# How do I solve this problem using net present value and internal rate of return methods?

Julie Kowalis, an investment analyst, wants to know if her investments during the past four years have earned at least a 12% return. Four years ago, she had the following investments:She purchased a small building for \$50,000 and rented space in it. She received rental income of \$8,000 for each of the four years and then sold the building this year for \$55,000

Assuming there were no expenses related to being a landlord, which is almost never the case, the \$8000 annual rental income was equivalent to earning 16% interest. There was also a capital gain of \$5000 at the end of four years, which is equivalent to about 2.45% interest compounded annually. That is a net annual return of 18.45%, which is much better than 12%.

In real life, however, the tax, insurance and upkeep expense of owning the building would probably be in a 3%-5% range per year, with an additional 6-7% (1.2% annualized) cost of selling, making it a slightly better than 12% yield investment.

## To solve this problem using the net present value (NPV) and internal rate of return (IRR) methods, you need to calculate the cash flows and apply the formulas for these financial measures.

1. Calculate the cash flows:
- Initial investment: Julie purchased the building for \$50,000.
- Rental income: Julie received \$8,000 per year for four years.
- Sale of the building: Julie sold the building for \$55,000 at the end of the four years.

2. Calculate the NPV:
The NPV measures the present value of all cash flows associated with an investment. To calculate the NPV, you need to discount the future cash flows back to their present value. The formula for NPV is as follows:

NPV = CF0 + (CF1 / (1+r)^1) + (CF2 / (1+r)^2) + ... + (CFn / (1+r)^n)

Where:
- CF0: Initial investment
- CF1, CF2, ... , CFn: Cash flows in different periods
- r: Discount rate (in this case, the desired rate of return is 12%)

In this case, consider the following cash flows:
CF0 = -\$50,000 (Initial investment)
CF1 = \$8,000 (First-year rental income)
CF2 = \$8,000 (Second-year rental income)
CF3 = \$8,000 (Third-year rental income)
CF4 = \$8,000 + \$55,000 (Fourth-year rental income + Sale proceeds)

Calculate the present value of each cash flow and sum them up to find the NPV. If the NPV is positive, it indicates that the investment has generated a return higher than the desired rate, in this case, 12%.

3. Calculate the IRR:
The IRR is the discount rate at which the NPV of cash flows becomes zero. It represents the annualized rate of return on an investment. To calculate the IRR, you need to find the discount rate that makes the NPV equal to zero.

Using the same cash flow values as in step 2, you have:
NPV = CF0 + (CF1 / (1+IRR)^1) + (CF2 / (1+IRR)^2) + ... + (CFn / (1+IRR)^n)

Find the value of IRR that makes NPV = 0 using trial and error or by using Excel or financial calculators.

Comparing the NPV and IRR values to the desired rate of return (12%) will help determine if the investment has performed well, considering the given cash flows.

Note: The additional information about expenses in the problem description suggests that the actual yield may be slightly better than 12% after factoring in those costs. However, this explanation focuses on the basic calculation using NPV and IRR methods.