ESTIMATING INPUTS FOR VALUATION
ESTIMATION OF DISCOUNT RATES

* Critical ingredient in discounted cashflow valuation. Errors in estimating the discount rate or mismatching cashflows and discount rates can lead to serious errors in valuation.

* At an intutive level, the discount rate used should be consistent with both the riskiness and the type of cashflow being discounted.

I. Cost of Equity

The cost of equity is the rate of return that investors require to make an equity investment in a firm. There are two approaches to estimating the cost of equity;

* a risk and return model

* a dividend-growth model.

Models of Risk and Return
The Capital Asset Pricing Model

* Measures risk in terms on non-diversifiable variance

* Relates expected returns to this risk measure.

* It is based upon several assumptions -

(a) that investors have homogeneous expectations about asset returns and variances

(b) that they can borrow and lend at a riskfree rate

(c) that all assets are marketable and perfectly divisible

(d) that there are no transactions costs and that there are no restrictions on short sales.

Risk in the CAPM

Beta: The non-diversifiable risk for any asset can be measured by the covariance of its returns with returns on a market index, which is defined to be the asset's beta.

The cost of equity will be the required return,

Cost of Equity = Rf + Equity Beta * (E(Rm) - Rf)

where,

Rf = Riskfree rate

E(Rm) = Expected Return on the Market Index

Using the Capital Asset Pricing Model

* Inputs required to use the CAPM -

(a) the current risk-free rate

(b) the expected return on the market index and

(c) the beta of the asset being analyzed.

Practical issues in using the CAPM --
1. Measurement of the risk premium

* It is generally based upon historical data, and the premium is defined to be the difference between average returns on stocks and average returns on riskfree securities over the measurement period.

 Historical Period Stocks - T. Bills Stocks - T.Bonds Arithmetic Geometric Arithmetic Geometric 1926-1990 8.41% 6.41% 7.24% 5.50% 1962-1990 4.10% 2.95% 3.92% 3.25% 1981-1990 6.05% 5.38% 0.13% 0.19%

* Generally, geometric averages provide better estimates of risk premiums in valuation.

• The risk premiums will vary across markets, depending upon their riskiness. While historical data can be used to estimate premiums outside the United States, it is not very reliable. An alternative way of estimating premiums is to use country bond ratings to estimate these premiums relative to the U.S. premium. For instance, the risk premiums for South American countries can be estimated as follows:

 Country Rating Risk Premium Argentina BBB 5.5% + 1.75% = 7.25% Brazil BB 5.5% + 2% = 7.5% Chile AA 5.5% + 0.75% = 6.25% Columbia A+ 5.5% + 1.25% = 6.75% Mexico BBB+ 5.5% + 1.5% = 7% Paraguay BBB- 5.5% + 1.75% = 7.25% Peru B 5.5% + 2.5% = 8% Uruguay BBB 5.5% + 1.6% = 7.1%

A Warning: If you add a default premium to the risk premium, do not add a risk premium to the risk free rate. That would be double counting.

II. Riskfree Rate

The long-term government bond rate is the appropriate riskfree rate. When the long term government bond rate is not available, it may make sense to look at the rate at which large corporations can borrow in the local market.

Illustration 3: Using the CAPM to calculate cost of equity

Glaxo Holdings had an estimated beta of 1.10 at the end of 1994. At the same point in time, the thirty-year treasury bond rate in the United States was 8.00%. The estimated cost of equity for Glaxo in December 1994 was ñ

Cost of Equity = 8.00% + 1.10 (5.50%) = 14.05%

This is the cost of equity, if cash flows are estimated in dollars. The same estimation can be done in British Pounds, using the long term Government Bond rate in the U.K., which was 8.50% at the same point in time ñ

Cost of Equity (British Pounds) = 8.50% + 1.10 (5.50%) = 14.55%

This difference reflects differences in expected inflation in the two markets.

• Wellcome is planning an acquisition of Glaxo. Wellcome has a beta of 0.65. Which beta would you use in valuing Glaxo?

Determinants of Betas

* The more sensitive a business is to market conditions, the higher is its beta. Thus cyclical firms can be expected to have higher betas, other things remaining equal, than non-cyclical firms.

* The beta of a firm is the weighted average of the betas of its different business lines.

