Interest Rate Put Payoff: CFAI Topic Test "Silva"

Can someone please explain why the put payoff of \$875,000 results in a positive for the numerator? Wouldn’t the put payoff DECREASE the cost of the loan, therefore, resulting in a negative \$875,000 in the numerator? When I did it the way that makes sense to me (which is obviously the exact opposite of the correct way), I get a reduced interest rate of 2.76% rather than the answer of 6.38%?

Question

On 16 March 2012, First Citizen Bank (FCB) approached Silva for advice on a loan commitment. At that time, FCB had committed to lend \$100 million in 30 days (on 15 April 2012), with interest and principal due on 12 October 2012, or 180 days from the date of the loan. The interest rate on the loan was 180-day Libor + 50 bps, and FCB was concerned about interest rates declining between March and April. Silva advised FCB to purchase a \$100 million interest rate put on 180-day Libor with an exercise rate of 5.75% and expiring on 15 April 2012. The put premium was \$25,000. Libor rates on 16 March 2012 and 15 April 2012 were 6% and 4%, respectively. The option was exercised on 15 April 2012, and the payoff was received on 12 October 2012. FCB has asked for a written evaluation of the success of the strategy.

The effective annual rate is calculated as follows:

Future value of put premium on 15 April:

\$25,0001+(0.06+0.005)30360=\$25,135.42" name=“QMK33379-E” resolution=“300” src=“https://ondemand.questionmark.com/resources/399691/topicresources/1264820388/QMK33379-E.png” width=“315” />

Effective loan outlay = \$100,000,000 + \$25,135.42 = \$100,025,135.42.

Loan interest is calculated as;

\$100,000,000(0.04+0.005)180360=\$2,250,000" name=“QMK33379-E___2” resolution=“300” src=“https://ondemand.questionmark.com/resources/399691/topicresources/1264820388/QMK33379-E___2.png” width=“311” />

Put Payoff:

\$100,000,000[max0,0.0575−0.04180360=\$875,000" name=“QMK33379-E___3” resolution=“300” src=“https://ondemand.questionmark.com/resources/399691/topicresources/1264820388/QMK33379-E___37fe66983-4e29-4e41-a229.jpg” width=“813” />

Effective interest = \$2,250,000 + \$875,000 = \$3,125,000.

Effective annualized loan rate:

[100,000,000+3,125,000100,025,135]365180-1=0.0638" name=“QMK33379-E___5” resolution=“300” src=“https://ondemand.questionmark.com/resources/399691/topicresources/1264820388/QMK33379-E___5.png” width=“310” />

"On 16 March 2012, First Citizen Bank (FCB) approached Silva for advice on a loan commitment. At that time, FCB had committed to lend \$100 million in 30 days "

Put option on IR protects lender (The Bank) from IR decrease thus if executed increases collected interests and finally increases EAR. They even said further in question “was concerned about interest rates declining”.

Call option on IR protects borrower.

Lender

Put premium increases value of NP in denominator thus, all else equal, decreases earnings in the form of EAR for lender.

Borrower

Call premium decreases value of NP in denominator thus, all else equal, increases cost of borrowing in the form of EAR for borrower.

You use a put when you MAKE a loan, so you add the put payoff (assuming there is one) to your interest proceeds.

It’s all coming together now.

Thanks, Flashback and Jay.