I.  The Valuation Technology

The general objectives of business valuation from a standpoint of corporate managers, external financiers and investors, and valuation specialists include: 1) the identification and measurement of value-driving activities, the opportunity costs for undertaking such activities, and their associated risk exposures both on a stand-alone and a portfolio basis; and 2) the application of alternative valuation methodologies on strategic, managerial, and operational decision-makings to optimize the risk-return tradeoffs in an uncertain and dynamic framework.

In practice, users of valuation methodologies seek to utilize all available knowledge to the extent that their resources, efforts, and exposures can be minimized.  With unlimited access to the sources of internal information, corporate managers have employed an economic-based valuation approach to capture direct contributions of certain operating, investment, and financing activities on firm value.  In contrast, creditors and shareholders who lack a direct control over the firm’s operations often rely on an accounting-based valuation approach to evaluate financial statements and establish the linkage between the firm’s performance and its market value.  Valuation specialists such as corporate lenders, investment bankers, professional fund managers, and portfolio investment advisors have utilized a wide range of sophisticated valuation techniques subject to their unique analytical styles and resource limitations.  Some prefer a “technical analysis” in which historical market data can be used as a basis for estimating value; others adopt a “fundamental analysis” to valuation either from a top-down or a bottom-up perspective.  With an advance in computer technology, however, many valuation specialists have turned their attention to a quantitative-based valuation approach that allows them to not only accurately measure the independent and joint impacts of internal business activities and external market events on firm value based on sophisticated mathematical models but also identify the mispriced assets or “valuation gaps” that may occur from persistent market imperfections and the sub-rational behavior of some groups of market participants.

From a perspective of decision-makers to invest in or dispose of different asset classes, the objectives of valuation become more complicated as there are various hybrid characteristics and multi-layered issues involved.  This is due to the fact that many of financial securities and contracts have been engineered to produce specific payoffs during a specified period that meet each user’s unique investment, financing, and hedging requirements.  The basic valuation methodologies that are geared towards different asset classes include project valuation, equity valuation, fixed-income valuation, and contingent-claim valuation.  The aggregation and unbundling of these underlying assets and hybrid securities would unavoidably lead to more complications.  Nevertheless, the choice regarding which asset-based valuation methodologies should be used depends largely on the user-based valuation approaches to be discussed below and is dictated by the users’ different objectives and constraints.

1.1  Economic-based Valuation Approach

The valuation approach that appeals most to economic, accounting, and finance practitioners is the so-called Discounted Cash Flow (DCF) methodology.  It rests on the postulate that “the intrinsic value of any asset or security is equivalent to the discounted value of all streams of future cash flow it could generate and/or realize.”  This postulate indicates that the sources of value come from either the internally generated income stream or the externally realized changes in market value of the asset itself between the acquisition date and the liquidation date, or from both.

The construct of the DCF model is built on the concept of Time Value of Money (TVM) that involves the use of appropriate discount rates to derive the present value of each cash flow generated during each period from the current valuation date (on or after the acquisition date) to the future horizon date (on or before the liquidation date) plus the present value of the horizon payoff realized on the liquidation date.  An algebraic form of the DCF model is given by:

Intrinsic Value = S Cash Flow/(1 + Discount Rate) ^{Period Number} + Payoff/(1+ Discount Rate) ^{Horizon Period}

In this DCF model, the cash flow in each period has a direct contribution to value creation.  The discount rate represents the opportunity cost for committing to such value creation activities with an inverse contribution to value.  Our task here is clear: 1) identification of all cash flow sources that result directly and indirectly from operating, investment, and financing activities and 2) determination of the appropriate cost of capital that is commensurate with the risks of undertaking those activities.

Sources of Cash Flow Identification

To identify the sources of business cash flow, a manager may begin by looking at the revenues that can be generated from utilizing the assets specific to that business unit.  In general, business cash flow comes from two sources – investment and operation.  In the first source, two classes of assets under the firm’s balance sheet are involved: 1) net working capital (consisting of receivables and inventories less payables); and 2) net fixed assets (acquisition costs less accumulated depreciations).  An increase in either net working capital or net fixed assets would represent a cash outflow while a decrease in either of them leads to the opposite effect.  Any change in the value of these two asset classes during a period can affect the cash flows in either direction depending on their net effect.  Therefore, cash flow from investment is an outflow by default.

