Retirment problem

[barn]

I am having a hard time to understand a question in my fin math class, i am not sure what to do with it.

Calculating Annuity Values
Nelson wants to save money to meet 3 objectives. First , he would like to be able to retire in 30 years from now with retirement incomeof $25,000 per month for 20 years, with the first payment received in 30 years and 1 month from now. Second, he would like to purchase a cabin in 10 years at an estimated cost of $350,000. Third, after he passes on at the end of the 20 years of withdrawls, he would like to leave an inheritance of $750,000. He can afford to save $2,00 per month for thenext 10 years. If he canearn an 11% EAR before he retires and an 8% EAR after he retires, how much will he have to save each month in years 11 through 30?

I have converted 11% EAR to 10.48 APR and 8% EAR to 7.72 APR, I think these are right.
Next I figured out that after 10 years of saving 2100 a month he will have $442,201.15 less the $350,000 for the cabin he will have $92,201.15 left over.
I am stuck at this point an unsure what to do next?

 

[Denis]

Your rate conversions are correct: 1200[1.11^(1/12) - 1] = 10.4815.... and 1200[1.08^(1/12) - 1] = 7.7208....
After the 350,000 withdrawal, he'll have 92,239.6906... left over; your amount is ok if you use 10.48 exactly.

The "trick" is to go backward, from the 700,000 ending requirement.
1: calculate present value (240 months) of 700,000
2: calculate present value (240 months) of 240 payments of 25,000; both at lower rate, of course

These 2 amounts will total ~3,202,127

Now you need to calculate the monthly deposit from year 11 to year 30 that will permit the account to
accumulate to 3,202,127 remembering that the 92,239 also accumulates interest.

The 92,239 will accumulate to ~743,668.
So the monthly deposits must accumulate to 3,202,127 - 743,668 = ~2,458,459
The required monthly deposit becomes ~3,041

I'm assuming you're familiar with the formulas to calculate this stuff...
Give it a shot; come back if anything not clear.

 

[barn]

Can you please post the formulas used for the calculation, thank you

 

[Denis]

Can you please post the formulas used for the calculation, thank you

You're in a "fin math class", given this problem, and don't know/have the financial formulas? :shock:

i = monthly interest factor ; as example (11% annual), 1.11^(1/12) - 1 = .008734...
n = number of months

FV of monthly deposit D : D[(1 + i)^n - 1] / i ; to get future value of the 120 deposits of 2100 [1]

FV of amount A: A(1 + i)^n ; to get future value of [1] - 350,000 [2]

PV of amount F: F / (1 + i)^n ; to get the present value of 700,000 [3]

PV of monthly payment P: P[1 - 1 / (1 + i)^n] / i ; to get present value of the 240 payments of 25000 [4]

Required deposit to accumulate to F: F(i) /[(1 + i)^n - 1] ; to calculate the 240 deposits [5]

You should get these results (closest dollar):
[1]: 442240 - 350000 = 92240
[2]: 743665 : FV of 92240, 240 months
[3]: 150184 : PV of 700000, 240 months (lower rate)
[4]: 3051943 : PV of 240 payments of 25000 (lower rate)
[5]: 3041 : required deposit to reach [3]+[4]-[2], 240 months

 

[jonah]

Next I figured out that after 10 years of saving 2100 a month he will have $442,201.15 less the $350,000 for the cabin he will have $92,201.15 left over.

Rounding difference. A closer approximation:

\(\displaystyle 2,100\frac{{\left( {1.11} \right)^{10} - 1}}{{\left( {1.11} \right)^{{\textstyle{1 \over {12}}}} - 1}} - 350,000 \approx {\rm{\$ 92,239}}{\rm{.69068}}...\)

Use the end of 30 years from now as the comparison date.
Thus,

\(\displaystyle \left[ {2,100\frac{{\left( {1.11} \right)^{10} - 1}}{{\left( {1.11} \right)^{{\textstyle{1 \over {12}}}} - 1}} - 350,000} \right]\left( {1.11} \right)^{20} + R\frac{{\left( {1.11} \right)^{20} - 1}}{{\left( {1.11} \right)^{{\textstyle{1 \over {12}}}} - 1}}\)

\(\displaystyle = 25,000\frac{{1 - \left( {1.08} \right)^{ - 20} }}{{\left( {1.08} \right)^{{\textstyle{1 \over {12}}}} - 1}} + 750,000\left( {1.08} \right)^{ - 20}\)


Now solve for R.
Might differ with Sir Denis’ result due to exact formulas used and due to incorrect use of $700,000 instead of $750,000.

 

[Denis]

Yep, I erroneouly ended with 700,000 where the afore mentionned amount should be 750,000

As penance, I'm making up a "statement format" showdown of the results that will be obtained using
them there formulas, to 6 decimal places thereby outdoing Sir Jonah by 1. Here tizz, in its initial glory:

MONTH          TRANSACTION                   INTEREST                     BALANCE 
(Part 1; 11%)
    0                                                                      .000000
    1         2,100.000000                    .000000                 2,100.000000
    2         2,100.000000                  18.342647                 4,218.342647
 ......
  120         2,100.000000               3,811.152554               442,239.690680
  120      -350,000.000000                    .000000                92,239.690680
(Part 2; 11%)
   1          3,053.867773                 805.676233                96,099.234676
   2          3,053.867773                 839.387781                99,992.490220
 ......
  240         3,053.867773              27,793.538749             3,212,854.413383
(Part 3; 8%)
   1        -25,000.000000              20,671.602035             3,208,526.015418
   2        -25,000.000000              20,643.752992             3,204,169.768410
 ......
  240       -25,000.000000               4,954.495959               750,000.000000