Approaches to valuing real optionsApproaches to valuing real options

Analytical:Analytical: Binomial modelBinomial model Black-Scholes formulaBlack-Scholes formula

SimulationsSimulations

Binomial modelBinomial model

V – the gross value of the project (expected value of V – the gross value of the project (expected value of subsequent CF)subsequent CF)d = 1/ud = 1/uThere exists a “twin” security that can be traded, which There exists a “twin” security that can be traded, which price S is perfectly correlated with V.price S is perfectly correlated with V.If there is an option on the project, we use “replicating If there is an option on the project, we use “replicating portfolio” technique (or risk neutral probabilities, which is portfolio” technique (or risk neutral probabilities, which is the same) to determine its valuethe same) to determine its value

V

V+=uV

V-= dV

V++=u2V

V--=d2V

V+-=udV

q

1-q

1-q

1-q

q

q…

EE++ = NS = NS++ - (1+ - (1+rr)B)BEE-- = NS = NS-- - (1+ - (1+rr)B)BN = (EN = (E++ - E - E--)/(S)/(S++ - S - S--))B = (NSB = (NS-- – E – E--)/(1+)/(1+rr))

““risk-neutral” valuation:risk-neutral” valuation:EE00 = NS – B = (pE = NS – B = (pE++ + (1-p)E + (1-p)E--)/(1+)/(1+rr))where p = ((1 + r)S – Swhere p = ((1 + r)S – S--)/(S)/(S++ - S - S--)= (1 + )= (1 + rr – d)/(u - d) – d)/(u - d)We have u=1.8, d=0.6, r=0.08, hence p=0.4, EWe have u=1.8, d=0.6, r=0.08, hence p=0.4, E00 = 25.07 = 25.07

V=100, S=20I0=104

V+=180, E+=67.68 S+=36q=0.5

0.5 V-=60, E-=0S-=12

I1=112.32

At the end At the end pp depends only on u, d and depends only on u, d and rr::pp = (1 + = (1 + rr – d)/(u - d) – d)/(u - d)In fact, In fact, pp can be found from the following: can be found from the following:S = (S = (ppuS + (1-uS + (1-pp)dS)/(1+)dS)/(1+rr),),i.e. i.e. pp must be such that the risk-neutral valuation of the must be such that the risk-neutral valuation of the “twin” security yields its actual price – “twin” security’s “twin” security yields its actual price – “twin” security’s value in the “risk-neural world” must be the same as in value in the “risk-neural world” must be the same as in the “real” world the “real” world

Thus:Thus: pp does not depend on the actual probability of going does not depend on the actual probability of going

up up qq. Reason: . Reason: qq is already incorporated in the price S. is already incorporated in the price S. Given the tree,Given the tree, p p does not depend on the particular does not depend on the particular

option (in particular on where we are in the tree)option (in particular on where we are in the tree)

Example: Option to abandon for Example: Option to abandon for salvage value or switch usesalvage value or switch use

We should switch at such points (If the option is to We should switch at such points (If the option is to switch any time we want, we switch the first time we switch any time we want, we switch the first time we get to such a node)get to such a node)

100

180

60

324

36

108 85

127.5

68

191

54.4

102

Current project. Values of V Alternative use. Values of V

E0=?

E+=180

E-=68

What is the value of the option to switch in year 1?

E0 = ((ppEE++ + (1- + (1-pp)E)E--)/(1+)/(1+rr) - I) - I00 = 0.44 = 0.44

(we can use the same probabilities (we can use the same probabilities pp as before) as before)

If we had no option to switch, the project would have NPV = -4 (also as before)

Hence, the value of the option is 4.44

Black-Scholes Pricing FormulaBlack-Scholes Pricing Formula(no dividend case, call option)(no dividend case, call option)

C0 = the value of a European option at time t = 0r = the risk-free interest rateS = the price of the underlying asset (or “twin” security)E – exercise price (e.g. investment required)N(.) – cumulative standard normal distribution functionσ – standard deviation of the underlying asset return

)N()N( 210 dEedSC rT

T

Tσ

rESd

)2

()/ln(2

1

Tdd 12

Adjusting for dividends (i.e. if the project Adjusting for dividends (i.e. if the project generates cash flows before the option generates cash flows before the option

“expiration” date)“expiration” date)

)'N()'N( 210 dEedSeC rTDT

T

Tσ

DrESd

)2

()/ln('

2

1

Tdd '' 12

Assume a constant dividend yield (i.e. constant cash flow) every year. Then:

Some caveats about the real Some caveats about the real options approachoptions approach

Black-Scholes formula presumes a diffusion Black-Scholes formula presumes a diffusion Wiener process for underlying (“twin”) security:Wiener process for underlying (“twin”) security:

Is it always the case?Is it always the case?

Can we always find a “twin” security? If not, Can we always find a “twin” security? If not, people do “market asset disclamer” assumption: people do “market asset disclamer” assumption: the project itself is a “twin” security as if it could the project itself is a “twin” security as if it could be traded.be traded.

