Give answer in two steps and explanation when required and final answer
A project requires an initial investment of $
1
0
0
,
0
0
0
and is expected to produce a cash inflow before tax of $
2
6
,
1
0

Question

Give answer in two steps and explanation when required and final answer
A project requires an initial investment of $
1
0
0
,
0
0
0
 ﻿and is expected to produce a cash inflow before tax of $
2
6
,
1
0
0
 ﻿per year for five years. Company A has substantial accumulated tax losses and is unlikely to pay taxes in the foreseeable future. Company B pays corporate taxes at a rate of 
2
1
%
 ﻿and can claim 
1
0
0
%
 ﻿bonus depreciation on the investment. Suppose the opportunity cost of capital is 
8
%
.
 ﻿Ignore inflation. Required: Calculate project NPV for each company. What is the IRR of the post
-
tax cash flows for each company?

VIPSolutions · Accepted Answer

Let's proceed step by step as requested.

---

## Step 1: Calculate NPV for Each Company

### **Company A (No Taxes Due to Losses)**
- **Initial investment:** $100,000 (outflow)
- **Annual pre-tax cash inflow:** $26,100 (no taxes, so after-tax = $26,100)
- **Life:** 5 years
- **Discount rate (r):** 8%

**NPV formula:**
$$
NPV = \sum_{t=1}^{5} \frac{26,100}{(1.08)^t} - 100,000
$$

Calculate present value of inflows:
$$
PV = 26,100 \times \left[\frac{1 - (1 + r)^{-n}}{r}\right]
$$
$$
PV = 26,100 \times \left[\frac{1 - (1.08)^{-5}}{0.08}\right]
$$

First, compute $(1.08)^{-5}$:
$$
(1.08)^{-5} = 1 / (1.08^5) \approx 1 / 1.4693 \approx 0.6806
$$

So,
$$
\left[\frac{1 - 0.6806}{0.08}\right] = \frac{0.3194}{0.08} = 3.9925
$$
$$
PV = 26,100 \times 3.9925 = 104,225.25
$$

$$
NPV_A = 104,225.25 - 100,000 = \boxed{4,225.25}
$$

---

### **Company B (21% Tax Rate, 100% Bonus Depreciation)**
- **Initial investment:** $100,000
- **Annual pre-tax cash inflow:** $26,100
- **Tax rate:** 21%
- **Bonus depreciation:** $100,000 claimed in year 1

**Year 1:**
- **Taxable income:** $26,100 (cash inflow) - $100,000 (depreciation) = **$-$73,900$**
- **Tax shield in year 1:** $-$73,900 \times 21\% = -$15,519$ (tax saving, i.e., refund/offset)
- **Cash flow in year 1:** $26,100 (inflow) + $21,000 (depreciation shield) = $47,100

But let's clarify:
- The **total depreciation tax shield** in year 1 is $100,000 × 21% = **$21,000**.
- The taxable income is negative, so all tax shield is realized in year 1.
- **After-tax cash flow each year:**
    - Year 1: $26,100 - 0.21 × ($26,100 - $100,000) = $26,100 - 0.21 × (-$73,900) = $26,100 + $15,519 = $41,619$
    - Years 2-5: No depreciation, so after-tax cash flow is $26,100 × (1 - 0.21) = $20,619$

**So, cash flows:**
- Year 0: -$100,000
- Year 1: $41,619
- Years 2-5: $20,619 each

**Now, NPV:**
$$
NPV_B = \frac{41,619}{1.08} + \frac{20,619}{(1.08)^2} + \frac{20,619}{(1.08)^3} + \frac{20,619}{(1.08)^4} + \frac{20,619}{(1.08)^5} - 100,000
$$

Let's calculate each term:
- Year 1: $41,619 / 1.08 = 38,537.04$
- Year 2: $20,619 / (1.08)^2 = 20,619 / 1.1664 = 17,685.08$
- Year 3: $20,619 / (1.08)^3 = 20,619 / 1.2597 = 16,373.26$
- Year 4: $20,619 / (1.08)^4 = 20,619 / 1.3605 = 15,162.33$
- Year 5: $20,619 / (1.08)^5 = 20,619 / 1.4693 = 14,040.62$

