Question

An investor plans the following investments for the next 20 years: 12.00 years of $10,200.00 per year, and 8.00 years of $13,825.00 per year.

She thinks his investments will earn 6.00% a year for the first 12.00 years, and then earn 11.00% per year for the last 8.00 years. How much would the investor have to set aside today if she wants to fund the entire account?

Answer 1

The amount to be set aside = Present value of investment to be made.

Step1.

find the present value of investment to be made at $10,200 for first 12 years

Present Value = $10,200 X PV Annuity factor 12 years at 6%,

= (((1+i)ⁿ)-1) / (((1+i)ⁿ)(i)),

Where 'i' is the interest rate = 6% or 0.06,

n is the number of years = 12,

Applying values in formula, ((1.06¹²)-1) / ((1.06¹²)(0.06),

8.38384394038333,

Present Value = $10,200 X 8.38384394038333,

= $85,515.20819191

Step2.

find the present value of investment to be made at $13,825 for 8 years.

Present Value = $13,825 X PV Annuity factor 8 years at 11%,

PV Annuity factor 8 years at 11% = (((1+i)ⁿ)-1) / (((1+i)ⁿ)(i)),

Where 'i' is the interest rate = 11% or 0.11,

n is the number of years = 8,

Applying values in formula, ((1.11⁸)-1) / ((1.11⁸)(0.11),

PV Annuity factor 8 years at 11% = 5.14612276092708,

Present Value = $13,825 X 5.14612276092708,

Present Value =$71,145.1471698169

Step 3.

Discound the amount in step2 at 12th year discount factor.

The amount get from step 2 is the present value at the year 12 of the investment from year 13 to year20. The amount to set aside to fund the entire account at the year0 i.e, now. So the amount in step2 is to be discounted to arrive its year0 value by using the discount factor of year12.

the formula for discount rate = 1/ (1+i)ⁿ, where i is the existing interest rate = 6% or 0.06,

n = 12

the discount factor of year12 = 1/ (1.06¹²),

= 0.496969363577001

The Year0 value of investment from 13^th year through 20^th year = Step2 Value X the discount factor of year12,

=$71,145.1471698169 X 0.496969363577001,

= $35,356.9585105759

Step 4.

Add step1 value and step3 value. this is the present value of total investment to be made annually. this is the amount to be setaside to fund the entire account.

Total Year0 Value = Step1 value + Step3 Vaue,

= $85,515.20819191 + $35,356.9585105759,

= $120,872.166702486, Rounded to 2 decimal places = $120,872.17.

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