View Full Version : challenging annuity problem

May 21st 2009, 08:41 PM
I came across this problem today and cannot figure out how the answer is obtained.

It says:

Sigmund and Karl each borrowed an identical amount from Ludwig at a nominal rate of discount of 5.4% convertible quarterly. Sigmund repays his loan by making payments of $2,000 at the end of each year for six years. Karl makes payments of $3,200 at four equally spaced times T, 2T, 3T, and 4T. Find T. [HINT: you will need to find the interest rate I for a period of length T.]

The answer is supposed to be 1.89171 years.

Does anyone know how to attack this problem? I once received an explanation last year from someone but it didn't make sense to me and now I don't remember what the explanation is.

May 21st 2009, 10:59 PM
Since discount is 0.0054 convertible quarterly, annual effective discount rate would be: d = 1 - (1 - 0.0054/4)^4 = 0.0529163. Therefore, annual effective interest rate is 1 / (1 - 0.0529163) - 1 = 0.0558729.

So, now we can find the original loan amount (make sure your calculator is set to END):
N = 6
I/Y = 5.58729
PMT = 2000
FV = 0
=> PV = 9963.476

We can then find Karl's effective interest rate of period T:
N = 4
PV - -9963.476
PMT = 3200
FV = 0
=> I/Y = 10.83231

So, to find the length of period T, we must set the following equation:
(1.0558729)^T = 1.1083231

(The above statement is true since 0.0558729 is the annual effective interest rate, 0.1083231 is the effective interest rate of period T, and we are using compound interest.)

T = 1.89171

May 24th 2009, 10:06 PM
Interesting problem..I was close up until the end, but it makes perfect sense now !