An investment pays you $20,000 at the end of this year, and $10,000 at the end of each of the four following years. What is the present value (PV) of this investment, given that the interest rate is 4% per year?
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PV = 20000(1.04)^-1 + 10000(1.04^-2 + 1.04^-3 + 1.04^-4 + 1.04^-5)
= ...
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To calculate the present value (PV) of the investment, we need to discount each future cash flow to its present value and then sum them up.
Step 1: Calculate the present value of each cash flow using the formula: PV = CF / (1 + r)^n, where CF is the cash flow, r is the interest rate, and n is the number of years.
PV of $20,000 at the end of this year:
PV1 = $20,000 / (1 + 0.04)^1 = $20,000 / 1.04 = $19,230.77
PV of $10,000 at the end of each of the four following years:
PV2 = $10,000 / (1 + 0.04)^2 = $10,000 / 1.0816 = $9,236.76 (for year 2)
PV3 = $10,000 / (1 + 0.04)^3 = $10,000 / 1.1259 = $8,888.47 (for year 3)
PV4 = $10,000 / (1 + 0.04)^4 = $10,000 / 1.1699 = $8,555.52 (for year 4)
PV5 = $10,000 / (1 + 0.04)^5 = $10,000 / 1.2167 = $8,248.46 (for year 5)
Step 2: Sum up the present values of all cash flows:
PV = PV1 + PV2 + PV3 + PV4 + PV5
= $19,230.77 + $9,236.76 + $8,888.47 + $8,555.52 + $8,248.46
= $54,159.98 (rounded to two decimal places)
Therefore, the present value (PV) of this investment, given an interest rate of 4% per year, is approximately $54,159.98.
To calculate the present value (PV) of this investment, we need to discount each future cash flow to its present value. The present value is the current worth of a future sum of money, taking into account the time value of money.
In this case, the investment pays $20,000 at the end of this year (Year 0), and $10,000 at the end of each of the next four years (Year 1 to Year 4). The interest rate is given as 4% per year.
To calculate the present value, you can use the formula for the present value of an annuity:
PV = C * (1 - (1+r)^(-n)) / r
Where:
PV = Present Value
C = Cash flow (annual payment)
r = Interest rate
n = Number of periods
Let's calculate the present value step by step:
1. Calculate the present value of the $20,000 cash flow at the end of Year 0:
PV_Year0 = $20,000 / (1 + 0.04)^0 = $20,000
2. Calculate the present value of the $10,000 cash flow at the end of Year 1:
PV_Year1 = $10,000 / (1 + 0.04)^1 = $10,000 / 1.04 = $9,615.38 (rounded)
3. Calculate the present value of the $10,000 cash flow at the end of Year 2:
PV_Year2 = $10,000 / (1 + 0.04)^2 = $10,000 / 1.0816 = $9,254.90 (rounded)
4. Calculate the present value of the $10,000 cash flow at the end of Year 3:
PV_Year3 = $10,000 / (1 + 0.04)^3 = $10,000 / 1.1249 = $8,897.11 (rounded)
5. Calculate the present value of the $10,000 cash flow at the end of Year 4:
PV_Year4 = $10,000 / (1 + 0.04)^4 = $10,000 / 1.1699 = $8,544.14 (rounded)
6. Finally, calculate the total present value by summing up all the present values across all years:
PV_total = PV_Year0 + PV_Year1 + PV_Year2 + PV_Year3 + PV_Year4
PV_total = $20,000 + $9,615.38 + $9,254.90 + $8,897.11 + $8,544.14 = $56,311.53 (rounded)
Therefore, the present value (PV) of this investment, given an interest rate of 4% per year, is approximately $56,311.53.