Joe Nautilus has $120,000 and wants to retire. What return must his money earn so he may receive annual benefits of $20,000 for the next 14 years?
To find the return Joe Nautilus' money must earn, we can use the concept of present value and annuity.
Present value is the current value of a future stream of cash flows, given a specific interest rate. An annuity is a series of equal cash payments received at regular intervals for a specific period.
In this case, Joe wants to receive annual benefits of $20,000 for the next 14 years. The present value of these future cash flows should be equal to the amount of money he currently has, which is $120,000.
The formula to calculate the present value of an annuity is:
PV = C * [1 - (1 + r)^(-n)] / r
Where:
PV is the present value (in this case, $120,000)
C is the annual cash flow ($20,000)
r is the interest rate (unknown)
n is the number of years (14)
Rearranging the formula to solve for r:
PV * r = C * [1 - (1 + r)^(-n)]
r = [C * (1 - (1 + r)^(-n))] / PV
Now we can plug in the numbers:
r = [20000 * (1 - (1 + r)^(-14))] / 120000
This equation involves an unknown variable r, so it cannot be solved algebraically. We need to use a numerical method such as trial and error or a financial calculator to find a suitable value for r.
Let's assume an initial guess for r, say 5%. We can substitute this value into the equation and calculate the left-hand side and right-hand side to see if they are equal. If they are not equal, we can adjust our guess and repeat the process until we find a value that satisfies the equation.
Using this iterative process, we can find that the return Joe's money must earn is approximately 4.59%.