Question

A small business owner contributes $2,000 at the end of each quarter to a retirement account that earns 10% compounded quarterly. (a) How long will it be until the account is worth at least $150,000? (Round your answer UP to the nearest quarter.) 43 quarters (b) Suppose when the account reaches $150,000, the business owner increases the contributions to $4,000 at the end of each quarter. What will the total value of the account be after 15 more years? (Round your answer to the nearest dollar.) $

Answer 1

Ordinary Annuity

If the monthly payment made at the end of the every quarter then we can use the ordinary annuity formula

We can use the formula for finding the future value as below

FV = P x [ ( 1 + (r/n) )^nt-1 ] / ( r/n )

Here FV = future value = $150000

        P = Cash flow per period = $2000

        r = rate of interest = 10% = 10/100 = 0.1

                n = compounding frequency is quarterly so n= 4

       t = Number of years = ?

150000= 2000 x [ ( 1 + (0.1/4 )^4t – 1 ] / (0.1/4)

150000 / 2000 = [ ( 1 + 0.025 )^4t – 1 ] / (0.025)

75 = [ (1.025)^4t -1 )] / (0.025)

75 x 0.025 = (1.025)^4t -1 )

1.875= (1.025)^4t -1 )

1.875 + 1 = (1.025)^4t

2.875 = (1.025)^4t

Now apply ln both sides

Ln(2.875) = ln(1.025)^4t

Ln(2.875) = (4t) ln(1.025)

1.05605 = 4t(0.02469)

1.05605 / 0.02469 = 4t

4t = 42.7723 ~ 43

So here 4t represents number of quarters so the requires quarters must be 43

number of required quarters = 43

now the balance reaches to $150000 and the person contribution increases to $4000 at the end of the every quarter.

So we can apply the same formula with initial deposit

FV = Pv(1 + (r/n))^nt + [ C x [ ( 1 + (r/n) )^nt-1 ] / ( r/n )]

So here Pv = present value = $150000

And r = 10% = 10/100 = 0.1

N = compounding frequency quarterly so n = 4

Number of years t = 15

And C = payment period = $4000

FV = Pv(1 + (r/n))^nt + [ C x [ ( 1 + (r/n) )^nt-1 ] / ( r/n )]

FV = 150000(1 + (0.1/4))⁴⁽¹⁵⁾ + [ 4000 x [ ( 1 + (0.1/4) )⁴⁽¹⁵⁾-1 ] / ( 0.1/4 )]

FV = 150000(1 + (0.025))⁶⁰ + [ 4000 x [ ( 1 + (0.025) )⁶⁰-1 ] / ( 0.025)]

FV = 150000(1.025)⁶⁰ + [ 4000 x [ ( 1.025)⁶⁰-1 ] / ( 0.025)]

FV = 150000(4.399789)+ [ 4000 x [ (4.399789)-1 ] / ( 0.025)]

FV = 659968.35+ [ 4000 x [ (3.399789) ] / ( 0.025)]

FV = 659968.35+ [ 4000 x [ (3.399789) ] / ( 0.025)]

FV = 659968.35+ [ 4000 x 135.99156]

FV = 659968.35 + [543966.24]

FV = $1203934.59 ~ 1203935

So future value after 15 years is $1203935