• Pharmaceutical companies are increasingly looking at expanding into biotechnology, either through acquisitions or through projects. Biotechnology firms have an average beta of 1.8. What would the impact of these actions be on pharmaceutical firmsí betas?

II. Degree of Operating Leverage:

* DOL is a function of the cost structure of a firm, and is usually defined in terms of the relationship between fixed costs and total costs.

* A firm which has high operating leverage -> higher variability in earnings before interest and taxes (EBIT) -> higher beta

• Sticking again with pharmaceutical firms, there are some firms which have very high fixed costs, either because they focus heavily on research, or because of the nature of their business. (Synergen, for instance, had a beta of 1.65 in 1994. It had research and development expenses which were 670% of sales in 1994.) These firms will have higher betas than other firms which have more traditional cost structures. (Wellcome is a good example.)

III. Financial Leverage:

* An increase in financial leverage will increase the equity beta of a firm.

* If all of the firm's risk are borne by the stockholders, i.e., the beta of debt is zero, and debt has a tax benefit to the firm, then,

Levered Beta = Unlevered Beta (1 + (1-t) (D/E))

where,

t = Corporate tax rate

D/E = Debt/Equity Ratio

Illustration 4: Effects of Financial Leverage on Betas

Glaxo, with a beta of 1.10, had a debt/equity ratio of 4% in 1994. If Glaxo were to raise its debt/equity ratio to 20%, its beta would be much higher (Tax rate was 30%) ñ

Current Beta (Levered) = 1.10

Unlevered Beta = 1.10 /(1+0.7*0.04) = 1.07

New Levered Beta = 1.07 (1+0.7*0.2) = 1.22

Other approaches to estimating Betas

I. Using comparable firms:

* Use the betas of publicly traded firms which are comparable in terms of business risk and operating leverage.

* Correct for differences in financial leverage between the firm being analyzed and the comparable firms.

Illustration 5: Using comparable firms to estimate betas

Assume that you are trying to estimate the beta for a private firm that is in the business of manufacturing, selling and servicing fax machines. The betas of publicly traded firms involved in office equipment and supplies are as follows (They face an average tax rate of 40%). The private firm had a debt/equity ratio of 30%.

 Firm Beta Debt/Equity General Binding 1.00 0.20 Hunt Manufacturing 0.80 0.03 Moore Coroporation 0.95 0.05 Nashua Company 0.90 0.10 Pitney Bowes 1.20 0.45 Average 0.97 1.17

Unlevered Beta of office supply firms = 0.97 / (1 + (1-0.4) (0.17)) = 0.88

Beta for private firm involved in office supplies = 0.88 (1 + (1-0.4) (0.3)) = 1.04

II. Using fundamental factors:

* Combines industry and company-fundamental factors to predict betas.

* Income statement and balance sheet variables are important predictors of beta

* Following is a regression relating the betas of NYSE and AMEX stocks in 1991 to five variables - dividend yield, coefficient of variation in operating income, size, debt/equity and growth in earnings.

BETA = 0.9832 + 0.08 CV in Operating Income - 0.126 Dividend Yield + 0.15 Debt/Equity Ratio + 0.034 Growth in Earnings per Share - 0.00001 Total Assets

where,

CV in Operating Income = Coefficient of Variation in Operating Income

= Standard Deviation in Operating Income/ Average Operating Income

Illustration 6: Using fundamental information to predict betas

Assume that you are trying to estimate the beta for a private firm, with the following financial characteristics (defined consistently with the regression):

Coefficient of Variation in Operating Income = 2.2

Dividend Yield = 0.04

Debt/Equity Ratio = 0.30

Growth in Earnings per share = 0.30

Total Assets = \$ 10,000 (in thousands)

The estimated beta for this firm,

BETA = 0.9832 +0.08*2.2 - 0.126* 0.04 + 0.15* 0.30 + 0.034 * 0.30 - 0.00001*10,000 = 1.19

The Arbitrage Pricing Model

* Logic behind the arbitrage pricing model (APM) same as the logic behind the CAPM, i.e., investors get rewarded for taking on non-diversifiable risk.

* Measure of this non-diversifiable risk in the APM, however, is not a single factor but is determined by an asset's sensitivity to various economic factors that affect all assets.

* The number and the identity of the factors are determined by the data on historical returns.

