Please help me with the following problem. I'm not able to get the right answer! Please provide a full solution that I can follow... thank you.
How much money should be invested now at 7% to obtain $9,000 in 5 years if interest is compounded:
a) Quarterly
b) Continuously
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My work:
a)
B = 9000(1 + (0.07/4)^20
B = 12 733
b)
B(5) = 9000e^0.07
B(5) = 0652.57
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Textbook answer:
a) $6 361.42
b) $6 342.19
Quarterly:
For an initial balance B, you want B*(1+.07/4)^20 = 9000.
(1+.07/4)^20 = 1.0175^20 = 1.414778
And 9000./1.414778 = 6361.42 QED
Continuously, you want B*e^(.07*5) = 9000.
e^(.35) = 1.4190675 -- take it from here.
what is the present value of nine annual cash payments of 4 000 to be paid at the end of each year using an interest rate of 6%.
To find the correct answers for both parts (a) and (b), let's break down the process step by step:
a) Quarterly compounding:
This means the interest is calculated and added to the amount every quarter. To find the correct amount to invest, we can use the following formula:
B = P(1 + r/n)^(nt)
Where:
B is the future value (in this case, $9,000)
P is the principal amount (the amount to be invested)
r is the annual interest rate (7% or 0.07)
n is the number of compounding periods per year (quarterly, so 4)
t is the number of years (5)
Substituting the given values, we have:
9000 = P(1 + 0.07/4)^(4*5)
To solve for P, we can divide both sides of the equation by (1 + 0.07/4)^(4*5) and then multiply by 9000 to isolate P:
P = 9000 / (1 + 0.07/4)^(4*5)
Evaluating the expression, we find:
P ≈ $6,361.42
So, approximately $6,361.42 should be invested at a 7% interest rate compounded quarterly to obtain $9,000 in 5 years.
b) Continuous compounding:
Continuous compounding means the interest is recalculated and added to the amount an infinite number of times. The formula for continuous compounding is:
B = Pe^(rt)
Where:
e is the mathematical constant approximately equal to 2.71828
Substituting the given values, we have:
9000 = Pe^(0.07*5)
To solve for P, we can divide both sides by e^(0.07*5) and then multiply by 9000:
P = 9000 / e^(0.07*5)
Evaluating the expression, we find:
P ≈ $6,342.19
Therefore, approximately $6,342.19 should be invested at a 7% interest rate compounded continuously to obtain $9,000 in 5 years.
I hope this clears up any confusion and helps you arrive at the correct solution! Let me know if you have any further questions.