Question

Davis is setting an investment plan with $10000 in 445 years until his retirement. His plan has two phases. In the first 20 years the rate of return is 8% per year, compounding semi-annually. In the last 25 years, the rate of return is 10% per year compounding annually.   

Required :- A) Calculate the effective ANNUAL interest rate (EAR) Davis receives during the first 20 years of his investment. B) Assume that at the end of the first 20 years. Davis decides to withdraw $5000 from his investment. How much money Davis will have in 45 years? C) If DAvis wishes to have exactly $600,000 in his account when he is retired, which is the rate of return should he has in the first 20 years? D) Assuming that at the end of the first 20, Davis changes his investment strategy and puts exactly $700 into a superannuation account at beginning of each month for the interest rates of 9% for the left 25 years. How much would be accumulates when he retires by this cash flow only? E) How much money Davis could accumulates for this cash flow alone if he puts that $700 at the end of each month rather than at the beginning of each month for 25 years. F) If at the first day of Davis retirement the superannuation fund starts to pay him $80,000 per year forever, what is the implied rate of return if the present value of this cash flow is $800,000?

Answer 1

Fund Davis has $ 10000

Investment Horizon = 45 years (20 + 25)

A)

		Rate	compounding
Phase 1	20 years	8%	6 monthly
Phase 2	25 years	10%	yearly

EAR = ( 1 + i / n) ⁿ  - 1

where i = rate of interest

n = number of compounding periods

EAR = [ 1 + 8%/ / 2] ² - 1

= 1.04² - 1

= 1.0816 - 1 => 8.16%

B)

Davis invests $ 10000 for 20 years @ 8% compounded semi annually

Future value = P[ 1 + r% / n] ^{t *n}

r = rate of interest

n = compounding factor

t = time horizon

here p = 10000

n = 2 as interest is paid semi annually

t = 20 years

=100000 [ 1 + 8% / 2] ^{20 * 2}

= $48010.20

Davis withdraws $ 5000

Amount remains = 48010.2 - 5000 =$  43010.2

This will be invested for 25 years @ 9% compounded annually

FV = 43010.20 [ 1 + 9%] ²⁵

Davis will have in 45 years = $ 370880.42

C)

Davis need to have $ 600000 after 45 years

The FV factor for 25 years is = [1 + rate/100] ⁿ

here r =rate of interest = 9%

n =time horizon = 25 years

FV factor = [1 + 9/100 ] ²⁵

FV factor_{(9% , 25 years)} = 8.623

Let us make a equation for getting $ 6000000 after 45 years

assume rate of interest to be x% in first 20 years

Future value = P[ 1 + r% / n] ^{t *n}

r = rate of interest

n = compounding factor

t = time horizon

here p = 10000

n = 2 as interest is paid semi annually

t = 20 years

{10000[1 + x% / 2]^20*2} FV factor_(9%,25years) = 600000

{10000[1 + x% / 2]⁴⁰ } is the Future value after 20 years

FV factor_{(9% , 25 years)} = 8.623

10000[1 + x% / 2]⁴⁰ = 600000 / 8.623

10000[1 + x% / 2]⁴⁰ = 69581.35

[1 + x% / 2]⁴⁰ = 69581.35 / 10000

[1 + x% / 2]⁴⁰ = 6.958135

[ 1 + x% / 2] = 6.958135 ^ 1/40

Calculate 6.958135 ^ 1/40 = 1.0496

1 + x/2 = 1.0496

x/2 = .0496

x = 0.0992 or 9.92%

D)

Davis deposits $ 700 at the beginning  of each month for 25 years @ 9%

FV(rate, nper, pmt, pv,type)

where rate = 9% / 12 as monthly interest will be paid

nper = total periods for which interest will be received => 25 * 12 = 300

pmt = $ - 700

type = beginning or end of the period the investment is made 1 is for beginning and 0 is for end of the period

Calculate this in excel =FV(9%/12,25*12,-700,1)

=

$790,671.25

E)

Same formula of FV needs to be applied here except last variable type to be 0 as 0 is for end of the period

Calculate this in excel = FV(9%/12,25*12,-700,0)

=

$784,785.36

F)

Annual Payment = After retirement Davis is paid $ 80000 per year forever

PV of this cash flow is $ 800000

Present value of perpetuity

Rate = Annual Payment / Present value of annuity

Rate = 80000 / 800000

= 10%