In this workout, we are told that a company has borrowed $25 million at a rate of three months sofa, backward looking compounded term rate, plus a spread of 175 basis points.

They have bought a 5.2% cap on three months sofa to protect themselves against rates going higher.

We are asked to calculate the maximum rate they have capped their loan at and the loan interest caplet payoff and net payment made by the company.

If at the end of the current interest period, three months sofa is 5.42% and we are told to ignore the premium, we are also asked to check that the net payment on the loan corresponds to the capped loan rate.

We've got information on the period start date, the period end date, the accrual days, which is the number of days in between those two dates, the basis of 360 days, the loan amount or cap notional of $25 million.

The spread over so far on the loan of 175 basis points and the cap strike rate of 5.2%.

We also have three months sofa at 5.42%.

So what is the capped loan interest rate? Well, that would be the cap strike rate of 5.2% plus the 175 basis points, and that gives us 6.95% as the capped loan interest rate.

So this is the maximum the company would pay on the loan.

To work out the interest on the underlying loan, we need to take the loan amount of 25 million, multiply that by the interest rate of sofa, plus the 175 basis points spread, and then apportion this for the number of days we are looking at.

And that's 92 out of the 360 and that gives us $458,083.33.

Next to look at the caplet payoff, we are gonna take the difference between the three month sofa and the cap strike rate, but we're gonna build in a max function here because we only want an answer if there's a positive payoff on the caplet as this is an option.

If there's a negative payoff, we simply would not exercise the option.

So we need to take the maximum of zero and the difference between the three month sofa and the cap strike rate and then multiply that by the cap notional of $25 million and apportion for 92 of the 360 days.

And that gives us a Caplet payoff of $14,055.56.

Putting these two together, we get the net payment on the loan.

So we are gonna take the loan interest, which is a payment that is going to be reduced by the positive payoff on the caplet, and that gives us a net payment on the loan of $444,027.78.

To test. Our formula is working properly.

Let's say three months sofa.

Were lower than the cap strike rate, so imagine three months sofa were 5%.

We would then have a zero caplet payoff as the caplet is outta the money.

It would not be exercised.

And so the three month sofa rate of 5% would be used rather than the 5.2% cap rate.

So we can see that cap payoff is zero, let's just undo that.

And then we can finish up by working out the effective rate on the loan.

To work out the effective rate, we are going to express the net payment as a percentage of the loan amount of $25 million.

But then we want an annual rate.

So we're gonna gross this up by multiplying by 360 and dividing by 92.

And that gives us an effective rate of 6.95%, which correctly agrees to the capped loan interest rate that we calculated of 6.95%.

So this is the maximum we would pay on the loan if we bought the cap.