),0(~, dtNdzdzdtS

dS

Analogy between the Black-Analogy between the Black-Scholes and binomial modelsScholes and binomial models

At the limit, as the time period length in the binomial model goes to At the limit, as the time period length in the binomial model goes to zero, the binomial process converges to the corresponding Wiener zero, the binomial process converges to the corresponding Wiener process. Thus, the Black-Scholes formula is nothing else but a process. Thus, the Black-Scholes formula is nothing else but a binomial “risk-neutral pricing” formula but in continuous time (for binomial “risk-neutral pricing” formula but in continuous time (for comparison see e.g. Copeland-Weston, pp. 264 - 269)comparison see e.g. Copeland-Weston, pp. 264 - 269)

An example of two techniques yielding close results even when a An example of two techniques yielding close results even when a two-period binomial approximation is used: Copeland-Weston, pp. two-period binomial approximation is used: Copeland-Weston, pp. 269 – 273.269 – 273.

Note: to go from Black and Scholes to the binomial model you do Note: to go from Black and Scholes to the binomial model you do the following transformation (Cox, Ross, and Rubinstein, 79): the following transformation (Cox, Ross, and Rubinstein, 79):

udeu nT /1 ,/

Equity as a Call Option on the Firm Equity as a Call Option on the Firm

The equity in a firm is a residual claim, i.e., equity The equity in a firm is a residual claim, i.e., equity holders lay claim to all cash flows left over after other holders lay claim to all cash flows left over after other financial claim-holders (debt, preferred stock etc.) have financial claim-holders (debt, preferred stock etc.) have been satisfied.been satisfied.If a firm is liquidated, the same principle applies, with If a firm is liquidated, the same principle applies, with equity investors receiving whatever is left over in the firm equity investors receiving whatever is left over in the firm after all outstanding debts and other financial claims are after all outstanding debts and other financial claims are paid off.paid off.The principle of The principle of limited liabilitylimited liability, however, protects equity , however, protects equity investors in publicly traded firms if the value of the firm is investors in publicly traded firms if the value of the firm is less than the value of the outstanding debt, and they less than the value of the outstanding debt, and they cannot lose more than their investment in the firm.cannot lose more than their investment in the firm.

Equity as a call optionEquity as a call option

The payoff to equity investors, on liquidation, can The payoff to equity investors, on liquidation, can therefore be written as:therefore be written as:Payoff to equity on liquidation Payoff to equity on liquidation = V - D if V > D= V - D if V > D

= 0 if V = 0 if V D, D,wherewhereV = Value of the firmV = Value of the firmD = Face Value of the outstanding debtD = Face Value of the outstanding debt

This is a call option with a strike price of D, on an This is a call option with a strike price of D, on an asset with a current value of Vasset with a current value of V

Application to valuation: A simple Application to valuation: A simple exampleexample

Assume that you have a firm whose assets are Assume that you have a firm whose assets are currently valued at $100 million and that the currently valued at $100 million and that the standard deviation in this asset value is 40%.standard deviation in this asset value is 40%.

Further, assume that the face value of debt is Further, assume that the face value of debt is $80 million (It is zero coupon debt with 10 years $80 million (It is zero coupon debt with 10 years left to maturity).left to maturity).

If the ten-year treasury bond rate is 10%,If the ten-year treasury bond rate is 10%,

– – how much is the equity worth?how much is the equity worth?

– – how much is the debt worth?how much is the debt worth?

Model ParametersModel Parameters

Value of the underlying asset = V = Value of the firm = Value of the underlying asset = V = Value of the firm = $100 million$100 millionExercise price = D = Face Value of outstanding debt = Exercise price = D = Face Value of outstanding debt = $80 million$80 millionLife of the option = t = Life of zero-coupon debt =Life of the option = t = Life of zero-coupon debt =10 years10 yearsVariance in the value of the underlying asset = Variance in the value of the underlying asset = σσ22 = = Variance in firm value = 0.16Variance in firm value = 0.16Riskless rate = r = Treasury bond rate corresponding to Riskless rate = r = Treasury bond rate corresponding to option life = 10%option life = 10%

Valuing Equity as a Call OptionValuing Equity as a Call OptionBased upon these inputs, the Black-Scholes model provides Based upon these inputs, the Black-Scholes model provides the following:the following:

dd11 = 1.5994 = 1.5994 N(dN(d11) = 0.9451) = 0.9451

dd22 = 0.3345 = 0.3345 N(dN(d22) = 0.6310) = 0.6310

Value of the call = Value of equity:Value of the call = Value of equity:S = 100 (0.9451) - 80 exp(0.10*10) (0.6310) = $75.94 millionS = 100 (0.9451) - 80 exp(0.10*10) (0.6310) = $75.94 millionValue of the outstanding debt = $100 - $75.94 = $24.06 Value of the outstanding debt = $100 - $75.94 = $24.06 millionmillion

