Optimal Portfolio Strategy for Risk Management in Toll Road Forecasts and Investments Rohan Shah and Phani R. Jammalamadaka biases in traffic forecasts, in addition to other studies highlighting T&R forecast uncertainty (3, 4), that identify differences in forecast and actual traffic patterns along toll roads and other transportation systems, including public transit. As a result, these toll infrastructure practices are often confronted with traffic forecasting and investment decision making under various kinds of a priori uncertainties. The study leveraged modern portfolio theory and stochastic time series models to develop a risk management strategy for future traffic projections along brownfield toll facilities. Uncertainty in future traffic forecasts may raise concerns about performance reliability and revenue potential. Historical time series traffic data from brownfield corridors were used for developing econometric forecast estimates, and Monte Carlo simulation was used to quantify a priori risks or variance to develop optimal forecasts by using mean-variance optimization strategies. Numerical analysis is presented with historical toll transactions along the Massachusetts Turnpike system. Suggested diversification strategies were found to achieve better long-term forecast efficiencies with improved trade-offs between anticipated risks and returns. Planner and agency forecast performance expectations and risk propensity are thus jointly captured. Optimal Portfolio-Based Risk Management From a broader perspective, this study assumes that the aforementioned T&R forecasting and investment challenges have parallels with monetary investments in stock markets, real assets, and even commodities that are similarly sensitive to market volatility and trends. The study attempts to link T&R forecasting and toll road investment workflows with financial investment strategies for real assets, which leverages strategies from the modern portfolio theory (MPT), which was originally proposed by Markowitz (5). The MPT provides a set of effective tools widely applied in finance, markets, and investment economics. Qualitatively, MPT suggests diversification of all available investment assets, options, or both with a view to distributing the risks, which quantitatively enables optimal asset allocation by constructing portfolios that maximize expected returns on the basis of prevalent risks. Risk represents the likelihood that the returns increase or diminish in value and by how much and is represented by variance or standard deviation of return rate. Higher expected returns inherently carry higher risks. MPT also incorporates the correlation and covariance between assets to capture variations relative to each other. It also assumes “rational” investor behavior and where to minimize risks. However, investors willing to take additional risk often expect to be compensated with higher anticipated returns. In the same spirit, a variety of available forecast models are treated as planner assets or options, which can be combined into a hypothetical portfolio of sorts. Planners are faced with a decision to choose the best forecast estimates. Similar to investment port folios developed in financial MPT, this study proposes the notion of a forecast portfolio, which contains the set of forecast models and options. For this study, time series–based forecast models are used; however, they can be replaced by any stochastic forecasting tools available to practitioners. Each model projects its own set of forecasts, which are equivalent to expected returns. Because of the presence of future uncertainty and risks, they are equivalents of risky assets. A practical challenge arises in quantifying risks or variance from model forecasts. In this Transportation infrastructure and the tolling industry are witnessing a variety of project financing, procurement, and delivery methods that include design–build, build–operate–transfer, public–private partnerships, and concessionaires. Such alternatives often help meet financing gaps related to limited federal budgets. They are also sensitive to uncertainty and risks surrounding long-term performance and profitability. Toll road assets are generally part of revenue-dependent infrastructure systems that rely on traffic across the facility for revenue generation, investment returns, debt coverage, and operations and maintenance costs. In addition, concerns often surround the reliability of any forecasting activity originating from uncertainty about the future and, in the case of toll facilities, include several factors, such as market volatility, future economic growth, and operational factors such as toll revenue recovery. These concerns pose challenges in long-range toll network planning and policy making, as well as with determining financial feasibility owing to unpredictable future cash flows. In recent times, varying user-adoption levels of widespread conversion to electronic toll collection systems pose a challenge in estimating true market penetration and revenue yield. Jammalamadaka et al. provide further insight into sources of traffic and revenue (T&R) uncertainty from supply- and demand-side perspectives (1). Bain (2) presents a synthesis of potential errors and R. Shah, CDM Smith, Inc., Toll Finance and Technology Group, 12357-A Riata Trace Parkway, Suite 210, Austin, TX 78727. P. R. Jammalamadaka, CDM Smith, Inc., Toll Finance and Technology Group, 8140 Walnut Hill Lane, Suite 