Option Pricing Applications in Equity Valuation

A few caveats on applying option pricing models

1. The underlying asset is not traded

• Option pricing theory is built on the premise that a replicating portfolio can be created using the underlying asset and riskless lending and borrowing.
• The options presented in this section are on assets that are not traded, and the value from option pricing models have to be interpreted with caution.

2. The price of the asset follows a continuous process

• The Black-Scholes option pricing model is derived under the assumption that the underlying asset's price process is continuous, i.e., there are no price jumps.
• If this assumption is violated, as it is with most real options, the model will underestimate the value of deep out-of-the-money options.
  • One solution is to use a higher variance estimate to value deep out-of-the-money options and lower variance estimates for at-the-money or in-the-money options.
  • Another is to use an option pricing model that explicitly allows for price jumps, though the inputs to these models are often difficult to estimate.

3. The variance is known and does not change over the life of the option

• The assumption that option pricing models make, that the variance is known and does not change over the option lifetime, is not unreasonable when applied to listed short-term options on traded stocks.
• When option pricing theory is applied to long-term real options, there are problems with this assumption, since the variance is unlikely to remain constant over extended periods of time and may in fact be difficult to estimate in the first place.

4. Exercise is instantaneous

• The option pricing models are based upon the premise that the exercise of an option is instantaneous.
• This assumption may be difficult to justify with real options, where exercise may require the building of a plant or the construction of an oil rig, actions which are unlikely to happen in an instant.
• The fact that exercise takes time also implies that the true life of a real option is often less than the stated life.

The General Framework

• The equity in a firm is a residual claim, i.e., equity holders lay claim to all cashflows 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.

Payoff to equity on liquidation

= V - D if V > D

= 0 if V

where,

V = Value of the firm

D = Face Value of the outstanding debt and other external claims

• A call option, with a strike price of K, on an asset with a current value of S, has the following payoffs:

Payoff on exercise = S - K if S > K

= 0 if S

• If the debt in the firm is a single issue of zero-coupon bonds with a fixed lifetime, and the firm can be liquidated by equity investors at any time prior, the life of equity as a call option corresponds to the life of the bonds.

• Assume that you have a firm whose assets are currently valued at 100ドル million and that the standard deviation in this asset value is 40%.
• Further, assume that the face value of debt is 80ドル 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? What should the interest rate on debt be?

The parameters of equity as a call option are as follows:

Value of the underlying asset = S = Value of the firm = $ 100 million

Exercise price = K = 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 = s2 = Variance in firm value = 0.16

Riskless rate = r = Treasury bond rate corresponding to option life = 10%

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 1.5994 N(d1) = 0.9451

d2 = 0.3345 N(d2) = 0.6310

A. Valuing equity in a troubled firm

• The first implication is that equity will have value, even if the value of the firm falls well below the face value of the outstanding debt.
• Such a firm will be viewed as troubled by investors, accountants and analysts, but that does not mean that its equity is worthless.
• Just as deep out-of-the-money traded options command value because of the possibility that the value of the underlying asset may increase above the strike price in the remaining lifetime of the option, equity will command value because of the time premium on the option (the time until the bonds mature and come due) and the possibility that the value of the assets may increase above the face value of the bonds before they come due.

The parameters of equity as a call option are as follows:

Value of the underlying asset = S = Value of the firm = $ 50 million

Exercise price = K = 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 = s2 = Variance in firm value = 0.16

Riskless rate = r = Treasury bond rate corresponding to option life = 10%

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 1.0515 N(d1) = 0.8534

d2 = -0.2135 N(d2) = 0.4155

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Value of the call = 50 (0.8534) - 80 exp(-0.10)(10) (0.4155) = 30ドル.44 million

Value of the bond= 50ドル - 30ドル.44 = 19ドル.56 million

• The equity in this firm has substantial value, because of the option characteristics of equity.
• This might explain why stock in firms, which are in Chapter 11 and essentially bankrupt, still has value.

