# Chapter 33 Measuring investment return.

WHAT IS IT?

An analytical method of comparison is an objective way to assess the relative risks and potential rewards from alternative investments. Although the ultimate decision as to which (if any) investment should be made may be subjective, the assessment should initially be based on both factual and measurable information.

The techniques discussed in this chapter are all based on the time value of money concept. The general principles of present value and future value, discussed in Chapter 32, "Time Value Concepts," may be sufficient in themselves to evaluate the return on a particular investment. But, in many cases, the more sophisticated approaches described below may be appropriate or necessary.

HOW DOES IT WORK?

This chapter will examine five comparative techniques commonly used to evaluate alternative investment opportunities. All of these devices have the same goal: they are designed to measure the return on an investor's principal. This is the concept of return on investment (ROI).

1. Net Present Value (NPV) is the difference between the present value of all future benefits of an investment and the present value of all capital contributions. This method measures the tradeoff between the cash invested (outflows) and the benefits projected (inflows).

2. Internal Rate of Return is that discount rate at which the present value of all the future benefits an investor will receive from an investment exactly equals the present value of all the capital contributions the investor will be required to make. IRR is generally used to compare the effective interest rates of two or more investments, particularly with non-matching or irregular cash flows.

3. Modified Internal Rates of Return are IRR methods that adjust cash flows for realistic assumptions regarding reinvestment rates, borrowing rates, and the like.

4. Pay Back Period measures the relative periods of time needed to recover the investor's capital (income received after the pay back period will be considered gain).

5. Cash on Cash, as its name implies, analyzes an investment by dividing the annual cash flow by the amount of the cash investment in order to determine the cash return on the cash invested.

1. These methods of comparison offer a way to measure the potential return on alternative investment choices in a logical and consistent manner.

2. The use of mathematics (assuming the accuracy of the input) balances an investor's natural inclination to play a hunch by introducing objectivity into the decision-making process.

3. The use of more than one comparative method will help the investor more effectively recognize and evaluate risk/reward parameters.

1. There are a number of unknowns in the measurement process. For instance, an investor can only estimate or project how much money will be received from a given investment and when that cash inflow and outflow will occur. There is no way to completely guarantee either the amount or timing of cash flows.

These uncertainties, though drawbacks, should not preclude the use of analytical techniques. There is no single technique that can be applied to every case. The method that most closely reflects the investor's perception of how return should be measured is the one that should be used. It is important that the chosen method be used consistently. In many cases, more than one analytical technique should be used in order to corroborate the results of the other approaches.

2. Undue reliance on mathematical quantitative evaluation techniques, as noted in the previous chapter, may create a false sense of security. Such excessive reliance can both hinder the investor's ability to utilize appropriate subjective analytical skills and inhibit the consideration of external factors that may affect the viability of the investment.

3. It is possible that a particular investment may not be adequately or properly quantified by any of the evaluation techniques discussed here; an investor may not have objective tools available.

4. It is often difficult to know which measuring device to select. The use of an inappropriate technique will often result in drawing an inappropriate conclusion.

5. In practice it is often difficult to obtain accurate and comprehensive data.

WHEN IS THE USE OF THIS TECHNIQUE INDICATED?

The five techniques discussed in this chapter are all useful when an investor wants to compare alternative investments that are seemingly similar with respect to risk and it is desirable to distinguish the investments by measuring their relative rewards in terms of: (1) timing--when and/or how often cash must be invested in and/or may be taken out of the investment; and (2) quantity--the rate of interest the investment will earn or the rate at which it will grow.

A second reason for the use of these return-on-investment techniques is to evaluate one particular investment in comparison with a safe alternative, such as a U.S. Treasury Note. Used in this manner, the safe alternative serves as a benchmark to determine whether the potential reward from the investment under consideration is sufficient relative to the associated risks.

HOW IS IT IMPLEMENTED

Before examining these five comparative techniques, it may help to review a number of rate of return concepts. See "Rate of Return Concepts," on page 294.

An examination of the five comparative techniques for evaluating alternative investment opportunities [(1) net present value (NPV), (2) internal rate of return (IRR), (3) modified internal rate of return (MIRR), (4) pay back period, and (5) cash on cash] follows.

I. Net Present Value

Net present value is an extension of the present value concepts discussed in the previous chapter. Present value is the amount that must be invested now to produce a given future value. For instance, if an investor can invest money at 10%, the investor must have \$1,000 now in order to have it grow to \$1,100 one year from now. \$1,000 is the present value of \$1,100 to be received in one year. Obviously, the present value is affected by (1) the interest (investment analysts call this the discount) rate, as well as (2) the length of the investment period.

Present value is a simple means of comparing two investments. For example, an investor is considering an investment of \$1,000 that will pay \$1,200 three years from now. The investor can also invest the \$1,000 in an alternative investment of equal risk and earn 10% on the money. Which investment should the investor make?

An easy way to compare the investments is to compute the present value of the \$1,200 payable three years from now at a 10% discount rate. The present value of the first investment is only \$902; the present value of \$1,000 in hand today is, obviously, \$1,000. Therefore, from a pure present value standpoint, the proposed investment for \$1,200 in three years is inferior to the alternative of simply investing the \$1,000 at 10%.

Net present value is the net difference between (a) the present value of all future benefits to be realized from an investment and (b) the present value of all capital contributions into the investment. A negative net present value should result in an almost automatic rejection of the investment. A positive net present value indicates that the investment is worth further consideration since the present value of the stream of dollars that will be recovered exceeds the present value of the stream of dollars that will be paid out.

The difficulty is determining what discount rate should be used in computing the present values of the cash inflows and cash outflows. Usually this discount rate will be the minimal acceptable rate of return, found by determining the cost of capital or, as in the example above, determining the rate an alternative investment of similar quality/risk can earn. In the example above, the rate was 10%. Once this reinvestment rate is determined, it can be used as the discount factor to compute the present value of the money invested and the present value of the expected return.

These present value amounts are then netted against each other. If the result is positive, the investment will exceed the reinvestment rate and should be considered. If the net present value is a negative number and falls short of the reinvestment rate, the investment under consideration should be rejected.

To the extent that a proposed investment yields a positive net present value, the investment provides a potential cushion for safety. It may also allow the investor to incur certain additional costs (such as attorney's, accountant's, or financial planner's fees in connection with the analysis of the investment) and still achieve the desired

reinvestment rate.

An example of the use of net present value analysis may be helpful.

Assume that at the beginning of the year an individual has been shown an investment opportunity requiring a lump sum outlay of \$10,000. Currently the funds he would use for this investment are in a money market fund earning 6% annually, net after taxes. The investment proposal projects the following after-tax cash flows at the end of each year.
```               Amount

1              \$2,000
2               1,500
3                 750
4                 500
5              10,000
```

Based solely on net present value analysis, should he make the investment?

The first step in the analysis would be to determine the appropriate reinvestment rate. In this example, the investor currently is earning a net after tax return of 6% in what he believes is a safe investment. To warrant any further consideration, this proposed investment must have at least a positive net present value based on the benchmark reinvestment rate of 6%.

Does the proposed investment meet/beat that benchmark? The stream of dollars projected to be received from the proposed investment has a present value of \$11,720 assuming a 6% discount rate. The net present value is a positive \$1,720 (the difference between the present value of the future stream of cash inflows, \$11,720, and the \$10,000 present value of the lump sum outflow of \$10,000). (See below regarding how to compute net present value.) Therefore, the proposal does deserve further consideration.

But what if the investor demands a rate of return from the proposed investment higher than his benchmark rate of 6% in order to compensate for the additional risk? If he sets a 15% rate as his minimum, the stream of dollars projected to be received from the proposed investment has a present value of only \$8,624. The net present value is a negative \$1,376 (the difference between the present value of the future stream of cash inflows, \$8,624, and the \$10,000 present value of the lump sum outflow of \$10,000). (See below regarding how to compute net present value.) Therefore, the proposal would be rejected based upon the requirement of a 15% reinvestment rate.

What discount rate, when applied to the expected stream of cash inflows from the proposed investment, has a present value exactly equal to the \$10,000 lump sum investment (a net present value of \$0)? That discount rate is 10.5%. This computation illustrates the concept of Internal Rate of Return, discussed more fully below.

How to Compute Net Present Value

A. Lump Sum Investment, Single Future Receipt

Assume an individual makes a lump sum investment at the beginning of year one of \$10,000. The expected return on this investment is \$15,000 (after tax) to be received as a single amount at the end of year five. The investor's discount rate (for an alternative safe investment) is 6% after tax. What is the net present value of the investment under consideration?

To compute the net present value of the investment, the following basic steps are necessary:

1. Compute the present value of the \$10,000 investment. Since only one payment is required (immediately at the beginning of the cash flow period), the present value of that payment would be \$10,000.

2. Compute the present value of the \$15,000 future amount to be received (at the end of year five) using the 6% discount rate. Refer to the Present Value Table in Appendix A. The applicable factor for the present value of \$1 at the end of 5 years, using a 6% discount rate, is 0.7473. This factor is multiplied by the \$15,000 amount to be received in the future. The present value is therefore \$11,210 (\$15,000 x 0.7473). [Calculating the present value of a lump sum using a formula, rather than a table factor, is discussed in Chapter 32.]

3. Subtract the present value of the \$10,000 lump sum investment (\$10,000) from the present value of the \$15,000 single payment to be received (\$11,210). The net amount is +\$1,210; a positive net present value. [Note that if the \$15,000 were not received until the end of seven years, the present value of the receipt would be only \$9,977 (\$15,000 x 0.6651 discount factor), resulting in a negative net present value of \$23.]

B. Lump Sum Investment, Multiple Future Receipts

Assume an individual makes a lump sum investment at the beginning of year one of \$10,000, the present value of which is \$10,000 [Step (1)].

The expected return on this investment (received at each year end) and the present value of each receipt, discounted at 6%, are as follows [Step 2]:
```                      Amount              PV
Year                 Received            @ 6%

1                     \$ 2,000           \$ 1,887
2                       1,500             1,335
3                         750               630
4                         500               396
5                     \$10,000            \$7,473
Total Receipts        \$14,750           \$11,721
```

[The present value amounts in this table were computed using the Present Value Table in Appendix A and applying the factors to each particular cash flow as seen in the prior example. Computations made using different tables or software may vary slightly due to rounding differences, including the number of decimal places that factors are rounded.]

