4. A firm's net income last year was $1.5 million. Its net income grew 5 percent during the last 5 years. If that growth rate continues, how long will it take for the firm's net income to double?
Growth rate = 5%/5yrs. = 1%/yr.
A = Po + Po*r*t = 3 Million.
1.5 + 1.5*0.01*t = 3
1.5 + 0.015t = 3
0.015t = 1.5
t = 100 Years.
To determine how long it will take for the firm's net income to double, we need to calculate the annual growth rate. We can then use this growth rate to find the number of years required for the net income to double.
Step 1: Calculate the annual growth rate
The net income increased by 5 percent each year for the last 5 years. To find the growth rate, we can use the formula:
Growth Rate = [(Final Value / Initial Value)^(1/N) - 1] * 100
Where:
- Final Value = Net income at the end of the 5-year period = $1.5 million + 5 percent
- Initial Value = Net income at the start of the 5-year period = $1.5 million
- N = Number of years = 5
Using the formula, we can calculate the growth rate:
Growth Rate = [(1.5 million * 1.05)^(1/5) - 1] * 100
= [(1.575 million)^(1/5) - 1] * 100
Step 2: Calculate the number of years required for the net income to double
To find the number of years required for the net income to double, we can use the compound interest formula:
Final Value = Initial Value * (1 + Growth Rate/100)^N
Where:
- Initial Value = Net income at the starting point = $1.5 million
- Final Value = Net income at the doubling point = $1.5 million * 2
- Growth Rate = Calculated in step 1
We need to solve for N, the number of years.
$1.5 million * 2 = $1.5 million * (1 + Growth Rate/100)^N
Divide both sides by $1.5 million:
2 = (1 + Growth Rate/100)^N
Take the logarithm of both sides to solve for N:
log(2) = log[(1 + Growth Rate/100)^N]
Using logarithm properties, we can rewrite the equation as:
N = log(2) / log(1 + Growth Rate/100)
Finally, substitute the calculated growth rate into the equation and solve for N:
N = log(2) / log(1 + [(1.575 million)^(1/5) - 1] * 100)
By calculating the value of N, we can find out how long it will take for the firm's net income to double based on its historical growth rate.