INVESTMENTEVALUATION TECHNIQUES
A.Payback Period
Proj. A Proj. B
------
Year 0 (3,000) (3,000)
Year 1 1,000 2,000
Year 2 2,000 1,000
Year 3 3,000 4,000
Payback = 2 years Payback = 2 years
Both projects have a payback of two years, so the payback method indicates that the two projects are equally desirable.
Problems:
1)Ignores the Time Value of Money
2)Ignores cash flows beyond the payback period
Project B returns $1,000 a year earlier than Project A and also returns an additional $1,000 in the last year.
Present Value Payback, which utilizes the present value of each year's cash flow, overcomes the first problem, but not the second.
B)Net Present Value (NPV)
We need a methodology that takes into account all of the cash flows as well as the time value of money. Net Present Value is one such technique:
NPV = PV of Cash Inflows PV of Cash Outflows
Required Rate of Return = 10%
0 1 2 3
(4,000) 1,000 2,000 3,000
0.9091
909
0.8264
1,653
0.7513
2,254
NPV @ 10% = 816
NPV represents the increase in the value of the firm that occurs by accepting the project. In other words, it represents the amount by which the value of the project exceeds its cost.
Proof:
Year 0 Investment 4,000 Cash Flow Year 1 1,000
Return of Investment (600) Less: Interest (400) (10%*$4,000)
-- --
Year 1 Investment 3,400 Return of Investment 600
Return of Investment (1,660)
-- Cash Flow Year 2 2,000
Year 2 Investment 1,740 Less: Interest (340) (10%*$3,400)
Return of Investment (2,826) ---
-- Return of Investment 1,660
Surplus Return (1,086)
PVIF10%,3 0.7513 Cash Flow Year 3 3,000
--- Less: Interest (174) (10%*$1,740)
Present Value 816 --
Return of Investment 2,826
The problem with NPV is that there is no consideration of cost, or what is referred to as size disparity.
Proj. A Proj. B
--
Present Value of Inflows 1,050 125
Cost (1,000) (100)
-
Net Present Value 50 25
If these are mutually exclusive projects (i.e., choose one or the other, but not both), the NPV criterion says to choose Project A. While Project A increases the value of the firm by twice the amount of Project B, it costs ten times as much. The NPV does not indicate how efficiently money has been invested.
Capital Rationing - the allocation of a scarce resource, in this case money.
C)Internal Rate of Return (IRR)
Another measure of the efficiency of investment is the Internal Rate of Return. When someone asks what rate of return an investment is earning, they mean the Internal Rate of Return. The IRR can be defined as
PV of Inflows @ IRR = PV of Outflows @ IRR
or
NPV @ IRR = 0
This is the actual rate of interest that is being earned on the investment. While the present value and annuity tables can be used in certain cases, the more general situation of uneven cash flows requires that the IRR be found by trial and error.
From the previous example, it is clear that more than 10% is being earned, since the NPV is $816.
Try 20%
NPV @ 20% = 1,000*(.8333) + 2,000*(.6944) + 3,000*(.5787) 4,000
= 3,958 4,000 = (42)
Interpolating provides an estimate:
10% 816
Z 816
10% IRR 0 858
20% (42)
Z = 9.51%
+10.00
IRR = 19.51%
Year 0 Investment 4,000 Cash Flow -- Year 1 1,000
Return of Investment (220) Less: Interest (780) (19.5% * $4,000)
Year 1 Investment 3,780 Return of Investment 220
Return of Investment (1,262)
Year 2 Investment 2,518 Cash Flow -- Year 2 2,000
Return of Investment 2,509 Less: Interest (738) (19.5% * $3,780)
Surplus Return 9 Return of Investment 1,262
PVIF10%,3 0.7513
Present Value (7) Cash Flow -- Year 3 3,000
Less: Interest (491) (19.5% * $2,518)
Return of Investment 2,509
Hence, it is the rate of interest earned on the funds that remain invested within the project. This is the economic interpretation of the mathematical solution. Note that the "true" IRR is 19.44%. The error occurs because interpolation assumes linearity of a non-linear function.
RISK AND REQUIRED RATES OF RETURN
The second approach is to calculate an appropriate risk-premium that is required in addition to the risk-free rate of interest.
The risk-free rate of interest is a function of the rate of inflation and the "real" rate of interest. The risk-premium is dependent upon the business risk and financial risk. As previously seen, the financial risk that results from the use of debt simply magnifies the business risk that is associated with a project or company.
The Capital Asset Pricing Model is the most well-known of the risk-premium models. The CAPM assumes that the only relevant risk is the systematic risk and, therefore, the appropriate risk premium is a multiple of the market's risk premium.
The difficulties of implementing the CAPM include obtaining a valid estimate of β as well as estimating the market's expected rate of return. Typically, the latter problem is circumvented by using historical averages of the market's realized risk premium, although the appropriate measure of "the market" is debatable. Estimating β, on the other hand, is difficult even for a firm that is publicly traded, and impossible for a firm that is privately-held.