The net present value is therefore \$1,721 (\$11,721 present value of the future flow of receipts, less \$10,000 present value of the lump sum investment) [Step 3].

C. Multiple Investments, Multiple Future Receipts

Continuing the above example, assume that instead of one \$10,000 investment at the beginning of year one, there will be two \$5,000 payments, one at the beginning of year one, and the other at the beginning of year two. The present value of the investment, using the same 6% discount rate would be computed as follows [Step 1].
```                                             PV
Year     Payment        @ 6%

[Beginning]          1       \$ 5,000      \$ 5,000
[Beginning]          2         5,000        4,717
Total Payments               \$10,000      \$ 9,717
```

When multiple investment payments are required over a period of time, the present value of these payments must be determined. In this example, the present value of the first \$5,000 investment is \$5,000. The present value of the second \$5,000 is \$4,717, \$5,000 multiplied by the present value factor of 0.9434 (since the payment is made at the beginning of the second year, the present value factor for the end of the first year is appropriate). If it is assumed the present value of the receipts are the same as above, \$11,721 [Step 2], then the net present value of this investment is \$2,004 (\$11,721 - \$9,717) [Step 3]. This larger net present value is logical in this case, since, for an equivalent series of cash inflows, it is better to make the outflows/payments later rather than earlier, due to the time value of money.

II. Internal Rate of Return

In the discussion of the concept of net present value (NPV), it was defined as the difference between the present value of all future benefits to be realized from an investment and the present value of all capital contributions into the investment. The discount (interest) rate at which these two present values will be equal is the Internal Rate of Return (IRR) of the investment.

Stated in other terms, in computing IRR, the interest rate sought is that rate at which inflows of cash, discounted to present value, will equal the original (and subsequent, if applicable) principal contributions. It is a method of determining what percentage rate of return estimated cash inflows would provide based on a known investment (cash outflow). Actually, even the cash outflow must sometimes be estimated.

Internal rate of return is really the same as a present value computation except that the discount rate is either not known or not given. The financial advisor is therefore attempting to find that rate which will discount the future cash inflows so that they will precisely equal the investor's initial (and subsequent, if applicable) investment(s).

Confused? Here's an example.

Assume a client is considering the purchase of a \$100,000 unit of a limited partnership. The full \$100,000 is due at the beginning of the year. It is estimated that, after taxes, she should be receiving the following cash inflows at the end of each year.
```  End of
Year       In-Flow

1             \$10,000
2              10,000
3             120,000
```

A NPV analysis of this investment, looking at several alternative rates of return, would look as presented in Figure 33.1.

At an 8% rate, this investment has a positive net present value of \$13,092. That is, if \$100,000 was invested at 8% (for example, in a certificate of deposit), the present value of the future cash inflow should be \$100,000. But, in the investment above, the present value of the expected cash inflows was actually \$113,092, \$13,092 higher than it should be at 8%. Therefore, it is obvious that the investment is generating a significantly higher rate of return than 8%.

At a 10% rate, this investment has a positive net present value of \$7,513. Again, if \$100,000 was invested at 10%, the present value of the future cash inflow should be \$100,000. Since the present value of the expected cash inflows is actually \$107,513, it appears that the investment is generating a higher rate of return than 10% as well.

At a 20% rate, this investment has a negative net present value of \$15,279 - the present value of the expected cash inflows is only \$84,721, compared to the \$100,000 present value that would be anticipated if the invested \$100,000 actually earned 20%. Therefore, the investment must be generating a lower rate of return than 20%.

What is the actual rate of return on this investment? Clearly it lies somewhere between 10% and 20%. The exact rate is equivalent to the internal rate of return. The internal rate of return of the investment is approximately 12.9%. That is, the present value of the cash outflows (\$100,000) is roughly equal to the present value of the cash inflows (\$100,089) when discounted at a 12.9% rate.

How to Compute Internal Rate of Return

In an earlier example it was assumed a client was considering the purchase of a \$100,000 unit of a limited partnership, and that, after taxes, she should be receiving the following cash inflows:
```   End of
Year        In-Flow

1               \$10,000
2                10,000
3               120,000
```

In manually computing the IRR of this investment, the first step is to compute the NPV using a preliminary estimate of the IRR. If the first computation results in a positive NPV, a second calculation, using a higher discount rate, will be necessary. If a negative NPV is computed, the recalculation will require a lower discount rate. The process continues until a NPV of \$0 is reached (i.e., the rate at which the present value of the cash outflows equals the present value of the cash inflows).

The chart in Figure 33.2 provides an example of the series of iterative computations that might be necessary.

If 8% is used as the initial (test) discount rate, a positive net present value is found, and therefore a higher discount rate should be tried. On the second attempt, using 10%, there is still a positive NPV, although it is closer to \$0 as the internal rate of return if approached. The third computation, using 20%, yields a negative NPV. Therefore, the IRR must be between 10% and 20%.

After several attempts, a rate of 12.9% is tried, which results in a positive NPV of only \$89. It's getting close. For most planning purposes, this would be close enough. To do the job more accurately and efficiently, a business calculator or a computer could be used, which can repeat this iterative process hundreds of times for total accuracy in a matter of seconds.

Shortcomings of the IRR Method

What are the shortcomings of the IRR? There are several, but to some extent the shortcomings arise from widespread misconceptions and misunderstandings about the theoretical underpinnings as well as the proper application of the method, even among the most elite and sophisticated investors and investment advisers. Consequently, although IRR is one of the most commonly used tools; it is also the least understood and most misapplied method of evaluating alternative investments.

One of the most common misconceptions, even appearing in many, if not most, of the finance and investment textbooks used by the leading business schools, is that the IRR method inherently assumes that the cash flows from an investment being evaluated are implicitly reinvested at the computed internal rate of return of the investment itself. This is decidedly not the case, but this generally held misconception is one of the principal reasons the IRR is often misapplied.

The fact is that the IRR assumes that the cash flows are not reinvested, at any rate. Rather, it assumes cash flows from an investment are consumed when paid and never enter the analysis again. If it were assumed they are reinvested at the IRR, they would not be, or should not be, shown as cash flows.

The reality is that the IRR measures the rate of return on the unrecovered investment over time. Cash flows from an investment represent money taken out of the investment. Over time, only the investor's as-yet-unrecovered investment implicitly continues to earn the IRR. The following example shows five alternative investments and demonstrates the point. In each case, only the unrecovered investment earns the IRR; cash flows received back from the investments do not subsequently earn anything.
```                              Project Cash Flows

Year         A           B           C           D           E

0        (\$1,000)    (\$1,000)    (\$1,000)    (\$1,000)    (\$1,000)
1             100          50       (200)          200        600
2             100          50       (200)          200        600
3           1,100       1,215        1,793         869       (55)
IRR           10%         10%          10%         10%        10%
```

Each project has a 10% IRR.

Project A earns 10%, but also pays out \$100, or 10% of the initial \$1,000 investment, each year. Therefore, the unrecovered investment earning 10% remains at \$1,000 each year, which is similar to a 10% coupon bond purchased at par value.

Project B, in contrast, is similar to a bond that has been purchased at a discount. It earns 10% on the initial \$1,000 investment, but it only pays out \$50 each year. Therefore, Project B's unrecovered investment increases each year.
```                                                        Increase
(Decrease)
Unrecovered     Earnings @      (Pay-in)     In Unrecovered
Year     Investment      10% IRR         Payout        Investment

0-1          \$1,000        \$100.00         \$50.00            \$50.00
1-2           1,050         105.00          50.00             55.00
2-3           1,105         110.50       1,215.50        (1,105.00)
3                 0
```

Project B is the same as Project A with all but \$50 of the yearly cash flows essentially reinvested in the project at 10%. These implicitly reinvested earnings increase the unrecovered investment and earn the 10% IRR rate.

Project C has an even greater increasing unrecovered investment over time, as additional deposits are made to the investment and no withdrawals are taken.
```                                                       Increase
(Decrease)
Unrecovered    Earnings @     (Pay-in)     In Unrecovered
Year      Investment      10% IRR       Payout        Investment

0-1           \$1,000       \$100.00     \$(200.00)           \$300.00
1-2            1,300        130.00      (200.00)            330.00
2-3            1,630        163.00      1,793.00        (1,630.00)
3                  0
```

Since none of the earnings (or original investment) are taken out of Project C until the end, all of the implicit earnings are essentially reinvested in the project at 10%.

Project D has a decreasing unrecovered investment over time.
```                                                       Increase
(Decrease)
Unrecovered    Earnings @     (Pay-in)     In Unrecovered
Year     Investment      10% IRR       Payout        Investment

0-1           \$1,000       \$100.00       \$200.00         (\$100.00)
1-2              900         90.00        200.00          (110.00)
2-3              790         79.00        869.00          (790.00)
3                  0
```

In this case, the project throws off cash flows that are more than the implicit earnings each year, thereby reducing the unrecovered investment over time and, as a result, the implicit earnings.

Project E has decreasing and, ultimately, negative unrecovered investment over time.
```                                                      Increase
(Decrease)
Unrecovered    Earnings @     (Pay-in)     In Unrecovered
Year     Investment      10% IRR       Payout        Investment

0-1          \$1,000       \$100.00       \$600.00         (\$500.00)
1-2             500         50.00        600.00          (550.00)
2-3            (50)        (5.00)       (55.00)             50.00
3                 0
```

Project E can actually be viewed as a combination of investment and loan since it requires the investor to contribute additional dollars at the end to pay back early withdrawals in excess of the remaining unrecovered investment. Since there are no explicit payments on the loan, except to pay off the entire amount at the end, the implicit loan does, in fact, charge interest at a rate equal to the 10% IRR.

Although it is true that if the cash flows were reinvested at the IRR, the computed IRR would not change, this misconception about the IRR and reinvestment of cash inflows leads to errors in the application of the method. In some cases, it does not matter at what rate an investor can reinvest cash flows; in other cases, it makes all the difference in the world.

Specifically, when investors want to know what rate of return they will earn on an investment, their potential reinvestment rate for the cash flows is immaterial, generally. The IRR is the rate of return they will earn on the investment itself.

However, when investors want to use the IRR to compare investments that involve different initial outlays, cash flow patterns, and/or investment terms, they must explicitly account for the differences in cash flows, otherwise they, not the IRR method, are implicitly assuming cash flows are reinvested at the IRR. When reinvestment at the IRR is not realistic, the IRR method, improperly used, leads to poor choices among investments.