* The arbitrage pricing model relates expected returns to economic factors, with a beta specific to each factor. If these factor-specific betas and the factor risk premia can be estimated, the cost of equity can also be estimated. where,

Rf = Riskfree rate

bj = Beta specific to factor j

E(Rj) - Rf = Risk premium per unit of factor j risk

k = Number of factors

Using the Arbitrage Pricing Model

Illustration 7: Using the APM to estimate the cost of equity

Assume that the parameters for the arbitrage pricing model have been estimated, and that there are three factors,

Riskfree rate = 3.35%

E(R1) - Rf = 3% : Risk Premium for factor 1

E(R2) - Rf = 4% : Risk Premium for factor 2

E(R3) - Rf = 1.5% : Risk Premium for factor 3

Assume that the betas specific to each of these factors are estimated for Pepsi Cola in 1992 and that the estimates are as follows:

b1 = 1.20

b2 = 0.90

b3 = 1.10

Substituting into the APM,

Cost of Equity = 3.35% + 1.20 (3%) + 0.90 (4%) + 1.1 (1.5%) = 12.20%

Considerations on the use of the APM

* Capital asset pricing model can be considered to be a specialized case of the arbitrage pricing model, where there is only one underlying factor and that underlying factor is completely measured by the market index.

* In general, the CAPM has the advantage of being a simpler model to estimate and to use, but it will underperform the richer APM when the company is sensitive to economic factors not well represented in the market index.

Example: Oil companies, which derive most of their risk from oil price movements, tend to have low CAPM betas.

* The biggest intuitive block in using the arbitrage pricing model is its failure to identify specifically the factors driving expected returns.

Multi-factor Models for risk and return

* Unidentified factors in the arbitrage pricing model are replaced with macro-economic variables,

* For example, Chen, Roll and Ross (1986) suggest that the following macroeconomic variables are highly correlated with the factors that come out of factor analysis -- industrial production, changes in default premium, shifts in the term structure, unanticipated inflation and changes in the real rate of return. These variables can then be correlated with returns to come up with a model of expected returns, with firm-specific betas calculated relative to each variable.

* Costs of going from the arbitrage pricing model to a macro-economic multi-factor model can be traced directly to the errors that can be made in identifying the factors.

* The factors in the model can change over time, as will the risk premia associated with each economic factor.

* Using the wrong factor(s) or missing a significant factor in a multi-factor model can lead to inferior estimates of cost of equity.

Dividend Growth Model

For a firm which has a stable growth rate in earnings and dividends, the present value of a share of equity can be written as:

Po = Present Value of expected dividends = DPS1 / (ke - g)

where,

P0 = Price of the stock today

DPS1 = Expected dividends per share next year

ke = Cost of Equity

g = Growth rate in dividends (steady state)

A simple manipulation of this formula yields,

ke = DPS1 / P0 + g

= Expected Dividend Yield + Growth rate in earnings/dividends

More importantly, since the current price is a key input to the model, it is inappropriate to use this approach to value stock in a firm. There is a strong element of circular reasoning involved that will lead the analyst to conclude, using this cost of equity, that equity is fairly valued.

Illustration 8: Using the Dividend Growth Model to estimate the cost of equity: Southwestern Bell

In 1992, Southwestern Bell paid dividend per share of \$2.82 and the stock traded at \$66 in December 1992. The estimated growth rate in dividends is 5.5% and the firm is assumed to be in steady state. The dividend growth model can then be used to estimate the cost of equity;

Expected Dividends in 1993 = \$2.82 *1.055 = \$2.98

Cost of Equity = \$2.98 / \$66 + 5.5%

= 10%

To illustrate the circular reasoning involved in using this cost of equity to value stock,

Value of Equity = \$ 2.98 / (.10 - .055) = \$66

Not surprisingly, the stock is found to be fairly valued.

Weighted Average Cost of Capital (WACC)

Definition of the Weighted Average Cost of Capital (WACC)

The weighted average cost of capital is defined as the weighted average of the costs of the different components of financing used by a firm.

WACC = ke ( E/ (D+E+PS)) + kd ( D/ (D+E+PS)) + kps ( PS/ (D+E+PS))

where,

WACC = Weighted Average Cost of Capital

ke = Cost of Equity

kd = After-tax Cost of Debt

kps = Cost of Preferred Stock

E/(E+D+PS) = Market Value proportion of Equity in Funding Mix

D/(E+D+PS) = Market Value proportion of Debt in Funding Mix

PS/(E+D+PS) = Market Value proportion of Preferred Stock in Funding Mix

Illustration 9: Calculating the Cost of Capital: Genzyme Corporation.

