# Interest Rate Call Problem

In 40 days, a firm plans to borrow \$5 million for 180 days. The borrowing rate is LIBOR plus 300 b.p. Current LIBOR is 5%. The firm buys a call that matures in 40 days with a NP of \$5 million, 180 days in underlying (D = 180), and a strike rate of 4.5%. The call premium is \$8,000. Calculate the effective annual rate on the loan if at expiration LIBOR = 5%

7.488%

8.104%

I also get 8.103% it feels wrong for some reason…

feels right to me…

Call is in the money isn’t it? They would have to pay 8% if they did nothing. I have the call paying of \$12,500 but costing a FV of \$8,071.11. Did I mess something up? I can’t see how they would have to pay more effectively.

that’s why I was saying it feels wrong, intuitively. I also did not look in the book… but here’s how I got to that. Call option payoff is 12,500 FV of the premium is 8,044.44 ( @ 5% x 40/360) interest is 200,000 So 5,187,500/4,991,955.56 = .03917 then 1.0397^365/180 = .08103 but i really haven’t done this in a long time so i may be wrong. i always get confused.

I’ll post the solution when I finish reviewing SS 15 in a few hours

I was never good at these types of problems: somehow I got through the program without ever understanding interest rate options well. How do you get that the call pays 12,500 at expiration? I assume that the strike of 4.5% means that at expiration, you get to borrow \$5M for 180 days at 4.5% instead of at 5%+300bps=8%, but how does that turn into \$12,500?

i looked at the call option as a totally separate contract. so (.05-.045) 5,000,000 = 25,000* 180/360 = 12,500 you’re really paying 4.5% + 300 bps instead of 5% + 300bps. the momentum is mounting against me. i’m probably wrong.

cfasf1 Wrote: ------------------------------------------------------- > that’s why I was saying it feels wrong, > intuitively. I also did not look in the book… > > but here’s how I got to that. > > Call option payoff is 12,500 > FV of the premium is 8,044.44 ( @ 5% x 40/360) > interest is 200,000 > > So 5,187,500/4,991,955.56 = .03917 then > 1.0397^365/180 = .08103 > > but i really haven’t done this in a long time so i > may be wrong. i always get confused. I think you got it right. I added the FV of the call premium to the denom. So I was basically saying that I was able to borrow more because I had to pay a call premium. That doesn’t make sense.

8.1%, using the same calcs as cfasf

bchadwick Wrote: ------------------------------------------------------- > > How do you get that the call pays 12,500 at > expiration? I assume that the strike of 4.5% > means that at expiration, you get to borrow \$5M > for 180 days at 4.5% instead of at 5%+300bps=8%, > but how does that turn into \$12,500? Call contact is totally seperate from the loan. It is based on LIBOR. So here you strike is 4.5 when LIBOR is 5. Payoff = 5M (.05 - .045) (180/360) = \$12,500 You still have to borrow at LIBOR + 300bps. So you are borrowing at 8% but your call will offset some of the cost (but not effectively in this case it seems).

It makes sense now that I think about it. The LIBOR when the call was purcahsed was 5% and at expriation it was 5%. There is no way I could gain on the option without the underlying rate changing. Edit: my thinking may be flawed here, but I agree with the above answers now. Well done guys.

Thanks for the clarification, mwvt9. This is one of the reasons I still post here - hammering down stuff I didn’t really get when I took the exams.

In 40 days, a firm plans to borrow \$5 million for 180 days. The borrowing rate is LIBOR plus 300 b.p. Current LIBOR is 5%. The firm buys a call that matures in 40 days with a NP of \$5 million, 180 days in underlying (D = 180), and a strike rate of 4.5%. The call premium is \$8,000. Calculate the effective annual rate on the loan if at expiration LIBOR = 5% Call premium = 8000 * (1+5%+3%)^(40/360) = 8086 Actual Amount borrowed net = 5m - 8086 = 4991914 Interest at maturity = 5m * (5%+3%)^(180/360) = 200000 Payoff = 5m * (5%-4.5%)^(180/360) = 12500 Net CF = 5m + 200000 - 12500 = 51875000 EAR = (5187500/4991914)^(365/180) -1 = 8.105%

Proposed answer: Net loan amount = 5MM – 8M(1 + (0.08 * 40/360)) = 4,991,929 Call Payoff = 5MM(0.05 - 0.045) (180/360) = \$12,500 Dollar Cost of Loan =5MM * 0.08 * (180/360) – 12,500 = \$187,500 Effective Annual Rate = (\$5,187,500 / \$4,991,929)^(365/180) -1 = 0.081043 or 8.1% When I did this problem I calculated it the same way that Corrupted did. Can someone explain why the interest as 0.08 * 180/360 as opposed to 1.08^180/360?

Anything based on LIBOR uses the (X/360) convention.

1.08 = 1 (principle payment) + 0.08 (Interest) for the interest portion of calc, you should omit the principle repayment which would be added after.

Bankin’ Wrote: ------------------------------------------------------- > Proposed answer: > > Net loan amount > = 5MM – 8M(1 + (0.08 * 40/360)) > = 4,991,929 > > Call Payoff > = 5MM(0.05 - 0.045) (180/360) = \$12,500 > > Dollar Cost of Loan > =5MM * 0.08 * (180/360) – 12,500 = \$187,500 > > Effective Annual Rate > = (\$5,187,500 / \$4,991,929)^(365/180) -1 > = 0.081043 or 8.1% > > > When I did this problem I calculated it the same > way that Corrupted did. Can someone explain why > the interest as 0.08 * 180/360 as opposed to > 1.08^180/360? I think you were asking why just multiply instead of ^180/360, correct? does anyone know? i’m not sure. also. when calculating the fv of the call option, i just used current libor… i see you guys used the terms of the loan. (adding 300bps) which is correct? I always thought, at least through schweser, we just use current libor because the loan terms are separate from the call option…