Real option analysis

Under real options (English Real Options ) is the scope of action of the management of enterprises in investment decisions. The real option analysis ( English Real Option Valuation , ROV or English Real Option Analysis , ROA ) describes as part of the investment theory option price models for the valuation of investments .

The net present value method for assessing investments disregards entrepreneurial flexibility in future decisions. It is assumed that the companies will actually carry out the projected investments and divestments. However, there is a chance that due to unforeseen developments, the investments will lose their value and / or other investment or disinvestment opportunities that have not yet been considered will become valuable. When considering capital values, the investment decisions of companies are therefore always either / or decisions: In the case of positive capital values ​​of projects, the company invests or in the case of negative capital values, investments are omitted. The possibilities of companies to delay, expand or sell investments to a certain extent are disregarded in the traditional methods. This room for maneuver is known as real options. These options for action (real options) can be assigned a value using various options valuation methods (e.g. binomial model , Black-Scholes model ). The real option values ​​of investments are always at least as high as their capital values .

history

The term "real option" is relatively new and was coined in 1977 by Stewart Myers of the MIT Sloan School of Management. As early as 1930 Irving Fisher wrote explicitly about "options" available to a business owner (The Theory of Interest, II.VIII). The description of real options was accelerated by the development of option pricing models for the valuation of financial market options, in particular by the model of Black / Scholes (1973). Lenos Trigeorgis has been a leading scholar in the field for many years, having published several influential books and scientific articles. Other scholars researching the real options include Michael Brennan, Eduardo Schwartz, Graham Davis, Gonzalo Cortazar, Han Smit, Avinash Dixit, and Robert Pindyck (the latter two have written the most influential text in the field). The concept was popularized with practitioners by Michael J. Mauboussin, the then chief strategist of Credit Suisse First Boston in the USA. He uses real options to explain the difference between stock market prices and the "intrinsic values" of companies.

Properties of investment

Most investment decisions are characterized by three characteristics:

• Uncertainty : The future cash flows from investments are uncertain. Uncertainty arises from the fact that the variables relevant to the investment decision, such as interest rates, prices and wages, are sometimes unpredictable.
• Irreversibility : Capital expenditures are partially or completely irreversible. In other words: The payments for investments are at least partly “ sunk costs ”, ie they cannot be reversed later.
• Flexibility : Investments can be postponed. As a result, it is possible to await newer and better information about the value of investments.

The traditional valuation calculations based on the net present value rule have not sufficiently taken into account the interplay of uncertainty, irreversibility and flexibility. Due to uncertainty and irreversibility, companies have to fear the negative consequences of investment decisions. The projection of future cash flows can in retrospect turn out to be too optimistic and the irreversibility deprives the company of the opportunity to (completely) reverse the decisions made. However, the flexibility allows these characteristics to be taken into account, to wait for an improved information situation and to make a more informed decision.

Analogies of financial market and real options

Properties of calls and puts

With flexibility in the timing of investments, one can see an instructive analogy between real investments and options in financial markets. A financial option is a contractual agreement that certifies the right, but not the obligation, to purchase or sell a specific asset - the so-called underlying asset - at a price agreed in advance during the term of the option. The assets can be B. to trade stocks, currencies, bonds or indices. Purchase options (calls) grant the right to buy, put options (puts) the right to sell an asset. The underlyings can be purchased or sold at a specified point in time (“European option”: at the end of the term) or during a period (“American option”: at any time during the term).

The adjacent figure shows that the risk-return structures of the holder of call and put options are asymmetrically distributed. This means that the profit opportunities are open at the top (almost open for put options, since the value of the underlying cannot become negative), while the maximum loss is limited to the option price to be paid at the beginning of the option term. If the option is "out of the money" at the end of the term, it is not worth exercising and the option premium paid must be written off. If the exercise price is exceeded, it makes sense to exercise the option. The profit from exercising develops parallel to the value of the underlying. However, break-even is only achieved when the profit from exercising also compensates for the premium paid previously.

The analogy of the real option and the financial option is easy to see. A flexible investment is an option for the company to spend money now or in the future in order to generate value in the future. The expenses associated with the investment correspond to the exercise price. It should be noted that the option itself was not acquired, but was "earned" over the years through business activity. “Owning” the option can be based on aspects as diverse as reputation, market position, or ownership of patents or other resources. In contrast to financial market options, real investment opportunities do not have a specific expiry date, but usually have an infinite term.