In December 1994, Genzyme Corporation had

* a beta of 1.60. This results in

Cost of equity = 8.00%+1.60 (5.50%) = 16.80%

* an after-tax cost of debt of 6.30%. (Pre-tax cost of debt=9.00%; Tax rate=30%)

* Equity comprised 85.00% (in market value terms) of the funding mix and debt made up the remaining 15.00%.

* The cost of capital for Genzyme can then be calculated as follows:

WACC = 16.80% (0.85) + 6.30% (0.15) = 15.23 %

ESTIMATION OF CASH FLOWS

Cash flows to Equity for a Levered Firm

Revenues

- Operating Expenses

= Earnings before interest, taxes and depreciation (EBITDA)

- Depreciation & Amortization

= Earnings before interest and taxes (EBIT)

- Interest Expenses

= Earnings before taxes

- Taxes

= Net Income

+ Depreciation & Amortization

= Cash flows from Operations

- Preferred Dividends

- Capital Expenditures

- Working Capital Needs

- Principal Repayments

+ Proceeds from New Debt Issues

= Free Cash flow to Equity

Levered Firm at Desired Leverage

Net Income

- (1- DR) (Capital Expenditures - Depreciation)

- (1- DR) Working Capital Needs

= Free Cash flow to Equity

For this firm,

Proceeds from new debt issues = Principal Repayments +DR (Capital Expenditures - Depreciation + Working Capital Needs)

Illustration 10: Estimating the cash flow to equity for a firm at its desired leverage: Warner-Lambert

The following is an estimation of free cash flows to equity for Warner-Lambert in 1994 and for 1995 (projected).

• The company had \$1.5 billion in debt outstanding and \$ 9 billion in market value of equity, leading to a debt to capital ratio (ìdebt ratioî) of 14%. This debt ratio is assumed to be stable. (How does one know?)
• The company reported net income of \$695 million in 1994 and is projected to have a net income of \$765 million in 1995.
• The company reported depreciation of \$180 million in 1994 and is expected to have depreciation of \$200 million in 1995.
• The company had capital expenditures of \$362 million in 1994 and is expected to have capital expenditures of \$400 million in 1995.
• The companyís working capital increased to \$225 million in 1994 from \$203 million in 1993. It is expected to maintain working capital at the same percentage of sales in 1995, and sales are expected to increase from \$6420 million in 1994 to \$7100 million in 1995.

 1994 Estd 1995 Net Income \$ 695.00 \$ 765.00 - (Cap Ex - Depreciation) (1-DR) \$ 156.52 \$ 172.00 (DR = 14%) - ( Change in Working Capital) (1-DR) \$ 18.92 \$ 20.50 (DR = 14%) FCFE \$ 519.56 \$ 572.50

Proposition : The Free Cash Flow to Equity will increase as the amount of debt financing used by the firm increases. Thus FCFE will be an increasing function of _.

Illustration 11: Sensitivity to Debt Ratio - Warner Lambert Corporation

The following are the cash flows to equity for Warner Lambert, using a debt ratio of 40% instead of a debt ratio of 14%.
 1994 Estd 1995 Net Income \$ 695.00 \$ 765.00 - (Cap Ex - Depreciation) (1-DR) \$ 109.20 \$ 120.00 (DR = 40%) - ( Change in Working Capital) (1-DR) \$ 13.20 \$ 14.30 (DR = 40%) FCFE \$ 572.60 \$ 630.70

The following graph illustrates the effect on free cash flows to equity of changing the debt ratios from 0% to 100% ñ CASHFLOWS TO THE FIRM

EBIT ( 1 - tax rate)

+ Depreciation

- Capital Spending

- Change in Working Capital

= Cash flow to the firm

Illustration 12: Estimating the expected cash flow to the firm - Siemens

Siemens AG, as a German-based multinational involved in a wide range of businesses, reported earnings before interest and taxes of 3,482 million DM in 1992, prior to general provisions and extraordinary charges. It had depreciation of 4,613 million DM and capital expenditures of 5,560 million DM in 1992. In addition, the working capital increased from 14,306 million DM in 1991 to 15,405 million DM in 1992. The earnings before interest and taxes is expected to increase to 3,967 million DM in 1993. Capital expenditures, depreciation and working capital are all expected to increase by 5% in 1993. The firm had a tax rate of 38% in 1992. The free cashflows to the firm for 1992 and 1993 (estimated) are provided below.