For instance, an investor is offered an immediate annuity paying \$5,092.61 per year for 20 years for an initial investment of \$50,000. He computes the IRR and finds that it is 8%. Whether he spends the cash flows or reinvests them is immaterial; the IRR method does not care. Whatever rate he might earn by reinvesting the cash flows paid to him does not change the return on the investment in the annuity. The annuity investment is earning 8%, and that is a fact regardless of whether or not and at what rate the cash outflows are reinvested.

Now, however, assume he is also offered another annuity that will pay him \$6,021.02 per year for 15 years for the same \$50,000 initial investment. He computes the IRR for this annuity and finds that it is 8.5%. Is this the better deal? Not necessarily, even though it is, in fact, paying a half a percent more on his investment.

The 8.5%, 15-year annuity might be the better deal if he could reinvest part or all of the \$928.41 annual difference in cash flows for the first 15 years so that he could at least match the cash flows he would otherwise receive for the last five years of the 8%, 20-year annuity. But if he cannot, the 8% annuity would be the better deal, even though it really is paying one-half percent less on his money, because of the inadequate return that will be earned in the final 5 years, compared to the continued 8% growth available in the first scenario.

Assume the investor feels he could earn 5% by reinvesting the difference in cash flows in an investment with the same level of safety and security as the annuity. The \$928.41 annual difference invested each year at 5% would grow to \$20,033.81 in 15 years. If this amount continues to earn 5%, he could withdraw \$4,627.31 each year for the next 5 years before exhausting the fund. But that is \$465.30 per year less than he would receive from the 8%, 20-year annuity, despite the fact that the investor applied all of his excess growth savings to the side investment at 5%. So he would be worse off investing in the 8.5%, 15-year annuity than if he invested in the 8%, 20-year annuity.

Obviously, if his best alternative investment with the same level of safety and security earns only 5%, then both of these annuities are good investments for him. The IRR of either one is greater than his opportunity cost rate or hurdle rate of 5%. If he had \$100,000 to invest and was limited to a \$50,000 investment in each annuity, it would be reasonable to invest in both of them. However, if he ranked the annuities based on their IRRs in order to choose between them, he would have made the wrong choice. Later in the chapter, it is explained how to properly use the IRR to compare investments.

In addition to it being often misunderstood and misapplied, the IRR method has other real weaknesses, but they can be overcome with proper use of the method. These limitations include:

* As described above, investors cannot use an IRR method (unless modified, as discussed below) to compare (directly) mutually exclusive investments, particularly when they have different time periods and cash flow timings (the investment with the highest IRR is not necessarily the best investment among a mutually exclusive set);

* The unmodified IRR method does not consider realistic (or sometimes, any) reinvestment rates for positive cash flows or realistic borrowing rates for negative cash flows over the holding period;

* An investment project may have multiple IRRs; and

* Solving for the IRR often requires a series of iterative calculations to determine the IRR since, for many types of IRR calculations, there is no single, closed-end formula to compute the IRR. However, financial calculators and computer software programs (such as the common spreadsheet) have built-in functions that are adequate to solve for the IRR in most cases.

Let us start with the problem of multiple IRRs. The other issues will be discussed in later sections of this chapter.

How can an investment have more than one IRR?

Let us prove it first and then explain why. Figure 33.3 shows an admittedly odd hypothetical investment that has four IRRs. The investment involves an initial pay-in of \$1,000, a payout of \$4,700, another pay-in of \$8,277.50, another payout of \$6,356.75, and then one last pay-in of \$1,828.78. As the table shows, if this cash flow stream is discounted at -5%, 10%, 25%, and 40%, in each case the net present value is zero. By definition, an IRR is a discount rate that equates the net present value of the cash flows to zero, so these rates are all IRRs.

Potentially, an investment has as many different real IRRs as there are changes in the sign, or direction, of the cash flows over time. In most cases when there is a single initial investment or a series of investment deposits (pay-ins), and, subsequently, a series of cash disbursements/returns (payouts) from the investment, there is only one distinct IRR, since there is only one change in the direction of cash flows. However, if the investment involves a series of pay-in deposits mixed over time with a series of payout returns, then each time the cash flow stream changes from pay-ins to payouts, or vice versa, there could potentially be more IRRs.

One problem with multiple IRRs is that it is not always easy to find them all and most software solutions will find just one, or refuse to solve for any at all if there could be more than one IRR. Regardless of the initial guess or seed one picks to start the search for multiple IRRs, the calculator or software program may home in on a particular IRR, even if it happens to be the one furthest away from the initial guess. There is generally no assurance that picking a seed closer to one of the IRRs than the others will cause the solution algorithm to find the nearest IRR. Consequently, even if it is known that there is more than one IRR, after one is found, changing the guess or seed to find the next IRR will not always be successful with the use of calculator or computer solvers, unless the entire process is undertaken manually. Finding each successive IRR tends to become increasingly difficult. (6)

In addition, just because there could be several distinct real IRRs does not mean that there actually are several distinct real IRRs. In some cases, some of the IRRs are in the realm of imaginary numbers: those odd numbers that involve the square root of -1. Although such numbers have real-world significance in the arena of physics and electronics, for example, they have no practical significance in the realm of finance and investments.

Even if one finds all the real IRRs, the question remains: which is the right one? Mathematically, they are all correct, but which is the correct one financially? Suppose one finds that an investment has two IRRs, -10% and +25%. If one invests in the project, is the investor losing 10% or making 25%? Perhaps both? Could the investor be both making 25% and losing 10%, thereby averaging 7.5%? Hardly!

The only way to solve the dilemma is to use one of the modified IRR methods discussed in the next section. Depending on how the investor actually will handle the cash flows from the project, it can be determined which IRR is in fact closer to the rate the investor will actually earn on the investment. The added bonus of learning about modified IRR methods is that they are also necessary whenever a person wants to compare or rank investment alternatives, one against another. The proper procedures for applying net present value analyses and internal rate of return methods when comparing or ranking investments is discussed in the last section of this chapter.

III. Modified Internal Rate of Return Methods

As a result of the problems inherent in the use of the regular IRR method, a number of modified IRR methods have been devised to circumvent the problems. These modified methods adjust the cash flows of the investment to account for reinvestment, borrowings, setting up sinking funds to cover later cash outflows, and the like.

Debates have raged in the financial and investment community as to which is the best method. The debates are quite surprising since the answer to that question is quite simple and obvious. The best method is that which most closely approximates how, in fact, an investor is most likely to handle the cash flows of the investment. Any idea that a one-size-fits-all approach to the problem is the right method is off the mark. None of the modified methods is the best approach all of the time and each of the modified methods is the best approach some of the time. The following discussion explains and demonstrates the application of five different modified IRR methods that may be suitable depending on the nature of the cash flows and how the investor plans to handle them.

The IRR-reinvestment-rate method is used to solve the problem of intermediate cash inflows by making specific assumptions about how those cash inflows will be reinvested. For example, Figure 33.4 shows the case where the initial investment is \$10,000. The investor expects to receive annual cash inflows of \$2,000 per year for five years, at which time he expects to sell or liquidate the investment for \$11,000 (receiving a total of \$13,000 in the fifth year). For simplicity, it is assumed all cash flows are at the end of each year. The regular, unadjusted IRR for this investment is 21.31%.

The investor expects to be able to reinvest the cash inflows at 10% over the five-year period. If he wants to compare this investment with other potential investments, which probably will have a different cash flow pattern over the five-year period, he has to account for the intermediate cash flows. If he expects to reinvest the cash flows at 10%, he can take each of the cash inflows and accumulate them at 10% until the end of the fifth year (the future value of each cash inflow grown at the reinvestment rate from payout until the end of year 5). By making this adjustment he converts the cash flow stream from the original series of six cash flows in years 0 through 5 into just two cash flows: his initial \$10,000 investment in year 0 and the sales or liquidation value of \$11,000 plus the \$12,210 accumulated balance of his reinvested cash flows, or \$23,210, in year 5. The IRR-reinvestment-rate with adjusted cash flows is 18.34% - this is the IRR for a single initial deposit in year 0 of \$10,000 and a final \$23,210 inflow at the end of year 5.

[FIGURE 33.4 OMITTED]

If the investor were instead planning to consume \$500 of the intermediate cash flows each year and to reinvest only the remaining \$1,500 of each annual cash inflow, the appropriate way to adjust the cash flows under this method would be to leave \$500 of the yearly cash inflows where they are each year and to accumulate only the \$1,500 he plans to actually reinvest. In this case, the cash flow stream would be adjusted to show a \$10,000 initial investment and cash inflows of \$500 in years 1 through 4. Then in the fifth year, the total cash inflow would be composed of \$11,000 from sales proceeds, \$2,000 from the fifth year cash inflow, and \$7,658 from the accumulated value of the \$1,500 reinvested in years 1 through 4 at 10%, for a total of \$20,658. The IRR of this cash flow stream is 18.94% - more than when all the cash flows are reinvested, but still less, of course (since the reinvestment rate is lower than the pure unmodified IRR), than when none of the cash flows are assumed to be reinvested.

The IRR-safe-rate method is used to solve the problem of additional cash outflows (or additional investments required after the initial investment in year 0) by assuming additional funds are set aside up-front as a sinking fund that is treated as part of initial year-0 investment to cover later cash outflows. This method is demonstrated in Figure 33.5. In this case the initial investment is \$30,000, but the project requires additional outlays of \$11,000 in year 1, \$10,000 in year 2, and subsequent outlays decreasing each year by \$1,000 to \$7,000 in year 5. In year 5, the investor expects to sell the investment for \$140,000. This is the type of cash flow pattern one might see in developing rental or commercial real estate or in bringing a new drug to market. In each case, one would expect a number of years of continuing cash outlays while the real estate is being cleared, the buildings are built, and tenants are found, or while the drug is researched, developed, and tested. However, once the property is completed and rented or the drug is ready for market, the building or rights to the drug can be sold for large amounts.

In this case, the IRR-safe-rate method is an appropriate IRR method if the investor plans to set aside enough capital at the beginning to assure that he can make the cash outlays when they are needed in future years. If the investor assumes he can invest the sinking fund at a safe rate of 8%, the additional amount he would need to set aside the first year to cover the later year's cash outlays is \$36,546 (determined by summing each future required cash outlay discounted by the 8% investment rate). By doing so, the investor converts the original cash flow stream into just two cash flows: an initial total outlay of \$66,546 in year 0 and a \$140,000 cash inflow in year 5. Before adjusting the cash flows, the IRR was 19.05%. After the adjustment, the IRR fell to 16.04% (because some funds were specifically invested at a much lower rate (8%) for a period of time).

