, pays A

The two structures are 48 points of notional apart at that floating level. Whatever else it is, the misreading is not a rounding.

Where the leverage actually sits

Differentiate equation (1) once and then again:

A loses three points for every point the floating rate rises, so the deal carries exactly three times a single swap's exposure. That is worth stating plainly, because a term sheet with a multiplier in it invites people to hunt for the risk in the shape of the payoff. It is not there. The second derivative is zero, so there is no convexity, no optionality and nothing for a volatility number to price. The leverage is in the notional and nowhere else.

Hedging follows immediately. Three vanilla receive-floating swaps at eight per cent, or one on triple the notional, and the book is flat with nothing left over. No dynamic adjustment, no rebalancing, no residual.

One boundary is worth drawing before anyone over-reads equation (2). Being three vanilla swaps does not make the deal a fair one. It is worth three times whatever an eight per cent swap is worth, which depends on where the market swap rate sits: positive to A if the market rate is below eight per cent, negative above. The factorisation says what the deal is, and pricing it still needs a curve.

The version where the factorisation fails

Look at the leg B pays. At a floating rate above twelve per cent, $24 - 2L$ is negative, which means B stops paying and A starts. Real inverse floaters are usually written with a floor at zero on that coupon, so it never turns negative. That single clause destroys equation (2), and the difference can be written down exactly:

which is two caplets struck at twelve per cent, per period. So the floored deal is three vanilla swaps plus a strip of options that A is long, and the whole thing is no longer linear in $L$: it kinks at twelve, and above that level A pays only $L$ rather than $3L - 24$. At a floating rate of twenty-four the floored version costs A twenty-four points instead of forty-eight.

That is the version that genuinely needs a volatility surface, and telling the two apart is the real content of reading the term sheet. The deal in this problem has no floor, so it is three swaps and it is worth exactly three swaps.

The convention that has to hold

Equation (1) added the two legs on the same date, and that step is doing more work than it appears to. It needs both legs on the same schedule, the same notional and the same day-count basis. Change the payment frequency on one leg, or price one on an actual-over-360 basis and the other on thirty-over-360, and the cash flows no longer line up period by period. The equivalence then holds only in present value, and only once the two schedules have been discounted separately and properly.

When the conventions do line up, the statement is as strong as a statement in finance gets. The two structures pay A the same amount in every state, not on average and not under some model, which was checked in exact rational arithmetic over 241 floating rates including negative ones and over 20,000 random eight-period paths, with zero mismatches. Because the payoffs match state by state, the present values agree under any discount curve at all, and that was confirmed on 200 randomly generated positive curves. Matching payoffs is strictly stronger than matching prices, and it is why no volatility, no distribution and no curve assumption appears anywhere above.

The habit generalises past this deal. When a structure looks exotic, collect every leg into one expression in the underlying rate before doing anything else, then look at the expression. If it is linear, the deal is a portfolio of vanillas and the only question left is how many. If a $\max$ survives the collection, as in equation (6), that is where the model belongs.

Sources and further reading

• The instrument equation (2) factors into Interest rate swap
• The structure the original term sheet imitates Inverse floating rate note
• The options that appear in equation (6) Interest rate cap and floor
• The one operation the whole article performs Factorization
• Why linearity settles the pricing question Linear function

Both readings of the algebra were checked separately, collecting the legs and factoring the result, so a sign slip in either would have surfaced. The break-even is exactly eight, the slope is exactly minus three at every rate tested, the second difference is exactly zero, and the unfloored deal is strictly decreasing across all 241 rates with no kink anywhere on the grid.