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You plan to retire in 40 years and can invest to earn 6%. You estimate that you will need $38,000...

Question:

You plan to retire in 40 years and can invest to earn 6%. You estimate that you will need $38,000 at the end of each year during your 25 years of retirement. During your retirement years, you estimate you will be able to earn 5%. How much do you estimate you will need to set aside at the end of each year?

Annuity:

Annuity refers to the series of equal cash flow received or paid periodically. It is useful in financial planning purposes. Present value of annuity gives us the present value of all the cash flows at the beginning of the series and the future value of annuity gives us the value of the cash flow at the end of the series.

Answer and Explanation: 1

Let us note down the two formulas to be used in the question,

Present value of annuity
= Annuity x PVAF(r,n)

PVAF or present value annuity factor is the sum of discounting factors at a given periodic rate r for n number of periods

It can be calculated using the following formula:

{eq}\frac{1-(1+r)^{-n}}{r} {/eq}
r = Periodic Rate
n = Number of periods

{eq}\text{Future value of annuity = Annuity * ((1 + Periodic Rate)}^\text{Number of periods} - 1) / \text{Periodic rate} {/eq}

Now,

We will use Present value of annuity formula to get the value at the time of retirement and then use Future value of annuity formula to get the annuity required to be deposited each year.

Here,

Annuity = $38,000.00

r = 5% = 0.05

n = 25

So,

Present value of annuity
= Value required at time of retirement
= 38,000 x PVAF(5%,25 years)
= 38,000 x 14.0939
= $535,568.20

  • PVAF(5%, 25 years)
    = {eq}\frac{1-(1+r)^{-n}}{r} {/eq}
    = {eq}\frac{1-(1.05)^{-25}}{0.05} {/eq}
    = 14.0939

Now,

For Future Value of annuity:

Here,

  • Future value of Annuity = $535,568.20
  • Periodic rate = 6% = 0.06
  • Number of periods = 40

Substituting the values in the formula, we get

Annuity
= {eq}535,568.20 * 0.06 / (1.06^{40} - 1) {/eq}
= $3,460.59


Learn more about this topic:

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How to Find the Value of an Annuity

from

Chapter 21 / Lesson 15
9.4K

An annuity is a type of savings account that pays back the investor in the future. Learn the formula used to calculate an annuity's value, and understand the importance of labeling specific numbers to calculate an output over time.


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