. Sam won $150,000 in the Michigan lottery and decides to invest the money for retirement in 20
years. Find the accumulated value for Sam’s retirement for each of his options:
(a) a certificate of deposit paying 5.4% compounded yearly
(b) a money market certificate paying 5.35% compounded semiannually
(c) a bank account paying 5.25% compounded quarterly
(d) a bond issue paying 5.2% compounded daily
(e) a saving account paying 5.19% compounded continuously
a) 150,000 * 1.054^20 = 320,440.07 check
do the others the same way as I showed you in the previous problem below
To find the accumulated value for Sam's retirement for each option, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = accumulated value (total amount of money after the investment period)
P = principal or initial amount invested
r = annual interest rate (written as a decimal)
n = number of times interest is compounded per year
t = number of years
Let's calculate the accumulated value for each option.
(a) Certificate of Deposit:
P = $150,000
r = 5.4% = 0.054
n = 1 (compounded yearly)
t = 20 years
A = 150000(1 + 0.054/1)^(1*20)
A = 150000(1.054)^20
A ≈ $413,000.51
The accumulated value for option (a) is approximately $413,000.51.
(b) Money Market Certificate:
P = $150,000
r = 5.35% = 0.0535
n = 2 (compounded semiannually)
t = 20 years
A = 150000(1 + 0.0535/2)^(2*20)
A ≈ $412,449.68
The accumulated value for option (b) is approximately $412,449.68.
(c) Bank Account:
P = $150,000
r = 5.25% = 0.0525
n = 4 (compounded quarterly)
t = 20 years
A = 150000(1 + 0.0525/4)^(4*20)
A ≈ $412,044.38
The accumulated value for option (c) is approximately $412,044.38.
(d) Bond Issue:
P = $150,000
r = 5.2% = 0.052
n = 365 (compounded daily)
t = 20 years
A = 150000(1 + 0.052/365)^(365*20)
A ≈ $411,758.58
The accumulated value for option (d) is approximately $411,758.58.
(e) Savings Account (continuous compounding):
P = $150,000
r = 5.19% = 0.0519
n = continuous compounding
t = 20 years
A = 150000 * e^(0.0519*20)
A ≈ $411,708.66
The accumulated value for option (e) is approximately $411,708.66.
So, the accumulated values for each of Sam's options are:
(a) $413,000.51
(b) $412,449.68
(c) $412,044.38
(d) $411,758.58
(e) $411,708.66
To find the accumulated value for each of Sam's options, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = accumulated value
P = principal amount (the initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
For each option, we can plug in the given values and calculate the accumulated value.
(a) Option: Certificate of deposit paying 5.4% compounded yearly
Here, the interest is compounded once a year:
r = 5.4% = 0.054 (as a decimal)
n = 1
t = 20 years
A = 150,000(1 + 0.054/1)^(1*20)
= 150,000(1 + 0.054)^20
≈ 150,000(1.054)^20
≈ 150,000(1.485947)
≈ $222,892.05
The accumulated value for Sam's retirement with a certificate of deposit option is approximately $222,892.05.
(b) Option: Money market certificate paying 5.35% compounded semiannually
Here, the interest is compounded twice a year:
r = 5.35% = 0.0535 (as a decimal)
n = 2
t = 20 years
A = 150,000(1 + 0.0535/2)^(2*20)
= 150,000(1 + 0.0535/2)^40
≈ 150,000(1.02675)^40
≈ $225,107.62
The accumulated value for Sam's retirement with a money market certificate option is approximately $225,107.62.
(c) Option: Bank account paying 5.25% compounded quarterly
Here, the interest is compounded four times a year:
r = 5.25% = 0.0525 (as a decimal)
n = 4
t = 20 years
A = 150,000(1 + 0.0525/4)^(4*20)
= 150,000(1 + 0.0525/4)^80
≈ 150,000(1.013125)^80
≈ $225,807.32
The accumulated value for Sam's retirement with a bank account option is approximately $225,807.32.
(d) Option: Bond issue paying 5.2% compounded daily
Here, the interest is compounded 365 times a year (assuming no leap years):
r = 5.2% = 0.052 (as a decimal)
n = 365
t = 20 years
A = 150,000(1 + 0.052/365)^(365*20)
= 150,000(1 + 0.052/365)^7300
≈ 150,000(1.0001423)^7300
≈ $226,109.61
The accumulated value for Sam's retirement with a bond issue option is approximately $226,109.61.
(e) Option: Saving account paying 5.19% compounded continuously
For continuous compounding, we use the formula:
A = P * e^(r*t)
Where e is the base of the natural logarithm (approximately 2.71828):
P = 150,000
r = 5.19% = 0.0519 (as a decimal)
t = 20 years
A = 150,000 * e^(0.0519*20)
≈ 150,000 * e^(1.038)
≈ 150,000 * 2.82037
≈ $423,055.50
The accumulated value for Sam's retirement with a saving account option (compounded continuously) is approximately $423,055.50.
In summary:
(a) Certificate of deposit: $222,892.05
(b) Money market certificate: $225,107.62
(c) Bank account: $225,807.32
(d) Bond issue: $226,109.61
(e) Saving account (continuous compounding): $423,055.50
(a) $429,440.97; (b) $431,200.96; (c) $425,729.59; (d) $424,351.12;
(e) $423,534.64