[FIGURE 33.5 OMITTED]

The IRR-safe-rate method and the IRR-reinvestment-rate methods may be combined, if appropriate. This method would typically be used when cash flows continue to be flowing out for several years and then change to inflows during the term of the investment. Figure 33.6 shows how this method may be employed. In this case cash flows out through year 2 of the project and then becomes positive until year 5 when the investment is assumed to be sold for \$140,000. If the investor plans to set aside a sinking fund invested at a safe rate of 6% to cover the additional cash outflows in years 1 and 2 and also expects to be able to reinvest the cash inflows in years 3 and 4 at 8%, he can convert the original cash flow stream once again into just two cash flows in year 0 and year 5. The IRR of the unadjusted cash flows is 20.60%. After adjusting the cash flows, the IRR is 19.05%.

The IRR-borrowing-rate method solves the problem of later cash outflows by assuming money will be borrowed to cover future cash outflows and will be repaid out of later cash inflows. Figure 33.7 demonstrates the use of this method.

In this case the investor expects ongoing cash outflows through the first 3 years of the project. Instead of setting money aside up front to cover these later cash outflows, this investor plans to borrow money as needed from a line of credit and then repay the loans out of future cash inflows. The investor's borrowing rate is 12%. The \$10,000 borrowed in year 3, plus interest, is repaid in year 4 out of the \$20,000 cash inflow in that year, leaving \$8,800. This remaining \$8,800 cash inflow in year 4 is then applied to reduce the outstanding balance of the amount borrowed from year 2, leaving a remaining balance yet to be repaid of \$16,288. This amount plus the entire \$20,000 borrowed in year 1, plus the accruing interest are carried forward to year 5 at 12%. After applying the \$20,000 cash inflow from operations in year 5 against this loan balance, the remaining balance is \$29,713. This remaining amount is repaid from the net sale proceeds.

By applying this methodology, the investor has once again converted the cash flow stream from one involving a number of different payouts and pay-ins over a number of years into just two cash flows at the very beginning and the very end of the investment period.

The IRR before adjusting the cash flows is 17.08%. After adjusting the cash flows, the IRR jumps to 18.16%. What this result shows is the benefit of financial leverage. Any time a person can borrow at a rate less than the rate the person is earning on an investment, borrowing to finance part of the investment will increase return because the person is earning more on the amount he borrows than he is paying to borrow it. Conversely, any time a person reinvests cash flows at a rate less than the unmodified IRR on the investment, the modified IRR will be less than the unmodified IRR because the person is now earning less on this part of the investment than on the underlying investment itself, which reduces the overall return on the total investment.

[FIGURE 33.6 OMITTED]

The Adjusted Rate of Return (ARR) is the last modified IRR method that will be described here. The modified IRR methods described so far calculate the actual rate of return earned by an investor who really intends to handle cash flows in the manner applied by the chosen method. In contrast, the adjusted rate of return method does not.

The adjusted rate of return is the method preferred by many leading tax and investment specialists, especially in real estate. It seems to be preferred largely because its proponents believe that a one-size-fits-all method is an appropriate approach to modified IRR analyses. However, any methodology that adjusts cash flows in a manner starkly different than how the cash is actually likely to flow will derive IRR results that have as much bearing to the real IRR as the adjusted cash flows have to the real cash flows. In other words, one cannot pretend that cash is going to flow one way when it is actually going to flow another way and expect the results one gets from those assumptions to have any resemblance to actual values.

The whole point of time value analysis is that the amount and timing of cash flows matters. If a person moves a cash flow from one period to another in his methodology when the cash flow is not actually going to move from one period to another, the IRR computed will be distorted. The more a modified IRR method moves or adjusts cash flows in a manner that does not reflect the underlying reality, the greater will be the distortion and the more useless or worthless the results derived from the method employed.

The ARR is applied in the following manner:

1. discount all cash flows into the project (negative cash flows or investments into the project) at a safe rate of return back to period 0 (find the PV of the pay-ins);

2. compound all cash flows out of the project (positive cash flows or returns of investment) at a safe rate to the end of the holding period (find the FV of the cash flows from the project); and

3. The ARR is the discount rate that equates the PV of cash flows into the project with the discounted FV of cash flows out of the project.

Figure 33.8 shows each of the methods discussed in this section, as they would be applied to a common set of cash flows (including an additional method shown in pieces but not explicitly demonstrated thus far, the IRR-Borrow/Reinvestment Rate Method, which merges the already-covered IRR-Borrow and IRR-Reinvestment methods). To show how the results differ for each modified IRR method, the rate for all types of adjustments is the same, 6%, regardless of whether the adjustment is for reinvestment, borrowing, or for a sinking fund invested at a safe rate.

[FIGURE 33.7 OMITTED]

The great disparity in the computed IRRs--ranging from 14.4% to 30.66%--using the various methods for the same original cash flows, proves how important it is to use the appropriate method for the circumstances. Using an inappropriate method may lead an investor to rejecting an investment that is really an acceptable investment, or worse, accepting an investment that is really unacceptable.

It is perfectly possible that the same investment could be acceptable to one investor yet unacceptable to another, even if their opportunity cost rates or hurdle rates are identical. For example, assume two investors have identical hurdle rates of 22% for the investment shown in Figure 33.8. However, for investor 1, the IRR-safe-rate method most closely approximates how he will handle the cash flows from the investment while, for investor 2, the IRR-safe-rate/reinvestment-rate method better describes how he would handle the cash flows of the investment. In this case, the investment is acceptable to investor 1, since it earns 25.18% in his circumstances, but is unacceptable to investor 2, since it earns only 20.44% in his circumstances.

IV. Pay Back Period

Pay back period analysis is a time value of money concept. This method compares alternative investments by measuring the length of time required to recover the original investment. From this perspective, the investment that returns the original capital in the shortest period of time is the best investment.

The major flaw in this analytical technique is obvious: taken to its extreme, an indiscriminate investor would choose a deal requiring a \$10,000 investment which paid back \$15,000 in one year rather than an alternative which required the same capital outlay but returned \$25,000 in two years. In other words, this method fails to capture the potentially extremely favorable returns that may accrue under one investment beyond the end point of the pay back period of another investment choice.

V. Cash on Cash

Cash on cash analysis, as its name implies, focuses on the amount of cash generated by the investment. It ignores both taxes and the potential gain from any sale.

[FIGURE 33.8 OMITTED]

To compute cash on cash return, divide the annual cash flow by the cash investment. For example, let us assume that a woman plans to invest \$10,000 in the stock of Z-Rocks Corp. Each year she would receive a dividend of \$1,000, and she expects to receive \$15,000 upon the sale of the stock in three years.

She is considering an alternative investment for the \$10,000, Eye-B-Em Stock, which yields a cash distribution of only \$600 per year, but which she believes will be worth \$25,000 at the end of the three-year investment period. Under the cash on cash method her return on the first investment is 10% per year (\$1,000 dividend divided by the \$10,000 investment), while the return on the alternative is only 6% per year (\$600 divided by the \$10,000 investment).

Using only the cash on cash method to evaluate the investments, she would choose the Z-Rocks stock, because of the higher annual dividend. However, if she had compared the investments by looking at their relative internal rates of return or adjusted rates of return, she would probably have chosen the alternative. Ultimately, which is appropriate depends upon her needs and plans for use of the cash inflows from the investments.

VI. Using Analytical Methods to Compare Investments

The net present value method or the unmodified IRR method should virtually always give the same accept or reject decision regarding an investment when the investment is evaluated independently of other investments. For example, for investors whose opportunity cost rate or hurdle rate is 15%, all investments with an IRR above 15% will also have positive net present values (and thus should be acceptable). Conversely, all investments with positive net present values will have IRRs in excess of 15% (and thus should still be acceptable). Investments with IRRs less than 15% will also yield net present values that are negative, and should be unacceptable.

Independent Versus Mutually Exclusive Investments

If an investor is comparing investments and trying to choose the best among a number of investment alternatives, picking the investment with the highest NPV or the highest IRR may not lead to the best investment.

As long as investments are evaluated independently, the NPV and unmodified IRR will work just fine in determining whether an investment is acceptable or unacceptable. Two projects are considered independent if the acceptance or rejection of one project has no bearing on the acceptability or feasibility of investing in the other project. However, if the investments cannot be evaluated independently and an investor cannot choose to invest in both alternatives at the same time, then the NPV method and/or the IRR method must be applied in a more involved manner to determine the better alternative.

Two investments that are not independent are said to be mutually exclusive. That is, they are mutually exclusive if the acceptance of one project precludes the acceptance of the other.

Example 1. A plot of land can be used for farming or as the site for an office complex, but not for both.

Example 2. Corporate money allocated to payroll increases cannot be used to finance an advertising campaign.

Example 3. Like the annuity investor described earlier, the choice is between two annuities and the investor may choose to invest in one, or the other, or neither, but not both.

Table A shows the forecasted cash flows from two investment projects. The project with highest ranking relative to an independent standard (the NPV or IRR) is not necessarily the best alternative if the projects are mutually exclusive.

If evaluated independently, project 1 is acceptable if the investor's alternative rate on the use of funds is less than 40%. Project 2 is acceptable if the alternative rate on the use of funds is less than 25%.

However, if the two investments are evaluated as mutually exclusive investments, Project 1 may not be superior to Project 2.

Table B presents a terminal value analysis of projects 1 and 2 assuming a 10% cash reinvestment rate. The investor is better off after the eight years by about \$385.65 if he invests in Project 2 rather than in Project 1, despite Project 1's much higher rate of return. This result occurs because of the relatively low cash reinvestment rate.

As was explained earlier, the IRR is earned on the unrecovered investment. Project 1 correctly earns 40%, but it pays out all of its earnings. Therefore, the unrecovered investment remains at \$1,000 and the cash flows can only be reinvested at 10%, not 40%. In contrast, Project 2 earns 25% on an ever-increasing unrecovered investment, since it makes no cash payments until the end of year eight. In the meantime, the implicit earnings are effectively being reinvested in the project and earning 25%. As Table B demonstrates, earning 40% on \$1,000 for eight years while reinvesting the earnings at only 10% is less profitable than investing an initial amount of \$1,000 at 25%, but implicitly reinvesting all the earnings at that same 25% rate.