• Stockholders and bondholders have different objective functions, and this can lead to agency problems, where stockholders can expropriate wealth from bondholders.
• The conflict can manifest itself in a number of ways - for instance, stockholders have an incentive to take riskier projects than bondholders do, and to pay more out in dividends than bondholders would like them to.
• This conflict between bondholders and stockholders can be illustrated dramatically using the option pricing model.
• Since equity is a call option on the value of the firm, an increase in the variance in the firm value, other things remaining equal, will lead to an increase in the value of equity.
• It is therefore conceivable that stockholders can take risky projects with negative net present values, which while making them better off, may make the bondholders and the firm less valuable. This is illustrated in the following example.

Illustration 5: Effect on value of the conflict between stockholders and bondholders

• Consider again the firm described in illustration 16.1 , with a value of assets of 100ドル million, a face value of zero-coupon ten-year debt of 80ドル million, a standard deviation in the value of the firm of 40%. The equity and debt in this firm were valued as follows:

Value of Equity = 75ドル.94 million

Value of Debt = 24ドル.06 million

Value of Firm == 100ドル million

• Now assume that the stockholders have the opportunity to take a project with a negative net present value of -2ドル million, but assume that this project is a very risky project that will push up the standard deviation in firm value to 50%.

• The equity as a call option can then be valued using the following inputs:

Value of the underlying asset = S = Value of the firm = $ 100 million - 2ドル million = $ 98 million (The value of the firm is lowered because of the negative net present value project)

Exercise price = K = 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 = s2 = Variance in firm value = 0.25

Riskless rate = r = Treasury bond rate corresponding to option life = 10%

Based upon these inputs, the Black-Scholes model provides the following value for the equity and debt in this firm.

Value of Equity = 77ドル.71

Value of Debt = 20ドル.29

Value of Firm = 98ドル.00

• The value of equity rises from 75ドル.94 million to $ 77.71 million , even though the firm value declines by 2ドル million. The increase in equity value comes at the expense of bondholders, who find their wealth decline from 24ドル.06 million to 20ドル.19 million.

Illustration 6: Effects on equity of a conglomerate merger

You are provided information on two firms, which operate in unrelated businesses and hope to merge.

Firm A Firm B

Value of the firm 100ドル million $ 150 million

Face Value of Debt $ 80 million $ 50 million (Zero-coupon debt)

Maturity of debt 10 years 10 years

Std. Dev. in firm value 40 % 50 %

Correlation between firm

cashflows 0.4

The ten-year bond rate is 10%.

• The variance in the value of the firm after the acquisition can be calculated as follows:

Variance in combined firm value = w12 s12 + w22 s22 + 2 w1 w2 r12 s1 s2

= (0.4)2 (0.16) + (0.6)2 (0.25) + 2 (0.4) (0.6) (0.4) (0.4) (0.5)

= 0.154

The values of equity and debt in the individual firms and the combined firm can then be estimated using the option pricing model:

Firm A Firm B Combined firm

Value of equity in the firm 75ドル.94 134ドル.47 $ 207.43

Value of debt in the firm 24ドル.06 $ 15.53 $ 42.57

Value of the firm 100ドル.00 150ドル.00 $ 250.00

• The combined value of the equity prior to the merger is $ 210.41 million and it declines to 207ドル.43 million after.
• The wealth of the bondholders increases by an equal amount.
• There is a transfer of wealth from stockholders to bondholders, as a consequence of the merger. Thus, conglomerate mergers that are not followed by increases in leverage are likely to see this redistribution of wealth occur across claim holders in the firm.

The examples that have been used to illustrate the use of option pricing theory to value equity have made some simplifying assumptions. Among them are the following:

(1) There were only two claim holders in the firm - debt and equity.

(2) There is only one issue of debt outstanding and it can be retired at face value.

(3) The debt has a zero coupon and no special features (convertibility, put clauses etc.)

(4) The value of the firm and the variance in that value can be estimated.

Illustration 7: Valuing Equity as an option - The example of an airline

• Assume that you have been asked to value an airline. The airline owns routes in North America, Europe and South America, with the following estimates of current market value:

North America $ 400 million

Europe $ 500 million

South America $ 100 million

• The airline has considerable debt outstanding. It has four debt issues outstanding with the following characteristics:

Maturity Face Value Coupon Duration

20 year debt $ 100 mil 11% 14.1 years

15 year debt $ 100 mil 12% 10.2 years

10 year debt $ 200 mil 12% 7.5 years

1 year debt $ 800 mil 12.5% 1 year

• The stock of the airline has been trading on the NYSE. The annualized standard deviation in ln(stock prices) has been 25%, and the firm's debt has been approximately 90% of the firm value (during the variance estimation period).
• The firm is rated B. Though its bonds are not traded, other B rated bonds have had an annualized standard deviation of 10% (in ln(bond prices)). The correlation between B rated bonds and this airline's stock price is 0.3.
• The firm pays no dividends. The current T. Bond rate is 8%.