Another approach to calculating the required rate of return on equity is to separate the overall risk-premium into a risk-premium for debt of the company and a risk-premium for the equity of the company. In this manner, the debt's risk-premium can be objectively observed by the rate of interest that the market is charging on the firm's bonds (or the bank's effective interest rate on loans if the firm is privately-held). To this is added an equity risk-premium based upon historical stock (total) risk-premiums in comparison to historical debt risk-premiums.
The most widely referenced source for historical returns used in calculating risk-premiums is Ibbotson Associates' Stocks, Bonds, Bills, and Inflation annual Yearbooks. This source tracks the market performance of a variety of financial instruments from 1926 to the present. Their data include the market performances of government treasury bills, treasury notes, treasury bonds, inflation rates, company returns by decile, large company stocks, small company stocks and corporate bonds. Using data from the 2001 Yearbook, the average (geometric) annual return for corporate bonds can be calculated from the Yearbook's total return index for the past 75 years as
Corp. Large Small
Bondst-bills Stocks Stocks
Cumulative Wealth Index 200064.07716.5632586.5246402.228
Cumulative Wealth Index 1925 1.000 1.000 1.000 1.000
Geometric Average Yield = 5.70% 3.81% 11.05% 12.40%
Corporate Bond Premium = 5.7% - 3.81% = 1.89%
Equity Risk Premium = 11.05% - 5.7% = 5.34%
Small Firm Risk Premium = 12.4% - 11.05% = 1.35%
Consider now a company that has a cost of debt of 8%. Then, adding in the equity risk-premium of 5.34% yields a cost of equity of 13.34%. If the firm is relatively small, another 1.35% might be added to adjust for firm size to yield a required rate of return of 14.69%. Note that this would only truly be consistent with publicly traded small stocks if the risk-free rate of interest was 6.11%; i.e., the risk-premium for the debt was still 1.89%.
Suppose instead that the required return on equity is being estimated for a privately-held firm. The argument could be made that the historic risk-premiums for publicly traded debt and equity are not relevant, but that provides no insight into what an appropriate required rate of return might be. Empirical research has shown, and U.S. courts have acknowledged, that a discount is appropriate in the valuation of privately-held securities. A discount in value is equivalent to an increase in rate of return. Thus, it is possible to utilize the preceding methodology as a means of estimating the required rate of return for a small, privately-held firm as well.
Assume a company's debt carries an 11% rate of interest. If the risk-free rate is 5%, then the debt risk-premium is 6%. Compared to the historic average risk-premium of 1.89% for large, publicly-traded corporate bonds, the company's debt risk-premium is 3.18 times the historic average (6%/1.89% = 3.18). Even disregarding the public vs. private nature of the debt, this is not inconsistent with the risk of the debt being 3.18 times that of a large public company. That is, such a risk-premium would be justified if the βdebt of the company were 3.18 times that of the average large corporation. Assuming that the beta of the equity of was also 3.18 times that of a large, publicly traded company, then the appropriate equity risk-premium would be 16.96%, or 3.18 times the historic average of 5.34%. Adding this to the cost of the debt would yield a required return on equity of 27.96% for the firm.
While 27.96% appears to be very high, it is not inconsistent with required rates of return on private equity. The increased risk characteristics of the firm could be a result of a higher asset beta, higher leverage, or a combination of the two, as well as the risk of being privately-held (i.e., variability of obtainable prices for the securities). Thus, the resulting estimate of the required rate of return on equity is still consistent with financial theory. (It should be noted that no small firm premium should be included in this case since the debt and equity premiums calculated presumably would reflect this factor.)
A Note on Use of Historic Averages
Arguments have been put forth that using past, realized returns does not reflect investor expectations. For example, large company stock investors did not expect to lose 9% on their equity investments in 2000. Conversely, they in all likelihood did not expect to earn 37% in 1995, 23% in 1996, 33% in 1997, 29% in 1998 and 21% in 1999. Beninga and Sarig argue that historic averages, in the long-run, should be the same as investor expectations, however. If the long-run historic averages were different, then incentives would arise for risk-arbitrage in securities which would create the market forces that would result in market prices yielding historic averages consistent with expectations.
The other question that often arises over the use of historic averages is in regard to the appropriate period of time to consider for calculating the averages. It is generally considered to be more appropriate to use longer periods of time, rather than shorter periods, in order to capture the overall cyclical effects of the economy. The advantage of a longer time period also overcomes the inherent error of an historic estimate that can occur from choice of time periods, as the following graph depicts.
As may be observed, a short period of time can result in grossly underestimating (red line) or overestimating (blue line) the trend. A longer time period, even when measuring from a boom period (crest) to a recession (trough) results in a more representative estimate (gold line) of the true trend (dotted line).