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SUB :- finance
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TOPIC :-

Step -1
introduction with a concept about 50 words in non mathmatical then provide an explanation: about the in 2-3 lines.

Step:-2
li

Question

Tagging
SUB :- finance
SUB SUBJECT:-
TOPIC :-



Step -1 
introduction with a concept about 50 words in non mathmatical then provide an explanation: about the in 2-3 lines. 



Step:-2
list given :-

give FORMULAS ::- with where

Give a step-by-step solution with calculation, don't miss out intermediate calculations, give the correct calculations using the formula with step by step approach.



Give an explanation just after every step in the last 1-2 lines 

Give the final answer.     



Don't give any personal pronouns and give answer step by step with all calculations with any calculations missiingA new furnace for your small factory will cost $34,000 and a year to install, will require ongoing maintenance expenditures of $2,200 a
year. Butit is far more fuel-efficient than your old furnace and will reduce your consumption of heating oil by 3100 gallons per year.
Heating oll this year will cost $3 a gallon; the price per gallon is expected to increase by $0.50 a year for the next 3 years and then to
stabilize for the foreseeable future. The furnace will last for 20 years, at which point it will need to be replaced and will have no
salvage value. T

VIPSolutions · Accepted Answer

### Step 1: **Introduction (Non-mathematical, ≈50 words)**

Capital budgeting decisions help businesses evaluate the long-term value of large investments, such as new machinery or equipment, by comparing costs and future savings. Techniques like Net Present Value (NPV), Internal Rate of Return (IRR), and payback period are used to determine if an investment is worthwhile by analyzing expected cash flows and costs over time.

*Explanation: These methods ensure resources are invested in projects that maximize returns and minimize risks.*

---

### Step 2: **Solution**

#### **Given Data**
- Initial cost of furnace = $34,000  
- Annual maintenance = $2,200  
- Annual oil savings = 3,100 gallons × [Oil price per gallon]  
- Oil price this year = $3/gallon  
- Oil price increases by $0.50/gallon for next 3 years, then stabilizes  
- Furnace life = 20 years  
- Discount rate = 12%  
- No salvage value

---

#### **a. Net Present Value (NPV)**

**Formula:**  
$$
NPV = \sum \frac{C_t}{(1 + r)^t} - C_0
$$
Where:  
- $ C_t $ = Net cash flow at year $ t $  
- $ r $ = Discount rate (12%)  
- $ C_0 $ = Initial investment ($34,000)

---

**Step 1: Calculate annual oil savings (Years 1-3):**
- Year 1: $3.00/gallon × 3,100 = $9,300  
- Year 2: $3.50/gallon × 3,100 = $10,850  
- Year 3: $4.00/gallon × 3,100 = $12,400  
- Year 4-20: $4.00/gallon × 3,100 = $12,400 each year

*Explanation: The oil price increases for three years, then stays constant. Multiply each year's price by annual gallons saved.*

---

**Step 2: Calculate net annual cash flows (Savings - Maintenance):**
- Year 1: $9,300 - $2,200 = $7,100  
- Year 2: $10,850 - $2,200 = $8,650  
- Year 3: $12,400 - $2,200 = $10,200  
- Year 4-20: $12,400 - $2,200 = $10,200 each year

*Explanation: Subtract annual maintenance from savings for net cash flow.*

---

**Step 3: Calculate Present Value (PV) of each cash flow:**

$$
PV = \frac{C_t}{(1 + r)^t}
$$

- PV Year 1: $7,100 / (1.12)^1 = $6,339.29  
- PV Year 2: $8,650 / (1.12)^2 = $6,899.31  
- PV Year 3: $10,200 / (1.12)^3 = $7,266.91

