In 90 days, a firm wishes to borrow $10 million for 180 days. The borrowing rate is LIBOR plus 200 basis points. The current LIBOR is 4%. The firm buys an interest-rate call that matures in 90 days with a notional principal of $10 million, 180 days in underlying, and a strike rate of 4.1%. The call premium is $9,000. What is the effective annual rate of the loan if at expiration LIBOR = 4%? A) 0.0637. B) 0.0619. C) 0.0787. (this one confused me as to which convention to use)
isn’t this just the effective rate of libor plus 200 bps since your call will be out of the money. so b?
This is not precise… You should compute the FV of the interest rate call : 9000 *(1+(0.04+0.02)*90/360)). The result 9135$ shoould be substracted from the amount borrowed, which means the firm effectively borrowed 9,990,865$… The interest pais on the loan is 10,000,000*0.06*180/360=300,000$, while the interest rate call expires worthless and provides no payments… Effective Annual interest rate = (effective amount paid/amount effectively borrowed) ^ 180/365 - 1 = 0.0637 The answer is a)
sorry for the typing error above: the last calculation is : Effective Annual interest rate = (effective amount paid/amount effectively borrowed) ^ 365/180 - 1 (and not 180/365)… the rest is ok
CFALEB, 1. Why do you distract the 9000+ from the amount borrowed? You bought the call so shouldn’t you add it? 2. Why do you use 360-days for all calculations apart from the last one (EAR)? Thanks! (Ps. you are correct but those two things confused me)
mcpass, 1- You substract the call option premium from the amount borrowed because practically speaking, you borrowed 10 M dollars, but you wanted some protection against rising interest rates that could lead to higher payments on the loan. That is why some of the amount borrowed (9000 of the 10M) are used to pay for that protection. It would be clearer if you considered the investor as someone who has absolutely no money at the start. When he borrowed 10M, this is all he has. If he wants to buy protection, he should use some of those 10M he borrowed to pay for that protection, which leaves him with less than 10M. The residual is the amount he effectively borrowed and can actually use for whatever his reason is. Alternatively, in order not to forget this concept, you could think of the effective amount borrowed, as the actual net inflow of cash. The inflow is 10M while there is an outflow of 9000$. The net amount is the amount effectively borrowed. This is why you substract the call premium. If you were considering an investor who buys a put to protect against falling interest rates, this investor would be a lender. Thus consider he lent out 2M dollars to someone. In order to buy the put option, he should disburse more money (the cost of the premium). IN that case you ADD the put premium to the amount of money lent out, since both are actual outflows… 2- When you want to find the FV of the call or put interest rate option, you use 360 days as for when you want to calculate the interest paid/received on the loan for the period as well as for the payoff on those options (call or put). This is because you assume no daily compounding but only divide the annual interest rate by the period considered (in your case 180 days). The convention is to use 360 days for those calculations. For the EAR calculations, and whenever you compound, it is also conventional to use 365 days. It is written somewhere that some analysts might use different nuymber of days in both cases, so don’t worry too much about that. it is easy to remember that 365 is only used in the final step. Hope that helps… and sorry for writing too long 
I might add one more thing concerning point1: when you substract the call option price from the loan you took, you are actually increasing the EAR since you are dividing the interest paid by a lower denomiator (the 10M - 9000$). The increase of the EAR is a way to account for the extra cost of the option bought to protect your loan. Ame apply for when you add the put premium to the denominator. You lower the EAR since you divide the interest received by a larger number, factoring in th effect that the put you pay for lowers your total revenus (interests)
this is very helpful. thanks.
CFALEB Wrote: ------------------------------------------------------- > Effective Annual interest rate = (effective amount > paid/amount effectively borrowed) ^ 180/365 - 1 = > 0.0637 > > The answer is a) Agree with everything you said, just one small correction: Effective annual rate =(1+(effeictive amount paid/amount effective borrow))^(365/180)-1.
This is good stuff CFALEB, I’m perfectly okay with such long stories!
this is giving me nightmares from about 9 months ago
Hey ws, You are absolutely right about the 365/180 in the formula. Actually I corrected it above in my second post… As for the formula you rewrote : Effective annual rate =(1+(effeictive amount paid/amount effective borrowd))^(365/180)-1. You should not start it with 1+ because when you divide the amount paid by the amount borrowed, by definition it is supposed to be greater than 1, orelse the lender would be losing money since you would be returning him less money than he gave you… This is why you don’t add the 1 at the start, or your number raised to the power 365/180 would be greater than 2 and your EAR would be greater than 100%… the formula is : (effective amount paid/amount effective borrowed)^(365/180)-1 If I m wrong in my formula or I misunderstood your correction, please correct me… Cheers
CFALEB Wrote: ------------------------------------------------------- > > Effective annual rate =(1+(effeictive amount > paid/amount effective borrowd))^(365/180)-1. > > (effective amount paid/amount > effective borrowed)^(365/180)-1 > effective amount paid=$300,000; amount effective borrowed=$9,990,865; no questions about those numbers. I know where we misunderstand each other. In your formular, your effective amount paid is actually interest + principle ($300000+$9990865), that is exactly why you don’t need to add a 1 in front (I 100% agree) When I did my formular, I used (1+(300000/9990865))^(365/180)-1. My effective amount paid was only the interest portion, so what is why I added the 1 in front. Anyway, we were both on the same track…good point!!
So is the answer a or b? I did the math and I am getting 0.06183…or did i miss something? [1+(300,000/9,990,865)]^(365/180)-1 Thanks
Answer is B.
Thanks…and thanks CFALEB for the explanation… really helped a lot…
good question!
Few days ago, this is what i wanted to confirm when i asked if we are doing the optional reading section for reading 38, Swaps? i kind of would need a refresher to do this question!! all of you who answered remembered this? Anish
And ‘ws’ or ‘CFALEB’, could you show the change in calculations if at expiration LIBOR = 4.2% ? (Thanks in advance).
ebiegs327 Wrote: ------------------------------------------------------- > And ‘ws’ or ‘CFALEB’, could you show the change in > calculations if at expiration LIBOR = 4.2% ? > (Thanks in advance). effectively borrowed amount: 10 mil - 9,000*(1 + (4%+2%)/4)=9,990,865 effectively paid: 10 mil*(1+(4.1%+2%)/2)=10,305,000 effective interest rate = (10,305,000/9,990,865)^(365/180)-1=6.48%