I'm not sure how to approach this problem
What is the fair value of a 10-year annuity of $5,000 whose payment will start five years from now and last until fourteen years from now (10 payments), if the interest rate is 8%?
To find the fair value of a 10-year annuity, you can use the formula for the present value of an ordinary annuity. The formula is:
PV = PMT * (1 - (1 + r)^(-n)) / r
Where:
PV is the present value or fair value of the annuity
PMT is the payment per period ($5,000 in this case)
r is the interest rate per period (8% in this case)
n is the total number of periods (10 in this case)
In this problem, the payments start five years from now and last until fourteen years from now, which means there is a delay of five years before the annuity payments begin. So, we need to adjust the formula to account for this delay. We can use the future value of a single sum formula:
FV = PV * (1 + r)^n
Where:
FV is the future value of the annuity
PV is the present value or fair value of the annuity (what we are trying to find)
r is the interest rate per period (8% in this case)
n is the total number of periods (10 in this case)
Since we are trying to find the present value (PV) of the annuity, we rearrange the formula:
PV = FV / (1 + r)^n
Now, we can substitute the values into the formula:
FV = PMT * (1 - (1 + r)^(-n)) / r
FV = $5,000 * (1 - (1 + 0.08)^(-10)) / 0.08
Next, we need to calculate the future value (FV):
FV = $5,000 * (1 - (1.08)^(-10)) / 0.08
Finally, we can calculate the present value (PV):
PV = FV / (1 + r)^n
PV = [$5,000 * (1 - (1.08)^(-10)) / 0.08] / (1 + 0.08)^5
Calculating these values/results in:
FV = $35,429.89
PV = $21,403.11
Therefore, the fair value of the 10-year annuity is approximately $21,403.11.