Which investment is actually better depends critically on the assumed reinvestment rate. In this example, for reinvestment rates that are less than 12.22%, Project 2 is superior. Alternatively, if reinvestment rates are greater than 12.22%, Project 1 is superior.

Before proceeding on, let us consider another measure closely related to net present value. Net present value is the difference between the discounted benefits and the discounted costs of a project. If the present value of the benefits exceeds the present value of the costs, the project is an acceptable investment.

The Discounted Accounting Return Ratio (DARR) is the ratio of discounted benefits to discounted costs (also called the Discounted Profitability Index or DPI). It is a measure of the PV of benefits to the PV of the costs of an investment relative to each dollar of investment. A related value, called the Undiscounted Accounting Return Ratio (UARR), or Undiscounted Profitability Index (UPI), is essentially a version of the payback method described earlier.

Figure 33.9 presents an example of three projects, B, C, and D, with cash flows and both the DPI and UPI for each investment.

The decision rule when using the DPI is simple: a project is acceptable when the ratio is greater than one. It gives exactly the same accept and reject decisions as the net present value method.

The DPI is introduced because it is sometimes beneficial to view the feasibility of investments in relative terms (per dollar of investments), rather than in absolute terms, as is the case with the DPI's sister method, the net present value method.

Incremental Analysis Methodology

When evaluating mutually exclusive investments or comparing any two or more investments to determine the better or best among the alternatives, the traditional NPV and IRR evaluation methods are up to the task, but they must be applied incrementally. They have to be applied following the three basic rules, which state that the analysis must:

1. Take account of all costs and all sources of return;

2. Give more weight to early cash flows than later ones (time-value weighting); and

3. Standardize choices (compare apples with apples), which means they must account for differences in:

* Initial outlays;

* Holding periods;

* Cash flow payouts;

* Risk levels; and

* Time perspective.

Standardizing the choices means that if the initial outlays are different, the difference must be accounted for in the analysis. If the length or term of one investment is different than another investment, the analysis must account for how cash that is generated by the investment that ends first will be reinvested until the end of the term of the second investment. If the cash flow streams are different, the analysis must account for how those differences in cash flows will be handled. If the risk levels are different, appropriately different discount rates or hurdle rates must be employed for each of the investments. Finally, each of the investments or projects must be evaluated as of the same point in time. If a present value is used to evaluate one investment, present values must be used to evaluate all investments. It really makes no difference whether the investments are evaluated as of today. They could be evaluated as of a future date, such as the end of the term of the longest project. However, if one project is evaluated as of a future date, all the others must be evaluated as of that same future date.

An incremental analysis is applied as follows when one is using the NPV method, the DPI method, or the IRR method.

First, each of the investments in the comparison must be evaluated as if they were independent investments by applying any of the traditional NPV, DPI, or unmodified IRR methods (bearing in mind that, due to different risk levels, they may have different hurdle rates for initial rejection). Any investment that is unacceptable when evaluated as a completely independent investment can be disregarded from further consideration.

Second, for ease of analysis, the surviving investment alternatives are sorted by the size of the initial outlay, from smallest to largest. (This is not really necessary, but it makes the process less confusing.)

Third, starting with the investments with the smallest and next to smallest initial outlays, the investments are compared head-to-head based upon the NPV, DPI, or IRR of the additional or incremental investment in the larger outlay investment over the smaller outlay investment. The winner of the comparison goes on to challenge the next larger outlay project until all investments have been challenged and the winner determined.

The basic premise behind the incremental analysis is that, when one is selecting between two mutually exclusive investments (e.g., B and D), the mutually exclusive project involving the larger initial outlay (D) can be evaluated as if it were a package of two separate projects. The cash flows of the investment with the smaller outlay (B) are subtracted period by period from the cash flows of the larger outlay investment (D), leaving a residual or incremental series of cash flows in the larger outlay investment. Therefore, since cash flows are carved out of the larger investment (D) that exactly match the cash flows of the smaller investment (B), those cash flows negate each other and one is left with just the difference in cash flows between the two investments, which can then be evaluated.

The fourth and final step at each stage of the incremental analysis is to apply the NPV, DPI, or IRR analysis to the incremental cash flows. The decision rule is simple. If, after applying the chosen methodology, the incremental project (D - B) is an acceptable investment in its own right, then the larger outlay project is the better investment. If the incremental investment in the larger outlay investment is rejected by the application of the chosen methodology, then the smaller outlay project is the better investment. By design, the cash flows of the larger project (D) are segregated into two separate cash flow pools. One of the pools is exactly equal to the cash flows from the smaller investment (B) and the other is equal to the difference in cash flows between the larger and smaller investment (D-B). Since the B pool of D obviously matches the real B project dollar for dollar, only the difference pool of cash flows, or the incremental investment in D over B, matters. If the D-B pool is an acceptable investment in its own right, then by selecting D, the investor gets everything he would have had by investing in B plus a profitable additional investment.

Figure 33.10 shows an example applying the incremental analysis to two projects B and D and using the DPI criteria to pick the winner. The analysis assumes an alternative rate of 10%.

Project B has the greater DPI, so if the investments were ranked by their DPI, project B would be the winner. However, Project D is the superior project. Although Project B has a DPI (1.331) that is greater than Project D's DPI (1.325), the incremental investment in Project D is itself still an acceptable investment (DPI 1.314). Therefore, given a mutually exclusive choice between B and D the investor would be better off choosing D despite B's higher DPI.

A terminal value comparison (see Figure 33.11) confirms that D is superior to B. Since D involves a \$500 greater initial outlay than project B, Project B must be supplemented with a side fund of \$500 to account for the money that otherwise would have been invested in Project D if Project D had been the selected investment. With a reinvestment rate of 10%, the future values of D's cash flows are \$251.58 greater than Project B's.

The following section presents an evaluation or comparison of five mutually exclusive investments using the IRR method in an incremental analysis. Once again, a hurdle rate or opportunity cost of 10% is assumed for the analyses. The five projects F through J have the following cash flows.
```Projected Cash Flows of Potential Projects

Project/
Year           F           G           H           I           J

0          (\$1,100)    (\$1,500)    (\$2,100)    (\$2,400)    (\$2,400)
1               400         550       1,000       1,120         200
2               440         700       1,100       1,000         200
3               444         633         760       1,079       3,156
```

Each project is first evaluated independently to weed out those that might be unacceptable before comparing them to the others.
```Project F

Year      Cash Flow    PV @ 6%     PV @ 8%    PV @ 10%

0          (1,100)     (1,100)     (1,100)     (1,100)
1              400         377         370         364
2              440         392         377         364
3              444         373         352         334

PV of Cash Flows:           42         (1)        (38)
IRR of Project F is 8%: Unacceptable

Project G

Year     Cash Flow     PV @ 10%     PV @ 12%     PV @ 14%

0         (1,500)      (1,500)      (1,500)      (1,500)
1             550          500          491          482
2             700          579          558          539
3             633          476          451          427

PV of Cash Flows:           55            0          (52)
IRR of Project G is 12%: Acceptable if Independent

Project H

Year      Cash Flow     PV @ 17%     PV @ 18%     PV @ 19%

0           (2,100)      (2,100)      (2,100)      (2,100)
1             1,000          855          847          840
2             1,100          804          790          777
3               760          475          463          451

PV of Cash Flows:             34            0         (32)
IRR of Project H is 18%: Acceptable if Independent

Project I

Year      Cash Flow     PV @ 15%     PV @ 16%     PV @ 17%

0           (2,400)      (2,400)      (2,400)      (2,400)
1             1,120          974          966          957
2             1,000          756          743          731
3             1,079          709          691          674

PV of Cash Flows:             39            0         (38)
IRR of Project I is 16%: Acceptable if Independent

Project J

Year      Cash Flow     PV @ 14%     PV @ 15%     PV @ 16%

0           (2,400)      (2,400)      (2,400)      (2,400)
1               200          175          174          172
2               200          154          151          149
3             3,156        2,130        2,075        2,022

PV of Cash Flows:             59            0         (57)
IRR of Project J is 15%: Acceptable if Independent
```

When each of the five projects, F through J, is evaluated independently, all but Project F are acceptable given a hurdle rate of 10%. The following table presents the independent rankings based upon their IRRs.
```Summary of Independent Rankings

Decision
Project     IRR    Rank     (10% Hurdle Rate)

H           18%      1          Acceptable
I           16%      2          Acceptable
J           15%      3          Acceptable
G           12%      4          Acceptable
F           8%       5         Unacceptable
```

As the subsequent analyses will show, Project H has the highest IRR, 18%, but it would not be the best investment if these five projects were not independent (that is, if they were mutually exclusive and the investor could choose one, and only one, of these alternatives).

Rather, Project J is the best alternative. The following pair-wise comparisons lead to the selection of Project J as the best alternative despite the fact that both Projects H and I have higher IRRs. The pair-wise comparisons begin with an incremental analysis of H relative to G, which are the two lowest outlay projects with an acceptable independent IRR.
```Incremental Analysis (H-G)

Cash Flows

Year                                H           G         (H-G)

0                                (2,100)     (1,500)      (600)
1                                  1,000         550        450
2                                  1,100         700        400
3                                    760         633        127
Present Value of Cash Flows:

Present Value of (H-G)

Year                               30%         36%         40%

0                                 (600)       (600)       (600)
1                                   346         331         321
2                                   237         216         204
3                                    58          50          46
Present Value of Cash Flows:         41         (3)        (29)
```

IRR of the incremental investment (H-G) is about 36%, which is acceptable; therefore H is superior to G.

It is probably not surprising that H beats G soundly, earning a whopping 36% on the \$600 additional investment in H over the investment in G, since H has an independent IRR of 18% and G's independent IRR is just 12%. Next, the winner of round 1, H, is compared to the next higher outlay project, I.
```Incremental Analysis (I-H)

Cash Flows

Year                   I          H        (I-H)

0                                (2,400)    (2,100)      (300)
1                                  1,120      1,000        120
2                                  1,000      1,100      (100)
3                                  1,079        760        319
Present Value of Cash Flows:

Present Value of (I-H)

Year                   4%         5%         6%

0                                  (300)      (300)      (300)
1                                    115        114        113
2                                   (92)       (91)       (89)
3                                    284        276        268
Present Value of Cash Flows:           7        (1)        (8)
```

IRR of the incremental investment (I-H) is about 5%, which is unacceptable; therefore I is not superior to H (i.e., I is rejected and H is retained as the best yet project).

