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Perils of the Internal Rate of Return

By Kyle Ferguson,2014-11-25 11:06
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Perils of the Internal Rate of Return

Sage Note: NPV and IRR Page 1

What’s it Worth?

    The two most-used measures for evaluating an investment are the net present value

    (NPV) and the internal rate of return (IRR).

    It is often assumed that higher is better for both the net present value and the internal rate of return. In particular, it is usually stated that investments with higher internal rates of return are more profitable than investments with lower internal rates of return. However, this is not necessarily so. In some situations, an investment with a lower internal rate of return may be better, even judged on narrow financial grounds, than an investment with a higher internal rate of return. This note explores why and when this reversal takes place.

    To review, both the net present value and the internal rate of return require the idea of an income stream, so let's start there. An income stream is a series of amounts of

    money. Each amount of money comes in or goes out at some specific time, either now or in the future. The income stream represents the investment; the income stream is all you need to know for financial evaluation purposes.

    In real life, individuals, charitable institutions, and even for-profit businesses have social or other goals when selecting investments. For businesses, the benefits of community good will are no less real for being difficult to measure precisely. For enterprises with social as well as financial goals, the measures discussed here are still useful: They tell you how much it costs you to advance your social goals. Here is an income stream example:

Year 0 1 2 3 4 5 6

    Income amounts -$1000 $200 $200 $200 $200 $200 $200

    Here we see seven points in time and, for each, a dollar inflow or outflow. At year 0 (now), the income amount is negative. Negative income is cost, or outgoing, or investment. In this example, the negative income amount in year 0 represents the cost of buying and installing the machine.

    In the future, at years 1 through 6, there will be net income of $200 each year. All of the amounts in the income stream are net income, meaning that each is income

    minus outgoings, or revenue minus cost. In year 0, the cost exceeds the revenue by $1000. In years 1 though 6, the revenue will exceed the cost by $200. This investment evidently has no salvage value. That is, there is nothing that can be sold in year 6, the last year. If there were, the amount that could be realized from the sale would be added to the income amount for year 6.

    For simplicity, all these examples have the incomes and outgoings at one-year intervals. Real-life investments can have income and expenses at irregular times, but the principles of evaluation are the same.

    Now let's discuss our two measures in connection with this income stream: Net Present Value

    The net present value of an income stream is the sum of the present values of the

    individual amounts in the income stream. Each future income amount in the stream is discounted, meaning that it is divided by a number representing the cost of capital from now (year 0) until the year when income is received or the outgoing is spent. For an individual , the cost of capital can either be how much you would have earned investing the money someplace else, or how much interest you would have had to pay

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Sage Note: NPV and IRR Page 2

    if you borrowed money. For a business the cost of capital is generally measured as the Weighted Average Cost of Capital, which is the weighted average cost of shareholders equity and debt.

    The word "net" in "net present value" indicates that our calculation includes the initial costs as well as the subsequent profits. It also reminds us that all the amounts in the income stream are net profits, revenues minus cost. In other words, "net" means the same as "total" here.

    The net present value of an investment tells you how this investment compares with your cost of capital. A positive net present value means this investment return is above your cost of capital. . A negative net present value means your return is below your cost of capital. .

    Consider again this income stream:

Year 0 1 2 3 4 5 6

    Income amounts -$1000 $200 $200 $200 $200 $200 $200

    Let's assume that the discount rate (the interest rate that you could earn elsewhere or at which you could borrow) will not change over the life of the project. This makes the calculation simpler. With this assumption, we can use the usual formula: Present Value of any one income amount = (Income amount) / ( (1 + Discount Rate) th power) to the a

    a is the number of years into the future that the income amount will be received (or spent, if the income amount is negative).

