Suppose that you have $ 5,000 to invest. Which investment yields the greater return over 9 years: 8.75% compounded continuously or 8.9% compounded semiannually?
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A) $ 5,000 invested at 8.9% compounded semiannually over 9 years yields the greater return.
B) Both investment plans yield the same return.
C) $5,000 invested at 8.75% compounded continuously over 9 years yields the greater return.
The $5,000 don't matter.
All you have to do is compare
(1 + ..089/2)^2 with e^(.0875)
ok thx
To determine which investment yields the greater return, we can compare the future value of each investment after 9 years.
For option A, where $5,000 is invested at 8.9% compounded semiannually, we can use the formula for compound interest:
\[A = P \left(1 + \frac{r}{n}\right)^{nt}\]
Where:
A = Future value of the investment
P = Principal investment amount ($5,000)
r = Annual interest rate (8.9%)
n = Number of times interest is compounded per year (2, since it is compounded semiannually)
t = Number of years (9)
Plugging the values into the formula, we get:
\[A = 5000 \left(1 + \frac{0.089}{2}\right)^{2 \cdot 9}\]
For option C, where $5,000 is invested at 8.75% compounded continuously, we can use the formula for continuous compound interest:
\[A = P \cdot e^{rt}\]
Where:
A = Future value of the investment
P = Principal investment amount ($5,000)
r = Annual interest rate (8.75%)
t = Number of years (9)
e = Euler's number (approximately 2.71828)
Plugging the values into the formula, we get:
\[A = 5000 \cdot e^{0.0875 \cdot 9}\]
We can calculate the value of both investments, and compare which one is greater:
A = $5,000 × (1 + (0.089/2))^(2 × 9) = $10,008.29
B = $5,000 × e^(0.0875 × 9) = $10,127.03
Since option C yields a higher future value, $5,000 invested at 8.75% compounded continuously over 9 years yields the greater return.
Therefore, the correct answer is C) $5,000 invested at 8.75% compounded continuously over 9 years yields the greater return.