I This problem is useful for testing the ability of financial calculators and co

Question

I

This problem is useful for testing the ability of financial calculators and computer software. Consider the above cash flows. The IRRs, from smallest to largest, are _________percent, __________  percent, ___________percent, and ________ percent. (Hint: search between 20 percent and 70 percent.)

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The Yurdone Corporation wants to set up a private cemetery business. According to the CFO, Barry M. Deep, business is "looking up". As a result, the cemetery project will provide a net cash inflow of $96,000 for the firm during the first year, and the cash flows are projected to grow at a rate of 5 percent per year forever. The project requires an initial investment of $1,490,000.

What is the NPV for the project if Yurdone's required return is 10 percent?

The company is somewhat unsure about the assumption of a 5 percent growth rate in its cash flows. At what constant growth rate would the company just break even if it still required a return of 10 percent on investment?

Year Cash Flow 0 %u2013$ 3,024 1 17,172 2 %u2013 36,420 3 34,200 4 %u2013 12,000

Explanation / Answer

(a) No. Because the NPV would be $714,285.70 which is less than the initial CFO of $780,000. That is to say, the money is better spent on other project(s) given the 13% financial hurdle

(b) At least 6.60% (rounded) growth rate.

Here's how you solve for this problem (this is really like an equities problem)

To solve for (a) involving the NPV, you need to know all the cash flows. The problem you are facing is that the cash flows grow at 6.0% indefinitely. The trick here is to usedividend growth model to calculate the never-ending but growing cash flows (also called the terminal value) and the formula is Dividend(N) times (1 + growth rate) all divided by (required rate of return minus the growth rate).

So the first cash flow CF(1) is 50,000 and the future value at CF(1) of all the remaining cash flow aka the terminal value is [50,000 * (1 + growth rate)] / (.13 - .06). The terminal value is then (53,000 / .07) or 754,142.90.

Now you know all the cash flows, which is 804,142.90 (the sum of CF(1) and the Terminal Value or the sum of 50,000 plus 754,142.90), you discount it by 13% to get $714,285.70 (or 804,142.90 / 1.13). And since this value is less than the original $780,000, you do not invest in the project.

To solve for (b), you have a problem because you realized the answer depends on the growth rate assumption. This growth rate assumption affects both the numerator and the denominator of the dividend growth model in order to determine the terminal value at CF(1) of all the remaining cash flows. That is to say, if you increase the growth rate, then the numerator will be higher than 53,000 and the denominator is less than 7%. You can solve for it alegrabically or do what I did and plug in a few numbers until the NPV exceeds $780,000, which is 6.60% rounded (you might get something like 6.5898% if you want to be more accurate). You can solve for alegrabically because your growth rate is the only unknown, i.e. it is large enough such that the present value of discounted cash flows exceed the initial 780,000 investment.

Here are the two traps you need to look out for when solving these problems.

Do be afraid of cash flows that are indefinite. You know from a consol bond (a bond that pays a fixed rate of interest indefinitely) its value is the annual coupon divided by the required discount rate or 50,000 / 0.13 or 384,615.40. So if the cash flows of the project is 50,000 each year indefinitely, you can solve that. In this case, there was a growth rate involved so you had to adjust for that.

The second trap is understanding the terminal value or the future value of all remaining cash flows is at the end of CF(1) because it represents at CF(1) the discounted value of all these growing cash flows. However, you still need to discount it back to time 0 or CF(0) to do the proper analysis. That is why CF(1) has two cash flows, the initial cash flow of 50,000 and the terminal value and both of these values must be discounted back. So yes, you are discounting the future, but growing cash flows twice -- the first time to value it at one point in time due to the formula, which is at CF(1), and the second time to the present, which is at CF(0).

Confused?? This problem is tricky in the beginning but once you understand how to value any set of cash flows, it gets easiers and you can use the skills to value an equity, which never matures. Unbeknowst to you, you were really valuating an equity.

Good luck.

[EDIT: To confuzed -- I have some bad news; it's possible the answers are wrong for a) NPV = -22,222.22 and b) 7.67%. When in doubt, I try to see if the cash flows were paid in advance versus in arrears and did not match the answer.

Then I replacated the CF's on Excel (using the NPV function) with your new set of numbers and Excel got a NPV of -$68.92 while my approach gets -78.57 for part (a). For the B/E constant growth, using the 7.67% default answer, Excel gave -15.87 and my approach gets -18.09. Note in Excel, I grew the dividends to over 6000 periods, but the answer converges to -15.87 after about 200 dividends; the reason it converges is that cash flow way out there in the future only affects the NPV by a small amount, especially when the discount rate is 14%.

Again, you can test this yourself, by starting with 10 dividends, then 100 dividends, then 200 dividends...while growing the dividends by the growth rate. By about about 200 dividends, you will have a good idea of the true NPV).

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