    The net present value (NPV) of a whole income stream is the sum of these present values of the individual amounts in the income stream. If we still assume that income comes or goes in annual bursts and that the discount rate will be constant in the future, then the NPV has this formula:

     2nNPV = I + I/(1+r) + I/(1+r) + … + I/(1+r) 012n

    The I s are income amounts for each year. The subscripts (which are also the exponents in the denominators) are the year numbers, starting with 0, which is this year. The discount rate - assumed to be constant - is r. The number of years the

    investment lasts is n. (This formula simplifies somewhat if the cashflows are all identical.)

    Three properties of the net present value of an income stream are:

    1. Higher income amounts make the net present value higher. Lower

    income amounts make the net present value lower.

    Try it yourself.

    Double-click on the table below, then Click on a box in the Income row. Edit the number there, deleting or adding some digits. Then press Enter.

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Sage Note: NPV and IRR Page 3

Year0123456

    Income -$1,000$200$200$200$200$200$200

    amounts

    Rate5%

    Present value-1000190181173165157149

    Net Present

    Value15

    The Net Present Value box shows the net present value, which is the total of the amounts in the boxes in the Present value row. (The total may be slightly off, due to rounding.)

    2. If profits come sooner, the net present value is higher. If profits come

    later, the net present value is lower.

    Try it yourself in the spreadsheet fragment above. You can see how moving the amounts further out changes the net present value of the income stream.

    3. Changing the discount rate changes the net present value. For an

    investment with the common pattern of having costs early and profits

    later, a higher discount rate makes the net present value smaller.

    Again, try it yourself. You can also change the income amounts, if you want.

    To summarize what was just illustrated, the net present value is higher if the income amounts are larger, or if they come sooner, or if the discount rate is lower. The net present value is lower if the income amounts are smaller, or if they come later, or if the discount rate is higher.

    Internal Rate of Return

    In the example we've been using, if you keep the income amounts at their original values: -1000, 200, 200, 200, 200, 200, and 200, and set the discount rate to 0.0547, the net present value becomes 0. This discount rate, 0.0547 or 5.47%, is the internal

    rate of return for this investment - it is the discount rate that makes the net present value equal to 0. You can try this in the spreadsheet fragment, by setting the discount rate to 0.0547 or 5.47%.

    If you now raise any of the income amounts in years 1 through 6 (feel free to edit an income amount and see for yourself), you will need a higher discount rate to bring the net present value back to 0. That would seem to imply that projects with higher incomes have higher internal rates of return.

    Similarly, if you lower any of the income amounts in years 1 through 6, then a lower discount rate will be needed to bring the net present value back up to 0. That would seem to imply that projects with lower incomes have lower internal rates of return. These seeming implications are actually often true, if the projects being compared have about the same “shape”, with the costs coming early and the benefits coming late, and if the projects being compared switch from net outgoing to net income at about the same time. Otherwise, though, the implications might not be true. Before we go on to that, a little review:

    Which of these measures (net present value and internal rate of return) requires you to know the future income and outgoing amounts?

    Which of the measures requires you to know what the discount rate will be in the future?

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Sage Note: NPV and IRR Page 4

    The internal rate of return does not require you to predict future discount rates. That would seem to make the internal rate of return the more useful (or less uncertain) measure. Sometimes, though, the internal rate of return can fool you. Contradictory Results

    A few years ago, the New England Journal of Medicine published a study that

    evaluated various types of professional education as if they were financial investments.

    The article is: Weeks, W.B., Wallace, A.E., Wallace, M.M., Welch, H.G., "A

    Comparison of the Educational Costs and Incomes of Physicians and Other

    Professionals," N Engl J Med, May 5, 1994, 330(18), pp. 1280-1286.

    The idea was to see if doctors were overpaid, by considering primary and specialty medical education as investments and comparing them with investing in education in business, law, and dentistry. Adjustments were made for differences in average working hours. The authors found that primary medicine was the poorest investment of all of these. Specialty medicine did better, but was not out of line with the other professions.

    In the results was this oddity: by the criterion of the net present value of lifetime educational costs and income benefits, specialist physicians tied for highest with attorneys. Both were ahead of business school graduates. However, by the criterion of the internal rate of return, specialty physicians, with a 21% average return, were well behind the attorneys‟ 25% average return, while the business school graduates‟ 29% average return was the highest of all. The present value and the internal rate of return ranked the alternatives differently!