The second source of cash flow comes from operating activities (conversion processes) that add value to the acquired asset classes mentioned above.  The manager can obtain the firm’s operating information from a profit and loss statement that accounts for both variable and fixed expenses incurred during an operating period.  This information is summarized in an accounting measure known as the earnings before interest and taxes (EBIT).  Unlike cash flow from investment activities, cash flow from operating activities is by default an inflow rather than an outflow.  By subtracting the planned investment cash outflows from the expected operating cash inflows, the manager can capture the stream of future net cash flows to be discounted back to the present time.

Risk-adjusted Discount Rate Determination

Now that value creation can be directly measured based on how the firm acquires and expends its assets to generate EBIT and net cash inflows, the next crucial task of the manager is to determine the discount rate.  This is easier said than done because the net cash flows must at least be equal to the cash flows that can be otherwise earned from the next best alternative investment with an equal risk exposure.  By comparison, if a business project cannot outperform a simple investment in bank accounts or government securities in terms of cash flows, then it should not be undertaken at all.  In other words, the discount rate to use in the DCF analysis must at least be the same as the highest rate of risk-free investment available to the firm.  This is a fundamental argument for the so-called Capital Asset Pricing Model (CAPM), which says that the required rate of return on a risky investment must be based on the equilibrium market risk-free rate plus the “market risk premium” to serve as an incentive for an investor to bear additional risk when undertaking a proposed investment.  We call this risk-adjusted rate of return as the “market capitalization rate,” which is the appropriate discount rate to use in this economic-based valuation.

The result of DCF valuation is an economic or “intrinsic” value of the business-unit investment.  This is equivalent to the stand-alone market value of its assets including both net working capital and net long-lived assets if the firm wants to sell the business unit today.  It is also equivalent to the value of equity if the firm decides to issue common stocks to raise external funds to finance this business unit.  And it should be the fair value for which the potential acquirers of the business unit are willing to pay.  If the firm decides to issue both common stocks and debt securities, then the market value of debt plus the market value of equity must be the same as the economic value of this business unit.  When this is the case, the discount rate would vary depending on the ratio between debt and equity as well as the difference between the cost of debt and the cost of equity capital.  Consequently, the DCF analysis would call for the weighted average cost of capital (WACC), which is a more appropriate rate to discount the business unit’s cash flows.  We shall later learn in detail how to statistically estimate the appropriate cost of equity capital using the observable market data.  This would lead us into the realm of quantitative-based valuation.

1.2  Accounting-based Valuation Approach

In comparison with the business-unit managers who know inside out about their firms’ investments and operations, the external claimholders and stakeholders typically know relatively less about the same activities.  The only convenient and cost-effective way these outsiders can measure the performance of those managers and the value of the firms or its equity is through the analysis of financial statements data, which are publicly available and required from those listed corporations.

Dividend Discount Method

External claimholders can observe and elicit relevant information about the firm value by focusing on some key performance indicators (KPIs) such as return on equity, dividend payout rate, and the growth rate in earnings and dividends.  These KPIs become the building blocks for the so-called Dividend Discount Method (DDM), which is a reduced-form DCF valuation approach discussed earlier.  DDM states that “the intrinsic value of a stock is the sum of all projected dividends that grow indefinitely at a constant rate discounted back to the current period at the cost of equity.”  This approach relies on the TVM concept of the present value of perpetuity wherein:

Intrinsic Equity Value = Projected Dividends/(Cost of Equity – Dividend Growth Rate)

The cost of equity can be obtained exogenously from the CAPM.  The dividend growth rate, on the other hand, is derived from accounting measures and is a product of return on equity and earnings retention rate, which is equal to (1 – dividend payout rate).  The shortcoming of DDM approach lies on this very input variable—the dividends cash flow—since not all firms promise to pay dividends to their shareholders.  The alternative models to DDM have been developed in consistent with the DCF method.  The first one is called the Residual Cash Flow (RCF) method, and the second one, the Operating Cash Flow (OCF) method.

Residual Cash Flow Method

RCF method recognizes that even though the firm need not pay dividends, it still generates the cash flow to which its shareholders are entitled.  Fortunately, accounting data allow valuation analysts to measure the fair value of equity through the determination of projected free cash flow to equity (FCFE).  By definition, FCFE is a residual cash flow after the investment cash flow and the debt-financing cash flow have been deducted from the operating cash flow.  Therefore, the intrinsic equity value is derived as:

Intrinsic Equity Value = Projected FCFE/(Cost of Equity – FCFE Growth Rate)

Operating Cash Flow Method

OCF method includes cash flow to debt holders in the valuation process by deriving the projected after-tax operating cash flow before any current debt repayments plus interest and new debt issues are taken into account, technically called operating cash flow to the firm (OCFF).  In other words, OCFF is a difference between after-tax operating cash flow and investment cash flow.  Discounting OCFF give us the intrinsic value of the firm:

 Intrinsic Firm Value = Projected OCFF/(Weighted Average Cost of Capital – OCFF Growth Rate)

To determine the intrinsic equity value based on OCF method, the fair value of debt must also be determined.  This is obtained by observing the market value of the firm’s debt securities or by discounting their face values at the appropriate yields to maturities.

