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 Financial Mathematics Old FM Forum

#1
08-09-2011, 12:25 PM
 charli175 Member SOA Join Date: Jul 2010 Location: NJ College: Georgetown University Posts: 118
Interest Rate Question

A 10-year loan of \$20,000 is repaid with payments at the end of each year. It can be repaid under the following options:
(X) Equal payments at the annual effective interest of 8%;
(Y) Installments of \$2,000 plus interest on the unpaid balance at an annual effective rate of i.
The sum of the payments under option (X) equals the sume of the payments under option (Y). Determine i.

I guess part of my problem was interpreting what they are asking. For (Y), paying installments of \$2,000 plus interest where the total payments equal (X) would simply mean it is the same series of payments with the same interest rate as option (X) right? So that doesnt make much sense. If it means that \$2,000 is paid each year then at the end of the ten years, an additional lump sum is paid to cover the unpaid balance, then this is as far as i got.

Option X pays \$2,980.589 each year for a total payment of \$29,805.89.
So for option Y:
20,000(1 + i)^10 = 2000s_10_i + 9805.89
Not exactly sure how to solve for the interest rate here. Any help?
#2
08-09-2011, 02:03 PM
 Hack Diesel Join Date: Jun 2011 Studying for Exam MLC/3L Posts: 15

You already stated that the total accumulated payment value is \$29,805.89, so there's no need for 20,000(1 + i)^10 or 9805.89 in your equation because s-angle-n/i already incorporates that interest is being paid with the \$2000 payments.

Your AV/FV is \$29,805.89, you have 10 periods, with payments of 2000. A simple plug and chug with a financial calculator will determine your interest rate, which you know needs to be higher than 8% because the payment \$2000 is obviously less than \$2980.589.

(I apologize if I'm completely wrong, I got a 4 and not an 8 or 9 like seemingly everyone on this forum gets on their first try of FM, so any verification is appreciated..)
#3
08-09-2011, 03:30 PM
 andyphillips Member Join Date: Sep 2010 Studying for MFE College: 0-1 Favorite beer: Mojo Risin' Posts: 777

This is how I did it:

You know the sum of the equal annual payments for X (29805.89) and the sum of the principal payments for Y (20000.00). This means that for option Y, you pay 9805.89 in interest. Your interest payments are

(20000i + 18000i+...+2000i) = 2000i(10+9+...+1) = 2000(55)i = 110000i

9805.89 = 110000i
i = .08914

Hope that helps!
#4
08-09-2011, 05:42 PM
 charli175 Member SOA Join Date: Jul 2010 Location: NJ College: Georgetown University Posts: 118

@HackDiesel: the 29805.59 is just the sum of the payments, so because there is interest, it doesnt represent the accumulated value.

@andyphillips: so you're saying that every year you're paying off 2000 of the principle plus the interest that was accumulated that year?
#5
08-09-2011, 06:09 PM
 andyphillips Member Join Date: Sep 2010 Studying for MFE College: 0-1 Favorite beer: Mojo Risin' Posts: 777

Quote:
 Originally Posted by charli175 @andyphillips: so you're saying that every year you're paying off 2000 of the principle plus the interest that was accumulated that year?
Yup. The question says that the payment is 2000 plus interest on the unpaid balance. This means P(1) = 2000, I(1) = 20,000*i, and B(1) = 20,000-2000 = 18000. Since P(n) will always be 2000, that part of your total payment is simply 20,000. Then, to find the total interest paid, notice that the unpaid balance decreases by 2000 each period. This means your total interest payments are (20000i + 18000i + ... + 2000i) = 9805.89, and solve for i.
#6
08-09-2011, 06:30 PM
 charli175 Member SOA Join Date: Jul 2010 Location: NJ College: Georgetown University Posts: 118

That makes sense. Thanks!

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