Question
A new machine requires an investment of $630,000 and will generate $100,000 in cash inflows for 7 years, at which time the salvage value of the machine will be $130,000. Using a discount rate of 10%, the net present value of the machine (when rounded to nearest whole dollar) is closest to: (Base your answer on the tables in the appendix).
Multiple choice question.
$70,000
($70,000)
($76,610)
$76,610
Multiple choice question.
$70,000
($70,000)
($76,610)
$76,610
Answers
GPT-4o mini
To calculate the net present value (NPV) of the machine, we must consider the initial investment, the cash inflows, and the salvage value discounted at the given rate (10%). Here’s a step-by-step calculation:
1. **Initial Investment**:
\[
\text{Initial Cash Outflow} = -630,000
\]
2. **Annual Cash Inflows**:
The machine generates $100,000 per year for 7 years. We will first calculate the present value of these cash inflows:
We can use the Present Value of Annuity formula:
\[
PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- \(C = 100,000\) (annual cash inflow)
- \(r = 0.10\) (discount rate)
- \(n = 7\) (number of years)
Plugging in the values:
\[
PV = 100,000 \times \left( \frac{1 - (1 + 0.10)^{-7}}{0.10} \right)
\]
Now calculate:
\[
PV = 100,000 \times \left( \frac{1 - (1.10)^{-7}}{0.10} \right)
\]
First, calculate \((1.10)^{-7}\):
\[
(1.10)^{-7} \approx 0.5132
\]
Then:
\[
\frac{1 - 0.5132}{0.10} \approx \frac{0.4868}{0.10} = 4.868
\]
Therefore, we have:
\[
PV \approx 100,000 \times 4.868 \approx 486,800
\]
3. **Salvage Value**:
The salvage value of $130,000 needs to be discounted to present value. The formula for calculating the present value of a single sum is:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where \(FV = 130,000\):
\[
PV = \frac{130,000}{(1 + 0.10)^{7}} = \frac{130,000}{(1.10)^{7}}
\]
First, calculate \((1.10)^{7}\):
\[
(1.10)^{7} \approx 1.9487
\]
Then:
\[
PV \approx \frac{130,000}{1.9487} \approx 66,743
\]
4. **Calculate Total NPV**:
Now combine the present values:
\[
NPV = \text{PV of Cash Inflows} + \text{PV of Salvage Value} + \text{Initial Investment}
\]
\[
NPV = 486,800 + 66,743 - 630,000
\]
\[
NPV \approx 486,800 + 66,743 - 630,000 \approx -76,457
\]
Now, rounding this to the nearest whole dollar, the NPV is approximately \(-76,457\), which is closest to \((-76,610)\).
Thus, the answer is: **($76,610)**.
1. **Initial Investment**:
\[
\text{Initial Cash Outflow} = -630,000
\]
2. **Annual Cash Inflows**:
The machine generates $100,000 per year for 7 years. We will first calculate the present value of these cash inflows:
We can use the Present Value of Annuity formula:
\[
PV = C \times \left( \frac{1 - (1 + r)^{-n}}{r} \right)
\]
Where:
- \(C = 100,000\) (annual cash inflow)
- \(r = 0.10\) (discount rate)
- \(n = 7\) (number of years)
Plugging in the values:
\[
PV = 100,000 \times \left( \frac{1 - (1 + 0.10)^{-7}}{0.10} \right)
\]
Now calculate:
\[
PV = 100,000 \times \left( \frac{1 - (1.10)^{-7}}{0.10} \right)
\]
First, calculate \((1.10)^{-7}\):
\[
(1.10)^{-7} \approx 0.5132
\]
Then:
\[
\frac{1 - 0.5132}{0.10} \approx \frac{0.4868}{0.10} = 4.868
\]
Therefore, we have:
\[
PV \approx 100,000 \times 4.868 \approx 486,800
\]
3. **Salvage Value**:
The salvage value of $130,000 needs to be discounted to present value. The formula for calculating the present value of a single sum is:
\[
PV = \frac{FV}{(1 + r)^n}
\]
Where \(FV = 130,000\):
\[
PV = \frac{130,000}{(1 + 0.10)^{7}} = \frac{130,000}{(1.10)^{7}}
\]
First, calculate \((1.10)^{7}\):
\[
(1.10)^{7} \approx 1.9487
\]
Then:
\[
PV \approx \frac{130,000}{1.9487} \approx 66,743
\]
4. **Calculate Total NPV**:
Now combine the present values:
\[
NPV = \text{PV of Cash Inflows} + \text{PV of Salvage Value} + \text{Initial Investment}
\]
\[
NPV = 486,800 + 66,743 - 630,000
\]
\[
NPV \approx 486,800 + 66,743 - 630,000 \approx -76,457
\]
Now, rounding this to the nearest whole dollar, the NPV is approximately \(-76,457\), which is closest to \((-76,610)\).
Thus, the answer is: **($76,610)**.