Project H wins again! Project I involves a greater outlay than project H, so if an investor picked project I, he would be making an incremental investment in Project I of \$300 above the amount otherwise invested in project H. Since cash flows from I have been carved out to exactly match the cash flows of Project H, that part of the investment in project I is assumed to earn 18%, exactly what Project H earns. However, the additional \$300 invested in project I only earns about 5%. Therefore, the investor would better off choosing to invest in Project H and to instead invest the additional \$300 at his alternative rate of 10%, which is 5% more than he would earn on that money if he chose to invest in Project I.

Finally, Project H is compared to its last remaining competitor, Project J.
```Incremental Analysis (J-H)

Cash Flows

Year                    J           H         (J-H)

0                                 (2,400)     (2,100)       (300)
1                                     200       1,000       (800)
2                                     200       1,100       (900)
3                                   3,156         760       2,396
Present Value of Cash Flows:

Present Value of (J-H)

Year                   10%         11%         12%

0                                   (300)       (300)       (300)
1                                   (727)       (721)       (714)
2                                   (744)       (730)       (717)
3                                   1,800       1,752       1,705
Present Value of Cash Flows:           29           1        (26)
```

IRR of the incremental investment (J-H) is about 11%, which is acceptable; therefore J is superior to H.

It may seem feasible that Project J could be superior to Project H, despite Project H's higher IRR, because Project J involves an additional outlay of \$300 that need only earn more than the 10% hurdle rate to be an acceptable investment. The analysis has been constructed so that the first \$2,100 of Project J's investment and the corresponding cash flows exactly matches Project H's investment and cash flows. Therefore, this portion of Project J's investment exactly matches Project H's performance. The additional \$300 investment in Project J then produces incremental cash flows that are in themselves acceptable relative to the hurdle rate of 10%. Based upon the incremental analysis, the first \$2,100 of the investment in Project J matches Project H's 18% rate of return. The additional \$300 invested in Project J earns about 11%, which is still greater than the hurdle rate. In other words, Project J exactly matches Project H's performance with an additional kicker of about an 11% return on the additional \$300 investment.

So, despite Project H's 3% advantage in IRR when evaluated independently, Project J is still the better investment.

The reason for this apparent contradiction is that the IRR is an annual rate of return on the unrecovered investment in Project J. Although Project J earns only a 15% rate of return on investment, versus 18% for Project H, the funds invested in Project J earn the 15% rate of return for a longer period of time than the funds earn the 18% rate of return in Project H. An investor in Project H earns 18%, but recovers the investment relatively quickly and then may reinvest those cash flows at only 10%. In contrast, since the payout in Project J comes later, an investor makes up for the lower 15% internal rate of return by effectively earning it for a longer period.

The discount rate, hurdle rate, alternative rate, or opportunity cost rate (whatever it is called) plays a critical role in the battle for investment dollars. Table NPV shows the results of using different discount rates in conjunction with the NPV method in an incremental analysis of Projects F through G. For discount rates below 11% (the rate earned on the incremental investment in Project J over Project H), Project J wins the right to the investor's money. But for discount rates above 11%, Project H jumps to the lead.
```Table NPV
NPV of Projects F, G, H, I, and J Assuming Various
Alternative Rates of Return

Hurdle Rate        F             G             H

0%                184.00        383.00        760.00
5%                 63.59        205.54        506.63
10%              (39.14)         54.09        289.18
12%              (76.06)        (0.34)        210.72 *
15%             (127.53)       (76.23)        101.04 *
18%             (174.78)      (145.91)          0.02 *

Hurdle Rate        I             J

0%                799.00      1,156.00 *
5%                505.78        698.15 *
10%               255.30        318.26 *
12%               165.20        184.39
15%                39.52          0.26
18%              (79.95)      (166.03)

* The superior investment if the projects are mutually
exclusive.
```

WHERE CAN I FIND OUT MORE?

1. Frank Fabozzi, Fixed Income Mathematics: Analytical and Statistical Techniques (Chicago, IL: Irwin Professional Publishers, 1996).

2. David Spaulding, Measuring Investment Performance: Calculating and Evaluating Investment Risk and Return (New York, NY: McGraw-Hill, 1997).

3. Robert Rachlin, Return on Investment Manual: Tools and Applications for Managing Financial Results (Armonk, NY: Sharpe Professional, 1997).

4. Arefaine Yohannes, The Irwin Guide to Risk and Reward (Chicago, IL: Irwin Professional Publishers, 1996).

5. Birrer & Carrica, Present Value Applications for Accountants and Financial Planners (New York, NY: Quorum Books, 1990).

6. Charles Akerson, The Internal Rate of Return in Real Estate Investments (Chicago, IL: American Inst. of Real Estate Appraisers, 1988).

7. David Leahigh, A Pocket Guide to Finance (Fort Worth, TX: Dryden Press, 1996).

8. Calculator Analysis for Business and Finance (New

York, NY: McGraw-Hill).

9. HP-12C Owner's Handbook and Problem-Solving Guide (Hewlett-Packard Co.).

Question--What is meant by sensitivity analysis?

Answer--In any quantitative analysis, it is necessary to make a number of assumptions and predictions (and at times, just some guesstimates). These inputs include: (1) yields, (2) rates of inflation, (3) growth rates, (4) effective tax rates, and (5) timing of inflows and outflows.

Obviously, these variables, by definition, cannot be predicted with certainty. Therefore, it is important to determine the sensitivity of the results of the analysis to fluctuations (upward or downward) in any of the variables relevant to decision-making. Sensitivity analysis can be defined as a procedure by which relevant variables are changed one at a time to determine the effect on the overall results.

Sensitivity analysis utilizes the mathematical methods of comparison discussed above. It enables the planner to ascertain which assumptions or predictions must be most-closely-monitored, emphasized, or modified, in order to obtain the most realistic or accurate conclusion possible.

Sensitivity analysis can be used to establish parameters of potential results such as worst down-side risk, highest upside potential, and most probable result. Once the worst case, best case and most probable scenarios have been developed, the investor can then make subjective judgments as to whether or not the investment is appropriate.

Question--In analyzing the viability of a real estate investment, how does one determine what rental revenues, expenses, and final selling price to use in the analytical computations?

Answer--Forecasting revenues, expenses, and final selling price in a real estate investment involves a multiplicity of factors, such as: (1) market demand for the specific location, (2) price and availability of comparable properties, (3) technological and other factors which may result in a greater or lesser property value, (4) government restrictions on alternative uses, and (5) neighborhood influences. The determination of what values to use will be based primarily on available historical data that will require modifications based on expected future trends.

Question--How can a planner help an investor determine how long to hold an investment?

Answer--The internal rate of return and adjusted rate of return concepts can be used to mechanically determine the most desirable holding period. For instance, IRR analysis could be used to ascertain the rate if the investment were held for 5 years, 6 years, 7 years, etc. Mathematically, the number of years that produces the highest IRR would be the one selected.

Some investment analysts ask, "What would the incremental effect be if the investment were held one additional year?" This one-year-more concept is often called the marginal method. It compares the benefit of waiting one more year to the benefit of investing the cumulative benefits received in an alternative investment.

Basing the decision solely on the mathematical results of this marginal analysis, it will make sense to remain in the original investment for at least one more year as long as the marginal IRR is greater than what the investor can receive from an alternative investment.

RATE OF RETURN CONCEPTS

Rate of return means different things in different contexts, depending upon a host of factors. Consequently, financial economists and investors use a wide variety of terms to describe returns in these different contexts, such as total return, holding-period return, annualized return, simple return, compound return, arithmetic average return, geometric average return, time-weighted return, dollar-weighted return, nominal return, real (inflation-adjusted) return, risk-adjusted return, after-tax return, taxable-equivalent return, internal rate of return, and various modified internal rate of return measures. Although financial professionals do not always define and use these terms in exactly the same way, the following descriptions are commonly accepted definitions for the various measures.

Total Return

The term total return means the total gain or loss over a specified period relative to an initial dollar investment at the beginning of the period. All investments potentially have some combination of six components or sources of return for which a proper analysis must account:

* Cash inflows from the investment (outflows to the investment); plus (minus)

* Price or capital appreciation (price or capital depreciation); plus (minus)

* Debt amortization (negative amortization); plus (minus)

* Tax shelter (tax payments); plus (minus)

* Other tangible property or boot received (paid); plus (minus)

* Intangible but measurable benefits or services received (paid), if any.

In addition, since the ultimate goal of investing is to enhance real future wealth, investors cannot ignore the potential erosion of their future purchasing power as a result of inflation.

Cash flows in the form of interest payments, dividends, rents, royalties, and the like are the most obvious and ubiquitous source of return. Also, for many investments, changes in the market value of the investment each period, such as a stock's price increase or decrease, are part of the investor's total return for each period. When an investment is leveraged through the use of borrowing, the analysis must account for changes in the debt balance. If an investment generates a net cash flow of \$100, but the investor's debt is also reduced (debt amortization) during the period by \$10, the investor is in fact \$110 wealthier than at the start of the period.

Although total returns are frequently computed without accounting for taxes, investors can use and spend only the amount they get to keep after paying taxes. Consequently, any income, capital gain, personal property, or other tax payments attributable or allocable to the investment reduce the investor's total return. Conversely, if interest payments on debt used to finance investments are tax-deductible or the investment generates depreciation deductions, real estate tax deductions, or other tax deductions, the tax savings so generated each period increase the investor's return above and beyond the amount measured solely by cash flow, price changes, and debt reduction.

If an investor receives any other tangible property or boot, such as shares of stock in lieu of cash interest payments, the total return includes the market value of the property or boot received. Finally, the total return must include the value of any intangible but measurable benefits received. For example, if an investor purchases a commercial office building and in the transaction receives an easement for a private access road, the market or use value of the easement for each period should be included in the total return. (1)

To compute the rate of return on investment for any period one simply divides the total return by the investor's equity in the investment at the beginning of the period.