1000, Dallas, TX 75231. Corresponding author: R. Shah, firstname.lastname@example.org. Transportation Research Record: Journal of the Transportation Research Board, No. 2670, 2017, pp. 83–94. http://dx.doi.org/10.3141/2670-11 83 84 study, standard deviation or variance in forecasts is used to quantify risks. Expected values (or returns) and variance (or risks) in forecasts from each model ultimately contribute to the returns, risks, or both of the combined forecast portfolio. An optimal forecast portfolio provides a forecast value that is derived from an optimal combination of multiple forecast options, as opposed to a singular, undiversified forecast estimate. In line with the merits of MPT strategies, optimal investment allocations and expectations across individual forecast assets in the portfolio can collectively manage risks more effectively. They can also support financing decisions that depend on anticipated forecasts. The allocation or diversification across forecast assets, each of which has a unique combination of risk and return, can be case specific. In other words, it can be tailored to the preferred degrees of risk averseness or risk affinity of the concessionaire, tolling agency, or financier, and in certain cases even project- or system-specific conditions. Study Applicability and Contributions The current study leverages econometric time series tools to develop the various forecast options making up the portfolio. Two independent classes of univariate time series models are estimated, including autoregressive integrated moving average (ARIMA) and Brownian motion mean reversion (BMMR) models. Time series models typically build on historical data and past trends, and hence the applicability of the current study also relies on historical toll transactions data on existing toll facilities. Accordingly, the current approach and framework are suitable for existing open facilities with rich historical data. These models learn discernible patterns in input data and previous growth rate trends and are adaptive to any extreme events (such as a major economic recession, such as in 2008) that may be reflected from data movements in the past. The risk analysis workflow includes Monte Carlo simulation–based techniques that quantify the variance in forecast estimates, which helps evaluate relative risk levels for each model forecast, as well as identify the largest contributors to the overall risk of the forecast portfolio. The study’s contributions span across academic research and industry and can be used to jointly manage forecast expectations and risks, as well as assist with policies and decision making to mitigate the effects of uncertainty. The study leverages well-known MPT economic strategies to propose a risk management approach to fit toll T&R forecasting. From the practice standpoint, single-point forecasts from demand-model-based T&R studies or forecast streams developed through other means can be overlaid with probable streams from stochastic time series models. This approach helps develop an early understanding of a priori risks associated with long-term traffic forecasts and toll revenue potential. These can also be engaged to proactively plan for risks and design more adaptive toll policies. Forecast risks also raise concerns about financial returns on infrastructure investments, future toll policy development, and long-term toll network expansion. Comprehensive long-range T&R studies in toll industry practice typically use a travel demand–toll diversion modeling framework for forecasting. The methods presented in this paper can supplement and cross-check forecasts developed in traditional long-range T&R forecast studies. The proposed methods can help benchmark forecasts and lay the groundwork for additional analyses and sensitivity testing as part of the comprehensive T&R study, as well as develop an envelope of reasonableness around results from a conventional four-step demand model. Transportation Research Record 2670 Toll Transactions Forecasting with Time Series Models The study considered annual transactions to avoid seasonality effects. The subsections below describe the models, cover brief numerical experiments validating predicted forecasts over observed field data, and produce long-term forecasts. Several studies have demonstrated applications of the time series models in traffic predictions (6–9). Model Descriptions ARIMA Model ARIMA is a combination of an autoregressive and a moving average model. The term “integrated” represents additional operations to stationarize the data. Stationarity of a time series variable indicates that its mean and variance do not change with time, which can be achieved with differencing operations. Autoregressive components indicate dependence of a variable on its own prior values. It is conceivable from practical experience and has been demonstrated in previous studies that urban corridor traffic patterns evolve over time and have interdependencies with previous patterns. In the same spirit, this notion is extended to toll transactions. The moving average components model prediction error of the independent variable as a function of its own past errors. Such a scenario may be practically perceivable in cases in which forecast errors may accumulate and propagate through time. ARIMA models are specified as ARIMA ( p, d, q); arguments p, d, q represent the orders of the autoregressive, differencing, and moving average