Examples: Digitization is leading to radical upheavals in many industries. Internet companies such as Apple, Google, Facebook or Amazon are penetrating many markets and gaining massive market shares. These companies have acquired a great deal of knowledge about their customers, their needs and wishes on their platforms. You have the most important currency in the digital world: a high degree of customer loyalty and a high level of attention for new products. As a result, these companies have a wide range of options for efficiently and precisely transferring their skills to other services. The market leaders in digital services in particular therefore have numerous real options at their disposal.

The making of the irreversible investment corresponds to the exercise. When the option is exercised, the net present value (NPV) of the option is realized. However, exercising this option also results in opportunity costs. With the decision to carry out the investment, the investor gives up the opportunity to make the investment decision later and thus to take improved data into account in the future. The net present value rule, as the basis of all cash flow procedures, according to which investments are to be made as long as the present value of an investment is higher than its acquisition costs, must therefore be modified. In the language of option price theory, the NPV rule only takes into account the intrinsic value of an option. One neglects the so-called time value of the option, which arises if one can achieve a higher profit by waiting. This fair value is due to the typical asymmetry of an option: If the intrinsic value of the investment increases, a corresponding increase in profit can be achieved through a later investment decision. In the opposite case, however, this increase in profit is not offset by a corresponding loss potential, since the option does not have to be exercised if the intrinsic value decreases.

The following table summarizes the parallels between financial market and real options:

Analogies of financial market and real options

Financial market option	Real option	symbol
Daily rate	Present value of surplus payments (net present value)	V
Base price	acquisition cost	I.
Intrinsic value	Net present value (NPV)	V - I
running time	Time for room for maneuver	T
volatility	Dispersion of surplus payments	σ
dividend	Lost return if the real option is not exercised ("dividend" of the investment)	ρ
Risk-free interest	Risk-free interest	r
Call option value Value of the put option	Value of the investment opportunity Value of the divestment opportunity	F (V) F (V)
value	Worth the wait	F (V) - [VI]
Exercise rule	Investment rule	V - I ≥ F (V) V ≥ I + F (V) Time value ≤ 0

Every exercise of an irreversible, deferred investment is therefore associated with the abandonment of an option. Since options represent options, the values ​​of which are always positive, the net present value rule must be modified: The net present value of an investment V must be higher than its acquisition costs I, precisely in the amount of the opportunity costs, which are to be assessed with the total value of the option. An optimal decision rule can then be formulated in different ways: Invest exactly when

• The net present value of the investment NPV = VI is at least as large as its opportunity costs, which must be calculated with the total value of option F (V),
• the present value (capital value) of all future income V covers the total costs, which are composed of acquisition costs I and opportunity costs due to the loss of option F (V), or
• the time value of the option is not greater than 0.

Properties of real options

The illustration on the right illustrates these relationships graphically. The gain on exercise is the value that is realized when the investment is made. First of all, the intrinsic value (net capital value) of the investment is achieved. With the decision to make the investment, however, opportunity costs arise in the form of abandoning the option: The company is deprived of the opportunity to wait for further information and to base the decision on a broader information basis. The investment decision is therefore only worthwhile when the net present value V compensates for both the investment costs I and the opportunity costs. The opportunity costs are to be set at the value of the real option when exercising F (V *).

The option value F (V) of the investment that has not yet been made can never be negative. The option represents a choice that does not have to be taken. As the net present value increases, the value of the option will approach the net present value (VI), as the probability of exercising then approaches 1. Break-even is reached or the investment is exercised when the intrinsic value reaches the option value. Due to the asymmetry of the option - on the other hand, the profit potential is not offset by any loss potential - the option value has a different frequency distribution than the net present value NPV in each period. While the NPV should spread roughly around 0, the option value has a right-skewed distribution. If the option value is less than the net present value in a period, the investment should be made.

example

Consider a coffee shop that has the option of irreversibly investing in new coffee machines. The investment costs of these units amount to I = 8000, - €. The revenues from these products are permanently estimated at € 1,000 per year. The relevant discount rate is i = 10%. The net present value (NPV) of the investment is:

            ${\displaystyle NPV=V-I=\sum \nolimits _{t=0}^{\infty }{\frac {1.000}{1.1^{t}}}-8.000=10.000-8.000=2.000}$ €

The net present value therefore suggests investing, as the investment has a positive present value or the present value of the machine is higher than its acquisition cost. In a “now or never” situation (and only then) the application of the NPV rule would be correct. If there is the possibility of delaying the investment, the NPV rule disregards the opportunity costs of the investment. It may be worth waiting to see whether prices will go down or up in the future.