1992 Projected 1993

EBIT (1 - tax rate) 2,159 mil DM 2,460 mil DM

+ Depreciation 4,613 mil DM 4,844 mil DM

- Capital Expenditures 5,560 mil DM 5,838 mil DM

- Change in Working Capital 1,098 mil DM 770 mil DM

Free Cashflow to firm 114 mil DM 696 mil DM

INFLATION AND VALUATION

Consistency Principle 1: Nominal cash flows should be discounted at nominal discount rates; Real cash flows should be discounted at real discount rates.

Illustration 13: The Effect of Inflation on Cash flows and Discount Rates

Consider a firm which has cash flows to equity currently of \$100 million and is expected to grow, in real terms, at 5% a year for the next three years and 3% a year after that. The firm has a beta of 1.0 and the current T.Bond rate is 6.5%. The expected inflation rate is 3% in both the firm's cash flows and the general economy. The valuation of this firm can be done on either a real or a nominal basis:

The estimates of growth on a real and nominal basis are done first

Real Nominal

Growth Rate in first 3 years 5% (1.05)(1.03)-1 = 8.15%

Growth Rate after year 3 3% (1.03)(1.03)-1 = 6.09%

The discount rate can similarly be estimated on a real and nominal basis:

Real Nominal

Discount Rate 1.12/1.03 -1 = 8.74% E(R) =6.5% +1(5.5%) =12%

The expected nominal return is estimated using the CAPM, with a historical premium earned by stocks over T.Bonds of 5.5%.

Using these growth rates the cash flows can be generated in both nominal and real terms:
 Real Cash flows Nominal Cash flows Year CF to Equity Terminal Value Nominal Cash flows Terminal Value 1 \$105 \$108 2 \$110 \$117 3 \$116 \$2,078 \$126 \$2,271

The terminal values are calculated as follows for real and nominal cash flows:

Terminal value (Real Cash flows) = \$ 116 * 1.03 / (.0874 - .03) = \$ 2,078

Terminal value (Nominal Cash flows) = \$ 126 * 1.0609 / (.12 - .0609) = \$ 2,271

This calculation assumes that the growth rates after year 3 are steady state growth rates and will continue through infinity.

The present values of the cash flows to equity and the terminal value can then be calculated using the appropriate discount rate:

Present value (using real cash flows) = \$105/1.0874 + \$ 110 / 1.08742 + (\$116 + \$2,078) / 1.08743 = \$ 1,896

Present value (using nominal cash flows) = \$108/1.12 + \$ 117 / 1.122 + (\$126 + \$2,271) / 1.123 = \$ 1,896

Illustration 13A: The Effect of mismatching cash flows and discount rates
 Real Cash flows@ Nominal rate Nominal Cash flows@ Real rate Year CF to Equity Terminal Value Nominal Cash flows Terminal Value 1 \$105 \$108 2 \$110 \$117 3 \$116 \$1,325 \$126 \$5,068

The terminal values are miscalculated because cash flows and discount rates are not matched.

Terminal value (using nominal rate and real cash flows) = \$ 116 *1.03 / (.12-.03) = \$1,325

Terminal value (using real rate and nominal cash flows) = \$ 126 * 1.0609 / (.0874 -.0609) = \$ 5,068

If real cash flows are discounted at the nominal discount rate, the present value is:

Present value (real cash flows discounted at nominal rates) = \$105/1.12 + \$ 110 / 1.122 + (\$116 + \$1,325) / 1.123 = \$ 1,207 < True value of \$1,896

If nominal cash flows are discounted at the real discount rate, the present value is:

Present value (nominal cash flows discounted at real rates) = \$108/1.0874 + \$ 117 / 1.08742 + (\$126 + \$5,068) / 1.08743 = \$4,293 > True value of \$1,896

TAXES AND VALUATION

Consistency Principle 2: Pre-tax cash flows should be discounted at pre-tax discount rates; After-tax cash flows should be discounted at after-tax discount rates.

