The chief financial officer of a home health agency needs to determine the present value of a $10,000 investment received at the end of year 10. What is the present value if the discount rate is: a. 6 percent? b. 9 percent? c. 12 percent? d. 15 percent?
To determine the present value of a future investment, you can use the formula for present value (PV) of a single cash flow:
PV = CF / (1 + r)^n
Where:
PV = Present Value
CF = Cash Flow
r = Discount Rate
n = Time Periods
In this case, the cash flow (CF) is $10,000 received at the end of year 10. Let's calculate the present value for each discount rate:
a. Discount Rate = 6%
PV = $10,000 / (1 + 0.06)^10
b. Discount Rate = 9%
PV = $10,000 / (1 + 0.09)^10
c. Discount Rate = 12%
PV = $10,000 / (1 + 0.12)^10
d. Discount Rate = 15%
PV = $10,000 / (1 + 0.15)^10
Now, let's calculate the present value for each discount rate:
a. Discount Rate = 6%
PV = $10,000 / (1 + 0.06)^10
PV = $10,000 / (1.06)^10
PV ≈ $5,931.97
b. Discount Rate = 9%
PV = $10,000 / (1 + 0.09)^10
PV = $10,000 / (1.09)^10
PV ≈ $4,570.88
c. Discount Rate = 12%
PV = $10,000 / (1 + 0.12)^10
PV = $10,000 / (1.12)^10
PV ≈ $3,401.02
d. Discount Rate = 15%
PV = $10,000 / (1 + 0.15)^10
PV = $10,000 / (1.15)^10
PV ≈ $2,691.42
So, the present value for each discount rate is:
a. $5,931.97
b. $4,570.88
c. $3,401.02
d. $2,691.42
To determine the present value of the $10,000 investment at the end of year 10, we need to use the formula for present value:
Present Value = Future Value / (1 + Discount Rate)^N
Where:
- Future Value is the amount of money received in the future ($10,000 in this case).
- Discount Rate is the rate of return required by the investor (6%, 9%, 12%, or 15% in this case).
- N is the number of years in the future (10 years in this case).
Let's calculate the present value for each discount rate one by one:
a. Discount Rate = 6%
Present Value = $10,000 / (1 + 0.06)^10
b. Discount Rate = 9%
Present Value = $10,000 / (1 + 0.09)^10
c. Discount Rate = 12%
Present Value = $10,000 / (1 + 0.12)^10
d. Discount Rate = 15%
Present Value = $10,000 / (1 + 0.15)^10
Now, we can calculate the present value for each discount rate:
a. Discount Rate = 6%
Present Value = $10,000 / (1 + 0.06)^10 = $6,209.08
b. Discount Rate = 9%
Present Value = $10,000 / (1 + 0.09)^10 = $4,258.97
c. Discount Rate = 12%
Present Value = $10,000 / (1 + 0.12)^10 = $3,105.30
d. Discount Rate = 15%
Present Value = $10,000 / (1 + 0.15)^10 = $2,270.68
Therefore, the present value of the $10,000 investment received at the end of year 10 is:
- $6,209.08 if the discount rate is 6%,
- $4,258.97 if the discount rate is 9%,
- $3,105.30 if the discount rate is 12%,
- $2,270.68 if the discount rate is 15%.