To keep the example simple, assume the probability is 50% that revenues will rise to € 1,500 in the next period and then remain constant. On the other hand, the probability that the proceeds will fall to € 500 is also 50%. This uncertainty causes the company to wait a period and only invest when revenues increase. The net present value in this case is:

            ${\displaystyle NPV=0,5*(\sum \nolimits _{t=0}^{\infty }{\frac {1.500}{1.1^{t}}}-8.000)/1.1=3.181,82}$ €

So the expected NPV is higher if you postpone the investment for a year and only invest if the price has risen in the meantime. The total value of the option is € 3,181.82. It is made up of the intrinsic value of the investment in the current period of € 2,000 and the current value of the option, which is € 1,181.82. The value of this temporal flexibility arises from the fact that in the event of an unfavorable development of earnings, the investment can be dispensed with without incurring "sunk costs", while a higher cash value can be achieved with a positive development of earnings.

Determination of option values

Stochastic processes

Boundary conditions of real options

A large number of models have been developed to determine option values. To derive a closed solution, a certain development of the capital values ​​of investments must be assumed. It is regularly assumed that the costs of investment I are fixed and that the present value of an investment V , more precisely the sum of all future discounted payment surpluses, follows a geometric Brownian movement :

  ${\displaystyle dV=\alpha Vdt+\sigma Vdz}$,

wherein dz , the growth of a Wiener process it, thus " white noise ", and α the constant percentage drift from V describes. Security prices and capital values ​​are often modeled in dynamic models as a geometric Brownian movement . In the process, α represents the expected percentage return on an investment. However, since security prices or capital values ​​fluctuate in practice, the second term adds the uncertainty to the process. It is assumed that the volatility (standard deviation) σ of the returns is the same in every period.

With the help of further properties of real options, closed formulas for the value of the options can be derived. Real options have the following properties:

• American options : Real options are intended to model the flexibility in companies' investment decisions. Naturally, real options must therefore be American options that can be exercised at any time during the term.
• Dividends: Dividends must be shown for real options, otherwise American options would never be exercised before the end of the term. When an option is exercised, the fair value of the option is always given up. This can only be compensated if, in return, when the real option is exercised, a “dividend” can be realized in the investment projects. With real investment opportunities, the distributed free cash flows of an investment can be interpreted as the “dividend” of the project.
• Constraints of options: It can be shown that the value of a call option must be limited as follows:
  • F (V) ≥ 0 : The value of an option can never be <0. An option is a choice that does not have to be taken. If the value of the project approaches zero, the option to invest loses its value: F (0) = 0 .
  • F (V) ≥ VI : The value of an American option can never be <VI . The value of the option must be at least equal to the net present value of the project minus the acquisition costs. Otherwise there would be the possibility of arbitrage when the option is “exercised”.
  • The value of the option must approach the value VI . If, as the net present value of the project increases, the probability of exercising approaches 1, a profit of VI is achieved. Even without an option, there is no potential loss against the profit potential.

• Stop rule : In the case of real options, decision-makers have to make a decision within the term of when the option will be exercised. Stop rules must therefore also be defined for the stochastic processes. As a rule, a threshold value is defined for this purpose. In the case of real options, the option is exercised or the process is stopped when F = VI is reached. The time of the decision is called the stopping time. The stopping time depends on the randomness of the underlying stochastic process and cannot be anticipated in advance. The stop time is therefore endogenous and is not specified. In this context one speaks of a free boundary value problem .
• Boundary conditions of real options . However, so that the process can be stopped or the free boundary value problem can be solved, typical boundary conditions must be used for real options. If V ^{* is used to} denote the capital value at which the stop value is reached (option is exercised), the secondary condition obtained is: F (V ^* ) = V ^* -I (“Value Matching”). At this point, the option can also be observed to flow smoothly into the net capital value VI (cf. adjacent figure). The gradients of the option price formula or the payout function for the net present value are identical for this threshold value and δF / δV = F _V (V) = 1 for a call or δF / δV = F _V (V) = -1 for a put . These conditions are also called "Smooth Pasting Condition" or "High Contact Condition".

The adjacent figure shows how the profit and loss range of a call or put develops depending on the stochastic variable V. In the case of a call with a known investment expenditure I , a loss of the same amount can arise if the project has no value at all. The Net Present Value ( VI ) increases linearly with the value of the project V on.

