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Sagot :
To determine the present value of the given investment project's cash flows at different discount rates, we need to use the Present Value (PV) formula for each cash flow. The formula to calculate the present value of a future cash flow is given by:
[tex]\[ \text{PV} = \frac{\text{Cash Flow}}{(1 + \text{discount rate})^n} \][/tex]
where [tex]\( n \)[/tex] is the year in which the cash flow occurs.
Let's calculate the present value for each discount rate step-by-step.
### Part (a): Present Value at 10 Percent
Given:
- Cash Flows: [tex]\( \$1,270, \$1,210, \$1,580, \$1,940 \)[/tex]
- Discount Rate: 10%
Using the formula, we calculate each year's present value:
1. Year 1: [tex]\( \frac{1270}{(1 + 0.10)^1} = \frac{1270}{1.10} \approx 1154.55 \)[/tex]
2. Year 2: [tex]\( \frac{1210}{(1 + 0.10)^2} = \frac{1210}{1.21} \approx 1000.00 \)[/tex]
3. Year 3: [tex]\( \frac{1580}{(1 + 0.10)^3} = \frac{1580}{1.331} \approx 1187.97 \)[/tex]
4. Year 4: [tex]\( \frac{1940}{(1 + 0.10)^4} = \frac{1940}{1.4641} \approx 1324.15 \)[/tex]
Adding these individual present values gives:
[tex]\[ 1154.55 + 1000.00 + 1187.97 + 1324.15 \approx 4666.67 \][/tex]
### Part (b): Present Value at 18 Percent
Given:
- Cash Flows: [tex]\( \$1,270, \$1,210, \$1,580, \$1,940 \)[/tex]
- Discount Rate: 18%
Using the formula, we calculate each year's present value:
1. Year 1: [tex]\( \frac{1270}{(1 + 0.18)^1} = \frac{1270}{1.18} \approx 1076.27 \)[/tex]
2. Year 2: [tex]\( \frac{1210}{(1 + 0.18)^2} = \frac{1210}{1.3924} \approx 869.19 \)[/tex]
3. Year 3: [tex]\( \frac{1580}{(1 + 0.18)^3} = \frac{1580}{1.6427} \approx 961.87 \)[/tex]
4. Year 4: [tex]\( \frac{1940}{(1 + 0.18)^4} = \frac{1940}{1.9374} \approx 1000.21 \)[/tex]
Adding these individual present values gives:
[tex]\[ 1076.27 + 869.19 + 961.87 + 1000.21 \approx 3907.54 \][/tex]
### Part (c): Present Value at 24 Percent
Given:
- Cash Flows: [tex]\( \$1,270, \$1,210, \$1,580, \$1,940 \)[/tex]
- Discount Rate: 24%
Using the formula, we calculate each year's present value:
1. Year 1: [tex]\( \frac{1270}{(1 + 0.24)^1} = \frac{1270}{1.24} \approx 1024.19 \)[/tex]
2. Year 2: [tex]\( \frac{1210}{(1 + 0.24)^2} = \frac{1210}{1.5376} \approx 787.06 \)[/tex]
3. Year 3: [tex]\( \frac{1580}{(1 + 0.24)^3} = \frac{1580}{1.9076} \approx 828.32 \)[/tex]
4. Year 4: [tex]\( \frac{1940}{(1 + 0.24)^4} = \frac{1940}{2.3610} \approx 820.82 \)[/tex]
Adding these individual present values gives:
[tex]\[ 1024.19 + 787.06 + 828.32 + 820.82 \approx 3460.39 \][/tex]
### Summary
\begin{tabular}{|l|cr|}
\hline a. Present value at 10 percent & [tex]$\$[/tex][tex]$ & $[/tex]4,666.67[tex]$ \\ \hline b. Present value at 18 percent & $[/tex]\[tex]$ & $3,907.54$ \\ \hline c. Present value at 24 percent & $\$[/tex] & [tex]$3,460.39$[/tex] \\
\hline
\end{tabular}
[tex]\[ \text{PV} = \frac{\text{Cash Flow}}{(1 + \text{discount rate})^n} \][/tex]
where [tex]\( n \)[/tex] is the year in which the cash flow occurs.
