Thread: Constant percentage of annual salary: Your father has just turned 50 (t = 0)...

1. Constant percentage of annual salary: Your father has just turned 50 (t = 0)...

Hello, this is my first post. I need to solve some problems for my class, and I got stuck with this one.

The problem (this is a translation):

Your father has just turned 50 (t = 0) and wants to retire in 15 years (t = 15). He thinks he will live 25 years after retirement, until he is 90 years old. He wants an annual amount indexed to the cost of inflation that will give him a purchasing power at age 65 equivalent to $50,000 today (the amount of the benefit will therefore vary each year). His retirement benefits will start on his retirement day in 15 years (t = 15) and he will receive a total of 25 indexed benefits (from t = 15 to t = 39). Over the next 40 years, inflation will be 3% per year. Your father currently has savings of$ 300,000 and expects to earn 8% per year on his investments over the next 40 years. He would also like to give you $25,000 when he turns 90. a) How much should he save over the next 15 years (with equal deposits made at the beginning of the year, so from t = 0 to t = 14) to reach his retirement goals (ie what is the deposit amount annual)? I calculated the value of$ 50,000 in 15 years, with an inflation rate of 3%. Which gives me :
FV15 = 50000(1,03)15
FV15 = 77898,37$Then, I used this formula : PV0 (at t=15) = A1/(r-g) x [1 - (1+g)/(1+r)n] to find the PV (t=15) of the annuity with constant growth (77898,37$)
Which gave me :
PV0 = (77898,37/0,08-0,03)x[1-[(1+0,03)/(1+0,08)]25]
PV0 = 1081651,48$Then, I updated the value of 25000$, to have its PV at t = 15
25000/(1,08)25 = 3650,45$Then I added those numbers to have the amount he needs to have at t=15 --> 1081651,48 + 3650,45 = 1 085301,93$

I capitalized the $300,000 to have its value at t = 15. And I subtracted this amount from the previous sum. 300000(1,08)15= 951650,73$
1 085301,93 - 951650,73 = 133 651,20$--> so this is the amount he needs to have at t = 15. (FV) Then, I calculated the PMT with my calculator : [FV = 133 651,20] ; [N = 15] ; [I = 8] ; [PV = 0] ; [PMT = 4922,31] So he needs to save 4922,31$ every year to reach his retirement goals.

I spent so much time trying to find a solution that I do not even know if I did the math. Which prevents me from solving the next exercise.

(b) Pension contribution amounts are generally expressed as a percentage of annual salary, which means that the amount of contributions varies each year. If your father is currently earning $75,000 (at t = 0) and his salary will increase with inflation each year, what is the constant percentage of his annual salary that your father should save each year (from t = 0 to t = 14) to reach your retirement goals? So here I just do not know which formula to apply, or what is the logic behind this problem. It would be really nice if someone could help me! thank you in advance 2. Agree with your calculations. In Bank statement format, the picture looks like this: (omitted the pennies!) Code: AGE TRANSACTION INTEREST(8%) BALANCE 50 300,000 51 4,922 24,000 328,922 52 4,922 26,314 360,158 ..... 65 4,922 80,028 1,085,302 66 -77,898 86,824 1,094,228 67 -80,235 87,538 1,101,531 .... 75 -101,640 87,035 1,073,328 .... 89 -153,739 23,964 169,769 90 -158,351 13,582 25,000 : your gift! Good luck! 3. Originally Posted by Denis Agree with your calculations. In Bank statement format, the picture looks like this: (omitted the pennies!) Code: AGE TRANSACTION INTEREST(8%) BALANCE 50 300,000 51 4,922 24,000 328,922 52 4,922 26,314 360,158 ..... 65 4,922 80,028 1,085,302 66 -77,898 86,824 1,094,228 67 -80,235 87,538 1,101,531 .... 75 -101,640 87,035 1,073,328 .... 89 -153,739 23,964 169,769 90 -158,351 13,582 25,000 : your gift! Good luck! Thank you very much!! So for question (b) I know I have to use the PMT found at (a) = 4922$ but I don't really know what formula to use. I tried with the same one used the same one that I used in (a) = A1/(r-g) x [1 - (1+g)/(1+r)n] but it's not right.

Would you have any idea of what I have to do? Thank you very much!

4. Originally Posted by MariaAM
Would you have any idea of what I have to do? Thank you very much!
Sorry Maria, but I'm stuck like you are:
I simply do not understand what is being asked...
Seems to be something like (not related to part a):
salary increase = 3% annually
contribution = 5% of salary
Would look like this after 4 years:
Code:
SALARY   CONTRIBUTION  TOTAL-SO-FAR
75000       3750           3750
77250       3863           7613
79568       3978          11591
81955       4098          15689
The problem would then be worded this way:
if present salary is $75000, and salary increases by 3% annually, and goal is to have$15689 after 4 years....
and the constant 5% of salary would need to be calculated.

5. Originally Posted by MariaAM
(b) Pension contribution amounts are generally expressed as a percentage of annual salary, which means that the amount of contributions varies each year. If your father is currently earning $75,000 (at t = 0) and his salary will increase with inflation each year, what is the constant percentage of his annual salary that your father should save each year (from t = 0 to t = 14) to reach your retirement goals? Re-read problem a few more times! Seems to be lacking information...due to translation? We know, from part a), that "the retirement goal" is to have$1,085,302
when father turns 65 years old.
But that involves initial savings of \$300,000 plus annual interest of 8%:
is that still applicable in part b) ?

Is there a given answer to part b) ?
If so, what is it? We can work in reverse using that...

6. HOKAY...finally "saw the light": all other conditions in part a) remain same; so:

I get ~5.5198%

Formula: F*(R - G) / {75000*[(1 + R)^N - (1 + G)^N]}
where :
F = Future value (1,085,302)
R = annual rate (.08)
G = inflation rate (.03)
N = number of years (15)

Creates this picture:
Code:
AGE TRANSACTION INTEREST(8%) BALANCE
50                           300,000
51    4,140       24,000     328,140
52    4,264       26,251     358,655
53    4,392       28,692     391,739
.....
63    5,902       67,672     919,474
64    6,080       73,558     999,111
65    6,262       79,929   1,085,302
Interesting problem (I think!).

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