Question

Problem 4-34 Growing Annuity

Your job pays you only once a year, for all the work you did over the previous 12 months. Today, December 31, you just received your salary of $59,000 and you plan to spend all of it. However, you want to start saving for retirement beginning next year. You have decided that one year from today you will begin depositing 10 percent of your annual salary in an account that will earn 9.9 percent per year. Your salary will increase at 2 percent per year throughout your career.

How much money will you have on the date of your retirement 35 years from today? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Future value           $

Answer 1

Given information:

Today is Dec 31

Current salary S = $59000, salary increase rate = g = 2%

Deposit rate d = 10% . starting end of year 1 from now.

interest rate = r = 9.9%

Let us determine the cash flow from depositing at the end of each year:

End of year 1 = 59000 will grow by 2% and you will save 10% of it = S*(1+g)*d = 59000*(1+2%)*10%.

This deposit will earn interest starting from the end of 2nd year till end of 35th year. That will be a total of 34 years.

Hence the future value (FV) of the first deposit will be S*(1+g)*d*(1+r)³⁴ = 59000*(1+2%)*10%*(1+9.9%)³⁴.

At the end of year 2, salary will have grown by another 2 % i.e. S*(1+g) *(1+g) = S(1+g)².

Salary = S*(1+g)²

Amount depostied = Salary * deposit rate = S*(1+g)²d

This amount will grow at interest rate r every year from the end of year 3, till end of 35th year That will be a total of 33 years.

FV at the end of 35th year = Amount deposited * (1+r)³³.= S*(1+g)²*d*(1+r)³³ = 59000*(1+2%)²*10%*(1+9.9%)³³.

Continuing the same approach, at the end of year 35, salary will have grown by another 2 % i.e. =  S(1+g)³⁵.

Salary = S*(1+g)³⁵

Amount depostied = Salary * deposit rate = S*(1+g)³⁵*d

This amount will not grow as this is the last day.

FV = same as the Amount deposited = S*(1+g)³⁵*d = 59000*(1+2%)³⁵*10%

Now, we sum it all together:

FV at the end of 35th year = S*(1+g)*d*(1+r)³⁴ + S*(1+g)²*d*(1+r)³³ + ..........+ S*(1+g)³⁵*d

This sum is a geometric progression with common ratio = (1+g) / (1+r)

First term = S*(1+g)*d*(1+r)³⁴ +

Number of terms = 35

Alternatively, for simplification, we can arrange this with the last term as first and first term as last

FV at the end of 35th year = S*(1+g)³⁵*d + S*(1+g)³⁴*d*(1+r) + ......... + S*(1+g)²*d*(1+r)³³ .+ S*(1+g)*d*(1+r)³⁴

his sum is a geometric progression with common ratio = (1+r) / (1+g)

First term = S*(1+g)³⁵*d

Number of terms = 35

Now let us apply the geometric progression sum forumula:

a + ar + ar² + ar³ + .... + arⁿ = a (rⁿ-1) / (r-1) where a is the first term, r is the common ratio and there are n terms.

FV = S*(1+g)³⁵*d * ( [(1+r) / (1+g) ]³⁵ - 1) / ( [(1+r) / (1+g) ] - 1)

Substitute the given values:

FV = 59000*(1+2%)³⁵*10%*(((1+9.9%)/(1+2%))³⁵ -1) / (((1+9.9%)/(1+2%))-1) = $1,921,346.34

money you will have on the date of your retirement 35 years from today = $1,921,346.34