3: Interest Rate Cap and Floor

Mini Chapter Three

Interest Rate Caps

• Interest Rate Caps – are a certain type of option on interest rates that cap the exposure to higher rates i.e. they pay off when market rate (or the relevant reference rate) is above the cap strike. As intuition would suggest they are frequently used by borrowers to hedge their funding costs. As an example a cap on an interest rate swap is the specified strike above which if the reference rate trades in the market, the buyer of the cap would be compensated the excess (Market rate – Cap Strike) yield on the contract notional.

• From a borrower’s perspective, you can think of a certain floating rate loan of say SOFR + 50bps for a 1 year tenor with quarterly payments. The interest cost on the liability can be hedged (especially in a rising rates environment) by buying a cap for a notional of USD 25 mio on SOFR at let’s say 5% (cap rate) i.e. the borrower would get compensated the excess (above the cap rate) by the dealer when SOFR fixes above 5%.

• Snapshot of the cash flows below suggests that the effective cost of funding for the borrower in a rising rates environment would be lower given the gains from the purchase of the cap.

Table 1. Cash Flows of a 5% Cap Strike

Source: Pandemonium.

• Cap payment is made by the dealer in the following quarter using an actual/360 day count convention. For instance the gain of 150bps as per the April 10 fix would be realised on July 10 as:

\[ [ ( 6.5\% - 5\% ) \times ( \dfrac{91}{ 360} )] \times 25,000,000 = USD 94,792 \]

• An interest cap therefore is a series of payments within the option contract, the payments decided as per the previous period floating rate reset i.e. if the reset rate is below the cap strike no payment would be made. This contingent payment period is referred to as the caplet. Also note that the first reset is excluded from payment considerations as the fixing is already known at the time of entering the contract.

• So a cap on a 5 year swap that resets quarterly, would comprise of 19 payment periods or caplets with each caplet (think of single period interest rate swap) being a call option on the underlying interest rate with the cap being the strike.

Connecting the Dots.

Caplets (or call option for a single period swap) on interest rates can also be understood as a put option on a coupon bearing bond with a same tenor as that of the caplet, strike price at par and cap rate as its coupon.

Consider a 5y cap with a strike of 5% and a quarterly reset. The caplet that starts in 12 months can be understood as a European put option on a 3 month bond exercisable at par 12m forward with face value at par (100 as per general references) and a coupon of 5% payable quarterly (i.e. single coupon for a 3m bond). If the 3m rate at the end of 12m is at 4% then the optionalised bond will be trading at a price of:

\[ \dfrac{ 100 \times (1 +\dfrac{5\%}{4} ) }{ (1 +\dfrac{4\%}{4} ) } \]

, where

\[ \dfrac{ 1 }{ (1 +\dfrac{4\%}{4} ) } \]

is the discount factor for the 3m period.

The price of the bond is above par and the put option at par is worthless. If however the 3m rate at the end of 12m is higher than 5% (say 6%) then the price of the bond will be

\[ \dfrac{ 100 \times (1 +\dfrac{5\%}{4} ) }{ (1 +\dfrac{6\%}{4} ) } \]

i.e. well below 100 and the option would be exercised for a payoff:

100 (Put Strike) –

\[ \dfrac{ 101.25 }{ 1.015 } \]

• Notice that this is the same as the present value of the excess (relative to cap strike) payable yield at the end of the 3m period i.e. the option payoff:

\[ \dfrac{ 100 \times (\dfrac{6\% - 5\% }{4} ) }{ (1 +\dfrac{6\%}{4} ) } \]

• In terms of cash flows therefore, a cap is a series of put options with expiries progressively extending with every reset but the same underlying bond tenor for each option; tenor equal to the reset period. In our example of a 5yr cap the different option tenors/expiries are 3m, 6m, 9, 12m…57m (total 19) with strike at par on a 3m underlying bond having a coupon rate of 5% p.a. payable quarterly.

• Floor on interest rates – are an option expression to protect oneself from a drop in interest rates by flooring them at a specific strike. A seller of a floor would pay the buyer the difference between the floor strike and market rate (as percentage of the notional) during a reset period for which the market rate was set below the floor strike. In the example above, if we set a floor on a fixed rate SOFR loan at 2% – the lending bank would earn a floor rate of 2% irrespective of how low interest rates go. For the buyer, a floor is a put option on interest rates or (as explained for caps above) a call option on floating rate bonds with tenors same as the floorlets.

Trivia. Loan documents/Term sheets of Banks sometimes do not explicitly mention a floor on the benchmark lending index but may define interest payments as unidirectional i.e. to be made only by the borrower to the lender. Floating rate loans – benchmark index + a fixed margin (akin to a credit spread) – are exposed to risks of negative interest rates but the one sided obligation to pay interest implies a long exposure to a 0% (or sometimes higher) floor on the floating index for the lending bank, sold to them by the corporate borrower. This also keeps the floor market perpetually bid by corporates as they rush to hedge their liabilities especially in a low rates backdrop, creating significant downside skews towards 0 and lower strikes. While in a high interest rate backdrop when it’s cheaper to buy ~0% floors, borrowers wouldn’t anticipate a collapse in rates as big as to warrant these hedges. For banks however, the value of their long floor positions appreciates as rates collapse/flirt with negative values (observed post GFC and Covid rates cuts + liquidity stimulus) which have been monetized using trading of these derivatives separately from the loans itself.

• Swap hedges for Loans – In case the borrower converts the floating loan to a fixed rate one through a vanilla swap, they need to be mindful of replicating the swap terms to match those of the loan that’s being hedged. If the swap isn’t structured to have the floor on the floating rate as that on the loan it could dangerously build up large payment obligations for the borrower in case rates plunge below the loan floor strike. For instance, in case of 0% loan floors and a paid fixed swap position (to hedge the floating rate loan) of the borrower, negative floating benchmark rate would imply cash outgo for the borrower on both the swap legs if the 0% floor isn’t embedded in the swap. This would render the supposed swap hedge ineffective and from an accounting standpoint the mark to market losses on the swap would need to be passed through to the borrower’s PnL.

Connecting the dots. Caps and Floor Parity – akin to a put call parity this too follows the same logic of being long a cap (call) and short a floor (put) at the same strike and the difference between the two equating to a short interest rate swap (outright forward in case of put-call parity) position. To see how this is true just simply think of the same strike cash (K) flows of a long cap to be complementary of a short floor (only one of them materialise), and the difference between the two would amount to Market rate – K being paid at all times, whether strike is above or below the market rate. This is equal to being paid a fixed rate swap at a coupon of K, alongside receiving the market floating rate.

\[ P V_{\text{Call}} - P V_{\text{Floor}} = P V_{\text{PaidSwap}} \]