Real Options in finanza aziendale: valutazione e gestione del rischio

Real Options

Riccardo Calcagno 2024-25 Politecnico di Torino - Corporate Governance and Finance - 01TUOPHAgenda

• What is a real option and its valuation using decision trees

· Analogy with financial options and general methods of valuation of real options · Some examples · Stage financing · Valuing growth potential · Multiperiod real options

• How to use Black and Scholes to evaluate real options (hints)

Definizione di Real Option

What is a real option · A real asset (not a financial security) that gives the owner the right to make a business decision in the future

• Real option means flexibility = the possibility to take state-contingent decisions · Contrarily to financial options, most of the time they are not traded · Main categories of real options: · The option to abandon a project without suffering (all the) losses · The option to delay an investment · The option to expand an existing project · Portfolios of 'simple' real options, multiperiod real options

Esempio di Real Option

A real option 0 1 NPV (in 1 Year) > 0 Expand /Invest u New Info. Do not Invest 0 First Inv./ Wait d NPV (in 1 Year) < 0 Expana/ Invest Invest (all) Today NPV of Project Today Do not Invest 0

Real Options nelle Decisioni di Capital Budgeting

Real options in capital budgeting decisions

• The value of real options may be extremely relevant or negligible, depending on the project
• As a rule: the higher the uncertainty, the more valuable are real options · Real Option Analysis is at the heart of the financing of highly innovative projects (stage financing)

Approccio NPV e Flessibilità

Real options in capital budgeting decisions · Classic NPV calculation ignores flexibility · NPV implicitly assumes every decision is taken at the first stage, no matter what information will be acquired in the future

Metodologia NPV

The NPV approach 0 1 NPV (in 1 Year) > 0 Expand /Invest u New Info. Do not Invest 0 First Inv./ Wait d NPV (in 1 Year) < 0 Expana/ Invest Invest (all) Today NPV of Project Today Do not Invest 0

Valutazione con Alberi Decisionali: Esempio

Valuation of real options with decision trees: an example 0 1 NPV (in 1 Year) > 0 Expand /Invest u New Info. Do not Invest 0 First Inv./ Wait d NPV (in 1 Year) < 0 Expana/ Invest Invest (all) Today NPV of Project Today Do not Invest 0

Valore dell'Opzione con Albero Decisionale

Value of the option using the decision tree: example First Investment = I0 ≥ 0

• Second Investment = I1 > 0 FCFs first Investment = II0 ≥ 0
• FCFs second Investment = III,u , II1,d (with probabilities pu, 1-pu), with: II1,u > Ii > II1,d Discount rate = r
• Value of the real option at t = 0: Co = NPVo (with flexibility) - NPVo (without flexibility) Co = 1 1+r :[(1-Pu)(11-111,d)] > 0

Calcolo del Valore dell'Opzione

Value of the option using the decision tree In general: the second investment Ii and the FCFs III,u , III,d may differ in the two cases (you invest immediately, or you postpone investment) Still, the value of the real option at t = 0: Co = NPVo (with flexibility) - NPVo (without flexibility)

Esempio: Stage Financing nel Venture Capital

Example - Venture Capital staging · A Venture Capital (VC) fund is considering investing in Pied Piper, a start-up that is trying to develop a revolutionary technology based on a new compression algorithm

• If the VC invest in Pied Piper and the technology is successful, the VC could exit in 4 years and realize a 10 million EUR cash flow then. However, this is uncertain and depends on how soon the technology is ready. The VC might lose the entire amount invested
• As the risk of the investment is high, the VC firm considers staging it. The VC firm could provide 500.000 EUR today (round A) and additional 500.000 EUR in year 2 (round B)
• For year 2 a milestone is set: the completion of the algorithm. There is a 70% probability that Pied Piper will be able to reach this milestone in time. The VC has the option not to provide round B financing.

Scenario di Investimento e Uscita

Example - Venture Capital staging

• If the full investment is provided (both round A and round B), the chances of a successful exit for the VC in year 4 will be 15% if the milestone is reached, and 2% if the milestone is not reached. The chances of a successful exit are 0 if the round B is not provided by the VC
• Assume all agents are risk neutral and that the risk-free rate is 5% (assumed to be constant)
• These assumptions are needed to ensure that the decision tree analysis is correct
• What is the NPV of the project, considering (or not) the staging?

Diagramma di Staging del Venture Capital

Example: Venture Capital staging 0 2 4 Round B -500k 10M*15% Milestone reached 70% No II stage 0 Round A -500k Milestone missed 30% Round B -500k 10M*2% Do not invest No II stage $0

Soluzione: Valore del Progetto senza Staging

0Example solution - Venture Capital staging Value of the project without staging option (= VC precommmits to both stage A and B financing): NPV = - 0.5 + 70%* − 0.5 (1 + 5%)2 + 15% * (10) (1 + 5%)4 / + 30% * − 1 0.5 (1 + 5%)2 + 2% * (10) (1 + 5%)4 / =- 0.04 <0

Soluzione: Valore del Progetto con Staging

Example solution - Venture Capital staging Value of the project with staging option: 15%*10 (1+5%)2 = 0.86 > 0: (VC invests in Round B) If the milestone is reached: NPV(round B) = - 0.5 + If the milestone is not reached: NPV(round B) = - 0.5 + 2%*10 (1+5%)2 = - 0.318 < 0: (VC abandons) 0.86 NPV(with staging) = - 0.5 + 70% * (1 + 5%)2 = 0.046 > 0 Value of the option to abandon (staging): Po = 0.046 - (-0.04) = 0.086 > 0

Verso una Soluzione Generale

To find a more general solution, let us step back ...