After-tax version of the Capital Asset Pricing Model

E(Rj) = Rf + d0 + d1 bj + d2 (gj - Rf)

where,

E(Rj) = Expected pre-tax return on asset j

Rf = Riskfree rate

bj = Beta of asset j

gj = Dividend Yield of asset j

d0 = A constant term

d1 = Market premium for systematic risk

d2 = Influence of dividend payout on expected returns

Illustration 14: Effect of personal taxes on cash flows and discount rates: Eli Lilly

The expected dividends and the terminal price were estimated for Eli Lilly at the beginning of 1992 for the next five years (before personal taxes). The firm had a beta of 1.05 and the treasury bond rate wass 7.15%. The following example values Eli Lilly on the basis of cash flows after personal taxes for an investor with a tax rate of 40% for ordinary income and 28% for capital gains. The cash flows on a pre and post tax basis are as follows:
 Before personal taxes After personal taxes Year Dividends per share Terminal price Dividends per share Terminal price 1 \$2.27 \$1.36 2 \$2.60 \$1.56 3 \$2.98 \$1.79 4 \$3.41 \$2.05 5 \$3.91 \$93.43 \$2.35 \$84.41

The discount rate can be estimated before personal taxes and used to calculate the present value per share:

Discount rate before personal taxes = 7.15% + 1.05 (5.5%) = 12.93%

Present Value per share (based upon pre-tax cash flows and pre-tax discount rates)

=\$ 2.27/1.1293 + \$ 2.60 / 1.12932 + \$2.98/1.12933 + \$3.41/1.12934 + (\$3.91+\$93.45)/1.12935 = \$ 62.20

The initial price is assumed to be \$62.20. The cash flows after personal taxes are estimated after personal taxes as follows:

Dividends per share after taxes = Dividends per share before taxes * (1 - Ordinary tax rate)

Terminal price after taxes = Terminal price before taxes - (Terminal price - Initial price) * Capital Gains tax rate = \$ 93.43 - (\$93.43 - \$62.20) * 0.28 = \$84.41

The discount rate before personal taxes can be apportioned into dividend yield and price appreciation:

Expected Dividend Yield = Expected Dividends next year/ Initial Price

=\$2.27 / \$62.20 = 3.71%

Expected Price Appreciation = Expected Return - Expected Dividend Yield

= 12.93% - 3.71% = 9.22%

Discount rate after personal taxes = 3.71% (1 - 0.40) + 9.22% (1 - 0.28) = 8.86%

Present Value per share (based upon after-tax cash flows and after-tax discount rates)

=\$ 1.36/1.0886 + \$ 1.56 / 1.08862 + \$1.79/1.08863 + \$2.05/1.08864 + (\$2.35+\$84.41)/1.08865 = \$ 62.20

Thus the value is unaffected by whether cash flows are before or after personal taxes, as long as the discount rates are adjusted accordingly.

ESTIMATING GROWTH RATES

I. Estimating and Using Historical Growth Rates

• Historical growth rates can be estimated in a number of different ways -
• Arithmetic versus Geometric Averages
• Simple versus Regression Models
• Historical growth rates can be sensitive to
• the period used in the estimation
• In using historical growth rates, the following factors have to be considered
• how to deal with negative earning
• the effect of changing size

Illustration 15: Using arithmetic average versus geometric average: Autodesk

The following are the earnings per share at Glaxo Pharmaceuticals, starting in 1989 and ending in 1994:
 Year EPS Growth Rate 1989 \$0.66 1990 \$0.90 36.36% 1991 \$0.91 1.11% 1992 \$1.27 39.56% 1993 \$1.13 -11.02% 1994 \$1.27 12.39%

Arithmetic mean = (36.36%+1.11%+39.56%-11.02%+12.39%)/5 = 15.68%

Geometric mean = (1.27/0.66)1/5 -1 = 13.99%

Illustration 16: Sensitivity of historical growth rates to the length of the estimation period: Glaxo

The following table provides earnings per share at Glaxo, starting in 1988 instead of 1989 and uses six years of growth rather than five to estimate the arithmetic and geometric averages.
 Time (t) Year EPS 1 1988 \$0.65 2 1989 \$0.66 3 1990 \$0.90 4 1991 \$0.91 5 1992 \$1.27 6 1993 \$1.13 7 1994 \$1.27

Arithmetic average = 13.32%

Geometric mean = (1.27/0.65)1/6 -1 = 11.81%

Illustration 17: Linear and Log-linear models of growth: Glaxo Inc.

