Question

I. Find the present and future value of $1000 received every month end for 20 years if the interest rate is J12 = 12% p.a.
II. Find the present value of $10,000 received at the start of every year for 20 years if the interest rate is J1 = 12% p.a. and if the first payment of $10,000 is received at the end of 10 years.
III. John is currently 25 years old. He has $10,000 saved up and wishes to deposit this into a savings account which pays him J12 = 6% p.a. He also wishes to deposit $X every month into that account so that when he retires at 55, he can withdraw $2000 every month end to support his retirement. He expects to live up till 70 years. How much should he deposit every month into his account?

Answer 1

i)

PVOA is present value of an annuity

$PVOA= payment * {1-(1+r )^{(-n)} \above{2pt} r}$

$PVOA= 1,000 * {1-(1+0.01 )^{(-240)} \above{2pt} 0.01}$

$PVOA=90,819.42/$

FVOA is future value of an annuity

$FVOA= payment * {(1+r )^{n}-1 \above{2pt} r}$

$FVOA= 1,000* {(1+0.01 )^{240}-1 \above{2pt} 0.01}$

$FVOA=989,255.37/$

ii) Present value of deferred annuity due;

$PVOA= payment * {1-(1+r )^{(-n)} \above{2pt} (1+r )^{(t-1)}}$

$PVOA= 10,000 * {1-(1+0.12 )^{(-20)} \above{2pt} (1+0.12 ) ^{(10-1)}}$

$PVOA=26,934.56/$

iii)First we work out John's retirement plan;

Which is to have 2,000/ every month for 15 years

$2,000*12*15=360,000/$

We find future value of 10,000/ he has now

$FV = PV *(1+r)^n$

FV is future value

PV is present value

$FV = 10,000 *(1+.005)^{360}$

$FV = 60,225.75/$

Amount to invest;

$360,000−60,225.75=299,774.25/$

We now calculate monthly deposits;

$A= MD *{1-(1+r )^{(-n)} \above{2pt} r}$

A is Amount to invest

MD is monthly deposit

$299,774.25= MD *{1-(1+0.05 )\left(^{\smash{-360}}\right) \above{2pt} 0.05}$

$MD= 1,797.30/$