To the next whole year, how long will it take $200 to double if it is deposited

Question

To the next whole year, how long will it take $200 to double if it is deposited and earns the following rates? Round your answers up to the next highest year. [Notes: (1) If you are using a financial calculator, you can enter the known values and then press the appropriate key to find the unknown variable. Then, without clearing the TVM register, you can "override" the variable that changes by simply entering a new value for it and then pressing the key for the unknown variable to obtain the second answer. This procedure can be used in parts b and d, and in many other situations, to see how changes in input variables affect the output variable.) (2) This problem cannot be solved exactly with some financial calculators. For example, if you enter PV = -200, PMT = 0, FV = 400, and I = 7 in an HP-12C, and then press the N key, you will get 11 years. The correct answer is 10.2448 years, which rounds to 10, but the calculator rounds up. However, the HP-10B gives the correct answer.]

Explanation / Answer

THIS WILL HELP YOU

1) ln(2) / ln(1.042) = 16.84, round to 17 years...I'll let you do the rounding for the rest...
ln(2) / ln(1.097) = 7.487
ln(2) / ln(1.209) = 3.652
ln(2) / ln(2) = 1

2)
a) Future Value annuity due [FVad] = FVoa(1+i)
FVoa = PMT [((1 + i)^n - 1) / i]
semi-annual "i" = 0.08 / 2 = 0.04
first the FVoa...
=400 [((1.04^6) - 1) / 0.04]
= 400[6.63298]
=2,653.19...now the FVad
2,653.19 * 1.04 = $2,759.3176
...now there are 10*2 = 20 - 6pmts made = 14 periods remaining
FV = 2,759.3176(1.04^14) = $4,778.2453, round: $4,778.25
(your answer may vary depending on when you start rounding the #s)

b) using FVad...i = 0.10/4 = 0.025, total n = 10 * 4 = 40, but first you must find the balance after your 5 payments...
PV (at time = after first 5 payments) = FV/(1+r)^n, here n = 40 - 5 = 35
1,244.72/1.025^35 = 524.49 <amount in account after your 5 payments, now use FVad
recall FVad = FVoa(1+i), so FVad/(1+i) = FVoa...
524.49/1.025 = PMT [((1.025^5) - 1) / 0.025
511.69756 = PMT[5.25633]
511.69756 / 5.25633 = PMT
PMT = $97.34885, round to $97.35
(your answer may vary depending on when you start rounding the #s)

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