• In these two examples, it is easy to draw the decision tree and all the elements needed to compute the real option valuation
• This is not always feasible in real life cases · Another way is possible ...

Agenda

• What is a real option and its valuation using decision trees · Analogy with financial options and general methods of valuation of real options · Some examples · Stage financing · Valuing growth potential · Multiperiod real options
• How to use Black and Scholes to evaluate real options (hints)

Soluzione Generale: Opzioni Reali come Opzioni Finanziarie

A general solution

• Most real options can be seen as call or put financial options on an investment: · Option to abandon = put option on an existing investment . Option to expand = call option on the expansion investment
• The valuation of real options then follows the valuation of financial options, with some adjustments
• Intuitively: the main problem with real options (not traded) is to find the "underlying security"

Metodi di Valutazione: Assunzioni Aggiuntive

Valuation methods for real options: additional assumptions

• To evaluate real options we use the same methods as with financial options, with one additional assumption:

• As underlying security we use the project itself (assuming no flexibility): M.A.D

• As in the valuation of financial options we assume there are no arbitrage opportunities

Metodi di Valutazione delle Opzioni Finanziarie

Recall: Valuation methods of financial options 1. Replicating portfolio 2. Risk-neutral probabilities

• Now we apply these two methods in the previous VC-staging example

Esempio di Staging del Venture Capital (Ripetuto)

Example (again): Venture Capital staging 0 2 4 Round B -500k 10M*15% Milestone reached 70% No II stage 0 Round A -500k Milestone missed 30% Round B -500k 10M*2% Do not invest No II stage $0 In particular, recall that we assumed: 1. The VC is risk-neutral 2. The risk-free rate = 5%

Opzione di Abbandono a t=2: Una Put

0The option to abandon at t=2: a Put Payoff at expiration t=2 Sd = 2% (10) (1.05)2 = 0.182 Strike Price 15% (10) = 1.36 (1.05)2 0 Sd 0.5 Su Sx If VC abandons, it saves the round B-investment (0.5M) -> K = 0.5M If VC abandons, it foregoes the possible future profits from the sale at T=4. These are contingent on reaching/not the milestone, i.e. they are random at t=2 (we denote them with Sx)

Valore dell'Opzione con Probabilità Risk-Neutral

The value of the option using risk-neutral probability

• The put option has an exercise price of 0.5M, so at expiration (t=2) its payouts are: . Milestone not reached = Max{0.5 - 0.182,0} = 0.318M · Milestone reached = Max{0.5 -1.36,0} = 0
• Under VC-risk-neutrality, the risk-neutral probability coincides with the actual probability of the events (milestone being reached or not)
• The value of the put option at t=0 therefore is: 0.3 (0.318) (1.05)2 = 0.086

Portafoglio Replicante

Replicating portfolio

• At t = 0 we build a portfolio using: 1. The underlying project without flexibility 2. A risk-free bond
• This portfolio pays the same payoffs as the option at t=2 in both states
• Let A be the number of "shares" of the project purchased in t=0
• Let B be the investment in risk-free bonds made in t=0 1.364 + B(1.05)2= 0 0.1824 + B(1.05)2= 0.318 · △ =- 0.27 · B = 0.333

Valore dell'Opzione tramite Portafoglio Replicante

Replicating Portfolio: Value of the Option

• But "shares" of the project do not exist on the stock market! We do not know their price
• We need to find the value of the underlying (i.e. the project without flexibility) at t=0. VC is risk-neutral, so we discount the project expected payoffs at rf = 5% So = 70% (1.36) + 30% (0.182) (1.05)2 = 0.913
• Using the no-arbitrage principle we obtain the value of the put option (= option to abandon) at t=0: Po = AS0 + B = - 0.27 (0.913) + 0.333 = 0.086

Rischio e Avversione al Rischio

Risk-aversion and systematic risk

• Suppose now the project contains systematic risk
• AND: VC firm is risk-averse
• Suppose also that: · Beta project = 2 · Market risk-premium = 4% -> Cost of capital of the project = 5% + 2*(4%) = 13%
• Compute again the value of the put option using both methods
• Notice: now consider the t = 2 payoffs Su and Sa (1.36M and 0.182M) as payoffs certain at t = 2

Portafoglio Replicante con Rischio Sistematico

Replicating portfolio

• The replicating portfolio does not change w. r. to the risk-neutral case: 1.364 + B(1.05)2= 0 0.1824 + B(1.05)2= 0.318 4 = - 0.27, B = 0.333
• The value of the underlying project (without flexibility) at t = 0 now is obtained by discounting the expected payoffs at t = 2 using the WACC: So = 70% (1.36) + 30% (0.182) (1.13)2 = 0.788
• Value of the put option: Po = AS0 + B = - 0.27 (0.788) + 0.333 = 0.12

Valutazione Risk-Neutral

Risk-Neutral Valuation

• Risk-neutral probability = the probability distribution that keeps the current value of the underlying the same as with risk-neutral investors
• From the project value obtained above we can derive the risk-neutral probability: 1.36(1 - p) + 0.182p (1.05)2 = 0.788
• p =. 417
• Notice p > 0. 3: the risk-neutral probability gives more weight to the low state
• Value of the option = expected option payouts using (p, 1 - p) discounted at rf PO = - 0.318p (1.05)2 0.318(0.417) (1.05)2 = 0.12

Valore del Progetto con e senza Flessibilità

Value of the project without and with flexibility

• Value without flexibility: NPV =- 0.5 + 70%* (1+ 13%)2 -0.5 + 1.36 + 30% * -0.5 + 0.182 (1+ 13%)2 = - 0.08
• Value with flexibility = Value without flexibility + Value of the option to abandon = - 0.08 + 0.12 = 0.04