An investor's equity in an investment at any point in time is the sum of all cash contributions, plus debt amortization, plus price appreciation or less price depreciation. For example, assume an investor purchases a residential rental unit for \$100,000 with an initial cash outlay of \$20,000 and borrowing of \$80,000. The investor's initial equity is \$20,000 (the amount of cash paid, not the total purchase price of the property). After one year, the investor's net operating cash flow (after debt-repayment) is \$5,000, the debt balance has been reduced by \$1,500, and the market value of the property has risen to \$102,500. Net tax savings (on interest and depreciation deductions in excess of taxable rental income) equal \$1,000. The investor's total return for the year is \$10,000 (\$5,000 cash flow + \$1,500 debt reduction + \$2,500 price appreciation + \$1,000 net tax savings), which is a 50% return on the initial \$20,000 equity investment. At the end of the year, the investor's equity in the investment has increased to \$24,000 (\$20,000 original equity + \$1,500 debt reduction + \$2,500 price appreciation). Each year or period of evaluation may generate a different rate of return (as both the return for the time period, and the underlying equity, may change). Consequently, different measures of the rate of return are often used to generalize the results or performance of an investment over a number of holding periods.

Holding-Period Return

The holding-period return is simply the total return for a specified period over which an investment is held. For instance, suppose a person invested \$1,000 in the S&P 500 index at the beginning of 1997 and held the investment for 5 years until the end of 2001 while reinvesting the cash flows. This investor would have accumulated about \$1,662.45 by the end of 2001 (ignoring expenses). (2) The 5-year holding-period return is \$662.45 (\$1,662.45- \$1,000) and the 5-year holding-period rate of return is 66.245% (\$662.45 / \$1,000). In contrast, if the dividends were not reinvested, the value of the stocks would have grown to about \$1,550.03 at the end of 2001 and total dividend payments over all 5 years would have totaled about \$85.96, for a cumulative holding-period return of \$635.99 (\$550.03 + \$85.96) and a holding-period rate of return of about 63.6%. The larger value for the first holding-period return is attributable to the earnings on reinvested dividends within the holding period (the market having risen during this period). (3)

Annualized Returns

Since comparing investment returns over different holding periods is difficult, the standard practice is to express returns in annualized terms. One may express the annualized return in one of two ways--as a simple or a compound annual return. Which expression is the better measure depends on the context in which it is used, as is described below. Two closely related terms that are also applicable in these contexts are the arithmetic average annual return and the geometric average annual return.

Simple and Arithmetic Average Annual Returns

Investors compute the simple annual return by dividing the holding-period return by the number of years in the holding period. Essentially, the simple annual return is the rate of return computed by stretching or shrinking the holding period to be equal to just one year and stretching or shrinking the holding-period return proportionately. As a result, it computes the annual rate as if the holding period were actually one year without assuming any reinvestment of principal and interest. For instance, in the example above, the annualized return for the S&P 500 investment earning a 66.245% 5-year holding-period rate of return is 13.25% (66.245% / 5). In other words, by shrinking the holding period from five years to just one year, the rate of return similarly shrinks to just one fifth its original value of 66.245% to 13.25%.

If the holding-period is less than a year, say a month, then the simple annual return of the 1-month holding-period return is determined by dividing the 1-month holding-period return by 1/12--in other words, by multiplying or stretching the period by a factor of 12. For example, suppose an investor pays \$9,925 for a T-bill maturing at \$10,000 in three months. The 3-month (1/4th year) holding-period return is \$75 and the 3-month holding-period rate of return is 0.756% (\$75 / \$9,925). Since the holding period must be stretched by four times its original length to equal a year, the simple annual rate of return is equal to four times the 3-month holding-period rate of 0.756%, or about 3.024% (0.756% x 4).

The arithmetic average return corresponds conceptually to the simple return, only applied to a series of holding-period returns. If investors want to find the arithmetic average annual return of a series of 1-year holding-period returns they simply sum the 1-year holding-period returns and divide by the number of years of returns. For example, if an investment earns 10%, -5%, and 15% over three years, the arithmetic average annual rate of return is 6.67% [(10% - 5% + 15%) / 3 = 6.67%].

Compound and Geometric Average Annual Returns

In contrast with the simple annual return, one computes the compound annual return by assuming earnings and principal are being reinvested rather than by stretching or contracting the holding period. Investors calculate the compound annual return of an n-year holding-period rate of return by adding one to the holding-period return, taking the nth root of this sum, and subtracting one.

For example, the 3-month T-bill described earlier has a 3-month holding-period rate of return of 0.756%. The compound annual rate of return is the annual rate investors would earn if they could continue to reinvest the earnings and principal at a 3-month holding-period rate of 0.756% for each of the next three 3-month periods. If this were the case, each initial dollar of investment would grow to \$1.03058 [[(1.00756).sup.4]] by the end of one year. With this assumption, the compound annual rate of return is about 3.06% (1.03058 - 1 [approximately equal to] 3.06%), which is greater than the simple annual rate of 3.024%, as computed above.

The compound annual rate of return for the S&P 500 index investment with a 66.245% 5-year holding-period rate of return is 10.7% [[(1.66245).sup.1/5] - 1 = 1.107 - 1 = 10.7%]. In other words, in this case the compound annual rate of return is the annual rate of return that, if principal and cash flows were reinvested each year at this rate, would provide a 66.245% holding-period return over a 5-year period. This 10.7% compound annual rate is about 2.55% less than the simple annual rate of 13.25% computed above.

The geometric average return is related to the compound return in the same manner as the arithmetic average return is related to the simple return. To calculate the geometric average annual return for a series of 1-year holding-period returns, add one to each period's return, compute the product of these sums over all periods, extrapolate the nth root of the product, where n is the number of years, and subtract one. (4) For example, the 3-year accumulation of \$1 invested as described above (earning 10%, -5%, and 15%) with earnings and principal reinvested is 1.20175 [(1+10%) x (1-5%) x (1+15%)]. Taking the third root of 1.20175 and subtracting one, the resulting geometric average annual return is 6.32% [[(1.20175).sup.1/3] - 1 = 1.0632 - 1 = 6.32%)]. This result is less than the simple annual rate of 6.67%.

Simple Versus Compound and Arithmetic Versus Geometric

As the examples above demonstrate, simple (arithmetic average) annual rates differ from compound (geometric average) annual rates. Is there a systematic relationship and is one measure better than another?

First, for any investment providing a net positive return over a given holding period, the simple annual rate is always equal to or greater than the compound annual rate if the reference holding period is greater than a year and always equal to or less than the compound annual rate if the holding period is less than a year. Similarly, for any series of (on average, positive) annual holding-period returns, the arithmetic average annual return is always equal to or greater than the geometric average annual return. In addition, the greater is the variability of the annual holding-period returns, the larger the difference will be.

The logic of these results may not be immediately apparent, but, if one thinks about it, it takes a lower return to achieve the same terminal value if one assumes that cash flows are reinvested and compounded.

Second, whether one measure is better than another depends upon how one intends to use it. If the objective is to measure historical performance as, for example, to compare the past performance of two different mutual funds, then the compound or geometric average returns are the better measures. If one is looking to estimate future returns based upon historical performance, the simple or arithmetic average returns are the better measures for estimating returns in any given future year. This is because, in general, the simple or arithmetic average is an unbiased estimate of expected future returns for a single year. However, if investors are attempting to estimate the average annual rate at which they could expect their money to grow over a period of years in the future, the compound or geometric average returns are the ideal measures. In general, using the simple or arithmetic average return to estimate expected future average returns over a number of years would lead to significant overestimations of future wealth accumulations (because, as noted above, the arithmetic average tends to be higher than the geometric, and the difference is magnified by increased variability).

The reason for this is demonstrated by a simple example. Suppose a person invests \$100 and earns a 100% return the first year and a negative 50% return the second year. This investor has an average annual rate of return of 25%, right? [(100% - 50%) / 2 = 25%]. Wrong! The investment grows from \$100 to \$200 when it earns a 100% return the first year and then falls back to \$100 when it loses 50% of the \$200 accumulated investment the second year. The investor starts with \$100 and ends with \$100 two years later, so the actual average annual return is zero% (the geometric average annual return), not 25% (the arithmetic average annual return)! Although this is an extreme example, the effect is generally the same, albeit to a lesser degree, in all other cases. (5)

Nominal and Real (Inflation-Adjusted) Returns

Nominal returns are simply the actual returns earned over a given period computed without accounting for changes in the purchasing power of the dollar (inflation). The inflation-adjusted or real rate of return represents the return adjusted for changes in the general level of the prices of goods and services that investors could purchase if they liquidated their investments. The real rate of return is computed by dividing one plus the nominal rate of return by one plus the inflation rate, and then subtracting one:

Real Rate of Return = (1 + Nominal Rate of Return) - 1/(1 + Inflation Rate) = (Nominal Rate - Inflation Rate)/(1 + Inflation Rate)

For example, an investor buys 100 shares of XYZ Corp. for \$10,000 and holds the shares for 5 years. During the 5-year period, the stock pays no dividends but grows to a value of \$17,623.42. The nominal 5-year holdingperiod rate of return is 76.2342% (\$7,623.42 / \$10,000). Now assume that the general level of prices has increased by 21.6653% over the 5-year period. The real 5-year holding-period rate of return is 44.8517% [(1.762342) / (1.216653) - 1].

Investors must use care when adjusting real holding-period returns to real annualized returns. For instance, the simple annual real return computed by dividing the 44.8517% real 5-year holding-period return by 5 years is 8.9703%. However, if, instead, one first computes the simple annual nominal return by dividing the 5-year nominal holding-period return of 76.2342% by 5 years and then computes the simple annual inflation rate by dividing the 5-year holding-period inflation rate of 21.6653% by 5 years, the results are 15.4468% and 4.3306%, respectively. Computing the simple annual real rate of return using these rates results in a value of 10.6522% (1.154468 / 1.043306 - 1), not 8.9703%! So which is the "real" simple annual real return?

In contrast, if one computes the compound annual real return from the 5-year real holding-period rate of return of 44.8517%, the result is 7.6923% [(1.448517)1/5 - 1]. If, instead, one first finds the compound annual nominal return from the 5-year nominal holding-period rate of return of 76.2342% and then the compound annual inflation rate from the 21.6653% 5-year holding-period inflation rate, the results are 12% [(1.762342)1/5 -1] and 4% [(1.216653)1/5 - 1], respectively. Now, using these two annual rates to compute the compound annual real rate of return also gives a result of 7.6923% (1.12 / 1.04 - 1).

