Find Constant % of Annual Salary to Reach Retirement Goal

[MariaAM]

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 :D

 

[Wilmer]

Agree with your calculations.
The whole episode in Bank statement format:
(pennies dropped!)

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!

 

[tkhunny]

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 :D

Assuming No Inflation
Part A: Annual Contribution = P in year 1, P in year 2, P in year 3, etc.

Assuming Constant Inflation
Part B: Annual Contribution = P in year 1, P(1+Inflation) in year 2, P(1+Inflation)^2 in year 3, etc.

NonConstant Inflation is a little harder, but not much. It looks like you don't need that discussed, here.

 

[Wilmer]

Assuming Constant Inflation
Part B: Annual Contribution = P in year 1, P(1+Inflation) in year 2, P(1+Inflation)^2 in year 3, etc.

Shouldn't the 8% annual rate also be used, since it's part of case a) ?
Very confusing to me...

 

[tkhunny]

Shouldn't the 8% annual rate also be used, since it's part of case a) ?
Very confusing to me...

The ONLY difference in the two problems is the size of the annual payment. That's only the base and the inflation. It is still your job to apply all the accumulation, just like you did before.

There are a lot of moving parts, but you already handled them all. Part B adds the increasing payment amounts. That's the only difference. You may be thinking that you have to rethink the whole problem in this new setting. You don't. Just add the slight additional complication and follow your plan from Part A.

 

[Wilmer]

Gotcha! Target is still 1,085,302.
The steady 4,922 contribution is replaced
by an equivalent series of increasing contributions.

 

[tkhunny]

Gotcha! Target is still 1,085,302.
The steady 4,922 contribution is replaced
by an equivalent series of increasing contributions.

Wilmer?! Didn't even look at who posted. Assumed it was the OP. :-) May have rephrased a bit.

 

[Wilmer]

Wilmer?! Didn't even look at who posted. Assumed it was the OP. :-) May have rephrased a bit.

The OP only made 1st post; s'been me and you since!
Whatya mean "rephrase"?

 

[tkhunny]

The OP only made 1st post; s'been me and you since!
Whatya mean "rephrase"?

Not sure, really. I want to write to a specific audience and if I have the wrong person in mind, I may pick a different word or phrase or paragraph structure. The more I know a person, the more specific the comments can be. I've seen you many times, here. I've read your responses. I've started to get a feel how your best respond to others. What I wrote was more of a response of generalities to an unknown audience. Just a personal adaptation. I don't do it on purpose and it's not a repeatable experiment. :-) I get in trouble for it. I tend to be a little casual with my language, at first. This annoys some folks.

 

[Wilmer]

Not sure, really. I want to write to a specific audience and if I have the wrong person in mind, I may pick a different word or phrase or paragraph structure. The more I know a person, the more specific the comments can be. I've seen you many times, here. I've read your responses. I've started to get a feel how your best respond to others. What I wrote was more of a response of generalities to an unknown audience. Just a personal adaptation. I don't do it on purpose and it's not a repeatable experiment. :-) I get in trouble for it. I tend to be a little casual with my language, at first. This annoys some folks.

NO PROBLEMS!
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:

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

Agree?