Larry has $2600 to invest and needs $3000 in 11 years. What annual rate of return will he need to get in order to accomplish his goal, if interest is compounded continuously?
To calculate the annual rate of return needed for Larry to accomplish his goal, we can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the final amount ($3000)
P = the initial amount ($2600)
e = the base of the natural logarithm (approximately 2.71828)
r = the annual interest rate (unknown)
t = the number of years (11 years)
Substituting the given values into the formula, we have:
3000 = 2600 * e^(11r)
Dividing both sides of the equation by 2600, we get:
3000/2600 = e^(11r)
Simplifying further, we have:
1.1538461538461538 = e^(11r)
To solve for r, we need to take the natural logarithm of both sides:
ln(1.1538461538461538) = ln(e^(11r))
ln(1.1538461538461538) = 11r * ln(e)
Using the fact that ln(e) = 1, we can simplify the equation to:
ln(1.1538461538461538) = 11r
Now, we can solve for r by dividing both sides by 11:
r = ln(1.1538461538461538) / 11
Calculating this using a calculator or online tool, we find that the rate of return needed for Larry to accomplish his goal is approximately 0.0343, or 3.43% (rounded to two decimal places).
To determine the annual rate of return Larry will need to achieve in order to accomplish his goal, you can use the continuous compound interest formula:
A = P * e^(rt)
Where:
A = the final amount after t years
P = the principal investment amount (initial amount), which is $2600 in this case
e = the mathematical constant equal to approximately 2.71828
r = the annual interest rate expressed as a decimal
t = the number of years
In this scenario, Larry needs to accumulate $3000 in 11 years. So, we can rearrange the formula to solve for r:
A = P * e^(rt)
3000 = 2600 * e^(11r)
Next, divide both sides of the equation by 2600:
3000/2600 = e^(11r)
Simplify the left side:
1.1538 ≈ e^(11r)
To isolate the exponent of e, you can take the natural logarithm (ln) of both sides:
ln(1.1538) ≈ ln(e^(11r))
Using the logarithmic property ln(e^x) = x:
ln(1.1538) ≈ 11r
Solve for r by dividing both sides by 11:
r ≈ ln(1.1538) / 11
Using a calculator, find the natural logarithm of 1.1538:
ln(1.1538) ≈ 0.1417
Divide 0.1417 by 11:
r ≈ 0.1417 / 11 ≈ 0.0129
Finally, convert the decimal to a percentage by multiplying by 100:
r ≈ 0.0129 * 100 ≈ 1.29%
Therefore, Larry will need an annual rate of return of approximately 1.29% in order to achieve his goal of $3000 in 11 years, assuming interest is compounded continuously.
let
2600(e^(11i) = 3000
e^(11i) = 1.153846
11i = ln(1.153846)
i = .013
a mere 1.3%