CRITERIOS PARA LA SELECCIÓN DE INSTRUMENTOS

The standard technique for evaluating projects is the net present value of the expected cash flow from the investment project, discounted by an appropriate rate. In financial theory, the capital asset pricing model (CAPM) is used to determine a theoretically appropriate discount rate of an asset, though the drawback of this approach is widely acknowledged. One

disadvantage of this approach is that it neglects the stochastic nature of the project‘s cash flows. Although CAPM appears as the simplest technique to use, the high volatility of traffic demand induces some fundamental uncertainties which are difficult to solve using this traditional discount cash-flow method. The traffic demand is quite uncertain in the

transportation project because the traffic volume may swing 10–20% per year. Under such conditions, the valuation model for the transport project proposed by this research considers the traffic demand as a stochastic variable in which certain price is assumed to be non- stochastic.

The binomial lattice discrete-time valuation model can be used to accurately approximate solutions from the continuous-time valuation model. The application of the binomial model is widely used by researchers and practitioners for transportation project evaluation. Among them are Blank et al (2009), Pichayapan et al (2003) and Charoenpornpattana et al (2003). In their studies, traffic flow is considered to be a stochastic variable in the model. This research focuses on a transportation system valuation in which traffic flow is a critical parameter. As traffic flows evolve stochastically overtime, the value of the project will also vary in the same manner. The volatility of the traffic demand also increases the option value of the project. For transportation project evaluation, the traffic flow is assumed to follow the geometric

Brownian motion (GBM). This assumption is often adopted by many researchers such as Ho and Liu (2002), Garvin and Cheah (2004), Brandao and Saraiva (2008) and Blank et al (2009). This research starts with the general model for predicting traffic volume. In general, traffic volume is uncertain which means the traffic flow is a stochastic process. GBM represents an uncertainty in traffic flow as follow;

dV/V = µ dt +ζ dz

Where V is the monthly or yearly traffic volume and μ is a traffic volume growth rate which is assumed to be constant, ζ is the volatility of traffic equaled to √∑ ̅

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incremental to a geometric Brownian motion (GBM) which is equal to ϵ√ where ϵis a normally distributed random variable whose mean is 0 and variance is 1. It is observed that the traffic flow is dependent on volatility (ζ) and trend (µ). A small volatility and drift rate may have a major impact on the range of possible future outcomes. The assumption of a GBM means that traffic flow is normally distributed, which implies that future traffic is log- normally distributed as follow:

√

and denotes a lognormal µ, ζ2 _by x ∼ lognorm (µ,ζ2)

Since the traffic flows follow a GBM, they may apply a binomial tree to represent a traffic path (Blank et. al., 2009). The traffic volume is assumed to follow a GBM which for each node, the traffic ‗S‘ in the period ‗i' and state ‗s‘ can be increased to uSsi or decreased to dSsi. By performing backward calculation, the investment decision can be exercised in each node (figure 4.2).

where p is the risk-neutral probability of traffic demand r is a risk-free interest rate

σ is traffic volatility = √∑ ̅

Figure 4.2: The binomial tree of the traffic volume in the project

The simple binomial model is presented to simulate the traffic demand for the infrastructure project. This model used a binomial model with risk-neutral probabilities, representing a discrete time characteristic to approximate the uncertainty associated with the changes in the value of a traffic demand (or project‘s cash flow) over time. This model rather explicitly

p 1-p S0d 𝑢 𝑒𝜎√ 𝑡 𝑑 𝑒 𝜎√𝛥𝑡 𝑝 𝑒 𝑟 𝑡 _𝑑 𝑢 𝑑 p 1-p S0ud S0 S0u S0u2 S0u3 S0u2d S0d2 S0ud2 S0d3

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recognises the volatility of demand for an infrastructure asset. Figure 4.2 illustrated the binomial path set up in the risk-neutral world.

The main assumption of the binomial lattice used in transport project valuation is that the traffic flow or the present value of the project fluctuates randomly in a complete, efficient market. Under the random walk assumption, the traffic volume jump is characterised by a normal or lognormal probability distribution. This means that the logarithm of the change of traffic volume [Log (V2/V1)] follows a normal or bell-shaped curve. The statistical test of the proof that the change of traffic volume follows a lognormal distribution is given in appendix C. The result of the statistical test shows that annual traffic volume is well described by a lognormal distribution.

Volatility is the critical parameter for option pricing in which the option values are sensitive to change in volatility. The estimation of the option‘s volatility is ambiguous and has an effect on the option‘s value. It is known that greater uncertainty (volatility) increases real option value. Volatility of the financial option can be implied by the current market price. However for real asset trading, implied volatility is difficult to observe from the current market price. An alternative is to estimate it using historical data. Since volatility is a sensitive parameter, this research has completed the sensitivity test for different values of traffic volatilities (see section 4.6, 5.3, 5.8 and 6.5).

It is expected to have different outcomes of real option valuation from a discrete-time model and a continuous-time model with yearly or monthly time intervals. It is generally known that the binomial model converges to the continuous-time model (Black-Scholes model) when the number of time periods increases and the length of each time period is infinitesimally short. This proof was given by Cox, Ross and Rubinstein (1979). The binomial model that

calculates option values for the SES project is checked against values from a Black-Scholes model in appendix D.

To apply the binomial model for the SES project, it is important to expand on the

understanding of risk neutrality, the complete market and a replicating portfolio in the case of an infrastructure project. It has been argued that while many practitioners have applied real option analysis on an infrastructure investment problem using a standard financial option valuation model,(e.g., binomial lattice, Black-Scholes), the strict assumptions are rarely satisfied for real infrastructure projects, including the SES project. The assumption of risk neutrality is crucial in real option work. What discount rate should be used is the most critical

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task and the discount rate can also change over time which is a shortfall for the traditional discounted cash flow method. The assumption of risk neutrality resolves this shortfall by transforming the real world into a risk-neutral world. In a risk-neutral world, all assets are evaluated at the risk-free interest rate and the project uncertainty is captured in the evaluation process by the volatility measure (Garvin and Ford, 2012).

The main assumption behind risk neutrality is that the underlying asset is traded in a complete market. Financial options use a strong relation between the underlying asset and the option to replicate the results of a riskless asset portfolio, and the value of option is obtained from the observation of the capital market. This standard option valuation uses the known price of an underlying asset to estimate the relevant price of the derivative asset. Therefore, it seems that the use of a replicating portfolio requires that the markets where assets are traded be

complete. In a complete market, a replicating portfolio can be found among traded assets and the price resulting from when the replicating portfolio and derivative price are the same (no arbitrage principle). Indeed, these characteristics of the complete market are relatively rare for an infrastructure project including the SES project. Copeland and Antikarov (2001) proposed the use of the present value of the project as the best market value for the asset. With this method, it allows to use the project itself as the basic asset in the replicating portfolio with the assumption that a project without flexibility is highly correlated with the value of the project with options (a form of the marketed asset disclaimer: MAD). Therefore, cash flows/NPVs from the SES project can be used as trade proxies in the market. Moreover, cash flows/NPVs from the SES project can still be reasonably tracked through the infrequency of real asset trading in the market. Therefore, these assumptions of risk neutrality and the complete market are reasonable in the case of the SES project.