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 and perform the calculations step by step.

**Given:**

- Initial investment $ = \$715,000 $ (at time 0)
- Year 1 cash flow ($ C_1 $) $ = \$30,750 $
- Year 2 cash flow ($ C_2 $) $ = \$30,750 + \$846,000 = \$876,750 $

We are to calculate the NPV at discount rates of 5%, 10%, and 15%.  
Then, estimate the IRR (the rate at which NPV = 0).

---

### 1. **NPV Formula**

$$
NPV = -\text{Initial Investment} + \frac{C_1}{(1 + r)} + \frac{C_2}{(1 + r)^2}
$$

Where:
- $ r $ is the discount rate (as a decimal)

---

### 2. **NPV at 5%**

$$
NPV_{5\%} = -715,000 + \frac{30,750}{1.05} + \frac{876,750}{(1.05)^2}
$$
$$
= -715,000 + 29,285.71 + \frac{876,750}{1.1025}
$$
$$
= -715,000 + 29,285.71 + 795,525.34
$$
$$
= 109,811.05
$$

---

### 3. **NPV at 10%**

$$
NPV_{10\%} = -715,000 + \frac{30,750}{1.10} + \frac{876,750}{(1.10)^2}
$$
$$
= -715,000 + 27,954.55 + \frac{876,750}{1.21}
$$
$$
= -715,000 + 27,954.55 + 724,586.78
$$
$$
= 37,541.33
$$

---

### 4. **NPV at 15%**

$$
NPV_{15\%} = -715,000 + \frac{30,750}{1.15} + \frac{876,750}{(1.15)^2}
$$
$$
= -715,000 + 26,739.13 + \frac{876,750}{1.3225}
$$
$$
= -715,000 + 26,739.13 + 663,316.84
$$
$$
= -24,944.03
$$

---

### 5. **Approximate IRR (where NPV = 0)**

From above, at 10% NPV ≈ \$37,541  
At 15% NPV ≈ -\$24,944

We can interpolate:

$$
IRR \approx 10\% + \frac{37,541}{37,541 + 24,944} \times (15\% - 10\%)
$$
$$
= 10\% + \frac{37,541}{62,485} \times 5\%
$$
$$
= 10\% + 0.601 \times 5\%
$$
$$
= 10\% + 3.00\% \approx 13\%
$$

---

## **Summary Table**

| Discount Rate | NPV (\$)      |
|:-------------:|:-------------:|
| 5%            | 109,811       |
| 10%           | 37,541        |
| 15%           | -24,944       |

**Approximate IRR = 13%**

---

### **Final Answers**

- **NPV at 5%:** \$109,811
- **NPV at 10%:** \$37,541
- **NPV at 15%:** -\$24,944
- **The discount rate at which NPV = 0 is approximately 13%.**

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