Investment X offers to pay you $6,000 per year for nine years, whereas Investment Y offers to pay you $8,000 per year for six years. If the discount rate is 5 percent, Investment X has a present value of ? and Investment Y has a present value of ?. If the discount rate is 15 percent, Investment X has a present value of ? and Investment Y has a present value of ?
Assume a $4,000 investment and the following cash flows for two alternatives.
Year InvestmentX Investment Y
1 $1,000 $1,300
2 800 2,800
3 700 100
4 1,900
5 2,000
a. Under the payback method, which investment should be chosen? (Show your work/analysis/calculations for each investment).
b. Why do other methods allow for a better analysis?
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To calculate the present value of an investment, we use the formula:
Present Value = Cash Flow / (1 + Discount Rate)^n
Where,
Cash Flow = annual payment
Discount Rate = interest rate
n = number of years
Let's calculate the present values for both investments:
For Investment X with a discount rate of 5 percent:
Cash Flow = $6,000 per year
n = 9 years
Present Value (Investment X) = $6,000 / (1 + 0.05)^9
Present Value (Investment X) ≈ $6,000 / (1.05)^9
Present Value (Investment X) ≈ $6,000 / 1.54509
Present Value (Investment X) ≈ $3,881.85
For Investment Y with a discount rate of 5 percent:
Cash Flow = $8,000 per year
n = 6 years
Present Value (Investment Y) = $8,000 / (1 + 0.05)^6
Present Value (Investment Y) ≈ $8,000 / (1.05)^6
Present Value (Investment Y) ≈ $8,000 / 1.34010
Present Value (Investment Y) ≈ $5,970.14
For Investment X with a discount rate of 15 percent:
Cash Flow = $6,000 per year
n = 9 years
Present Value (Investment X) = $6,000 / (1 + 0.15)^9
Present Value (Investment X) ≈ $6,000 / (1.15)^9
Present Value (Investment X) ≈ $6,000 / 2.49949
Present Value (Investment X) ≈ $2,400.24
For Investment Y with a discount rate of 15 percent:
Cash Flow = $8,000 per year
n = 6 years
Present Value (Investment Y) = $8,000 / (1 + 0.15)^6
Present Value (Investment Y) ≈ $8,000 / (1.15)^6
Present Value (Investment Y) ≈ $8,000 / 1.98548
Present Value (Investment Y) ≈ $4,031.46
Therefore, at a discount rate of 5 percent, Investment X has a present value of approximately $3,881.85, and Investment Y has a present value of approximately $5,970.14.
At a discount rate of 15 percent, Investment X has a present value of approximately $2,400.24, and Investment Y has a present value of approximately $4,031.46.
To find the present value (PV) of an investment, you need to discount the future cash flows using the discount rate. The present value formula is:
PV = CF / (1+r)^n
where PV is the present value, CF is the cash flow, r is the discount rate, and n is the number of years.
Let's calculate the present values for each investment at a 5 percent discount rate:
For Investment X:
CF = $6,000 per year
n = 9 years
r = 5% or 0.05
Using the formula:
PV of Investment X = $6,000 / (1+0.05)^9 = $6,000 / 1.551 = $3,868.28
Now let's calculate the present value for Investment Y:
CF = $8,000 per year
n = 6 years
r = 5% or 0.05
Using the formula:
PV of Investment Y = $8,000 / (1+0.05)^6 = $8,000 / 1.348 = $5,930.69
Now let's calculate the present values for each investment at a 15 percent discount rate:
For Investment X:
CF = $6,000 per year
n = 9 years
r = 15% or 0.15
Using the formula:
PV of Investment X = $6,000 / (1+0.15)^9 = $6,000 / 3.172 = $1,891.32
Now let's calculate the present value for Investment Y:
CF = $8,000 per year
n = 6 years
r = 15% or 0.15
Using the formula:
PV of Investment Y = $8,000 / (1+0.15)^6 = $8,000 / 1.718 = $4,656.97
Therefore, the present values for each investment are:
At a 5 percent discount rate:
- Investment X has a present value of $3,868.28
- Investment Y has a present value of $5,930.69
At a 15 percent discount rate:
- Investment X has a present value of $1,891.32
- Investment Y has a present value of $4,656.97