Relative Valuation Method

Another valuation method that is closely related to the DCF framework is the Relative Valuation (RV) method.   This approach infers the intrinsic value of equity primarily from the price-to-earnings ratio (P/E) and the expected earnings of the firm.  The estimated value of P/E ratio, or commonly called “earnings multiplier,” is adjusted from the DDM as follows:

Price/Earnings = (Projected Dividends/Earnings) / (Cost of Equity – Dividend Growth Rate)

The intrinsic equity value can thus be estimated by multiplying the P/E ratio with the projected earnings based on the firm’s pro forma financial statements:

Intrinsic Equity Value = (P/E) x Projected Earnings

Notice from all of the above equations that claimholders, particularly the shareholders, can extensively utilize the firm’s KPIs and other market indicators to arrive at an estimate of fair value without having to know all details about the firm’s operating and investment activities beyond those provided by its published financial statements.  Alternative ratios of price to other accounting values such as cash flow, sales, and book equity can also be used by adjusting the FCFE and OCFF formulae in the same manner as we do with the DDM formula.

1.3     Quantitative-based Valuation Approach

The previous two valuation approaches have helped the users to evaluate firm value and its equity based on the internally generated earnings and cash flow estimates.  Yet, there remains another source of value that has not been quantified in those models, i.e., a payoff due to an uncontrollable change in market values of the firm and its equity.  This is quite problematic because we intend to measure those changes in market values themselves rather than using them in the models.  Valuation specialists have longed for some practical methodology that can reliably capture the movements in asset prices.  It is the quantitative methods based on mathematical and statistical modelling that have become a prominent part in today’s valuation technology.

The building blocks of quantitative-based valuation approach consist of Risk-Neutral Valuation (RNV) method and Stochastic Diffusion Processes (SDP).  RNV method imposes strong assumptions that markets are relatively complete and investors are rationally competitive.  These assumptions rule out any permanent arbitrage opportunity and give rise to the “Law of One Price” that reflects all publicly available information.  This Law means that without new information that can affect expected cash flows, discount rates, and growth rates, the value of any asset should not change.  RNV also implies that any two assets that generate identical future cash flows (payoffs) must have the same value (price).  As a result, when a portfolio is constructed from buying one asset and selling the other, both having the same payoffs, there should be no risk in holding this portfolio, as the payoffs from the two assets would perfectly offset each other.  Thus this riskless portfolio payoffs should be discounted at the risk-free rate and its discounted value must also be zero.  Nonetheless, RNV operates in the deterministic mode where every input variable is known or observable.

The SDP are used to enhance the RNV method by introducing stochasticity into the valuation environment.  SDP in their purest form are underscored by the martingale (i.e., a fair game of chance), of which the two popular cases are random walk and sub-martingale (or random walk with positive drift).  The random-walk property of SDP is assumed for informationally efficient markets where sequential changes in asset value occur only by a random chance following a stochastic process.  If there is a certain growth trend underlying the movement of asset value, then the sub-martingale property of SDP will apply.  In mathematical formulation, an SDP known as the Weiner Process (or Brownian Motion in its definite integral form) is often used as the based construct to evaluate the semi-stochastic changes in asset price:

dP/P =  (Drift)dt + (Volatility)dz

where:     Drift (a)           =  growth rate in asset price per unit of time interval

                 Volatility (s)   =  variability in asset yield per unit of random change

                 dP/P                 =  rate of incremental change in asset price

                 dt                     =  incremental change in the time interval

                 dz                     =  incremental random-walk change

Considering the Weiner SDP above, all except the random-walk variable are deterministic or observable from historical data.  Firstly, the incremental change in time interval (dt) is controllable as investors can plan how immediate or lengthy their investment horizons are from the current date.  Secondly, the drift (a) and volatility (s) rates can be empirically retrieved from historical data of asset price series prevailed in the markets.  If an asset is not traded in the marketplace, then the price series of comparable assets can be used as the proxies for estimation.  Finally and most notably, the incremental random walk process (dz) is a function of both a deterministic time change and a stochastic random error (e) whose mean is zero and variance is unity.  Variance in a stochastic process is directly influenced by time increments, i.e., a unit increase in time results in a corresponding unit increase in variance.  To make variance the same unit as drift rate, its square root (or standard deviation) is used, thereby requiring the square root of time interval (dt^0.5) as follows:

dz = edt^0.5

where:     e   =  random error following a standard normal distribution with mean = 0 and variance = 1

