1. Percent interest, which is compounded annually until the certificate matures. You invest $1000 in a cd that matures after 10 years and pays 5
a.How much interest will the saver earn if the interest is left to accumulate?
FV = $1,000 x (1.05)10
FV = $1,628.89
Interest = $1,628.89 – $1,000
Interest = $628.89
b.How much interest will the saver earn if the interest is withdrawn each year?
Interest = $1,000 x 5% x 10
Interest = $500
c. Why are the answers to a and b different?
Because in (a), interest is compounded, while (b) only earns simple interest.
2.A self-employed person deposits $3,000 annually in a retirement account (called a Keogh account) that earns 8 percent.
a.How much will be in the account when the individual retires at the age of 65 if the savings program starts when the person is age 40?
PV of Annuity = $3,000 x [1-(1.08)-25] / 0.08
PV of Annuity = $32,024.33
FV = $32,024.33 x (1.08)25
FV = $219,317.83
b.How much additional money will be in the account if the saver defers retirement until age 70 and continues the contributions?
PV of Annuity = $3,000 x [1-(1.08)-30] / 0.08
PV of Annuity = $33,773.35
FV = $33,773.35 x (1.08)30
FV = $339,849.63
Additional Money = $339,849.63 – $219,317.83
Additional Money = $120,531.80
c.How much additional money will be in the account if the saver discontinues the contributions at age 65 but does not retire until age 70?
PV of Annuity = $3,000 x [1-(1.08)-25] / 0.08
PV of Annuity = $32,024.33
FV = $32,024.33 x (1.08)30
FV = $322,249.84
Additional Money = $322,249.84 – $339,849.63
Decrease in Money = $17,599.79
Note: There will be a decrease in money from (b) to (c).
3.A 45-year-old woman decides to put funds into a retirement plan. She can save $2,000 a year and earn 9 percent on this savings. How much will she have accumulated if she retires at age 65?
PV of Annuity = $2,000 x [1-(1.09)-20] / 0.09
PV of Annuity = $18,257.09
FV = $18,257.09 x (1.09)20
FV = $102,320.24
At retirement how much can she withdraw each year for 20 years from the accumulated savings if the savings continue to earn 9 percent?
Payment = FV / [1-(1.09)-20] / 0.09]
Payment = $102,320.24 / 9.12855
Payment = $11,208.82
4.If a father wants to have $100,000 to send a newborn child to college, how much must he invest annually for 18 years if he earns 9 percent on his funds? (Any current student who subsequently becomes a parent and wants to send a child to college should make this calculation early in the child’s life.)
FV = PV of Annuity Due x (1.09)18
$100,000 = PV of Annuity Due x (1.09)18
PV of Annuity = $100,000 / (1.09)18
PV of Annuity = $21,199.37
PV of Annuity = Payment x [1-(1.09)-18] / 0.09
$21,199.37 = Payment x 8.7556
Payment = $21,199.37 / 8.7556
Payment = $2,421.24
5.A widow currently has a $93,000 investment yielding 9% annually. Can she withdraw $16,000 a year for the next 10 yrs?
PV = Payment x [1-(1.09)-10] / 0.09
Payment = $93,000 / [1-(1.09)-10] / 0.09
Payment = $14,491.26
No, she cannot since the allowable maximum withdrawal is only $14,491.26
6.An investment generates $10,000 per year for 25 yrs. If you can earn 10 percent on other investments, what is the current valued of the investment? If it’s current price is $120,000, should you buy it?
PV of Annuity = $10,000 x [1-(1.10)-25] / 0.10
PV of Annuity = $90,770.40
No, I will not buy it because the current price is higher than the present value of the investment.
7.You are 25 years old and inherit $65,000 from your grandmother. If you wish to purchase a $100,000 yacht to celebrate your 30th birthday, what compound annual rate of return must you earn?
FV = PV x (1+r)5
$100,000 = $65,000 x (1+r)5
1.53846 = (1+r)5
(1.53846) 1/5 = 1+r
1.08998 = 1+r
Annual Rate = 8.998%
8.An investment offers to pay you $10,000 a year for five years. If it cost $33,520, what will be your rate of return on the investment?
PV of Annuity = Payment x [1-(1+r)-5] / r
$33,520 = $10,000 x [1-(1+r)-5] / r
Find r: since it would be very difficult to solve this manually, we’ll seek the aid of financial calculator by inputting the following:
Period (nper) = 5
Payment (pmt) = $10,000
Present Value (PV) = $33,250
Future Value (FV) = $0
Result: Rate of Return = 15.00%