Question

Suppose you have $3000 to invest. Which investment yields the greater return over 10 years: 6.5% compounded semiannually or 6% compounded continuously? How much more (to the nearest dollar ) is yielded by the better investment?

Answer 1

The second investment, which yields a 6% interest rate compounded continuously, has a higher future value compared to the first investment, which yields a 6.5% interest rate compounded semiannually. The difference in yield between the two investments is $115.76.

Step-by-step explanation:

To compare the two investments, we need to calculate the future value of each investment after 10 years.

For the first investment, which yields a 6.5% interest rate compounded semiannually, we can use the formula:

FV = P(1 + r/n)^(nt)

Where FV is the future value, P is the principal amount, r is the annual interest rate, n is the number of times the interest is compounded per year, and t is the number of years.

For the second investment, which yields a 6% interest rate compounded continuously, we can use the formula:

FV = Pe^(rt)

Where e is the base of the natural logarithm.

Calculating the future value of each investment:

For the first investment:

1. Principal amount (P) = $3000
2. Annual interest rate (r) = 6.5%
3. Number of times compounded per year (n) = 2 (semiannually)
4. Number of years (t) = 10

Using the formula:

FV = 3000(1 + 0.065/2)^(2*10)

Calculating the future value:

FV ≈ $5085.53

For the second investment:

1. Principal amount (P) = $3000
2. Annual interest rate (r) = 6%
3. Number of years (t) = 10

Using the formula:

FV = 3000e^(0.06*10)

Calculating the future value:

FV ≈ $5201.29

The second investment, which yields a 6% interest rate compounded continuously, has a higher future value of approximately $5201.29, compared to the first investment with a future value of approximately $5085.53.

The difference in yield between the two investments is approximately $5201.29 - $5085.53 = $115.76.

Answer 2

For Investment A with a 6.5% interest compounded semiannually, the future value is $5377.98. For Investment B with a 6% interest compounded continuously, the future value is $5417.18. The better investment yields approximately $39.20 more.

Step-by-step explanation:

To compare which investment yields the greater return over 10 years, we need to calculate the future value using compound interest.

Investment A: 6.5% compounded semiannually

The formula for calculating compound interest is:

`A = P(1 + r/n)^(nt),`

where A is the future value, P is the initial principal, r is the annual interest rate, n is the number of compounding periods per year, and t is the number of years.

For Investment A: P = $3000, r = 0.065, n = 2 (semiannually compounded), and t = 10. Plugging in these values, we get:

`A = $3000(1 + 0.065/2)^(2*10)`

= $5377.98

Investment B: 6% compounded continuously

The formula for calculating continuous compound interest is:

`A = Pe^(rt),` where e is Euler's number (approximately 2.71828).

For Investment B: P = $3000, r = 0.06, and t = 10. Plugging in these values, we get:

`A = $3000 * e^(0.06*10)`

= $5417.18

Subtracting the initial investment, the better investment yields approximately $39.20 more (rounded to the nearest dollar).