Step 1: Estimate the value of the firm = Sum of the value of its assets = 400 +たす 500 +たす 100 =わ 1,000 million

Step 2: Estimate the average duration of the debt outstanding = (100/1200) * 14.1 + (100/1200) * 10.2 + (200/1200) * 7.5 + (800/1200) * 1 = 3.9417 years

Step 3: Estimate the face value of debt outstanding = 100 +たす 100 +たす 200 +たす 800 =わ 1,200 million

Step 4: Estimate the variance in the value of the firm = Weighted average of the variances in stock and bond prices. =

Variance of the firm = (E/(D+E))2 se2 + (D/(D+E))2 sd2 + 2 (E/(D+E)) (D/(D+E)) red sesd

= (.1)2 (.25)2 + (.9)2 (.10)2 + 2 (.1)(.9)(.3) (.25)(.10) = 0.010075

Step 5: Value equity as an option

d1 = 0.7671 N(d1) = 0.7784

d2 = 0.5678 N(d2) = 0.7148

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Illustration 8: Valuing Equity as an option - Cablevision Systems

• Cablevision Systems was a firm in trouble in March 1995.
  • The book value of equity in March 1995 was negative : - 1820 million
  • It lost 315ドル million in 1994 and was expected to lose equivalent amounts in 1995 and 1996.
  • It had $ 3000 million in face value debt outstanding
• The weighted average duration of this debt was 4.62 years

Debt Type Face Value Duration

Short term Debt $ 865 mil 0.5 years

Bank Debt $ 480 mil 3.0 years

Senior Debt $ 832 mil 6.0 years

Senior Subordinated $ 823 mil 8.5 years

Total $ 3000 mil 4.62 years

• The value of the firm estimated using projected cashflows to the firm, discounted at the weighted average cost of capital was 2,871ドル million. This was based upon the following assumptions ñ
  • Revenues will grow 14% a year for the next 5 years, and make a linear transition to 5% in 10 years.
  • The COGS (not including depreciation), which is currently 69% of revenues, will drop to 65% of revenues in year 5 and stay at that level. (68% in 1996, 67% in 1997, 66% in 1998, 65% in 1999)
  • Capital spending and depreciation will grow 10% a year for the next five yearsafter which the growth rate will drop to 5% a year.
  • Working capital will remain at 5% of revenues.
  • The debt ratio, which is currently 70.14%, will drop to 50% after year 10. The cost of debt is 10% in high growth period and 8.5% after that.
  • The beta for the stock will be 1.55 for the next five years, and drop to 1.1 over the next 5 years.
  • The treasury bond rate is 7.5%.

Cost of Capital in high growth period = 16.03% (0.2986) + 10% (1 - 0.36) (0.7014) = 9.27%

Cost of Capital in terminal period = 13.55% (0.50) + 8.50% (1 - 0.36) (0.50) = 9.34%

• The projected cash flows over the next ten years are summarized below ñ

• Terminal Value = 248ドル.16 / ( .0934 - .05) = 5725ドル million
• The stock has been traded on the NYSE, and the variance based upon ln (monthly prices) between 1990 and 1994 is 0.0133.
• There are Cablevision bonds, due in 2002, that have been traded from 1990 to 1994, and the variance in ln(monthly price)s for these bonds is 0.0012.
• The correlation between stock price and bond price changes has been 0.25. The proportion of debt in the capital structure during the priod (1990-94) was 70%.

The stock and bond price variance are first annualized:

Annualized variance in stock price = 0.0133 * 12 = 0.16 Standard deviation = 0.40

Annualized variance in bond price = 0.0012 * 12 = 0.0144 Standard deviation = 0.12

Annualized variance in firm value

= (0.30)2 (0.16) + (0.70)2 (0.0.0144) + 2 (0.3) (0.7)(0.25)(0.40)(0.12)= 0.02637668

• The five-year bond rate (corresponding to the weighted average duration of 4.62 years) is 7%.

