Suppose that you are the manager of a small company that is contemplating construction of a suburban office block. The cost of buying the land and constructing the building is $
7
1
5
,
0
0
0
.
Your

Question

Suppose that you are the manager of a small company that is contemplating construction of a suburban office block. The cost of buying the land and constructing the building is $
7
1
5
,
0
0
0
.
 ﻿Your company has cash in the bank to finance construction. You forecast a shortage of office space in the area and predict that you will be able to rent out the building for two years at $
3
0
,
7
5
0
 ﻿a year. You forecast that at the end of that time you will be able to sell the building for $
8
4
6
,
0
0
0
.
 ﻿Thus, there are now two future cash flows
-
-
a cash flow of C
1
 
=
 ﻿$
3
0
,
7
5
0
 ﻿at the end of one year and a further cash flow of C
2
 
=
 
(
$
3
0
,
7
5
0
 
+
 ﻿$
8
4
6
,
0
0
0
)
 
=
 ﻿$
8
7
6
,
7
5
0
 ﻿at the end of the second year. Calculate the NPV of the office building venture at interest rates of 
5
,
 
1
0
,
 ﻿and 
1
5
%
.
 ﻿At what discount rate 
(
approximately
)
 ﻿would the project have zero NPV
?

VIPSolutions · Accepted Answer

Let's break down the problem:

### Given:
- **Initial outlay (at time 0):** $ \$715,000 $
- **Cash flow at end of year 1 ($ C_1 $):** $ \$30,750 $
- **Cash flow at end of year 2 ($ C_2 $):** $ \$30,750 + \$846,000 = \$876,750 $

We are to find the **Net Present Value (NPV)** at discount rates of **5%, 10%, and 15%**, and the **discount rate where NPV = 0** (IRR).

---

## 1. **NPV Formula**
$$
NPV = - C_0 + \frac{C_1}{(1+r)} + \frac{C_2}{(1+r)^2}
$$
Where:
- $ C_0 = 715,000 $
- $ C_1 = 30,750 $
- $ C_2 = 876,750 $
- $ r $ = discount rate

---

## 2. **Calculate NPV at Each Rate**

### **a) At 5%:**
$$
NPV = -715,000 + \frac{30,750}{1.05} + \frac{876,750}{1.05^2}
$$
Calculate each term:

$$
\frac{30,750}{1.05} = 29,285.71
$$
$$
\frac{876,750}{1.1025} = 795,357.14
$$
$$
NPV = -715,000 + 29,285.71 + 795,357.14 = 109,642.85
$$

---

### **b) At 10%:**
$$
NPV = -715,000 + \frac{30,750}{1.10} + \frac{876,750}{1.21}
$$
$$
\frac{30,750}{1.10} = 27,954.55
$$
$$
\frac{876,750}{1.21} = 724,586.78
$$
$$
NPV = -715,000 + 27,954.55 + 724,586.78 = 37,541.33
$$

---

### **c) At 15%:**
$$
NPV = -715,000 + \frac{30,750}{1.15} + \frac{876,750}{1.3225}
$$
$$
\frac{30,750}{1.15} = 26,739.13
$$
$$
\frac{876,750}{1.3225} = 663,246.13
$$
$$
NPV = -715,000 + 26,739.13 + 663,246.13 = -25,014.74
$$

---

## 3. **Discount Rate for Zero NPV (IRR)**

Set NPV = 0 and solve for $ r $:

$$
0 = -715,000 + \frac{30,750}{1+r} + \frac{876,750}{(1+r)^2}
$$

This is a quadratic in terms of $ (1+r) $.

Let $ x = 1 + r $:

$$
0 = -715,000 + \frac{30,750}{x} + \frac{876,750}{x^2}
$$

Multiply both sides by $ x^2 $:

$$
0 = -715,000 x^2 + 30,750 x + 876,750
$$
$$
715,000 x^2 - 30,750 x - 876,750 = 0
$$

Use the quadratic formula:
$$
x = \frac{30,750 \pm \sqrt{(30,750)^2 + 4 \times 715,000 \times 876,750}}{2 \times 715,000}
$$

Calculate:
- $ (30,750)^2 = 945,562,500 $
- $ 4 \times 715,000 \times 876,750 = 2,509,305,000,000 $
- Sum: $ 945,562,500 + 2,509,305,000,000 = 2,510,250,562,500 $
- Square root: $ \sqrt{2,510,250,562,500} \approx 1,585,998 $
- Denominator: $ 2 \times 715,000 = 1,430,000 $

$$
x = \frac{30,750 + 1,585,998}{1,430,000} \approx \frac{1,616,748}{1,430,000} \approx 1.1306
$$

Ignore the negative root.

Thus,
$$
r \approx 1.1306 - 1 = 0.1306 \text{ or } 13.06\%
$$

---

## **Summary Table**

| Discount Rate | NPV           |
|---------------|---------------|
| 5%            | \$109,643     |
| 10%           | \$37,541      |
| 15%           | -\$25,015     |
| **IRR**       | **13.1%**     |

---

### **Final Answers**

- **NPV at 5%:** \$109,643
- **NPV at 10%:** \$37,541
- **NPV at 15%:** -\$25,015
- **Discount rate for zero NPV (IRR):** **about 13.1%**

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