would you purchase a $100,000 20 year annuity today for $15,000 given a required rate of return of 14.5%? Why, what is the value of the annuity?
To determine whether you should purchase a $100,000 20-year annuity today for $15,000, we need to calculate the present value of the annuity at the given required rate of return of 14.5%.
The present value of an annuity can be calculated using the following formula:
PV = CF x (1 - (1 + r)^(-n)) / r
Where:
PV = Present Value of the annuity
CF = Cash flow per period (in this case, $100,000)
r = Required rate of return (in decimal form, so 14.5% becomes 0.145)
n = Number of periods (in this case, 20 years)
Plugging in the values:
PV = $100,000 x (1 - (1 + 0.145)^(-20)) / 0.145
Now, let's solve this equation to find the present value of the annuity:
PV = $100,000 x (1 - (1.145)^(-20)) / 0.145
= $100,000 x (1 - 0.1646) / 0.145
= $100,000 x (0.8354) / 0.145
= $83,540 / 0.145
≈ $576,055.17
Therefore, the present value of the annuity is approximately $576,055.17.
Since the value of the annuity is greater than the purchase price of $15,000, it would be advantageous to purchase the annuity since you would be paying less than the present value of the cash flows.