                 Diffusion (b)  =  sdt^0.5  =  time-varying standard deviation in rate of return

                 (¶B/¶t)      =  Theta (q) parameter     =  sensitivity of bond price to a change in time unit

dB = (Dm + 0.5Gs² + q)dt + (Ds)dz

where:    (Dm + 0.5Gs² + q)  =  Drift function (a)

                           Ds               =  Diffusion function (b)

Unfortunately, there is more than one specification of stochastic interest rate model we can use in valuing fixed-income securities.  Serial movements in short-term interest rate empirically exhibit a reversion to the long-run expectations, which necessitate an alternative SDP model called the Ornstien-Uhlenbeck Process.  This mean-reversion SDP is given as:

dy =  a(r - y)dt + bdz

where:       a(r - y)  =  Drift function (a)

                     b         =  Diffusion function (b)

                     r          =  long-run interest rate

                     y         =  short-term interest rate

                     a         =  speed of mean reversion

                     dy       =  incremental change in bond yield

                     dt        =  incremental change in time interval

                     dz        =  incremental random-walk change

The knowledge of stochastic interest rates is essential for corporate managers, investing claimholders, financial institutions, and valuation specialists to recognize their importance and impacts on valuation and the measurement and management of risks involving financial claims and contracts.

Contingent-Claim Valuation and the Stochastic Firm Value

Another related application of SDP and PDE in finance is on the valuation of contingent claims such as option contracts and other asset-linked derivative instruments.  Contingent-claim valuation is based on the concept of RNV and arbitrage-free portfolio formation in which the price of two assets generating identical payoffs must be the same.  Option contracts have this kind of feature that they allow the investors to synthetically create a portfolio comprising a long position in stock and a short position in bond.  We can envisage this option-pricing problem as if we were pricing equity in the sense that the firm’s common stock is merely a call option on the firm’s asset value with the firm’s face value of debt as the option’s exercise price.

The continuous-time European-style option pricing model on non-dividend stock was conceived from the seminal collaboration of Fisher Black and Myron Scholes in 1973 and shortly after from a major contribution of Robert Merton in 1974 by extending the Black-Scholes model to value an American-style option with dividend-paying stock.  Merton also made extensive use of this option-theoretic framework in other quantitative valuation, for instance, to price defaultable corporate debt, which is the genesis of credit risk modelling.  The contingent-claim model for equity valuation is given as follows:

Equity Value = (Firm Value)N (d₁) – (Present Value of Bond)N (d₂)

where:      N (d₁)  =  standard normal distribution of d₁  =  sensitivity of equity to firm value

                 N (d₂)   =  standard normal distribution of d₂  =  risk-neutral probability of non-default

                    d₁      =  [ln (V₀/B₀) + 0.5s²dt] / sdt^0.5

                    d₂      =  [ln (V₀/B₀) – 0.5s²dt] / sdt^0.5

The present value of corporate debt (B₀) is simply a face value of corporate bonds (B_t) maturing within a known time interval (dt) continuously discounted at a risk-free bond yield (r_f).

With a parity condition among the firm value (underlying asset price), the face value of bond (exercise price of asset), the firm equity (call option price), and the expected credit loss of debt (put option price), we can rewrite this contingent-claim model for risky debt valuation as follows:

Risky Debt Value = (Present Value of Bond)N (d₂) + (Firm Value)N (-d₁)

where:     N (-d₁)  =  standard normal distribution of -d₁  =  sensitivity of bond to firm value

                 N (d₂)   =  standard normal distribution of d₂   =  risk-neutral probability of non-default

                    -d₁     =  [ln (B₀/V₀) – 0.5s²dt] / sdt^0.5

                     d₂     =  [ln (V₀/B₀) – 0.5s²dt] / sdt^0.5

Recall that the above contingent-claim models are based upon the original Black-Scholes European-style option where the exercise of option right can occur only at the debt maturity date.  This certainly imposes a great deal of restrictions to their actual applicability.  Nonetheless, we still employ such models to serve as a benchmark against which our alternative valuations are measured. 