    By the way, since this article's 1994 publication, managed care in the US has forced specialty physician incomes down by perhaps one-third. This has sharply lowered the investment value of a specialty medical education.

    The NPV Curve

    One way to understand how the net present value and the internal rate of return can give seemingly different advice is to consider the net present value curve, or NPV

    curve. The NPV curve shows the relationship between the discount rate and the net present value for a range of discount rates. The present value at a given discount rate, such as 5%, and the internal rate of return are each points on the NPV curve. The NPV curve, the relationship between the discount rate and the net present value, has the formula we saw already for the net present value, for annualized costs and revenues and a constant discount rate. Each I is an income amount for a specific year.

    The subscripts (which are also the exponents in the denominators) are the year numbers, starting with 0, which is this year. The constant discount rate is r. The

    number of years the investment lasts is n. In Weeks‟ study of professionals‟ incomes,

    n was about 44, because costs and incomes were calculated from age 21 to age 65. We‟ll use an example with an n of 6, corresponding to our machine investment

    example above. The NPV is a function of r. Graphed, it looks like this:

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Sage Note: NPV and IRR Page 5

    NPV Curve

    $200

    $150

    $100

    $50

    $0

    -$50Net Present Value

    -$100

    -$150

    00.010.020.030.040.050.060.070.080.090.1

    Discount Rate (r)

The curve shows the net present value for discount rates (r) from 0 to 0.1 (0% to 10%).

    The marked points are the two points we get from our measures. The left dot (a diamond shape) shows the net present value at the discount rate of 0.05 (5%). The right dot (a square outline) shows the internal rate of return, because it is where the curve crosses the horizontal line indicating an NPV of 0. That right dot is between the 0.05 and 0.06 marks on the r axis, so the internal rate of return is between 0.05 and 0.06. (The actual internal rate of return is about 0.0547, as we saw earlier.) Imagine we have another possible investment, which has cashflows of $220 at each of years 1 to 6.

    This investment is like the first, except that the net profit in years 1 through 6 is $220 per year, rather than $200. We say that this investment has a similar "shape" to the first, because the costs and profits come at the same times. Also, the size of the initial outlay is the same for both. The only difference is the amount of profit. Here‟s a graph with both investments on it:

    NPV Curves

    $300

    $250

    $200

    $150

    $100

    $50

    $0

    Net Present Value-$50

    -$100

    -$150

    00.010.020.030.040.050.060.070.080.090.1

    Discount Rate (r)

    The pink curve (the uppermost curve) is the second investment. It is above and

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Sage Note: NPV and IRR Page 6

parallel to the first investment‟s blue curve. The triangular dot shows the net present

    value of the second investment at the discount rate of 0.05. The net present value there is a little over $100. This is higher than the diamond-shaped dot for the first curve, so the net present value (at r = 5%) of the pink-line investment is higher than

    the net present value at r = 5% for the original investment.

    The right dot (large square dot) shows where the second investment‟s curve crosses the NPV = 0 line. This is well to the right of the first investment‟s internal rate of return dot. The internal rate of return for the second investment is much higher (further to the right).

    In this example, our two measures, the net present value at r = 0.05 and the internal

    rate of return, tell us the same thing. They both say the second investment is better. A look at the graph above confirms that the second investment is better at all discount rates, so it is fair to say that the second investment is unequivocally better than the first.

    Can You Do Both Investments?

    Making an investment increases your wealth if its net present value is greater than 0 at the discount rate relevant to you. If your discount rate is less than 5.47%, both NPV curves are in positive territory, and you should do both, if you can. Sometimes, though, the alternative investments are mutually exclusive. For example, there may be two ways to build a dam across a particular river. You can do one or the other, but not both. There may be several alternative ways to address a workplace safety problem. There is no point to doing more than one if any one way solves the problem. Deciding on a professional education involves somewhat mutually exclusive choices. A few people do go to medical school and then law school, but the additional return from the second degree is not the same as what someone going to law school fresh out of college would expect.

