- #1

MariaAM

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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 :D