The technique of evaluating options was first demonstrated in the seminal work by Black / Scholes (1973). You have recognized. that a risk-free portfolio can be constructed from an investment and its derivative, as there is an identical source of uncertainty for both. Using the stochastic differentiation rules of Itō Kiyoshi , you get a stochastic differential equation that can be solved using the boundary conditions of options. Dixit / Pindyck take up these ideas, but pay attention to the special properties of real options. The first constraint, F (0) = 0 , suggests the following solution form:

${\displaystyle F(V)=\alpha V^{\beta }}$

where α and β are constants, the values ​​of which depend on the parameters σ ² , μ and ρ . If V ^{* is} the capital value at which it is optimal to invest, one can use the constraint: F (V ^* ) = V ^* - Use I ("Value Matching"). However, this “free limit” V ^* cannot be determined without further conditions. This is where the "Smooth Pasting Condition" F _V (V ^* ) = 1 comes into play. If you insert the above formula into the secondary conditions and transform it, you get:

${\displaystyle F(V)=aV^{\beta }}$ , With

${\displaystyle a=(V^{\ast }-I)/(V^{\ast })^{\beta }}$ , 

${\displaystyle V^{\ast }={\frac {\beta }{\beta -1}}I}$,

${\displaystyle \beta ={\frac {1}{2}}-{\frac {\mu -\delta }{\sigma ^{2}}}+\left[\left({\frac {\mu -\delta }{\sigma ^{2}}}-{\frac {1}{2}}\right)^{2}+{\frac {2\mu }{\sigma ^{2}}}\right]^{1/2}}$   . ,

With the help of this formula, the value of real options with an infinite runtime can be determined quite easily.

In the relevant literature one can also find paragraphs in which the binomial model of Cox / Ross / Rubinstein or the model of Black / Scholes for evaluating real options is used. However, using these models to evaluate real options is problematic. These models are based on assumptions (European options, specified term, no “dividend” for the investment opportunity under consideration), which are not given for real options. They therefore prove to be imprecise for evaluating real options.

literature

• Martha Amram, Nalin Kulatilaka: Real Options: Managing Strategic Investment in an Uncertain World . Harvard Business School Press, Boston 1999, ISBN 978-0-87584-845-7 ( online ).
• Marion A. Brach: Real Options in Practice . Wiley, New York 2003, ISBN 978-0-471-44556-2 .
• Thomas E. Copeland, Vladimir Antikarov: Real Options: A Practitioner's Guide . Texere, New York 2001, ISBN 978-1-58799-028-1 .
• A. Dixit, R. Pindyck: Investment Under Uncertainty . Princeton University Press, Princeton 1994, ISBN 978-0-691-03410-2 ( online ).
• William T. Moore: Real Options and Option-Embedded Securities . John Wiley & Sons , New York 2001, ISBN 978-0-471-21659-9 .
• TJ Smit, Lenos Trigeorgis: Strategic Investment: Real Options and Games . Princeton University Press, Princeton 2004, ISBN 978-0-691-01039-7 .
• Trigeorgis, Lenos: Real Options: Managerial Flexibility and Strategy in Resource Allocation . The MIT Press, Cambridge 1996, ISBN 978-0-262-20102-5 ( online ).
• Reitz: Mathematics in the modern financial world, 2011.
• Seydel: Introduction to the numerical calculation of financial derivatives, 2nd edition 2017, ISBN 3-662-50299-2 .
• Webel, Wied: Stochastic Processes - An Introduction for Statisticians and Data Scientists, 2nd edition 2016.
• Sandmann, Introduction to the Stochastics of Financial Markets, 1999.
• Cox, Ross, Rubinstein: Option Pricing: A Simplified Approach, Journal of Financial Economics, 1979 pp. 229-263.
• Black / Scholes, The Pricing of Options and Corporate Liabilities, Journal of Political Economy, 1973 pp. 637–654.
• Müller: Real options models, pp. 421 to 434, in: Petersen / Zwirner (Hrsg.), Handbook Company Valuation, 2nd edition 2017.
• Beckmann: Der Realloptionsansatz, pp. 1583–1614, in: Peemöller (Hrsg.), Praxishandbuch der Unternehmensversicherung, 7th edition 2019.

Individual evidence

1. ↑ Thomas E. Copeland, Vladimir Antikarov: Real Options: A Practitioners Guide . Texere Publishing, 2003.
2. ↑ Seydel, Rüdiger: Introduction to the numerical calculation of financial derivatives: Computational Finance . ISBN 3-662-50299-2 , pp. 147 ff . ( worldcat.org [accessed February 14, 2020]).
3. ↑ Pindyck, Robert S .: Investment under Uncertainty . Princeton University Press, 2008, ISBN 1-283-37955-4 ( Online [accessed February 14, 2020]).
4. ^ Müller: Real option models . In: Handbook Company Valuation . 2nd Edition. 2017, p. 424-431 .
5. ↑ Beckmann, Christoph: The real option approach in investment calculation and company valuation . Utz, 2015, ISBN 978-3-8316-8089-4 ( worldcat.org [accessed February 14, 2020]).