Therefore, annualizing real holding-period returns using the compound-return methodology is consistent, regardless of whether one first computes the real holding-period return and then annualizes that real holding-period return or first annualizes the nominal holding-period return and holding-period inflation rate and then computes the real return. In contrast, if one is using the simple-return methodology, one will derive different annualized real returns, depending upon the order of the computations. Although there is no definitive consensus on which order of computations is the correct methodology for computing the simple annual real rate of return, the authors strongly favor first calculating the real holding-period rate of return and then deriving the simple annual real rate of return. Computing the simple annual real return in this way has the theoretical advantage of starting the determination with the conceptually correct real holding-period return and also the practical advantage of requiring one less step in the calculations.

Internal Rate of Return and Net Present Value

The internal rate of return and the net present value (NPV) are essentially two-sides of the same coin. To understand the internal rate of return, one must first understand net present value. Net present value is simply the current value of a stream of cash flows discounted at some appropriate rate of return representing the investor's opportunity cost rate (best alternative rate). For example, assume an investor's opportunity cost rate is 6%. Assume also that the investor is contemplating a 3-year investment involving a \$1,000 cash outlay at the beginning of year 1, a \$2,000 cash outlay at the beginning of year 2, a \$1,000 cash withdrawal at the beginning of year 3, and a terminal value of \$2,236.75 at the end of year 3 (beginning of year 4). What is the net present value of this investment, after the value of each cash flow item is discounted for its appropriate time period?
```NPV = - \$1,000/[(1.06).sup.0] - \$2,000/[(1.06).sup.1] + \$1,000/
[(1.06).sup.2] + \$2,236.75 [(1.06).sup.3]

= - \$1,000/1 - \$2,000/1.06 + \$1,000/1.1236 + \$2,236.75/1.910

= - \$1,000 - \$1,886.79 + \$890.00 + \$1,878.02 = - \$118.77
```

The net present value is - \$118.77. A negative net present value means that this investment does not earn the investor's required 6% rate of return. However, it is important to note that the key aspect of evaluating an investment in terms of NPV is whether it is positive or negative (whether it earns more or less than the investor's required rate of return). The magnitude of the NPV is not necessarily a valid way to select between two different investment choices (selecting between multiple investment options is covered in greater detail later in this chapter).

So what annual rate of return does this investment earn? The rate of return that equates the NPV to zero is the annual rate of return earned by the investment. That rate of return is called the internal rate of return. The following table shows that 3.823% is the internal rate of return.
```NPV = - \$1,000/[(1.03823).sup.0] - \$2,000/[(1.03823).sup.1] +
\$1,000/[(1.03823).sup.2] + \$2,236.75/[(1.03823).sup.3]

=  - \$1,000/1 - \$2,000/1.03823 + \$1,000/1.07792 + \$2,236.75/1.11913

=  - \$1,000 - \$1,926.36 + \$927.71 + \$1,998.65  = - \$0.00
```

Net present values and internal rates of return are essential tools for measuring investment returns and comparing and analyzing alternative investments. How these tools are used to measure returns and in investment analysis is discussed in considerable detail later in this chapter.

Time-Weighted and Dollar-Weighted Annual Returns

The time-weighted annual return is the geometric (compounded) annual return measured on the basis of periodic market valuations of assets. This is the preferred method for evaluating the performance of investment managers because it eliminates the impact of cash contributions and disbursements (inflows and outflows). However, in principle it requires valuations to be made on the occasion of each cash flow. Approximations to this measure can be obtained by prorating cash flows to successive valuation points or by computing internal rates of return between valuation points. If there are no interim cash flows, the time-weighted return, compounded annually, determines the ending value of an investment.

In contrast, the dollar-weighted annual return is the rate of return that discounts a portfolio's terminal value and interim cash flows back to its initial value. It is equivalent to a portfolio's internal rate of return. The dollar-weighted return can be misleading for purposes of comparative performance measurement, because it is influenced by the timing and magnitude of contributions and disbursements that are beyond the control of the portfolio manager. However, if investors want to compute the average annual return they actually have earned on an investment over a given period where they have made cash withdrawals and/or contributions during that period, they should use the dollar-weighted return (internal rate of return) method.

A simple example can demonstrate the difference. Above, the discussion described a 3-year investment that had 1-year holding-period returns of 10%, -5%, and 15%. The geometric average annual return or time-weighted return is computed by compounding these returns over the three-year period, resulting in a geometric or time-weighted average annual return of 6.32%, as shown earlier. However, assume that a person invests \$1 the first year, another \$2 the second year, and withdraws \$1 the third year. In this case, the dollar-weighted return is equivalent to the internal rate of return (which will be discussed further below) of a \$1 outflow at the beginning of the 1st year, a \$2 outflow at the start of the 2nd, a \$1 inflow at the beginning of the 3rd year, and a \$2.24 withdrawal at the end of the 3rd year (the terminal value). This series of cash flows has an internal rate of return (a dollar-weighted annual return) of only 3.87% (as compared to the 6.32% time-weighted annual return). This makes sense, since the greatest amount of money was deposited in the 2nd year (when the 1-year return was negative) and less money was invested during the years of positive returns. As a result, the negative returns received a greater dollar-weight than the positive returns (in comparison to equal weightings under the time-weighted return), resulting in a lower dollar-weighted than the time-weighted annual return.

After-Tax Return and Taxable-Equivalent Return

What investors get to keep after paying taxes is what matters. Virtually all financial planning issues and investment choices involve tax considerations. The real value of any financial planning strategy or tactic or investment choice relates to the real spendable dollars that each option provides relative to the alternatives. For instance, nominal yields on taxable bonds are virtually uniformly higher than returns on tax-free municipals of comparable maturity. However, for some taxpayers in high tax brackets, the tax-free yields of municipal bonds are higher than the after-tax yields from taxable bonds, so the tax-free municipals are the preferable investment.

If it is assumed that the investment return is entirely currently taxable, then the after-tax return (also called the tax-free-equivalent return) is simply equal to the before-tax return less the taxes on the return. Specifically, if the tax rate is assumed to be t and the before-tax rate of return is r, then the after-tax return, rat, is:

rat = r x (1 - t).

For example, investors earning 6% before tax, whose tax rate on that income is 28%, will earn 4.32% after tax [6% x (1 - 28%) = 4.32%].

The taxable-equivalent return is the reverse of the after-tax return; representing the before-tax return (r) an investor would have to earn to equal, after tax (t), the return on a tax-free investment (rat):

r = rat / (1 - t).

Although these formulas are frequently used to compute after-tax returns, they are often not an accurate measure. Only a relatively small class of investments, such as money market funds, bank accounts, and the like provide investment returns that are entirely currently taxable. Other investments, such as stocks, real estate, and the like provide some combination of currently taxable income and income, return, or gain on which tax is deferred. In addition, a whole host of vehicles such as qualified retirement plans, commercial annuities, life insurance, Section 529 plans, Coverdell educational savings accounts, and the like provide unique tax incentives that cannot be accounted for using the simple formulas above.

In many financial planning and investment situations, not only the level of taxation but also the timing of taxation is a critical factor. Between two investments providing identical before-tax returns and identical total tax burdens, the one that defers some or all of the taxation to a later date is generally preferable.

Appendix K, "Additonal Time Value Concepts," has an extensive discussion of after-tax returns and the concept of tax leverage and describes the tax-adjusted time-value tools financial advisers and investors must understand to properly handle financial and investment plans. That appendix discusses the tools necessary to account for the five most-prevalent types of after-tax returns and tax leverage, ranging from nondeductible fully currently taxable vehicles, such as T-bills, on one end of the spectrum, through stocks and other income and appreciation assets in the middle of the spectrum, to tax-deductible, fully tax-deferred vehicles, such as IRC Section 401(k) plans on the other end of the spectrum.

Risk-adjusted return measures were developed to help investors gauge how much return they were getting per unit of risk. Both volatility (level of risk) and return are combined in one measure, permitting a rank ordering of investments. Investments ranking high on a risk-adjusted return scale demonstrated a favorable trade-off between risk and reward: either the returns were high enough to compensate for the additional risk taken, or, the returns may not have been extraordinarily high, but the risk taken was much lower than expected.

Various popular methods exist to measure an investment's risk-adjusted return. Some use standard deviation; others use beta. Risk-adjusted returns are most appropriate when used in "apples to apples" comparisons. Measures using standard deviation, for example, should compare funds in the same category or peer group. Beta relates a fund's volatility to a benchmark--usually the S&P 500 Index--so it is important to use an appropriate benchmark for a fair comparison.

These risk-adjusted return measures are discussed in greater detail in Chapter 31, "Measuring Investment Risk."

CHAPTER ENDNOTES

(1.) In some cases such intangibles might be capitalized for accounting purposes and booked as a separate asset or liability or as an adjustment to the initial investment or basis. However, from an economic standpoint, such benefits or services should be included in return measurements at the time when, otherwise, money would be paid for these benefits or services had they not been included as part of the deal.

(2.) Data from Stocks, Bonds, Bills, and Inflation 2002 Yearbook (Chicago, IL: Ibbotson Associates).

(3.) Whenever cash returns, such as dividends, interest, or rent, are not reinvested, but rather withdrawn during the holding period, the holding-period rate of return computed by adding all cash returns to the capital appreciation over the holding period will mismeasure the actual return. The error will be greater the longer

the time period between non-reinvestment of cash income and the end of the holding period. Similarly, any other additions or withdrawals of principal other than at the beginning or end of the holding period will distort the calculation of the actual holding-period return. Adjustments for such cash withdrawals or additions are discussed under "III. Modified Internal Rate of Return Methods." . 1

(4.) Geometric average annual return = [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

(5.) If, on average, the returns are negative over the period of consideration, the effect is a mirror image on the negative side of zero. The geometric average annual return will be greater than the arithmetic average annual return (but still smaller in absolute value).

(6.) There is a procedure based upon Sturm's Theorem for determining both how many IRRs are within the realm of real numbers and for finding partitions within the overall range of possible rates within which one and only one of the real IRRs can be found. By restricting searches within these ranges, each and every real IRR generally can be found. The technical details of this method are far beyond the scope of this discussion. See, for example, Stephen G. Kellison, Fundamentals of Numerical Analysis, Richard D. Irwin, Inc. or almost any other text on Numerical Methods.
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