Value-at-Risk Measurement of Market Risk Exposure

Quantitative-based valuation approach has long evolved academically in the mainstream finance but its presence to the general public has still been in its infant stage due to the fact that it requires some rigorous backgrounds in quantitative techniques that are not so amicable to layman practitioners.  This is not entirely true for the development of Value-at-Risk (VaR) methodology to measuring the downside risk of portfolio loss due to market price fluctuations.  A team of financial practitioners at J.P. Morgan during 1995 had come up with this brilliant yet simplified idea on how to capture the worst-possible movement of portfolio value in a normal market circumstance within a certain timeframe at a given subjective probability belief.  Their VaR-based analytical product is commercially known as the RiskMetrics^TM model.  The CreditMetrics^TM for credit risk quantification came along soon after with the requirements to comply with the revised Basel Capital Accord II during the late 1990’s.

VaR method attempts to measure the maximum dollar loss that deviates from the expected portfolio value that moves within a designated time horizon.  The required input variables for the VaR model consist of initial portfolio value (IPV), portfolio annual drift rate (a), portfolio annual volatility rate (s), portfolio horizon (dt), and the subjective probability measure (p) that a given critical portfolio value can occur.

With these variables, we can further derive the necessary parameters for VaR calculation: the expected portfolio value (EPV), the deviated portfolio value (DPV), and the critical portfolio value (CPV) corresponding to the p measure.  EPV is the product of IPV and (1+adt) whereas DPV is the product of IPV and sdt^0.5.  The way to derive CPV is not quite straightforward, as it requires the use of integral calculus to convert p measure into the corresponding value of CPV along the Cumulative Distribution Function (CDF).  We can resort to an Excel spreadsheet function called NORMINV(Probability,Mean,Deviation) to obtain the CPV.

EPV = IPV(1+adt)

DPV = IPV(sdt^0.5)

CPV = NORMINV(p,EPV,DPV)

There are two types of VaR measures: 1) Absolute VaR, which is the dollar amount difference between IPV and CPV; and 2) Relative VaR, which is the dollar amount different between EPV and CPV.  Usually, relative VaR is larger than absolute VaR whenever the portfolio drift rate is positive because EPV is larger than IPV.

Absolute VaR = IPV – CPV

Relative VaR = EPV – CPV

Once both of these VaR measures have been derived, the portfolio managers will be able to gauge the downside risk of loss in their portfolio values and set up their exposure limits accordingly.  Notice here that without the knowledge of SDP, we are unable to obtain the appropriate value of portfolio annual drift rate (a) to implement in the VaR model.  Thus, it is quite imperative that VaR practitioners understand the scopes and limitations of quantitative-based valuation approach in order to implement the relevant models effectively.

II.  Estimating the Discount Rates and Market Capitalization Rates

For both economic-based and accounting based valuation approaches, the dominant methodology is DCF, which requires an extensive use of appropriate discount rates commensurate with the risk of future cash flow streams.  The quantitative-based valuation approach, on the other hand, models various discount rates, e.g., rates of return on common stocks and bonds, in the SDP environment.  For the purpose of estimating the discount rates based on historical information, the ex ante (prescriptive) CAPM still maintains its popularity in both academic and practitioner worlds despite some of its ex post (descriptive) shortcomings.

We now turn our attention to an explicit calculation method by utilizing the CAPM to obtain the market capitalization rates for our valuation exercises.  We first establish that the rate of return on equity (r_E) observed in the stock markets is equivalent to the cost of equity capital (k_E) the firms actually pay to their stockholders.  We also distinguish between historical and expected cost of equity.  Our objective is to infer the expected cost of equity, E[k_E], which will eventually become our risk-adjusted discount rate, from the historical cost of equity, k_E.

To uncover the risk of our firm’s equity that is actively traded in the market, it is more convenient to link its return pattern to that of the market portfolio.  Simply put, we are interested in measuring the cross-sectional sensitivity in the rate of equity return given the market rate of return.  Since we can obtain the historical data of our equity return and market return on any desirable time scales (e.g., daily, weekly, or monthly), we should be able to identify their sensitivity relationship and use it to estimate the expected equity return.

The tool that we use to accomplish this task is regression analysis.  Regression model attempts to find the most fitted linear curve that best explains the relationship between the equity and market rates of return that have occurred during the same period.  In effect, this linear curve should be able to describe the variability in historical equity return to a proportionate variability in historical market return in a more predictable manner.  This type of return variability is generally known as “market risk” of equity.  The unexplained variability that is not on the regression line is called “specific risk” of equity.  The combination of explained and unexplained variability is therefore the equity’s “total risk.”