Sum of present values:
$$
NPV_B = (38,537.04 + 17,685.08 + 16,373.26 + 15,162.33 + 14,040.62) - 100,000 = 101,798.33 - 100,000 = \boxed{1,798.33}
$$

---

## Step 2: IRR of the Post-Tax Cash Flows

### **Company A:**
Cash flows:
- Year 0: -100,000
- Year 1-5: 26,100 each

IRR is the discount rate that makes NPV = 0:
$$
0 = -100,000 + 26,100 \times \left[\frac{1-(1+IRR)^{-5}}{IRR}\right]
$$

You can use a financial calculator or Excel:
- In Excel: `=IRR([-100000, 26100, 26100, 26100, 26100, 26100])`

Let’s estimate manually:
- $26,100 \times 5 = 130,500$ (undiscounted), so IRR will be **greater than 8%**.
- Try 12%:
    - $26,100 \times [(1-(1.12)^{-5})/0.12] = 26,100 \times [1 - 0.5674)/0.12] = 26,100 \times 3.6118 = 94,268$ (less than $100,000$)
- Try 10%:
    - $26,100 \times [(1-(1.10)^{-5})/0.10] = 26,100 \times 3.7908 = 98,909$
- Try 9%:
    - $26,100 \times 3.8897 = 101,593$ (more than $100,000$)

Interpolate between 9% and 10%:
$$
IRR = 9\% + \frac{100,000 - 98,909}{101,593 - 98,909} \times (10\% - 9\%)
$$
$$
= 9\% + \frac{1,091}{2,684} \approx 9\% + 0.41\% = 9.41\%
$$

**So, IRR ≈ 9.4%**

---

### **Company B:**
Cash flows:
- Year 0: -100,000
- Year 1: 41,619
- Years 2-5: 20,619 each

In Excel: `=IRR([-100000, 41619, 20619, 20619, 20619, 20619])`

Estimate manually:
- The first year is high, so IRR will be above 8%.

Try 9%:
- Year 1: $41,619 / 1.09 = 38,193$
- Year 2: $20,619 / 1.1881 = 17,358$
- Year 3: $20,619 / 1.2950 = 15,927$
- Year 4: $20,619 / 1.4116 = 14,613$
- Year 5: $20,619 / 1.5386 = 13,400$
- Total PV = $38,193 + 17,358 + 15,927 + 14,613 + 13,400 = 99,491$

So, NPV is slightly negative.

Try 8.5%:
- Year 1: $41,619 / 1.085 = 38,368$
- Year 2: $20,619 / 1.176 = 17,541$
- Year 3: $20,619 / 1.275 = 16,175$
- Year 4: $20,619 / 1.381 = 14,936$
- Year 5: $20,619 / 1.496 = 13,786$
- Total PV = $38,368 + 17,541 + 16,175 + 14,936 + 13,786 = 100,806$

So, NPV is positive.

The IRR is slightly below 8.5%. Let's interpolate:

At 8.5%: NPV = $806$  
At 9%: NPV = $-509$

IRR = 8.5% + (806/(806+509))*0.5%  
IRR = 8.5% + (806/1315)*0.5%  
IRR = 8.5% + 0.306*0.5% ≈ 8.5% + 0.15% = **8.65%**

---

## **Final Answers**

### **NPV:**
- **Company A:** $4,225.25
- **Company B:** $1,798.33

### **IRR:**
- **Company A:** **9.4%**
- **Company B:** **8.65%**

---

**Explanation:**  
Company A gains more benefit because it pays no tax, whereas Company B pays tax but gets a benefit from the upfront depreciation tax shield. The IRR for Company A is higher because all cash flows are untaxed. For Company B, the front-loaded tax shield improves early cash flow, but the ongoing tax on future cash flows reduces IRR and NPV compared to Company A.

✨ VIPSolutions

Question:

Final Answer:

📢 Instructions:

Study Help, Fast Answers

✨ VIPSolutions

Question:

Final Answer:

📢 Instructions:

Related Answers

Study Help, Fast Answers