Answer in 3 steps: GIve explanantion at the end of each step and give final answer at the end of the answr:

Crabbe Steel owns a number of steel plants in Pennsylvania. The firm reported after
-
tax o

Question

Answer in 3 steps: GIve explanantion at the end of each step and give final answer at the end of the answr:

Crabbe Steel owns a number of steel plants in Pennsylvania. The firm reported after
-
tax operating income of $
4
0
 ﻿million in the most recent year on capital invested of $
4
0
0
 ﻿million. The firm expects operating income to grow 
7
%
 ﻿a year for the next three years, and 
3
%
 ﻿thereafter.
If the firm's cost of capital is 
1
0
%
 ﻿and yoe expect the firm's current return on capital to continue in perpetuity, estimate the value at the end of the third year.
b
.
 ﻿If you expect operating income to stay fived affer year 
3
 
(
what you earn in year 
3
 ﻿is what you will earn every year thereafter
)
,
 ﻿estimate the terminal value.
c
.
 ﻿If you expect operating income to drop 
5
%
 ﻿a year in perpetuity after year 
3
 
,
 ﻿estimate the terminal value.

VIPSolutions · Accepted Answer

Let's solve the problem step by step as requested:

---

### **Step 1: Calculate Operating Income for Each Year**

- **Given:**
  - After-tax operating income (OI) in most recent year = **$40 million**
  - Growth rate for next 3 years = **7%**
  - After Year 3: Scenario varies.
- **Year 1 Operating Income:**
  $$
  OI_1 = 40 \times 1.07 = 42.8 \text{ million}
  $$
- **Year 2 Operating Income:**
  $$
  OI_2 = 42.8 \times 1.07 = 45.796 \text{ million}
  $$
- **Year 3 Operating Income:**
  $$
  OI_3 = 45.796 \times 1.07 \approx 49.0017 \text{ million}
  $$
**Explanation:**  
We grew the operating income by 7% each year for three years.

---

### **Step 2: Calculate Terminal Value for Each Scenario**

#### **a) Operating income grows at 3% forever after year 3**

$$
\text{Terminal Value at end of Year 3} = \frac{OI_4}{r - g}
$$
Where:
- $OI_4 = OI_3 \times 1.03$
- $r = 10\% = 0.10$
- $g = 3\% = 0.03$
$$
OI_4 = 49.0017 \times 1.03 = 50.4718 \text{ million}
$$
$$
\text{Terminal Value} = \frac{50.4718}{0.10 - 0.03} = \frac{50.4718}{0.07} = 721.026 \text{ million}
$$

#### **b) Operating income stays fixed after year 3 ($OI_4 = OI_3$ forever)**

$$
\text{Terminal Value} = \frac{OI_3}{r}
$$
$$
\text{Terminal Value} = \frac{49.0017}{0.10} = 490.017 \text{ million}
$$

#### **c) Operating income drops 5% a year after year 3 (perpetual decline)**

Here, $g = -5\% = -0.05$:

$$
\text{Terminal Value} = \frac{OI_4}{r - g}
$$
$$
OI_4 = 49.0017 \times 0.95 = 46.5516 \text{ million}
$$
$$
\text{Terminal Value} = \frac{46.5516}{0.10 - (-0.05)} = \frac{46.5516}{0.15} = 310.344 \text{ million}
$$

**Explanation:**  
We applied the perpetuity growth formula for each scenario, using the appropriate growth rate after year 3.

---

### **Step 3: Final Answers**

**a.** If operating income grows at 3% forever after year 3:  
**Terminal Value at end of Year 3 = \$721.03 million**

**b.** If operating income stays fixed after year 3:  
**Terminal Value at end of Year 3 = \$490.02 million**

**c.** If operating income drops 5% a year after year 3:  
**Terminal Value at end of Year 3 = \$310.34 million**

**Explanation:**  
The terminal value represents the present value (at the end of year 3) of all future operating income under each scenario, discounted at the firm’s cost of capital and reflecting the specified growth/decline rate.

---

## **Final Answers**

a. **\$721.03 million**  
b. **\$490.02 million**  
c. **\$310.34 million**

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