# CAPITAL INVESTMENT by tyndale

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```									                           INVESTMENT EVALUATION 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    816

10%   858

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.

K s  R  RP
f
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.

K s  R   (R - R )
f   i M    f
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.

K s  R  RP      RP
f    debt     equity

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
Bonds          t-bills          Stocks          Stocks
Cumulative Wealth Index 2000           64.077          16.563         2586.524        6402.228
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 β debt 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).

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