The earnings per share from 1988 until 1994 is provided for Glaxo., and the linear and log linear regressions are done below:
 Time (t) Year EPS ln(EPS) 1 1988 \$0.65 -0.43 2 1989 \$0.66 -0.42 3 1990 \$0.90 -0.11 4 1991 \$0.91 -0.09 5 1992 \$1.27 0.24 6 1993 \$1.13 0.12 7 1994 \$1.27 0.24

Linear Regression : EPS = 0.5171 + 0.1132 t

Log-linear Regression: ln (EPS) = -0.5536 + 0.1225 t

Expected EPS (1995) : linear regression = 0.5171 + 0.1132 (8) = \$1.42

Expected EPS (1995): log-linear regression = e (-0.55536 + 0.1225 (8))

= \$ 1.53

Dealing with negative earnings

Calculating growth rates when earnings become negative is problematic, since the traditional growth rate measures often fail. For instance, if earnings per share goes from <\$1.00> to \$1.00, the traditional growth rate measures would be ñ

Arithmetic Growth Rate = = EPSt/EPSt-1 -1 = \$1.00/<\$1.00> - 1 = -200 %

Geometric growth rate = \$1.00/<\$1.00> = -100%

There is a solution to this problem,

Modified growth rate =(EPSt-EPSt-1)/Max(EPSt,EPSt-1)

Illustration 18: Dealing with negative earnings: Sterling Chemicals.

The following series lists earnings per share from 1988 to 1994 for Sterling Chemicalsñ
 Time (t) Year EPS log(EPS) Growth Rate Modified Growth Rate 1 1988 \$3.56 1.27 2 1989 \$1.77 0.57 -50.28% -50.28% 3 1990 \$1.07 0.07 -39.55% -39.55% 4 1991 \$0.67 -0.40 -37.38% -37.38% 5 1992 \$0.08 -2.53 -88.06% -88.06% 6 1993 (\$0.10) NMF -225.00% -225.00% 7 1994 \$0.34 -1.08 -440.00% 129.41%

Approach 1: Using the slope coefficient from the linear regression

EPS =3.1114 - 0.5139 t

Average EPS (1988-94) = \$ 1.06

Growth rate = - 0.5139 / 1.06 = -48.48%

Approach 2: Using the minimum or maximum of earnings as the denominator

Arithmetic average, using modified growth rates = - 51.81%

Illustration 19: The Effect of size on growth: Amgen

Amgen increased its net income from \$19.1 million in 1989 to \$430 million in 1994. The following table shows the growth in net income for Amgen from 1989 to 1994, in both percentage and dollar terms.
 Year Net Income % Growth Rate Æ Net Income 1989 \$19.10 1990 \$86.20 351.31% \$67.10 1991 \$186.30 116.13% \$100.10 1992 \$306.70 64.63% \$120.40 1993 \$354.90 15.72% \$48.20 1994 \$430.00 21.16% \$75.10

Geometric Average Growth Rate = 86.42%

Assuming that this growth rate continues for the next five years,
 Year Net Income % Growth Rate Æ Net Income 1995 \$801.60 86.42% \$371.60 1996 \$1,494.32 86.42% \$692.72 1997 \$2,785.67 86.42% \$1,291.35 1998 \$5,192.98 86.42% \$2,407.31 1999 \$9,680.63 86.42% \$4,487.65

II. Analystsí Forecasts Of Earnings: How Good Are They?

Studies indicate that analysts do better than mechanical models at forecasting earnings in the short term, but do not do much better than such models over the long term.

Mean Relative Absolute Error

Study Analyst Group Analyst Mechanical

Forecasts Models

Collins & Hopwood Value Line Forecasts 31.7% 34.1%

1970-74

Brown & Rozeff Value Line Forecasts 28.4% 32.2%

1972-75

Fried & Givoly Earnings Forecaster 16.4% 19.8%

1969-79

Are some analysts more equal than others?

Some analysts are better at predicting earnings than other analysts. They are also usually better at picking stocks. Studies examining how effective analysts are conclude the following ñ

• On average, analysts affect stock prices with their recommendations; Buy recommendations affect prices less than sell recommendations. The prices tend to drift back to their original levels.
• The recommendations made by the ëbestí analysts (Institutional Investorís All American Analysts) have a greater impact on stock prices (3% on buys; 4.7% on sells). For these recommendations the price changes are sustained, and they continue to rise in the following period (2.4% for buys; 13.8% for the sells).

