Final answer:
To calculate the present value of receiving $10,000 annually for 10 years at a 5% interest rate, we use the formula for the present value of an annuity. The result is $77,218, which is the value of the future cash flows in today's dollars.
Step-by-step explanation:
To determine the present value of $10,000 received annually for 10 years at an interest rate of 5%, we use the formula for the present value of an annuity. This calculation involves discounting each payment back to its present value at the given interest rate and then summing these values.
The formula for the present value of an annuity is as follows:
PV = Pmt × [(1 - (1 + r)^{-n}) / r]
Where:
- PV is the present value of the annuity.
- Pmt is the annual payment ($10,000).
- r is the annual interest rate (5% or 0.05).
- n is the total number of payments (10).
Plugging in the values, we get:
PV = $10,000 × [(1 - (1 + 0.05)^{-10}) / 0.05]
After calculating, the present value of the annuity is obtained:
PV = $10,000 × [(1 - (1.05)^{-10}) / 0.05]
PV = $10,000 × [(1 - 0.61391) / 0.05]
PV = $10,000 × [0.38609 / 0.05]
PV = $10,000 × 7.7218
PV = $77,218
So the present value of receiving $10,000 annually for 10 years at a 5% interest rate today would be $77,218.