Please help me ASAP!!!!!!!!!!!! I did the work on most of these problems but I am still confused please help.
19. Comparing Investments Russ McClelland, who is self-employed, wants to invest $60,000 in a pension plan. One investment offers 7% compounded quarterly. Another offers 6.75% compounded continuously.
(a) Which investment will earn more interest in 5 yr?
(b) How much more will the better plan earn?
An employee wants to invest $50 comma 000 in a pension plan. One investment offers 4% compounded semiannually. Another offers 3.75% compounded continuously.
(a) Which investment will earn more interest in 6 yr?
(b) How much more will the better plan earn?
a) first offer
amount = 60000(1 + .07/4)^20 = 84886.69
second offer:
amount = 60000 e^(.0675(5)) = 84086.38
b) subtract the two amounts
To compare the two investments, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A is the final amount
P is the principal (initial investment)
r is the interest rate
n is the number of times interest is compounded per year
t is the number of years
Let's calculate the interest earned by each investment in 5 years:
For the first investment at 7% compounded quarterly:
P = $60,000
r = 7% = 0.07
n = 4 (quarterly compounding)
t = 5
Using the formula: A = 60,000(1 + 0.07/4)^(4*5)
For the second investment at 6.75% compounded continuously:
P = $60,000
r = 6.75% = 0.0675
n = ∞ (continuous compounding)
t = 5
Using the formula: A = 60,000e^(0.0675*5)
Now, let's calculate both A values to find the better investment:
For the first investment:
A = 60,000(1 + 0.07/4)^(4*5)
A ≈ 60,000(1.0175)^(20)
A ≈ 60,000(1.418519)
For the second investment:
A = 60,000e^(0.0675*5)
A ≈ 60,000e^(0.3375)
A ≈ 60,000(1.402023)
(a) The first investment will earn approximately $84,551.14 in 5 years.
(b) The second investment will earn approximately $84,121.34 in 5 years.
The better plan will earn approximately $429.80 more than the other plan.
To compare the two investments and determine which one will earn more interest in 5 years, we can calculate the future value of each investment using the given interest rates and compounding periods.
Let's start with investment A, which offers 7% compounded quarterly. The formula to calculate the future value with compound interest is:
A = P(1 + r/n)^(n*t)
Where:
- A is the future value
- P is the principal (initial investment amount)
- r is the annual interest rate in decimal form
- n is the number of compounding periods per year
- t is the number of years
Using the given values:
- P = $60,000
- r = 7% = 0.07
- n = 4 (quarterly compounding)
- t = 5
We can substitute these values into the formula:
A = 60000(1 + 0.07/4)^(4*5)
= 60000(1 + 0.0175)^20
= 60000(1.0175)^20
≈ $75,503.42
So, investment A will grow to approximately $75,503.42 in 5 years.
Now let's calculate the future value of investment B, which offers 6.75% compounded continuously. The formula for continuous compounding is:
A = Pe^(r*t)
Where:
- A is the future value
- P is the principal (initial investment amount)
- e is the base of the natural logarithm (approximately 2.71828)
- r is the annual interest rate in decimal form
- t is the number of years
Using the given values:
- P = $60,000
- r = 6.75% = 0.0675
- t = 5
We can substitute these values into the formula:
A = 60000 * e^(0.0675*5)
= 60000 * e^(0.3375)
≈ $75,139.91
So, investment B will grow to approximately $75,139.91 in 5 years.
To answer the questions:
(a) Investment A will earn more interest in 5 years.
(b) The better plan (investment A) will earn approximately $75,503.42 - $75,139.91 = $363.51 more.
Please note that these calculations are approximations due to rounding.