Let's calculate the present value for each discount rate step-by-step.
### Part (a): Present Value at 10 Percent
Given:
- Cash Flows: [tex]\( \$1,270, \$1,210, \$1,580, \$1,940 \)[/tex]
- Discount Rate: 10%
Using the formula, we calculate each year's present value:
1. Year 1: [tex]\( \frac{1270}{(1 + 0.10)^1} = \frac{1270}{1.10} \approx 1154.55 \)[/tex]
2. Year 2: [tex]\( \frac{1210}{(1 + 0.10)^2} = \frac{1210}{1.21} \approx 1000.00 \)[/tex]
3. Year 3: [tex]\( \frac{1580}{(1 + 0.10)^3} = \frac{1580}{1.331} \approx 1187.97 \)[/tex]
4. Year 4: [tex]\( \frac{1940}{(1 + 0.10)^4} = \frac{1940}{1.4641} \approx 1324.15 \)[/tex]
Adding these individual present values gives:
[tex]\[ 1154.55 + 1000.00 + 1187.97 + 1324.15 \approx 4666.67 \][/tex]
### Part (b): Present Value at 18 Percent
Given:
- Cash Flows: [tex]\( \$1,270, \$1,210, \$1,580, \$1,940 \)[/tex]
- Discount Rate: 18%
Using the formula, we calculate each year's present value:
1. Year 1: [tex]\( \frac{1270}{(1 + 0.18)^1} = \frac{1270}{1.18} \approx 1076.27 \)[/tex]
2. Year 2: [tex]\( \frac{1210}{(1 + 0.18)^2} = \frac{1210}{1.3924} \approx 869.19 \)[/tex]
3. Year 3: [tex]\( \frac{1580}{(1 + 0.18)^3} = \frac{1580}{1.6427} \approx 961.87 \)[/tex]
4. Year 4: [tex]\( \frac{1940}{(1 + 0.18)^4} = \frac{1940}{1.9374} \approx 1000.21 \)[/tex]
Adding these individual present values gives:
[tex]\[ 1076.27 + 869.19 + 961.87 + 1000.21 \approx 3907.54 \][/tex]
### Part (c): Present Value at 24 Percent
Given:
- Cash Flows: [tex]\( \$1,270, \$1,210, \$1,580, \$1,940 \)[/tex]
- Discount Rate: 24%
Using the formula, we calculate each year's present value:
1. Year 1: [tex]\( \frac{1270}{(1 + 0.24)^1} = \frac{1270}{1.24} \approx 1024.19 \)[/tex]
2. Year 2: [tex]\( \frac{1210}{(1 + 0.24)^2} = \frac{1210}{1.5376} \approx 787.06 \)[/tex]
3. Year 3: [tex]\( \frac{1580}{(1 + 0.24)^3} = \frac{1580}{1.9076} \approx 828.32 \)[/tex]
4. Year 4: [tex]\( \frac{1940}{(1 + 0.24)^4} = \frac{1940}{2.3610} \approx 820.82 \)[/tex]
Adding these individual present values gives:
[tex]\[ 1024.19 + 787.06 + 828.32 + 820.82 \approx 3460.39 \][/tex]
### Summary
\begin{tabular}{|l|cr|}
\hline a. Present value at 10 percent & [tex]$\$[/tex][tex]$ & $[/tex]4,666.67[tex]$ \\ \hline b. Present value at 18 percent & $[/tex]\[tex]$ & $3,907.54$ \\ \hline c. Present value at 24 percent & $\$[/tex] & [tex]$3,460.39$[/tex] \\
\hline
\end{tabular}
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