For Years 4–20 (17 years), use the Present Value of an Annuity formula:  
$$
PV_{4-20} = PMT \times \frac{1 - (1 + r)^{-n}}{r}
$$
where PMT = $10,200, r = 12\%, n = 17  
$$
PV_{4-20} = 10,200 \times \frac{1 - (1.12)^{-17}}{0.12}
$$
First, calculate $(1.12)^{-17} \approx 0.1534$  
$$
PV_{4-20} = 10,200 \times \frac{1 - 0.1534}{0.12}
$$
$$
PV_{4-20} = 10,200 \times \frac{0.8466}{0.12}
$$
$$
PV_{4-20} = 10,200 \times 7.055
$$
$$
PV_{4-20} = 71,961
$$
This is the present value at year 3; discount to present (year 0):  
$$
PV_{4-20,0} = \frac{71,961}{(1.12)^3} = \frac{71,961}{1.404928} = 51,260.54
$$

*Explanation: Discount future cash flows back to present using the discount rate.*

---

**Step 4: Calculate NPV**
$$
NPV = -34,000 + 6,339.29 + 6,899.31 + 7,266.91 + 51,260.54
$$
$$
NPV = -34,000 + 71,766.05 = \$37,766
$$

**Final Answer (a):**  
**Net Present Value = $37,766** (rounded to nearest dollar)

---

#### **b. Internal Rate of Return (IRR)**

**Formula:**  
$$
0 = -C_0 + \frac{C_1}{(1+IRR)^1} + \frac{C_2}{(1+IRR)^2} + ... + \frac{C_n}{(1+IRR)^n}
$$

Trial and error or using a financial calculator is typical, but a close estimate can be made since NPV is quite high at 12%. Try higher rates (e.g., 25%) to see if NPV approaches zero.

- If NPV is positive at 12%, try 25%:

- Year 1: $7,100 / 1.25 = $5,680  
- Year 2: $8,650 / (1.25)^2 = $5,528  
- Year 3: $10,200 / (1.25)^3 = $5,241  
- PV Years 4-20 at 25%:  
$$
PV_{4-20} = 10,200 \times \frac{1 - (1.25)^{-17}}{0.25} = 10,200 \times \frac{1 - 0.0702}{0.25} = 10,200 \times \frac{0.9298}{0.25} = 10,200 \times 3.719
$$
$$
= 37,943
$$
Discount to present (year 0):
$$
PV_{4-20,0} = \frac{37,943}{(1.25)^3} = \frac{37,943}{1.953125} = 19,429
$$

NPV at 25%:  
$$
NPV = -34,000 + 5,680 + 5,528 + 5,241 + 19,429 = 1,878
$$

Try 30%:
- Year 1: $7,100 / 1.3 = $5,462  
- Year 2: $8,650 / (1.3)^2 = $5,115  
- Year 3: $10,200 / (1.3)^3 = $4,610  
- PV Years 4-20 at 30%:  
$$
PV_{4-20} = 10,200 \times \frac{1 - (1.3)^{-17}}{0.30}
$$
$(1.3)^{-17} \approx 0.0296$
$$
= 10,200 \times \frac{0.9704}{0.30} = 10,200 \times 3.235 = 33,000
$$
Discount back:
$$
PV_{4-20,0} = \frac{33,000}{(1.3)^3} = \frac{33,000}{2.197} = 15,028
$$

NPV at 30%:  
$$
NPV = -34,000 + 5,462 + 5,115 + 4,610 + 15,028 = -3,785
$$

Interpolate IRR between 25% (NPV = $1,878) and 30% (NPV = -$3,785):

$$
IRR = 25\% + \frac{1,878}{1,878 + 3,785} \times (30\% - 25\%)
$$
$$
= 25\% + \frac{1,878}{5,663} \times 5\%
$$
$$
= 25\% + 0.332 \times 5\% = 25\% + 1.66\% = 26.66\%
$$

**Final Answer (b):**  
**IRR = 26.66%**

---

#### **c. Payback Period**

Cumulative net cash flows:
- Year 1: $7,100  
- Year 2: $8,650 (cumulative: $15,750)  
- Year 3: $10,200 (cumulative: $25,950)  
- Year 4: $10,200 (cumulative: $36,150)

$34,000 is recovered between years 3 and 4.

$$
\text{Additional needed after 3 years: } 34,000 - 25,950 = 8,050
$$
\[
\text{Fraction of year: } \frac{8,050}{1

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