Fortunately, specific risk of any individual equity can be reduced or even eliminated in a well-diversified portfolio.  One formation of the well-diversified portfolios is the market portfolio.  If we hold a portfolio that mimics the market index, we will then bear only the market risk while the rest of all specific risks would be eliminated.  By other names, market risk is also called “non-diversifiable risk” or “systematic risk” while specific risk is labeled as “diversifiable risk” or “idiosyncratic risk.”

The pre-regression relationship between equity and market returns is given by:

r_E  =  br_M + e                                                                                           Equation 1

                e  =  error term

The post-regression relationship between equity and market returns is given by:

r_E  =  a + br_M                                                                                           Equation 2

where:    a  =  intercept term

                b  =  slope term

Notice the difference between the two equations.  Before the regression, the relationship is noisy as the random error term is scattering around the regression line.  After the regression, the relationship becomes neater as the random errors are ironed out, leaving only the two stable parameters, the intercept (a) and the slope (b) terms.  The parameter of interest here is the slope term since we will use it to estimate the expected cost of equity, E[r_E], based on our knowledge of expected market return, E[r_M], and the risk-free rate, r_f.

2.1     Estimating the Beta and Alpha Parameters

Our principal technique in the estimation of market beta is regression analysis, technically known as the Ordinary Least Square (OLS) method.  To numerically calculate for the beta and the alpha parameters, we use:

b = Covariance (r_E, r_M) / Variance (r_M)                                                                 Equation 3

and

a = E[r_E] – bE[r_M]                                                                                    Equation 4

We shall base our regression analysis on the empirical data that can be observed in the marketplace, which include the average monthly equity return and market portfolio return that spanned the period of, say, 12 months.   These input variables are shown in the example at the end of this instruction note.  We first assume for OLS purposes that both equity rate of return (y) and market rate of return (x) are random variables each with its own distribution.  Our task is to find their underlying relationship, if any, based upon their joint distribution with statistical parameters like covariance and correlation.  Before we arrive at that destination, we will need to manipulate these input data so that they provide us with workable numbers for our statistical formulation.

The first type of parameter we can derive is a simple arithmetic average.  We can do so just by dividing the sum of each variable y and x by the number of observations, which is 12 months in this case.  The results are the average value of equity return E[Y], and the average value of market return E[X].  Note that the use of expectation notation E[·] is not quite appropriate here as it usually represents a forward-looking estimate of an unknown value.  Nevertheless, we should leave the properness of notation usage to be conventionally understood here before we continue.

The second type of parameter is a covariance of equity and market returns Cov[Y,X], which will become a nominator in Equation 3 for our beta estimate.  The formula for covariance of a pair of random variables can be written in many ways, such as

Cov[Y,X] = SD[Y]*SD[X]*Corr[Y,X]                                                                 Equation 5

where:     SD[Y]         =  standard deviation of random equity return y

                 SD[X]        =  standard deviation of random market return x

                 Corr[Y,X]  =  correlation coefficient of random variables y and x

Cov[Y,X] = E[(y – E[Y])(x – E[X])]                                                                   Equation 6

and

Cov[Y,X] = E[YX] – E[Y]*E[X]                                                                      Equation 7

where:       E[YX]   =  average of the product of x and y

It seems that Equation 7 is the easiest formula to implement since it involves only the averaging of y, x, and their product.  Equation 6 is a bit more complicated as it requires us to produce the deviated values of both variables from their average values.  Equation 5 should be the hardest one to derive since we also need to generate the value of correlation coefficient between the two variables.

The third type of parameter is a variance of market returns V[X].  In deriving the beta, a variance of equity returns V[Y] is not required.  Two ways to write the variance equations are:

V[X] = E[(x – E[X])(x – E[X])] = E[(x – E[X])²]                                                       Equation 8

or

V[X] = E[XX] – E[X]E[X] = E[X²] – (E[X])²                                                          Equation 9

Notice the similarity between Equations 6 and 8 and Equations 7 and 9.  They mean that the variance of a random variable is simply the covariance of that variable with itself.

After we have computed the covariance and variance parameters, we now can determine the value of beta coefficient of the regression based on Equation 3.  Using Equation 2, we should be able to produce a linear relationship between equity return and market return without knowing the value of the intercept.  To find out the value of the regression’s intercept term, we might need to know what it means.  The intercept term is the difference between the expected equity return and the regression estimate, i.e., the beta times the expected market return.  Using Equation 3, the intercept term will be stable within the regression equation while the actual random errors (noise) disappear.

