Question

3. Calculating IRR(internal rate of return) We will attempt to calculate IRR for a would-be pediatrician using the following basic assumption. At age 22, the future pediatrician would start her 4 years of medical school, costing her $50,309/year. She would then complete 3 years of residency, during which she would earn an income of $60,094/year. Her salary as a pediatrician after her residency would be $181,000/year and she would work until age 65. For simplicity, we assume alternatively, she would go straight to find a job paying her $45,478/year starting age 22 if she did not pursue her pediatrician career, and she would work until age 65 as well. Verify that the IRR would be 17.77% by calculating that the present value of her investment and return over her opportunity cost would be zero at this rate. Now suppose the government want to lower the burden by providing student loans with zero interest rate to medical students. For simplicity, let us assume that the effect of such a policy means she would not incur any tuition costs during her medical schooling years. Instead, she a. b. needs to settle the total amount of 4 year's tuition of $201,236 (4x$50,309) at age 48. Verify that her IRR would now become 27.11%. 4. Problem 3 from Chapter 7 in Phelps. Suppose a society consists of three cities (A, B, an C), with populations of 99,000, 51,000, and 6,000 people, respectively. Suppose also that the society has a total of 15 doctors. What cities will have how many doctors? If the number of doctors doubles, how many will live in each city? a. b.

Answer 1

Internal rate of return (IRR) is the interest rate at which the net present value of all the cash flows (both positive and negative) from a project or investment equal zero.

Internal rate of return is used to evaluate the attractiveness of a project or investment. If the IRR of a new project exceeds a company’s required rate of return, that project is desirable. If IRR falls below the required rate of return, the project should be rejected.

IRR Formula & Example

You can use the following formula to calculate IRR:

0 = P0 + P1/(1+IRR) + P2/(1+IRR)2 + P3/(1+IRR)3 + . . . +Pn/(1+IRR)n

where P0, P1, . . . Pn equals the cash flows in periods 1, 2, . . . n, respectively; and

IRR equals the project's internal rate of return.

Let's look at an example to illustrate how to use IRR.

Assume Company XYZ must decide whether to purchase a piece of factory equipment for $300,000. The equipment would only last three years, but it is expected to generate $150,000 of additional annual profit during those years. Company XYZ also thinks it can sell the equipment for scrap afterward for about $10,000. Using IRR, Company XYZ can determine whether the equipment purchase is a better use of its cash than its other investment options, which should return about 10%.

Here is how the IRR equation looks in this scenario:

0 = -$300,000 + ($150,000)/(1+.2431) + ($150,000)/(1+.2431)2 + ($150,000)/(1+.2431)3 + $10,000/(1+.2431)4

The investment's IRR is 24.31%, which is the rate that makes the present value of the investment's cash flows equal to zero. From a purely financial standpoint, Company XYZ should purchase the equipment since this generates a 24.31% return for the Company --much higher than the 10% return available from other investments.

A general rule of thumb is that the IRR value cannot be derived analytically. Instead, IRR must be found by using mathematical trial-and-error to derive the appropriate rate. However, most business calculators and spreadsheet programs will automatically perform this function.

[See How to Calculate IRR Using a Financial Calculator or Microsoft Excel]

IRR can also be used to calculate expected returns on stocks or investments, including the yield to maturity on bonds. IRR calculates the yield on an investment and is thus different than net present value (NPV) value of an investment.

Why is IRR Important?

IRR allows managers to rank projects by their overall rates of return rather than their net present values, and the investment with the highest IRR is usually preferred. This easy comparison makes IRR attractive, but there are limits to its usefulness.

One downside for example: IRR works only for investments that have an initial cash outflow (the purchase of the investment) followed by one or more cash inflows. In addition, IRR does not measure the absolute size of the investment or the return. This means that IRR can favor investments with high rates of return even if the dollar amount of the return is very small. For example, a $1 investment returning $3 will have a higher IRR than a $1 million investment returning $2 million, but the latter brings in $1 million dollars instead of just $2.

Another short coming is that IRR can't be used if the investment generates interim cash flows. Finally, IRR does not consider cost of capital and can't compare projects with different durations.

Overall, IRR is best-suited for analyzing venture capital and private equity investments, which typically entail multiple cash investments over the life of the business, and a single cash outflow at the end via IPO or sale.