    If you can only make one investment, you should choose the one with the highest net present value at the discount rate appropriate to you. A problem with that advice, though, is that discount rates can change with general economic conditions. You are therefore more confident about choosing one investment over another if your chosen investment has a higher net present value over a broad range of possible discount rates. In our example so far, the pink-line investment has a higher net present value at all discount rates, so we would choose it with confidence. Regardless of what happens in the future to discount rates, we'll be better off with the pink-line investment than with the original investment. (We‟ll leave the question of the uncertainty of the future

    cashflows to later …)

Can NPV Curves Cross?

    Yes, they can. If the NPV curves cross, then the choice of investment depends on the discount rate.

    To create an example, we'll change the original investment so that its profits come much later. This increases the effect of the discount rate on the net present value. Below are the two income streams, now. Also shown are their net present values at a 5% discount rate and their internal rates of return.

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Sage Note: NPV and IRR Page 7

Year 0 1 2 3 4 5 6 NPV at Internal

    0.05 rate of

    discount return

    rate

    Pink line -$1,000 $220 $220 $220 $220 $220 $220 $117 0.086

    investment

    Blue line -$1,000 $0 $0 $0 $0 $0 $1,550 0.076 $157

    investment

    The pink line investment has the higher internal rate of return, but the blue line investment has the higher net present value at a 5% discount rate. Our two measures are giving us opposite advice!

    NPV Curves

    $500

    $400

    $300

    $200

    $100

    $0Net Present Value

    -$100

    -$200

    00.010.020.030.040.050.060.070.080.090.1

    Discount Rate (r)

    The graph shows what‟s going on, by showing the Net Present Value curves for both investments for discount rates between 0% and 10%. The curves cross at a discount rate of about 0.064, or 6.4%.

    Now, to choose which investment we want to make, assuming we cannot make both, we have to make a guess about what future discount rates will be. If we expect discount rates to be less than 6.4%, where the curves cross, we choose the blue line investment. For discount rates above 6.4%, but below 8.56% (the internal rate of return of the pink line investment), we choose the pink line investment. At higher discount rates than 8.56%, we don't do either, because the net present values are negative.

    If Costs Come Later Than Profits

    If costs come later than profits, the NPV curve can tilt the other way, making it even more problematic to use the internal rate of return to compare investments. Costs can come later than profits if an investment creates environmental problems that will have to watched or cleaned up later. Nuclear power plants are a good example: After about 40 years of service (sometimes less than that), they become too contaminated with radiation to continue in service. They must then be closed and

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Sage Note: NPV and IRR Page 8

    either guarded where they are for thousands of years or dismantled and moved to a disposal site. Forestry operations also often generate a large “clean-up cost”, and so

    on.

Consider this income stream:

    Year 0 1 2 3 4 5 6

    Income amounts $200 $200 $200 $200 $200 -$200 -$900

    I've reduced the initial cost, but added a big cost at the end. Let's see what difference this makes in how the NPV changes when the discount rate changes. If we substitute these values in the spreadsheet fragment above, with a starting discount rate 5%, the net present value (NPV) is -$6. That's negative six dollars, so if your discount rate really were 5%, you would not want to make this investment.

    Try changing the discount rate, to 0.04 or 0.03. In the examples above, the NPV goes up when the discount rate is lowered. Is that true for this project? Then try 0.06 or 0.07. What happens to the NPV?

    (Keep the discount rates reasonably small, say between 0.00, which is 0%, and 0.3, which is 30%.)

    The relationship between the discount rate and the NPV is the reverse of what we see with "normal" investments! With this kind of income stream, higher discount rates make the net present value bigger, and lower discount rates make the net present value smaller.

