You want to be able to withdraw $20,000 each year for 15 years. Your account earns 7% interest. a) How much do you need in your account at the beginning? $ b) How much total money will you pull out of the account? $ c) How much of that money is interest?

Answer: A) The amount needed at the beginning would be $146,342

B) The total money you would pull out would be $300,001

C) The interest earned would total $153,659

Explanation: If you planned on withdrawing 20000 per year for 15 years, that would be a total of 300000 at the end of 15 years. The interest you would have earned is calculated as;

I = PRT

Note that the addition of your interest and the sum invested (Principal) shall be a total of 300000. Hence,

I + P = 300000

Making P the subject of the equation,

P = 300000 - I.

So we have, R as 0.07, T as 15 and P as 300000 - I, we can now substitute for the values as follows;

I = PRT

I = (300000 - I) x 0.07 x 15

I = (300000 - I) x 1.05

I = 315000 - 1.05I

Collecting like terms we now have

I + 1.05I = 315000

2.05I = 315000

Divide both sides of the equation by 2.05

I = 153659 (approximately)

If the interest earned is $153,659, and the amount you wish to withdraw at the end of 15 years is a total of $300,000, then the principal (P) invested would be

I = PRT

I/RT = P

153659 / (0.07 x 15) = P

146341.9 = P

P≈ 146342

Therefore, interest earned = $153,659

Principal invested = $146,342

Total withdrawal = $300,001

a) How much do you need in your account at the beginning?

$182,158.28

b) How much total money will you pull out of the account?

$300,000

c) How much of that money is interest?

$117,841.72

Explanation:

We need to calculate the present value of the annuity:

PV of annuity = P x {[1 - (1 + r) ⁻ⁿ] / r}

P = periodic payment (annuity) = $20,000 r = interest rate = 7% n = number of periods = 15

PV of annuity = $20,000 x {[1 - (1 + 7%) ⁻¹⁵] / 7%}

= $20,000 x {[1 - 1.07⁻¹⁵] / 7%} = $20,000 x (0.63755398 / 7%) = $20,000 x 9.107914 = $182,158.28

total money pulled out = $20,000 x 15 payments = $300,000

total interest received = $300,000 - $182,158.28 = $117,841.72