Question

You plan to work for another 30 years at which point you plan to retire, after which you expect to live for another 30 years. During your retirement, you plan to make annual withdrawals at the beginning of each year where each withdrawal should have the same purchasing power as $100,000 today. You would also like to provide a bequest to your children (paid when you die) that has the same purchasing power as $1 million today. To provide for this, you plan to make annual contributions to an investment account that earns 10% p.a. The first contribution will be made today and the last will be made one year before you retire.  Inflation is expected to be 2.8% p.a. What is the amount of your first contribution?  Answer to the nearest whole dollar

Answer 1

1.PV of growing 2.8%

Yr. 31 to yr. 60--ie. 30 withdrawals at 10% p.a.

2.PV of $ 1000000 at end Year 60

1.PV of growing 2.8%p.a.

Year 31 to yr. 60 which is 30 withdrawals at 10% p.a.

to find the PV at end year.30 or Beginingyear 31

Formula for PV of growing annuity

PVGA=(Pmt./(r-g))*(1-((1+g)/(1+r))^n)*(1+r)

Pmt.=the annual withdrawal = $ 100000

r-interest rate used = 10% or 0.10 p.a.

g=the inflation rate = 2.8% or 0.028 p.a.

n=no.of compounding periods = 30

Now, we get
PVGA = (100000/(10%-2.8%)) * (1-((1+2.8%)/(1+10%))^30) * (1+10%)

= 1388888.89 * 0.8688 * 1.1

= 1,327,333.

Now, the present value of this sum

Year 0 at (10%+2.8%) = 12.8%

PV=FV/(1+r)^n

where,

FV = $1,327,333
r=rate of discount

Rate 10%+2.8%= 12.8 % or 0.128 p.a.
n=No.of years = 30

Plugging in the values, in the above formula,
PV= 1,327,333 / (1+12.8%)^30 = 35,785

Now, for the next part,
ie. PV of $ 1000000 at end Year 60
the present value of this single sum , at Year 0 at (10%+2.8%)=12.8%

PV=FV/(1+r)^n
where,

FV = future value at end year.30 or Beginingyear 31 = $ 1000000
r=rate of discount

10%+2.8%= 12.8% or 0.128 p.a.
n=No.of yrs.= 60

Plugging in the values, in the above formula,

PV=1000000/(1+12.8%)^60 = 727

So we get this 35,785+727=36512

This amt. is the PV of growing-at 2.8%p.a.
so, again, as in 1. above

Formula for PV of growing annuity

PVGA=(Pmt./(r-g))*(1-((1+g)/(1+r))^n)*(1+r)

r-interest rate used-- ie.10% or 0.10 p.a.
g=the inflation rate, ie. 2.8% or 0.028 p.a.
n=no.of compounding periods = 30
Plugging in the values, in the above formula,

36512=(Pmt./(10%-2.8%))*(1-((1+2.8%)/(1+10%))^30)*(1+10%)

==> $ 2751

the amount of your first contribution is $2751