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Valuation Models

Aswath Damodaran

Misconceptions about Valuation

Myth 1: A valuation is an objective search for “true” value

Truth 1.1: All valuations are biased. The only questions are how much and in which direction.

Truth 1.2: The direction and magnitude of the bias in your valuation is directly proportional to who pays you and how much you are paid.

Myth 2.: A good valuation provides a precise estimate of value

Truth 2.1: There are no precise valuations

Truth 2.2: The payoff to valuation is greatest when valuation is least precise.

Myth 3: . The more quantitative a model, the better the valuation

Truth 3.1: One’s understanding of a valuation model is inversely proportional to the number of inputs required for the model.

Truth 3.2: Simpler valuation models do much better than complex ones.

Approaches to Valuation

Basis for all valuation approaches

The use of valuation models in investment decisions (i.e., in decisions on which assets are under valued and which are over valued) are based upon

a perception that markets are inefficient and make mistakes in assessing value

an assumption about how and when these inefficiencies will get corrected

In an efficient market, the market price is the best estimate of value. The purpose of any valuation model is then the justification of this value.

Discounted Cash Flow Valuation

What is it: In discounted cash flow valuation, the value of an asset is the present value of the expected cash flows on the asset.

Philosophical Basis: Every asset has an intrinsic value that can be estimated, based upon its characteristics in terms of cash flows, growth and risk.

Information Needed: To use discounted cash flow valuation, you need

to estimate the life of the asset

to estimate the cash flows during the life of the asset

to estimate the discount rate to apply to these cash flows to get present value

Market Inefficiency: Markets are assumed to make mistakes in pricing assets across time, and are assumed to correct themselves over time, as new information comes out about assets.

Discounted Cashflow Valuation: Basis for Approach

where CFt is the cash flow in period t, r is the discount rate appropriate given the riskiness of the cash flow and t is the life of the asset.

Proposition 1: For an asset to have value, the expected cash flows have to be positive some time over the life of the asset.

Proposition 2: Assets that generate cash flows early in their life will be worth more than assets that generate cash flows later; the latter may however have greater growth and higher cash flows to compensate.

Equity Valuation versus Firm Valuation

Value just the equity stake in the business

Value the entire business, which includes, besides equity, the other claimholders in the firm

I.Equity Valuation

The value of equity is obtained by discounting expected cashflows to equity, i.e., the residual cashflows after meeting all expenses, tax obligations and interest and principal payments, at the cost of equity, i.e., the rate of return required by equity investors in the firm.

where,

CF to Equityt = Expected Cashflow to Equity in period t

ke = Cost of Equity

Forms: The dividend discount model is a specialized case of equity valuation, and the value of a stock is the present value of expected future dividends. In the more general version, you can consider the cashflows left over after debt payments and reinvestment needs as the free cashflow to equity.

II. Firm Valuation

Cost of capital approach: The value of the firm is obtained by discounting expected cashflows to the firm, i.e., the residual cashflows after meeting all operating expenses and taxes, but prior to debt payments, at the weighted average cost of capital, which is the cost of the different components of financing used by the firm, weighted by their market value proportions.

APV approach: The value of the firm can also be written as the sum of the value of the unlevered firm and the effects (good and bad) of debt.

Firm Value = Unlevered Firm Value + PV of tax benefits of debt - Expected Bankruptcy Cost

Generic DCF Valuation Model

Valuing Global Crossing with Distress

Probability of distress

Price of 8 year, 12% bond issued by Global Crossing = $ 653

Probability of distress = 13.53% a year

Cumulative probability of survival over 10 years = (1- .1353)10 = 23.37%

Distress sale value of equity

Book value of capital = $14,531 million

Distress sale value = 15% of book value = .15*14531 = $2,180 million

Book value of debt = $7,647 million

Distress sale value of equity = $ 0

Distress adjusted value of equity

Value of Global Crossing = $3.22 (.2337) + $0.00 (.7663) = $0.75

Adjusted Present Value Model

In the adjusted present value approach, the value of the firm is written as the sum of the value of the firm without debt (the unlevered firm) and the effect of debt on firm value

Firm Value = Unlevered Firm Value + (Tax Benefits of Debt - Expected Bankruptcy Cost from the Debt)

The unlevered firm value can be estimated by discounting the free cashflows to the firm at the unlevered cost of equity

The tax benefit of debt reflects the present value of the expected tax benefits. In its simplest form,

Tax Benefit = Tax rate * Debt

The expected bankruptcy cost is a function of the probability of bankruptcy and the cost of bankruptcy (direct as well as indirect) as a percent of firm value.

Excess Return Models

You can present any discounted cashflow model in terms of excess returns, with the value being written as:

Value = Capital Invested + Present value of excess returns on current investments + Present value of excess returns on future investments

This model can be stated in terms of firm value (EVA) or equity value.

