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 :

FV_{15 }= 50000(1,03)^{15}

FV_{15 }= 77898,37$

Then, I used this formula : PV_{0}(at t=15) = A_{1}/(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 :

PV_{0}= (77898,37/0,08-0,03)x[1-[(1+0,03)/(1+0,08)]^{25}]

PV_{0 }= 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

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