The parameters of equity as a call option are as follows:

Value of the underlying asset = S = Value of the firm = $ 2871 million

Exercise price = K = Face Value of outstanding debt = $ 3000 million

Life of the option = t = Weighted average duration of debt = 4.62 years

Variance in the value of the underlying asset = s2 = Variance in firm value = 0.0264

Riskless rate = r = Treasury bond rate corresponding to option life = 7%

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 0.9910 N(d1) = 0.8391

d2 = 0.6419 N(d2) = 0.7391

Cablevision's equity was trading at 1100ドル million in March 1995.

II. Valuing Natural Resource Options/ Firms

The General Framework

• In a natural resource investment, the underlying asset is the resource and the value of the asset is based upon two variables - the quantity of the resource that is available in the investment and the price of the resource.
• In most such investments, there is a cost associated with developing the resource, and the difference between the value of the asset extracted and the cost of the development is the profit to the owner of the resource.
• Defining the cost of development as X, and the estimated value of the resource as V, the potential payoffs on a natural resource option can be written as follows:

Payoff on natural resource investment = V - X if V > X

= 0 if V

Illustration 9 : Application to valuation: A gold mine

• Consider a gold mine with an estimated inventory of 1 million ounces, and a capacity output rate of 50,000 ounces per year.
• The price of gold is expected to grow 3% a year.
• The firm owns the rights to this mine for the next twenty years.
• The present value of the cost of opening the mine is 40ドル million, and the average production cost of 250ドル per ounce. This production cost, once initiated, is expected to grow 4% a year.
• The standard deviation in gold prices is 20%, and the current price of gold is 350ドル per ounce. The riskless rate is 9%, and the cost of capital for operating the mine is 10%. The inputs to the model are as follows:

Value of the underlying asset = Present Value of expected gold sales (@ 50,000 ounces a year) = (50,000 * 350) * (1- (1.0320/1.1020))/(.10-.03) - (50,000*250)* (1- (1.0420/1.1020))/(.10-.04) = $ 42.40 million

Exercise price = PV of Cost of opening mine = 40ドル million

Variance in ln(gold price) = 0.04

Time to expiration on the option = 20 years

Riskless interest rate = 9%

Dividend Yield = Loss in production for each year of delay = 1 / 20 = 5%

(Note: It will take twenty years to empty the mine, and the firm owns the rights for twenty years. Every year of delay implies a loss of one year of production.)

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 1.4069 N(d1) = 0.9202

d2 = 0.5124 N(d2) = 0.6958

Call Value= 42.40 exp(-0.05)(20) (0.9202) -40 (exp(-0.09)(20) (0.6958)= $ 9.75 million

The value of the mine as an option is $ 9.75 million, in contrast to the static capital budgeting analysis which would have yielded a net present value of $ 2.40 million (42ドル.40 million - $ 40 million). The additional value accrues directly from the mine's option characteristics.

• Consider an offshore oil property with an estimated oil reserve of 50 million barrels of oil, where the present value of the development cost is 12ドル per barrel and the development lag is two years.
• The firm has the rights to exploit this reserve for the next twenty years and the marginal value per barrel of oil is 12ドル per barrel currently (Price per barrel - marginal cost per barrel).
• Once developed, the net production revenue each year will be 5% of the value of the reserves. The riskless rate is 8% and the variance in ln(oil prices) is 0.03.

Given this information, the inputs to the Black-Scholes can be estimated as follows:

Current Value of the asset = S = Value of the developed reserve discounted back the length of the development lag at the dividend yield = 12ドル * 50 /(1.05)2 = $ 544.22

(If development is started today, the oil will not be available for sale until two years from now. The estimated opportunity cost of this delay is the lost production revenue over the delay period. Hence, the discounting of the reserve back at the dividend yield)

Exercise Price = Present Value of development cost = 12ドル * 50 = 600ドル million

Time to expiration on the option = 20 years

Variance in the value of the underlying asset = 0.03

Riskless rate =8%

Dividend Yield = Net production revenue / Value of reserve = 5%

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 1.0359 N(d1) = 0.8498

d2 = 0.2613 N(d2) = 0.6030

Call Value= 544 .22 exp(-0.05)(20) (0.8498) -600 (exp(-0.08)(20) (0.6030)= $ 97.08 million

This oil reserve, though not viable at current prices, still is a valuable property because of its potential to create value if oil prices go up.