Relative Valuation

What is it?: The value of any asset can be estimated by looking at how the market prices “similar” or ‘comparable” assets.

Philosophical Basis: The intrinsic value of an asset is impossible (or close to impossible) to estimate. The value of an asset is whatever the market is willing to pay for it (based upon its characteristics)

Information Needed: To do a relative valuation, you need

an identical asset, or a group of comparable or similar assets

a standardized measure of value (in equity, this is obtained by dividing the price by a common variable, such as earnings or book value)

and if the assets are not perfectly comparable, variables to control for the differences

Market Inefficiency: Pricing errors made across similar or comparable assets are easier to spot, easier to exploit and are much more quickly corrected.

Variations on Multiples

Equity versus Firm Value

Equity multiples (Price per share or Market value of equity)

Firm value multiplies (Firm value or Enterprise value)

Scaling variable

Earnings (EPS, Net Income, EBIT, EBITDA)

Book value (Book value of equity, Book value of assets, Book value of capital)

Revenues

Sector specific variables

Base year

Most recent financial year (Current)

Last four quarters (Trailing)

Average over last few years (Normalized)

Expected future year (Forward)

Comparables

Sector

Market

Definitional Tests

Is the multiple consistently defined?

Proposition 1: Both the value (the numerator) and the standardizing variable ( the denominator) should be to the same claimholders in the firm. In other words, the value of equity should be divided by equity earnings or equity book value, and firm value should be divided by firm earnings or book value.

Is the multiple uniformally estimated?

The variables used in defining the multiple should be estimated uniformly across assets in the “comparable firm” list.

If earnings-based multiples are used, the accounting rules to measure earnings should be applied consistently across assets. The same rule applies with book-value based multiples.

An Example: Price Earnings Ratio: Definition

PE = Market Price per Share / Earnings per Share

There are a number of variants on the basic PE ratio in use. They are based upon how the price and the earnings are defined.

Price: is usually the current price

is sometimes the average price for the year

EPS: earnings per share in most recent financial year

earnings per share in trailing 12 months (Trailing PE)

forecasted earnings per share next year (Forward PE)

forecasted earnings per share in future year

Descriptive Tests

What is the average and standard deviation for this multiple, across the universe (market)?

What is the median for this multiple?

The median for this multiple is often a more reliable comparison point.

How large are the outliers to the distribution, and how do we deal with the outliers?

Throwing out the outliers may seem like an obvious solution, but if the outliers all lie on one side of the distribution (they usually are large positive numbers), this can lead to a biased estimate.

Are there cases where the multiple cannot be estimated? Will ignoring these cases lead to a biased estimate of the multiple?

How has this multiple changed over time?

PE Ratio: Descriptive Statistics

PE: Deciphering the Distribution

8 Times EBITDA is not cheap…

Analytical Tests

What are the fundamentals that determine and drive these multiples?

Proposition 2: Embedded in every multiple are all of the variables that drive every discounted cash flow valuation - growth, risk and cash flow patterns.

In fact, using a simple discounted cash flow model and basic algebra should yield the fundamentals that drive a multiple

How do changes in these fundamentals change the multiple?

The relationship between a fundamental (like growth) and a multiple (such as PE) is seldom linear. For example, if firm A has twice the growth rate of firm B, it will generally not trade at twice its PE ratio

Proposition 3: It is impossible to properly compare firms on a multiple, if we do not know the nature of the relationship between fundamentals and the multiple.

Relative Value and Fundamentals

What to control for...

Multiple Variables that determine it…

PE Ratio Expected Growth, Risk, Payout Ratio

PBV Ratio Return on Equity, Expected Growth, Risk, Payout

PS Ratio Net Margin, Expected Growth, Risk, Payout Ratio

EVV/EBITDA Expected Growth, Reinvestment rate, Cost of capital

EV/ Sales Operating Margin, Expected Growth, Risk, Reinvestment

Application Tests

Given the firm that we are valuing, what is a “comparable” firm?

While traditional analysis is built on the premise that firms in the same sector are comparable firms, valuation theory would suggest that a comparable firm is one which is similar to the one being analyzed in terms of fundamentals.

Proposition 4: There is no reason why a firm cannot be compared with another firm in a very different business, if the two firms have the same risk, growth and cash flow characteristics.

Given the comparable firms, how do we adjust for differences across firms on the fundamentals?

Proposition 5: It is impossible to find an exactly identical firm to the one you are valuing.

Comparing PE Ratios across a Sector

PE, Growth and Risk

Dependent variable is: PE

R squared = 66.2% R squared (adjusted) = 63.1%

Variable Coefficient SE t-ratio prob

Constant 13.1151 3.471 3.78 0.0010

Growth rate 121.223 19.27 6.29 ≤ 0.0001

Emerging Market -13.8531 3.606 -3.84 0.0009

Emerging Market is a dummy: 1 if emerging market

0 if not

Is Telebras under valued?

