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

MariaAM

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Feb 26, 2018
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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.:confused::confused:

(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 :D
 
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!

 
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