Extending the option pricing approach to value natural resource firms

• Since the assets owned by a natural resource firm can be viewed primarily as options, the firm itself can be valued using option pricing theory.
• The preferred approach would be to consider each option separately, value it and cumulate the values of the options to get the value of the firm.
• Since this information is likely to be difficult to obtain for large natural resource firms, such as oil companies, which own hundreds of such assets, a variant of this approach is to value the entire firm as one option.
• A purist would probably disagree, arguing that valuing an option on a portfolio of assets (as in this approach) will provide a lower value than valuing a portfolio of options (which is what the natural resource firm really own). Nevertheless, the value obtained from the model still provides an interesting perspective on the determinants of the value of natural resource firms.

Input to model Corresponding input for valuing natural resource firm

Value of underlying asset Value of cumulated estimated reserves of the resource owned by the firm, discounted back at the dividend yield for the development lag.

Exercise Price Estimated cumulated cost of developing estimated reserves

Time to expiration on option Average relinquishment period across all reserves owned by firm (if known) or estimate of when reserves will be exhausted, given current production rates.

Riskless rate Riskless rate corresponding to life of the option

Variance in value of asset Variance in the price of the natural resource

Dividend yield Estimated annual net production revenue as percentage of value of the reserve.

Illustration 11: Valuing an oil company - Gulf Oil in 1984

• Gulf Oil was the target of a takeover in early 1984 at 70ドル per share (It had 165.30 million shares outstanding, and total debt of 9ドル.9 billion).
• It had estimated reserves of 3038 million barrels of oil and the average cost of developing these reserves was estimated to be 10ドル a barrel in present value dollars (The development lag is approximately two years).
• The average relinquishment life of the reserves is 12 years.
• The price of oil was 22ドル.38 per barrel, and the production cost, taxes and royalties were estimated at 7ドル per barrel.
• The bond rate at the time of the analysis was 9.00%.
• Gulf was expected to have net production revenues each year of approximately 5% of the value of the developed reserves. The variance in oil prices is 0.03.

Value of underlying asset = Value of estimated reserves discounted back for period of development lag= 3038 * ($ 22.38 - 7ドル) / 1.052 = 42,380ドル.44

Exercise price = Estimated development cost of reserves = 3038 * 10ドル = 30,380ドル million

Time to expiration = Average length of relinquishment option = 12 years

Variance in value of asset = Variance in oil prices = 0.03

Riskless interest rate = 9%

Dividend yield = Net production revenue/ Value of developed reserves = 5%

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 1.6548 N(d1) = 0.9510

d2 = 1.0548 N(d2) = 0.8542

Call Value= 42,380.44 exp(-0.05)(12) (0.9510) -30,380 (exp(-0.09)(12) (0.8542)= $ 13,306 million

• This represents the value of the undeveloped reserves of oil owned by Gulf Oil.
• In addition, Gulf Oil had free cashflows to the firm from its oil and gas production of 915ドル million from already developed reserves and these cashflows are likely to continue for ten years (the remaining lifetime of developed reserves).
• The present value of these developed reserves, discounted at the weighted average cost of capital of 12.5%, yields:

Value of already developed reserves = 915 (1 - 1.125-10)/.125 = 5065ドル.83

Adding the value of the developed and undeveloped reserves of Gulf Oil provides the value of the firm.

Value of undeveloped reserves = $ 13,306 million

Value of production in place = $ 5,066 million

Total value of firm = $ 18,372 million

Less Outstanding Debt = $ 9,900 million

Value of Equity = $ 8,472 million

Value per share = $ 8,472/165.3 = 51ドル.25

This analysis would suggest that Gulf Oil was overvalued at 70ドル per share.