Predicted PE = 13.12 + 121.22 (.075) - 13.85 (1) = 8.35

At an actual price to earnings ratio of 8.9, Telebras is slightly overvalued.

PE Ratio without a constant - US Stocks

Relative Valuation: Choosing the Right Model

Contingent Claim (Option) Valuation

Options have several features

They derive their value from an underlying asset, which has value

The payoff on a call (put) option occurs only if the value of the underlying asset is greater (lesser) than an exercise price that is specified at the time the option is created. If this contingency does not occur, the option is worthless.

They have a fixed life

Any security that shares these features can be valued as an option.

Option Payoff Diagrams

Underlying Theme: Searching for an Elusive Premium

Traditional discounted cashflow models under estimate the value of investments, where there are options embedded in the investments to

Delay or defer making the investment (delay)

Adjust or alter production schedules as price changes (flexibility)

Expand into new markets or products at later stages in the process, based upon observing favorable outcomes at the early stages (expansion)

Stop production or abandon investments if the outcomes are unfavorable at early stages (abandonment)

Put another way, real option advocates believe that you should be paying a premium on discounted cashflow value estimates.

Three Basic Questions

When is there a real option embedded in a decision or an asset?

There has to be a clearly defined underlying asset whose value changes over time in unpredictable ways.

The payoffs on this asset (real option) have to be contingent on an specified event occurring within a finite period.

When does that real option have significant economic value?

For an option to have significant economic value, there has to be a restriction on competition in the event of the contingency.

At the limit, real options are most valuable when you have exclusivity - you and only you can take advantage of the contingency. They become less valuable as the barriers to competition become less steep.

Can that value be estimated using an option pricing model?

The underlying asset is traded - this yield not only observable prices and volatility as inputs to option pricing models but allows for the possibility of creating replicating portfolios

An active marketplace exists for the option itself.

The cost of exercising the option is known with some degree of certaint

Putting Natural Resource Options to the Test

The Option Test:

Underlying Asset: Oil or gold in reserve

Contingency: If value > Cost of development: Value - Dev Cost

If value < Cost of development: 0

The Exclusivity Test:

Natural resource reserves are limited (at least for the short term)

It takes time and resources to develop new reserves

The Option Pricing Test

Underlying Asset: While the reserve or mine may not be traded, the commodity is. If we assume that we know the quantity with a fair degree of certainty, you can trade the underlying asset

Option: Oil companies buy and sell reserves from each other regularly.

Cost of Exercising the Option: This is the cost of developing a reserve. Given the experience that commodity companies have with this, they can estimate this cost with a fair degree of precision.

Bottom Line: Real option pricing models work well with natural resource options.

The Real Options Test: Patents and Technology

The Option Test:

Underlying Asset: Product that would be generated by the patent

Contingency:

If PV of CFs from development > Cost of development: PV - Cost

If PV of CFs from development < Cost of development: 0

The Exclusivity Test:

Patents restrict competitors from developing similar products

Patents do not restrict competitors from developing other products to treat the same disease.

The Pricing Test

Underlying Asset: Patents are not traded. Not only do you therefore have to estimate the present values and volatilities yourself, you cannot construct replicating positions or do arbitrage.

Option: Patents are bought and sold, though not as frequently as oil reserves or mines.

Cost of Exercising the Option: This is the cost of converting the patent for commercial production. Here, experience does help and drug firms can make fairly precise estimates of the cost.

Bottom Line: Use real option pricing arguments with caution.

The Real Options Test for Growth (Expansion) Options

The Options Test

Underlying Asset: Expansion Project

Contingency

If PV of CF from expansion > Expansion Cost: PV - Expansion Cost

If PV of CF from expansion < Expansion Cost: 0

The Exclusivity Test

Barriers may range from strong (exclusive licenses granted by the government) to weaker (brand name, knowledge of the market) to weakest (first mover).

The Pricing Test

Underlying Asset: As with patents, there is no trading in the underlying asset and you have to estimate value and volatility.

Option: Licenses are sometimes bought and sold, but more diffuse expansion options are not.

Cost of Exercising the Option: Not known with any precision and may itself evolve over time as the market evolves.

Bottom Line: Using option pricing models to value expansion options will not only yield extremely noisy estimates, but may attach inappropriate premiums to discounted cashflow estimates.

Summarizing the Real Options Argument

There are real options everywhere.

Most of them have no significant economic value because there is no exclusivity associated with using them.

When options have significant economic value, the inputs needed to value them in a binomial model can be used in more traditional approaches (decision trees) to yield equivalent value.

The real value from real options lies in

Recognizing that building in flexibility and escape hatches into large decisions has value

Insights we get on understanding how and why companies behave the way they do in investment analysis and capital structure choices.

Which approach should you use? Depends upon the asset being valued..

And the analyst doing the valuation….