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?
simply solve
1(1.05)^t = 2
t log 1.05 = log 2
t = log2/log1.05 = appr 14.2 years
To determine how long it will take for the firm's net income to double, we need to calculate the annual growth rate required for the net income to double.
First, we need to find out the target net income for the firm to double, which would be twice its current net income:
Target Net Income = 2 * Current Net Income
Target Net Income = 2 * $1.5 million
Target Net Income = $3 million
Next, we will use the formula for compound interest to calculate the growth rate:
Future Value = Present Value * (1 + Growth Rate)^Time
In this case, the present value is the current net income, the future value is the target net income, and the time is the number of years required for the net income to double.
Now we can rearrange the formula to solve for the growth rate (r):
Growth Rate = (Future Value / Present Value)^(1 / Time) - 1
Plugging in the values we know:
Future Value = $3 million
Present Value = $1.5 million
Time = unknown (to be calculated)
Growth Rate = ($3 million / $1.5 million)^(1 / Time) - 1
To solve for Time, we need to isolate it by taking the logarithm of both sides:
log(Growth Rate + 1) = (1 / Time) * log(2)
Rearranging and solving for Time:
Time = log(2) / log(Growth Rate + 1)
Now we can substitute the given growth rate of 5% into the formula:
Growth Rate = 0.05
Time = log(2) / log(0.05 + 1)
Calculating Time using a calculator:
Time = log(2) / log(1.05)
Time ≈ 14.21
Therefore, it will take approximately 14.21 years for the firm's net income to double if the growth rate continues at 5% annually.