At the end of each year a self-employed person deposits $1,500 in a retirement a
ID: 2901657 • Letter: A
Question
At the end of each year a self-employed person deposits $1,500 in a retirement account that earns 10 percent annually.
a) How much will be in the account when the individual retires at the age of 65 if the contributions start when the person is 45 years old?
b) How much additional money will be in the account if the individual stops making the contribution at age 65 but defers retirement until age 70?
c) How much additional money will be in the account if the individual continues making the contribution but defers retirement until age 70?
d) Compare the answers to (b) and (c). What is the effect of continuing the contributions? How much is the difference between the two answers?
5. Graduating seniors may earn $45,000 each year. If the annual rate of inflation is 2 percent, what must these graduates earn after 20 years to maintain their current purchasing power? If the rate of inflation rises to 4 percent, will they be maintaining their standard of living if they earn $100,000 after 20 years?
8. You are offered $900 five years from now or $150 at the end of each year for the next five years. If you can earn 6 percent on your funds, which offer will you accept? If you can earn 14 percent on your funds, which offer will you accept? Why are your answers different?
Explanation / Answer
At the end of each year a self employed person deposits $1500 in a retirement account that earns 10 percent annually.
a.) How much will be in the account when the individual retires at the age of 65 if the contributions start when the person is 45 years old?
Amount in account at end of year when age is 45 =
A(45) = $1500
A(46) = $1500(1+10%) + $1500 = $1500(1.1) + $1500
A(47) = $1500(1.1)^2 + $1500(1.1) + $1500
A(48) = $1500(1.1)^3 + $1500(1.1)^2 + $1500(1.1) + $1500
...
A(65) = $1500(1.1)^20 + ... $1500(1.1)^3 + $1500(1.1)^2 + $1500(1.1) + $1500
(1.1)A(65) = $1500(1.1)^21 + ... $1500(1.1)^4 + $1500(1.1)^3 + $1500(1.1)^2 + $1500(1.1)
(1.1)A(65)-A(65) = $1500(1.1)^21 - $1500 = $1500 ((1.1)^21 - 1)
A(65) = $1500 ((1.1)^21 - 1) / (1.1 - 1)
A(65) = $1500 ((1.1)^21 - 1) / (0.1)
A(65) = $15000 ((1.1)^21 - 1)
A(65) ? $96,003.75
b.) How much ADDITIONAL money will be in the account if the individual stops making the contribution at age 65 but defers retirement until age 70?
A(65) (1.1)^5 - A(65) = A(65) ( (1.1)^5 - 1 )
= $15000 ((1.1)^21 - 1) ((1.1)^5 - 1)
? $58,611.25
c.) How much ADDITIONAL money will be in the account if the individual continues making the contribution but defers retirement until age 70?
A(65) = $15000 ((1.1)^21 - 1) ? $96,003.75
A(70) = $15000 ((1.1)^26 - 1)
A(70) ? $163,772.65
A(70) - A(65) ? $67,768.90
d.) Compare the answers to b and c.
Answer to b: $58,611.25
Answer to c: $67,768.90
What is the effects of continuing the contributions?
You get more money in the account at age 70 if you continue the contributions.
How much is the difference between the two answers?
$9,157.65; but $7500 of that are your contributions and only $1,657.65 is interest earned.
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a.) A = P (1 + r)^t
A = 45000 (1 + 0.02)^20
A ? use you calculator.
b.) A = 45000 (1 + 0.04)^20
$100,000 will maintain (or exceed) their initial standard of living if:
100000 ? 45000 (1 + 0.04)^20
100000 ? 98600.54
So their standard of living will be slightly better.
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A = 150 + 150(1+r) + 150(1+r)^2 + 150(1+r)^3 + 150(1+r)^4
(1+r)A = 150(1+r) + 150(1+r)^2 + 150(1+r)^3 + 150(1+r)^4 + 150(1+r)^5
A - (1+r)A = 150 - 150(1+r)^5
A = 150 (1 - (1+r)^5) / (1 - 1 - r)
A = 150 ( (1+r)^5 - 1) / r
For what r is A > 900?
150 ( (1+r)^5 - 1) / r > 900
( (1+r)^5 - 1) / r > 6
(1+r)^5 > 6 r + 1
---check for r = 6%:
(1.06)^5 >? 6(0.06) + 1
1.3382255776 >? 1.36
No.
So 6% would not give you more than $900 in 5 years.
---check for r = 14%:
(1.14)^5 >? 6(0.14) + 1
1.925414582400001 >? 1.84
Yes.
So 14% would give you more than $900 in 5 years.
P.S.: You can use a graphing calculator or GeoGebra to find the break-even rate is approximately 9.1281%.
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