There is also a short-cut way to directly solve for the beta and alpha estimates without going through the variance and covariance calculation procedures.  Observe the following formulae:

b = [NSyx – SySx] / [NSxx – Sx Sx]                                                            Equation 10

or

b = [NE[Y]E[X] – Syx] / [NE[X]E[X] – Sxx]                                                      Equation 11

where:       N  =  total number of periods under observation

and

a = E[Y] – bE[X]                                                                                  Equation 12

We can obtain exactly the same estimates of beta and alpha as we do from using the detailed procedures.  However, Equations 10 and 11 for the beta are quite difficult to memorize.  We just have to know the definitions and properties of variance and covariance to appreciate these formulae.  On the brighter side, Equation 10 does not require us to find the average values of y and x, which should be the quickest way to derive the beta estimate.  The formula to estimate the alpha based on Equation 12 is easier to memorize but cannot be used on a stand-alone basis without a prior estimate of the beta.

2.2     Estimating the Ex Ante Cost of Equity

We now use our ex post beta to predict an ex ante cost of equity, or before-the-fact k_E.  The finance model that allows us to estimate the cost of equity is CAPM.  CAPM relates the expected return on equity to the “risk-free rate” and the “risk premium” (i.e., the difference between the expected market return and the risk-free rate) through the market beta.  The model is formally expressed as:

E[k_E]  =  r_f + (E[r_M] – r_f)b

where:    E[k_E]   =  expected cost of equity

                r_f         =  risk-free rate of return on riskless security

                E[r_M]  =  expected rate of return on market portfolio (index)

                b         =  ex-post beta parameter 

Suppose that the risk-free rate is 5 percent and the expected rate of return on market portfolio is 25 percent.  The risk premium is therefore 20 percent.  Adjusting this risk premium with the market beta of 1.256526 gives us the estimate of market risk premium of the equity in question of 25.13 percent.  Adding it to the risk-free rate will result in an estimate for the expected rate of return on equity, or the ex ante cost of equity (E[k_E]), of 30.13 percent.  We have utilized this E[k_E] in our valuation methods.

III.  Interrelationships among Working Capital, Profit, and Equity Value

Thus far, we have seen in at least two valuation approaches above that their required input data are derived or extracted from the firm’s financial statements based heavily on accounting information.  The drivers for firm value and therefore its equity value are from both operating and investment performance, which are directly represented by earnings and changes in asset values.  These two components are translated into operating cash flows to be discounted at the firm’s cost of capital to determine the firm value pursuant to the DCF methodology.

The significance of working capital in value creation is through increased productivity, i.e., the turnover of current expenditures into revenues.  By definition, working capital turnover measures the dollar amount of revenues generated per one dollar of spending to accumulate accounts receivable and inventories.  The higher the working capital turnover during the operating period, the higher the productivity the firm has experienced.  However, an increase in working capital over a period represents a drain of cash while its decrease effectively means a cash inflow.  It is not unusual to see a high-growth firm spending more cash on working capital to exploit its currently strong productivity momentum.

High working capital turnover will not necessarily be translated into high value unless the firm is able to manage its operating costs effectively.  There are two levels of profit after the operating costs and expenses have been deducted: 1) operating income and 2) net income.  Operating income, or earnings before interest and taxes (EBIT), becomes the basis of operating cash flow where non-cash depreciation charges and changes in net working capital (a difference between changes in current assets and changes in current liabilities) are added to it.  This results in a set of Operating Cash Flow to the Firm (OCFF) as we have learned before.  At this level, the present value of OCFF discounted by the weighted average cost of capital (WACC) becomes the fair value of the firm’s assets.

Net income, on the other hand, contributes only to the firm’s equity since it represents the residual earnings after all fixed financial obligations with all external creditors (i.e., net financing expenses) have been met.  Using the same method to convert profit into cash flow as employed with operating income, we can add depreciation charges and changes in net working capital to net income.  By discounting this cash flow with the cost of equity (k_E), we are able to derive the fair value of equity.  As discussed earlier, we call the cash flow at this level as Free Cash Flow to Equity (FCFE).

By now, a linkage between economic value of the firm (V) and the accounting-based OCFF has been established through WACC after adjusted for a desirable capital structure.  Likewise, a direct relationship between the firm’s equity value (E) and the accounting-based FCFE is linked by the market risk-adjusted k_E.  Corporate managers, internal stakeholders, external claimholders, and valuation specialists should be conversant with their similarity and distinction so that any misrepresented analyses and false conclusions can be avoided.

