You plan to retire in exactly 20 years. Your goal is to create a fund that will
ID: 2802769 • Letter: Y
Question
You plan to retire in exactly 20 years. Your goal is to create a fund that will allow you to receive $23,000 at the end of each year for the 35 years between retirement and death (a psychic told you that you would die exactly 35 years after you retire). You know that you will be able to earn 11% per year during the 35-year retirement period.
A) How large a fund will you need when you retire in 20 years to provide the 35-year, $23,000 retirement annuity?
B) How much will you need today as a single amount to provide the fund calculated in PART A if you earn only 9% per year during the 20 years preceding retirement?
C) What effect would an increase in the rate you can earn both during and prior to retirement have on the values found in PARTS A and PARTS B?
D) Now assume that you will earn 10% from now through the end of your retirement. You want to make 20, end of year deposits into your retirement account that will fund the 35-year stream of $23,000 annual annuity payments.How large do your annual deposits have to be?
Explanation / Answer
A)
Formula for present value of ordinary annuity:
PV = P x [1 – (1 + r)-n/r]
Where,
PV = Present value of ordinary annuity
P = Payment per period = $ 23,000
r = rate per period = 11 % = 0.11
n = no. of periods = 35
PV = $ 23,000 x [1-(1+0.11)-35/0.11]
= $ 23,000 x [1-(1.11)-35/0.11]
= $ 23,000 x [1- 0.025924/0.11]
= $ 23,000 x [0.974076/0.11]
= $ 23,000 x 8.85524
= $ 203,670.51
$ 203,670.51 is required at the time of retirement to facilitate 35 year, $ 23,000 retirement annuity.
B)
As we know the maturity amount needed, we can calculate principal amount using the formula for compound interest as:
A = P x (1+i)n
P = A/(1+i)n
Where,
P = Principle amount, the amount require today
A = Amount on maturity after 20 years = $ 203,670.51
i = Interest rate = 9 % or 0.09 p.a.
n = Number of periods = 20
P = $ 203,670.51 / (1 + 0.09)20
= $ 203,670.51 / (1.09)20
=$ 203,670.51 / 5.604411
= $ 36,341.11
C)
If interest rate is increased, fund needed on retirement will be decreased.
Hence we need fewer amounts to reach the target fund.
Let’s check it by increasing rate by 1%.
Fund needed for 35 years, $ 23,000 annuity @ 12 % is:
[Formula same as above A]
PV = $ 23,000 x [1-(1+0.12)-35/0.12]
= $ 23,000 x [1-(1.12)-35/0.12]
= $ 23,000 x [1- 0.01893953/0.12]
= $ 23,000 x [0.98106047/0.12]
= $ 23,000 x 8.175503913
= $ 188,036.59
Amount require today for the fund of $ 188,036.59 by increasing interest rate to 10 %:
[Formula for compound interest is same as above in B]
P = $ 188,036.59/ (1 + 0.1)20
= $ 188,036.59/ (1.1)20
=$ 188,036.59/ 6.727499949
=$ 27,950.44
It is observed that on increasing rate by 1%, we need to pay $ 27,950.44 in place of $ 36,341.11 yearly to get the desired $ 23,000 annuity on retirement.
D)
We need a fund of $ 203,670.51 at the end of retirement which is the future value of annuity.
Formula for future value of ordinary annuity:
FV = P x [(1+r)n -1/r]
P = FV/ [(1+r)n -1/r]
Where,
FV = Future value of ordinary annuity = $ 203,670.51
P = Payment per period
r = rate per period = 10 % = 0.1
n = no. of periods = 20
P = $ 203,670.51 / [(1+0.1)20-1/0.1]
= $ 203,670.51 / [(1.1)20-1/0.1]
= $ 203,670.51 / (6.7275-1/0.1)
= $ 203,670.51 / (5.7275/0.1)
= $ 203,670.51 / 57.275
= $ 3,556.011
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