1. Approaches to valuing real options Analytical: Binomial model Binomial model Black-Scholes formula Black-Scholes formulaSimulations

2. Binomial model V – the gross value of the project (expected value of subsequent CF) d = 1/u There exists a “twin” security that can be traded, which price S is perfectly correlated with V. If there is an option on the project, we use “replicating portfolio” technique (or risk neutral probabilities, which is the same) to determine its value V V + =uV V - = dV V ++ =u 2 V V -- =d 2 V V +- =udV q 1-q q q …

3. E + = NS + - (1+r)B E - = NS - - (1+r)B N = (E + - E - )/(S + - S - ) B = (NS - – E - )/(1+r) “risk-neutral” valuation: E 0 = NS – B = (pE + + (1-p)E - )/(1+r) where p = ((1 + r)S – S - )/(S + - S - )= (1 + r – d)/(u - d) We have u=1.8, d=0.6, r=0.08, hence p=0.4, E 0 = 25.07 V=100, S=20 I 0 =104 V + =180, E + =67.68 S + =36 q=0.5 0.5 V - =60, E - =0 S - =12 I 1 =112.32

4. At the end p depends only on u, d and r: p = (1 + r – d)/(u - d) In fact, p can be found from the following: S = (puS + (1-p)dS)/(1+r), i.e. p must be such that the risk-neutral valuation of the “twin” security yields its actual price – “twin” security’s value in the “risk-neural world” must be the same as in the “real” world Thus: p does not depend on the actual probability of going up q. Reason: q is already incorporated in the price S. p does not depend on the actual probability of going up q. Reason: q is already incorporated in the price S. Given the tree, p does not depend on the particular option (in particular on where we are in the tree) Given the tree, p does not depend on the particular option (in particular on where we are in the tree)

5. Example: Option to abandon for salvage value or switch use We should switch at such points (If the option is to switch any time we want, we switch the first time we get to such a node) 100 180 60 324 36 108 85 127.5 68 191 54.4 102 Current project. Values of VAlternative use. Values of V

6. E 0 =? E + =180 E - =68 What is the value of the option to switch in year 1? (pE + + (1-p)E - )/(1+r) - I 0 = 0.44 E 0 = (pE + + (1-p)E - )/(1+r) - I 0 = 0.44 (we can use the same probabilities p 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

7. Black-Scholes Pricing Formula (no dividend case, call option) C 0 = the value of a European option at time t = 0 r = the risk-free interest rate S = the price of the underlying asset (or “twin” security) E – exercise price (e.g. investment required) N(.) – cumulative standard normal distribution function σ – standard deviation of the underlying asset return

8. Adjusting for dividends (i.e. if the project generates cash flows before the option “expiration” date) Assume a constant dividend yield (i.e. constant cash flow) every year. Then:

9. Some caveats about the real options approach Black-Scholes formula presumes a diffusion Wiener process for underlying (“twin”) security: Is it always the case? Can we always find a “twin” security? If not, people do “market asset disclamer” assumption: the project itself is a “twin” security as if it could be traded.

10. Analogy between the Black- Scholes and binomial models At the limit, as the time period length in the binomial model goes to zero, the binomial process converges to the corresponding Wiener process. Thus, the Black-Scholes formula is nothing else but a binomial “risk-neutral pricing” formula but in continuous time (for comparison see e.g. Copeland-Weston, pp. 264 - 269) An example of two techniques yielding close results even when a two-period binomial approximation is used: Copeland-Weston, pp. 269 – 273. Note: to go from Black and Scholes to the binomial model you do the following transformation (Cox, Ross, and Rubinstein, 79):

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

12. Equity as a call option The payoff to equity investors, on liquidation, can therefore be written as: Payoff to equity on liquidation = V - D if V > D = 0 if V  D, where V = Value of the firm D = Face Value of the outstanding debt This is a call option with a strike price of D, on an asset with a current value of V

13. Application to valuation: A simple example Assume that you have a firm whose assets are currently valued at $100 million and that the standard deviation in this asset value is 40%. Further, assume that the face value of debt is $80 million (It is zero coupon debt with 10 years left to maturity). If the ten-year treasury bond rate is 10%, – how much is the equity worth? – how much is the debt worth?

14. Model Parameters Value of the underlying asset = V = Value of the firm = $100 million Exercise price = D = Face Value of outstanding debt = $80 million Life of the option = t = Life of zero-coupon debt = 10 years Variance in the value of the underlying asset = σ 2 = Variance in firm value = 0.16 Riskless rate = r = Treasury bond rate corresponding to option life = 10%

15. Valuing Equity as a Call Option Based upon these inputs, the Black-Scholes model provides the following: d 1 = 1.5994N(d 1 ) = 0.9451 d 2 = 0.3345N(d 2 ) = 0.6310 Value of the call = Value of equity: S = 100 (0.9451) - 80 exp(0.10*10) (0.6310) = $75.94 million Value of the outstanding debt = $100 - $75.94 = $24.06 million

16. Valuing Business Methods of valuation DCF valuation (e.g. using WACC) DCF valuation (e.g. using WACC) Relative valuation (comparables) Relative valuation (comparables) Cost-based valuation Cost-based valuation

17. Relative valuation Based on comparison with similar firms on the market Uses ratios (multiples) of similar firms to estimate the share price or EV of a given firm Uses ratios (multiples) of similar firms to estimate the share price or EV of a given firm Most commonly used multiples: Earnings multiples Earnings multiples P/E – price to earnings ratio (share price / earnings per share ≡ Market Cap / Net Income) EV/EBITDA Revenue multiples Revenue multiples P/S – price to sales ratio EV/S – enterprise value to sales ratio Book (or replacement) Value multiples Book (or replacement) Value multiples P/BV – price to book value ratio EV/BV

18. Example. Valuing Ideko Corporation Line of business: designing and manufacturing sports eyewear Estimated 2006 Income Statement and Balance Sheet:

19. Sales = 75,000 EBITDA = 16,250 Net Income = 6,939 Debt = 4,500 Imagine you are considering acquire this company at a price of $150 mln. Is it a fair price? At this price: P/E = 21.6 P/E = 21.6 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 at a market rate of return) EV = 150 + 4.5 – 6.5 = $148 mln 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 at a market rate of return) EV = 150 + 4.5 – 6.5 = $148 mln EV/Sales = 2 EV/Sales = 2 EV/EBITDA = 9.1 EV/EBITDA = 9.1

20. Ideko Financial Ratios Comparison 150 looks like a reasonable price We can get a further idea by looking at the range of prices implied by the range of multiples for comparable firms (next slide)

22. Limitations of relative valuation Difficult to find truly good matches (even if they do the same business, your firm may be at a different stage of development, have different growth prospects, different business risk, different capital structure, etc.) What if the market is inefficient and incorrectly values your matches?

23. Correcting for Growth Rate Assume firms similar to yours have different earnings per share (or Net Income) growth rates. Two firms with the same earnings but different expected growth rates should have different prices (a firm with a higher growth rate should be priced higher) You should use growth-adjusted P/E ratio to value your firm: PEG=(P/E)/g, where g is the expected growth in EPS

24. Imagine 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

25. Reasons to apply relative valuation You may need to get an estimate very quickly You may not have enough data to build a financial model of the firm Information is undisclosed Information is undisclosed The company is too young (start-up) to have a history of operations The company is too young (start-up) to have a history of operations It may be impossible to do accurate predictions of FCF for a long term Multiples are often used to estimate a terminal value Multiples are often used to estimate a terminal value Useful to verify an estimate obtained by DCF