Here is the NPV graph:

    NPV Curves

    $60

    $40

    $20

    $0

    -$20

    -$40

    -$60Net Present Value

    -$80

    -$100

    00.010.020.030.040.050.060.070.080.090.1

    Discount Rate (r)

    The net present value at a 5% (0.05) discount rate is at -$6 on the net present value scale, and where the curve crosses the discount rate axis, where the net present value is $0, is the internal rate of return, 0.054 (5.4%).

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Sage Note: NPV and IRR Page 9

    Or, at least, it fits the standard definition of internal rate of return. However, unlike the usual situation, this project is profitable at interest rates above this IRR and

    unprofitable at interest rates below this IRR.

    Suppose we have an alternative project that also has this shape, with a big cost at the end, but slightly lower profits in the intermediate years. I‟ll call the new alternative the "green line investment."

    Year 0 1 2 3 4 5 6 NPV at Internal

    0.05 rate of

    discount return

    rate

    5.40% Red line -$200 $200 $200 $200 $200 $200 -$900 -$6

    investment

    7.01% Green line -$200 $195 $195 $195 $195 $195 -$900 -$27

    investment

    The green line investment has a lower NPV than the red line investment at all discount rates, because it has lower profits in years 1 through 5, and the same costs in years 0 and 6. In particular, as the table above indicates, it has a lower NPV at the 0.05 discount rate. The graph below shows the NPV curves for both investments, with the green line lying below the red line at all discount rates.

    NPV Curves

    $150.00

    $100.00

    $50.00

    $0.00

    -$50.00

    Net Present Value

    -$100.00

    -$150.00

    00.010.020.030.040.050.060.070.080.090.1

    Discount Rate (r)

    The green line investment is clearly inferior, but it has the higher internal rate of return. The green line investment's IRR is 0.07. The red line investment's is 0.054. Thus, for projects with big late costs, the better projects will have lower internal rates

    of return, the opposite of the rule for normal projects that have their costs early and their positive returns later.

    Now let‟s discover something even more strange. Try the cashflows for the red line

    investment, and change the discount rate and see the effect on its value. In this case, though, take the discount rate over 0.3 (30%) and all the way up to 1.0 (100%). Those

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Sage Note: NPV and IRR Page 10

    rates are much higher than, hopefully, we will ever see, even in NZ, but they are theoretically possible, and they show a strange phenomenon.

    Try raising the discount rate to 0.3, and notice what happens to the net present value. Then, raise the discount rate some more above that. In which direction does the NPV move now?

See if you can find the second IRR, where the NPV is zero again!

    Here's the NPV curve for the red line investment for discount rates from 0% to 100%.

    NPV Curve

    $150

    $100

    $50

    $0

    Net Present Value-$50

    -$100

    00.10.20.30.40.50.60.70.80.91

    Discount Rate (r)

    At discount rates below 0.054, the NPV is negative, and this investment is worse than doing nothing.

    At a discount rate of 0.054, the NPV is 0. The first IRR for this investment is 0.054. If the discount rate rises above 0.054, the NPV turns positive, and this investment switches to being profitable.

    At a discount rate of 0.262 (26.2%), the NPV for this investment reaches its maximum. If the discount rate rises further than that, the NPV falls. The NPV reaches 0 again at a discount rate of 0.86. This is the second IRR for this investment.

    If the discount rate were to rise even more, above 0.86, the NPV turns negative again. This investment re-switches to being unprofitable.

    Lesson: The NPV curve gives better guidance than the IRR alone

    The lesson we should get from this is that the internal rate of return, by itself, can fool you. If the investments being considered have different shapes (that is, very different timing of costs and benefits) or if the project has large late cleanup costs, then the higher-IRR-is-better rule can steer you to the wrong investment. Ideally, you want the NPV curve, if you want to evaluate an investment.

    A second lesson

    The NPV, and the IRR are summary statistics that describe a series of cashflows at specific times. There is no allowance for uncertainty in either of these. Reality, of course, is a little different: future discount rates are unknown, and the cashflows are

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