)N()N( 21 dDedVS rt

t

tσ

rDVd

)2

()/ln(2

1

tdd 12

Valuing BusinessValuing Business

Methods of valuationMethods of valuation DCF valuation (e.g. using WACC)DCF valuation (e.g. using WACC) Relative valuation (comparables)Relative valuation (comparables) Cost-based valuationCost-based valuation

Relative valuationRelative valuation

Based on comparison with similar firms on the marketBased on comparison with similar firms on the market Uses ratios (multiples) of similar firms to estimate the share price Uses ratios (multiples) of similar firms to estimate the share price

or EV of a given firmor EV of a given firm

Most commonly used multiples:Most commonly used multiples: Earnings multiplesEarnings multiples

P/E – price to earnings ratio (share price / earnings per share P/E – price to earnings ratio (share price / earnings per share ≡≡ Market Cap / Net Income)Market Cap / Net Income)EV/EBITDAEV/EBITDA

Revenue multiplesRevenue multiplesP/S – price to sales ratioP/S – price to sales ratioEV/S – enterprise value to sales ratioEV/S – enterprise value to sales ratio

Book (or replacement) Value multiplesBook (or replacement) Value multiplesP/BV – price to book value ratioP/BV – price to book value ratioEV/BVEV/BV

Example. Valuing Ideko CorporationExample. Valuing Ideko CorporationLine of business: designing and manufacturing sports Line of business: designing and manufacturing sports eyeweareyewear

Estimated 2006 Income Statement and Balance Sheet:Estimated 2006 Income Statement and Balance Sheet:

Sales = 75,000Sales = 75,000EBITDA = 16,250EBITDA = 16,250Net Income = 6,939Net Income = 6,939Debt = 4,500Debt = 4,500

Imagine you are considering acquire this company at a Imagine you are considering acquire this company at a price of $150 mln. Is it a fair price?price of $150 mln. Is it a fair price?At this price:At this price:

P/E = 21.6P/E = 21.6 EV = E + D – cash. Assume you estimate that Ideko holds $6.5 EV = E + D – cash. Assume you estimate that Ideko holds $6.5

mln in cash in excess of its working capital needs (i.e. invested mln in cash in excess of its working capital needs (i.e. invested at a market rate of return)at a market rate of return)EV = 150 + 4.5 – 6.5 = $148 mlnEV = 150 + 4.5 – 6.5 = $148 mln

EV/Sales = 2EV/Sales = 2 EV/EBITDA = 9.1EV/EBITDA = 9.1

Ideko Financial Ratios ComparisonIdeko Financial Ratios Comparison

150 looks like a reasonable price150 looks like a reasonable price

We can get a further idea by looking at the range We can get a further idea by looking at the range of prices implied by the range of multiples for of prices implied by the range of multiples for comparable firms (next slide)comparable firms (next slide)

Limitations of relative valuationLimitations of relative valuation

Difficult to find truly good matches (even if Difficult to find truly good matches (even if they do the same business, your firm may they do the same business, your firm may be at a different stage of development, be at a different stage of development, have different growth prospects, different have different growth prospects, different business risk, different capital structure, business risk, different capital structure, etc.)etc.)

What if the market is inefficient and What if the market is inefficient and incorrectly values your matches?incorrectly values your matches?

Correcting for Growth RateCorrecting for Growth Rate

Assume firms similar to yours have different Assume firms similar to yours have different earnings per share (or Net Income) growth rates.earnings per share (or Net Income) growth rates.

Two firms with the same earnings but different Two firms with the same earnings but different expected growth rates should have different expected growth rates should have different prices (a firm with a higher growth rate should be prices (a firm with a higher growth rate should be priced higher)priced higher)

You should use growth-adjusted P/E ratio to You should use growth-adjusted P/E ratio to value your firm: PEG=(P/E)/g, where g is the value your firm: PEG=(P/E)/g, where g is the expected growth in EPSexpected growth in EPS

Imagine a firm in the IT industry, InfoSoft, with Net Income = $977,300Imagine a firm in the IT industry, InfoSoft, with Net Income = $977,300

Based on Average PE its equity value (MCap) should be 977,300*28.41 = $27.765 mln

But imagine InfoSoft’s Net Income expected growth rate is 27.03%.

Then a more correct estimate of its equity value = 977,300*1.40*27.03 = $ 36.983 mln

Reasons to apply relative valuation Reasons to apply relative valuation

You may need to get an estimate very quicklyYou may need to get an estimate very quicklyYou may not have enough data to build a You may not have enough data to build a financial model of the firmfinancial model of the firm Information is undisclosedInformation is undisclosed The company is too young (start-up) to have a history The company is too young (start-up) to have a history

of operationsof operations

It may be impossible to do accurate predictions It may be impossible to do accurate predictions of FCF for a long termof FCF for a long term Multiples are often used to estimate a terminal valueMultiples are often used to estimate a terminal value

Useful to verify an estimate obtained by DCFUseful to verify an estimate obtained by DCF