III. Expected Growth And Fundamentals

Retention Ratio and Return on Equity

gt = Retained Earningst-1/ NIt-1 * ROE

= Retention Ratio * ROE

= b * ROE

Proposition 1: The expected growth rate in earnings for a company cannot exceed its return on equity in the long term.

Short term changes in ROE

Small changes in return on equity can lead to large shifts in the growth rate for some firms. The relationship between growth rates and changes in return on equity is as follows ñ

gt = [BV of Equityt-1 * (ROEt - ROEt-1) / NIt-1]+ b * ROE

Illustration 20: Changes in ROE and growth rates: Merck Inc.

The following table provides information on the retention ratio and the return on equity for Merck for 1994 and projections for 1995.

1994 1995

BV of Equity 11700

Net Income 3010

Retention Ratio 52.00% 52%

Return on Equity 26.0% 25.5%

Growth Rate = 0.52*0.26 = (11700*(.255-.26)/3010)

= 13.52% + 0.52*0.255 = 11.32%

A drop in the return on equity of 0.5% translates into a drop in the growth rate of 2.20%. If the ROE drops by 2% the expected growth rate in 1995 would be 4.72%.

ROE and Leverage

ROE = ROA + D/E (ROA - i (1-t))

where,

ROA = (Net Income + Interest (1 - tax rate)) / BV of Total Assets

= EBIT (1- t) / BV of Total Assets

D/E = BV of Debt/ BV of Equity

i = Interest Expense on Debt / BV of Debt

t = Tax rate on ordinary income

Note that BV of Assets = BV of Debt + BV of Equity.

Return on Assets, Profit Margin and Asset Turnover

ROA = EBIT (1-t) / Total Assets

= [EBIT (1-t) / Sales] * [Sales/Total Assets]

= Pre-interest profit margin * Asset Turnover

Illustration 22: Evaluating the effects of corporate strategy on growth rate and value: Procter and Gamble

Procter & Gamble decided to reduce prices on their disposable diapers in April 1993 to compete better with low-price private label brands. The pre-interest, after-tax profit margin is expected to drop from 7.43% to 7% as a consequence and the asset turnover is expected to increase from 1.6851 to 1.80. The following table provides projections in profit margins, asset turnover and growth rates after the shift in corporate strategy:

1992 After shift in strategy

EBIT (1- tax rate) \$ 2181

Sales \$ 29,362

Pre-interest, after-tax profit margin 7.43% 7.00%

Total Assets \$17,424

Asset Turnover 1.6851 1.80

Return on Assets 12.52% 12.60%

Retention Ratio 58.00% 58.00%

Debt/Equity 0.7108 0.7108

Interest rate on debt (1 - tax rate) 4.27% 4.27%

Growth Rate 10.66% 10.74%

Illustration 23: Adjusting inputs for firm type: Neutrogena Inc.

The following table provides estimates of return on assets, retention ratio and interest rates for Neutrogena, a cosmetics manufacturer. Neutrogena is expected to grow at an extraordinary rate in the first five years (growth phase) and at a stable rate after that (steady state).

Retention Ratio 76% 50%

Return on Assets 19.5% 15%

Debt/Equity 0% 25%

Interest rate on debt 10% 8.00%

Growth Rate 14.82% 8.375%

Illustration 24: Weighting based upon standard deviations: Autodesk Inc.

Consider Autodesk Inc,a software company, with earnings per share available from 1987 to 1992. The following table also provides analyst forecasts of expected growth over the next five years from nine analysts following Autodesk.

Historical EPS Analyst Estimates
 Year EPS Growth Rate Analyst Number Estimated Growth 1987 \$0.89 1 10.00% 1988 \$1.35 51.69% 2 10.50% 1989 \$1.91 41.48% 3 12.00% 1990 \$2.30 20.42% 4 12.50% 1991 \$2.31 0.43% 5 13.00% 1992 \$1.98 -14.29% 6 16.00% 7 16.00% 8 18.00%

Arithmetic mean = 19.95% Consensus Forecast = 14%

Standard Deviation = 27.49% Standard Deviation = 3.45%