The General Framework

• A product patent provides the firm with the right to develop the product and market it.
• It will do so only if the present value of the expected cash flows from the product sales exceed the cost of development.
• If this does not occur, the firm can shelve the patent and not incur any further costs.
• If I is the present value of the costs of developing the product, and V is the present value of the expected cashflows from development, the payoffs from owning a product patent can be written as:

Payoff from owning a product patent = V - I if V> I

= 0 if V



Obtaining the inputs for option valuation

Illustration 12: Valuing a product option

• Assume that a firm has the patent rights, for the next twenty years, to a product which
  • requires an initial investment of $ 1.5 billion to develop,
  • and a present value, right now, of cash inflows of only 1ドル billion.
• Assume that a simulation of the project under a variety of technological and competitive scenarios yields a variance in the present value of inflows of 0.03.
• The current riskless twenty-year bond rate is 10%.

The inputs to the option pricing model are as follows:

Value of the underlying asset = Present value of inflows (current) = 1,000ドル million

Exercise price = Present value of cost of developing product = 1,500ドル million

Time to expiration = Life of the patent = 20 years

Variance in value of underlying asset = Variance in PV of inflows = 0.03

Riskless rate = 10%

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 1.1548 N(d1) = 0.8759

d2 = 0.3802 N(d2) = 0.6481

Call Value= 1000 exp(-0.05)(20) (0.8759) -1500 (exp(-0.10)(20) (0.6481)= $ 190.66 million

• This suggests that though this product has a negative net present value currently, it is a valuable product when viewed as an option. This value can then be added to the value of the other assets that the firm possesses, and provides a useful framework for incorporating the value of product options and patents.

Illustration 13: Valuing a firm with only product options

• Consider a bio-technology firm, which has no cashflow-producing assets currently, but has one product in the pipeline that has much promise in providing a treatment for diabetes.
• The product has not been approved by the FDA, and, even if approved, it could be faced with competition from similar products being worked on by other firms.
• The firm, however, would hold the patent rights to this product for the next 25 years.
• After a series of simulations, under a variety of technological and competitive environments, the expected present value of the cash inflows is estimated to be 500ドル million, with a variance of 0.20 (signifying the uncertainty of the process).
• The expected present value of the cost of developing the product is estimated to be 400ドル million.
• The annual cashflows on the product, once developed, are expected to be 4% of the present value of the inflows. The twenty-five year bond rate is 7%.

The inputs to the option pricing model are as follows:

Value of underlying asset = Present value of expected cashflows = $ 500 million

Exercise price = Present value of cost of developing product for commercial use = 400ドル mil

Time to expiration on the option = Time to expiration on patent rights = 25 years

Variance in value of underlying asset = 0.20

Riskless rate = 7%

Dividend yield = Expected annual cashflow / PV of cash inflows = 4%

Based upon these inputs, the Black-Scholes model provides the following value for the call:

d1 = 1.5532 N(d1) = 0.9398

d2 = -0.6828 N(d2) = 0.2474

Call Value= 500 exp(-0.04)(25) (0.9398) - 400 (exp(-0.07)(25) (0.2474)= $ 155.66 million

• The estimated value of this firm, based upon an option pricing approach, is 155ドル.66 million. This is a more realistic measure of value than traditional discounted cashflow valuation (that would have provided a value of 100ドル million) because it reflects the underlying uncertainty in the technology and in competition.

• The firm is receiving royalties from Biogen discoveries (Hepatitis B and Intron) at pharmaceutical companies. These account for FCFE per share of 1ドル.00 and are expected to grow 10% a year until the patent expires (in 15 years). Using a beta of 1.1 to value these cash flows (leading to a cost of equity of 13.05%), we arrive at a present value per share:

• The firm also has a patent on Avonex, a drug to treat multiple sclerosis, for the next 17 years, and it plans to produce and sell the drug by itself. The key inputs on the drug are as follows:

Present Value of Cash Flows from Introducing the Drug Now = S = $ 3.422 billion

Present Value of Cost of Developing Drug for Commercial Use = K = $ 2.875 billion

Patent Life = t = 17 years Riskless Rate = r = 6.7% (17-year T.Bond rate)

Variance in Expected Present Values =s² = 0.224 (Industry average firm variance for bio-tech firms)

Expected Cost of Delay = y = 1/17 = 5.89%

d1 = 1.1362 N(d1) = 0.8720

d2 = -0.8512 N(d2) = 0.2076

Call Value per Share from Avonex = $ 907 million/35.5 million = $ 25.55

Biogen Value Per Share = Value of Existing Assets + Value of Patent = $ 12.14 + $ 25.55 = $ 37.69