Question

Excel Business & Finance

You are going to contribute some money to your retirement fund at the beginning of this year and each of the next 39 years. You have the following assumptions:
I. Initial balance for year 1 is assumed to be 0. II. Suppose during Year 1, your salary is $40,000 and your salary increases by 5 percent each year until your retirement. III. You want to contribute the same percentage of your salary each year while you are working for your retirement.
IV. When you retire in 40 years, you plan to withdraw $100,000 per year for 20 years (starting year 41) and will not make any contribution to your retirement during the 20 years.
You also Assume the following retirement investment portfolio:
I. First 20 years of your investing, the investments will earn 10 percent per year V. During all other years, your investments will earn 5 percent per year
Part I: Assume all contributions and withdrawals occur at the beginning of the year before investment returns are received, please set up the retirement problem to find out the minimum percentage of your salary you should save for your retirement in order for you to make the withdrawals.
Part II: Assume that withdrawals occur at the end of each year after the investment returns are received and contributions occur at the beginning of each year before the investment returns are received, please set up the retirement problem to find out the minimum percentage of your salary you should save for your retirement in order for you to make the withdrawals.

Answer 1

Let the amount that you contribute is $ x for the 1st year at the beginning of the year.

As the salary increases by 5% throughout the next 40 years and you also contribute the same % of your salary to this fund, your annual contribution will also increase by 5% every year

Now taking the end of 40th year as the reference point

Future value of your contributions= Present value of your expected withdrawals

Now, for first 20 years, the return is 10% and for the rest , it is 5%

Therefore,

a) Future value of 1st contribution = x * (1+0.1)²⁰ * (1+0.05)²⁰

Future value of 2nd contribution = x * 1.05* (1+0.1)¹⁹ * (1+0.05)²⁰

Future value of 3rd contribution = x * 1.05² * (1+0.1)¹⁸ * (1+0.05)²⁰

and so on till...

Future value of 20th contribution = x *1.05¹⁹ * (1+0.1)¹ * (1+0.05)²⁰

Future value of 21st contribution = x * 1.05²⁰ * (1+0.1)⁰ * (1+0.05)²⁰ = x *1.05²⁰ * (1+0.05)²⁰ = x *1.05⁴⁰

Future value of 22nd contribution = x * 1.05²¹ * (1+0.05)¹⁹ = x*1.05⁴⁰

and so on till....

Future value of 40th (last) contribution = x * 1.05³⁹ * (1+0.05)¹ = x* 1.05⁴⁰

Adding the above, we get the future value of all contributions

= {x * (1.1)²⁰ * (1.05)²⁰ + ....+ x * 1.05¹⁹ * (1.1)¹ * (1.05)²⁰​} + ​{​​​​​x * (1+0.05)⁴⁰+....+ x * (1+0.05)⁴⁰}

Applying GP formulas

= {x * (1.1)²⁰ * (1.05)²⁰ * (1-(1.05/1.1)²⁰) / (1-(1.05/1.1))} + {x * (1+0.05)⁴⁰ * 20}

=237.82 x + 140.80 x =378.62 x

Now, to calculate present value of your withdrawals of 100,000 per year for 20 years (at the beginning of year)

Present value of 1st withdrawal = 100,000

Present value of 2nd withdrawal = 100,000/(1+0.05)

and so on till

Present value of 20th withdrawal =100,000/(1+0.05)¹⁹

Therefore adding the above we get the present value of all withdrawals

= 100000 + 100000/1.05+ 100000/1.05²+....+100000/1.05¹⁹

By applying GP formula

= 100,000 * ( 1- (1/1.05)²⁰)/(1-(1/1.05))

= 1,308,532  

Therefore for PART I, the retirement problem is

378.62 x = 1308532.08

x = 3456

Therefore you have to contribute a proportion = 3456/40000 = 8.64% of your salary every year for retirement fund

For Part II

To calculate present value of your withdrawals of 100,000 per year for 20 years (at the end of year)

Present value of 1st withdrawal = 100,000/ (1+0.05)

Present value of 2nd withdrawal = 100,000/(1+0.05)²

and so on till

Present value of 20th withdrawal =100,000/(1+0.05)²⁰

Therefore adding the above we get the present value of all withdrawals

= 100000/1.05 + 100000/1.05²+ 100000/1.05³+....+100000/1.05²⁰

By applying GP formula

= 100,000/1.05 * ( 1- (1/1.05)²⁰)/(1-(1/1.05))

= 1,246,221.03  

Therefore for PART II, the retirement problem is

378.62 x = 1246221.03

x = 3291.48

Therefore you have to contribute a proportion = 3291.48/40000 = 8.23% of your salary every year for retirement fund