Determine the sum of money that must be invested today at 9% interest compounded annually to give an investor annuity (annual income) payments of $5,000 per year for 10 years starting 5 years from now.
To determine the necessary investment, we need to calculate the present value of an annuity. The present value is the current value of a future stream of cash flows.
In this case, we need to find the present value of the annuity payments. The formula for calculating the present value of an annuity is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value
PMT = Payment per period
r = Interest rate per period
n = Number of periods
Let's substitute the given values:
PMT = $5,000 (annual payment)
r = 9% = 0.09 (annual interest rate)
n = 10 (number of periods)
To calculate the present value, we also need to adjust the time period to account for the delay of 5 years. We subtract 5 from the number of periods, so n = 10 - 5 = 5.
Now, let's plug these values into the equation:
PV = $5,000 * (1 - (1 + 0.09)^(-5)) / 0.09
Using a calculator, we can evaluate this:
PV ≈ $19,533.03
Therefore, approximately $19,533.03 must be invested today to provide the investor with annual income payments of $5,000 per year for 10 years, starting 5 years from now.