IV.  Importance and Implications of Business Valuation on Lender’s Risk

The knowledge of firm value, its expectations and variability in relation to the value of debt contract is important to the firm’s lenders and creditors alike because it could trigger the firm to default on debt obligations whenever the varying firm value falls below the contractual debt value.  Essentially, the analysis of bank credits, debt securities, and fixed-income instruments can be seen as a by-product of the analysis of firm value and its equity because the difference between the two engenders the fair value of the firm’s debt.  This information not only indicates to lenders about the firm’s market value of debt at which to liquidate but also its current creditworthiness and future credit rating.  The lenders, in turn, can methodically evaluate default risk of their corporate borrowers before any loan application can be approved.

We already have a glimpse of the basic contingent-claim method of quantitative valuation approach.  Although the method is quite straightforward in terms of mathematical formulae, some computational complexity can arise, such as the way to derive a cumulative distribution function (CDF) for both the normal and standard normal variates.  However, such a method can be easily implemented on computers using standard spreadsheet programs.  With this computational convenience, it is advisable that the users of contingent-claim models explore further into their extensions with more realistic assumptions pertaining to different and unique characteristics of corporate borrowers.  Towards the end of this instruction note, we shall do some manual practice on a simple risky debt valuation model using hypothetical input and derived data.

V.  Requirements for Business Divestitures and Relevant Issues to Acquirers

Small businesses can expand by going public or being acquired by bigger and well-established firms.  In either case, the valuation of their assets and obligations is vital to both the current owner-managers and the corporate acquirers.  From an entrepreneur’s point of view, the track-record accounting performance of his/her business since inception would be used as the basis for determining the actual cost of capital, which is one of the major components for valuation.  From the standpoint of outside investors and acquirers, the earnings prospects, cash-flow growth potentials, and all other relevant risks after managerial restructuring (through corporatization) or reorganization (through merger and acquisition or M&A) become the more crucial factors.  By integrating the information from both parties, valuation specialists will be able to come up with a more realistic estimate of fair value for sales or trading.  And obviously, the most sensible valuation approach to use is the DCF methodology.

What valuation specialists should further investigate is the small business’s internal accounting systems to see how reliable the reported performance data and KPIs are.  Due to asymmetric information, the incumbent owner-managers always have an incentive to distort their historical performance by overestimating or revealing revenues and gains while underestimating or concealing expenses and losses.  If the internal accounting systems have not adequately been transparent, there is always a suspicion about a self-interest maximization behavior or “agency problem” of the current owners.  To counter the effects of agency problem, the offered price to acquire all the firm’s assets is often deeply discounted, which is conventionally known as “adverse selection” behavior of the potential public investors or corporate acquirers.

Once the M&A deal has been finalized, valuation specialists should be aware of the short- and long-term plans that the firms have about their acquired businesses.  Some firms strongly believe that under their larger corporate aegis, the newly acquired business unit would help enhance their revenues and earnings through synergy effects.  This is particularly true in the case of overhead-cost savings that the acquired business unit no longer needs to bear substantial fixed costs it used to have prior to the merger.  If the merger motive was based upon mispricing, the firms would decide sell their newly acquired business, in whole or in part, for short-term capital gains whenever there exist other willing buyers.  No matter what the motives of the acquirers are, the underlying exercise they have to perform is valuation to uncover any overlooked value drivers or diluters.

VI.  Conclusion

In this paper, we have progressively touched upon various indispensable toolkits for business valuation.  Relative valuation approach (e.g., the P/E method) is the easiest one to use for any individual public investors, as it requires nothing more than a good estimate of accounting earnings.  Intrinsic valuation approach (i.e., the DCF method) is more popular among corporate managers and fundamental analysts who possess a stockpile of internal and market data about the firm’s asset and equity value in order to derive the expected cash flows and the risk-adjusted rates of capitalization.  Quantitative valuation approach (i.e., the RNV method) has taken a step further than the DCF in that it no longer requires the use of risk-adjusted discount rate, but rather the risk-free rate, to evaluate both equity and debt of the firm.  However, the difficulty in implementing the RNV method such as those contingent-claim models is how to obtain a good estimate of volatility rates, which can be elusive and unstable over a longer period of time.

Needless to say, there are always tradeoffs between simplicity in model applications and accuracy in and predictive power of their results.  Once again, the users are urged to constantly keep track of new development in and venture into the more sophisticated valuation models while occasionally revisiting the basic ones to ensure that their professional valuation practices have been grounded on a solid and proven valuation technology.

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