components, respectively. Qualitatively p and q also indicate the number of historical data lags and forecast error terms affecting the present value. For the current study data set, individual correlation effects are found to be insignificant beyond the firstorder lags, and thus higher-order models are not found to be good fits to data. First-order model specifications, such as ARIMA (1, 1, 0), ARIMA (0, 1, 1), and ARIMA (1, 1, 1) are thus estimated on the differenced data set. For the purpose of comparison and expanding the diversity of the forecast portfolio, an additional second-order specification—ARIMA (2, 1, 0)—is also estimated. The forecasting operation under ARIMA is conducted by taking the expectation of the time series or expected values of transactions and error terms. This operation can also be referred to as “deterministic forecasting.” Under this case, error terms are eliminated since their expected values are zeros. BMMR Model BMMR models are part of the broader class of Brownian motion models, which are more commonly applied in financial economics to study patterns of particularly volatile or dynamic market variables, such as stocks (10), commodity or asset prices (11), and interest rates, as found in Dixit and Pindyck (12), among others. Previous applications in the transportation and traffic forecasting domain have been limited to a few studies (13, 14). Their application was also recently validated in Shah and Jammalamadaka (15). The subclass of mean reverting models, BMMR, is driven by a mean-reversion process that is used for modeling financial assets and commodities or generally quantities for which an infinite growth is not practically possible. Shah and Jammalamadaka 85 Risk Analysis and Monte Carlo Simulation Framework original data are kept aside exclusively for this purpose and are not used during model estimation (also called “out-of-sample testing”). Validation includes annual transactions data for 4 years (2010 to 2013), and the mean absolute percentage error (MAPE) is used to quantify prediction accuracy. The MAPE is a measure of accuracy of any forecasting method and is defined by the formula Both time series models used in this study have stochastic elements in the form of error terms that are present at every stage of forecasting. They are random variables representing the uncertain magnitudes and unpredictable trends of the deviations from actual values or expected values. Even though their expected value is statistically zero (they disappear under deterministic forecasting), under realistic conditions they may take finite measurable values. As part of the risk analysis approach, stochasticity is introduced with a Monte Carlo simulation of these error terms. It includes random sampling of the error terms that helps assign confidence bands around the single-point forecasts produced deterministically. The theoretical basis for this approach is that the randomness associated with every model forecast arises from the error term et, which follows a Gaussian (normal) distribution as follows: et ~ i.i.d. N (0, σ2). These error terms—sampled from a Gaussian distribution—give a range of errors and forecast estimates in the form of confidence bounds. As a result, the risk analysis treats the estimates as random variables themselves. Further mathematical description of error sampling can be found in the literature on theoretical time series, such as in Box et al. (16) and Enders (17). MAPE = 1 N xi − xˆι 100 ∑ N i =1 xi (1) where xi=actual observations of time series, x̂ι=estimated or forecast time series, and N=number of nonmissing data points. Validation results, including comparison with the actual transactions and the MAPE calculations, are summarized in Figure 1, a and b. The models are mobilized for long-term forecasting under a 30-year future planning horizon (through 2040). Planning and programming decisions about toll roads or their expansions are generally long term. They are also fiscally aligned with regional transportation or thoroughfare improvements dictated in long-range transportation plans for the area. Various degrees of confidence intervals (5%, 25%, 75%, 95%) are obtained, along with expected forecast values. However, for ease of representation in the plot area, they are plotted only for the preferred model specification (the one with the lowest MAPE during the forecast validation tests). Forecast streams are illustrated in Figure 1c. The long-range forecast streams reflect the validation period, indicating a mix of forecast performance by the models in the Empirical Validation A case study of the Massachusetts Turnpike corridor is used as a test bed application of the proposed method (18). Historical systemwide toll transactions dating back to the 1950s were acquired, and model validation tests for the two time series forecast models were conducted on observed transaction data. In addition, a fraction of the 80 Annual Total Transactions (millions) 78 76 74 72 70 68 66 64 62 60 2010 2011 2012 Forecast Model Validation Year Observed ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (2, 1, 0) BMMR ARIMA (1, 0, 1) 2013 ARIMA (1, 1, 0) (a) FIGURE 1 Toll transaction forecast streams with various time series model options: (a) short-term forecasts, comparison with actuals. (continued on next page) 86 Transportation Research Record 2670 6 5 MAPE (%) 4 3 2 1 0 ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0) Forecast Assets Models BMMR ARIMA (1, 0, 1) Annual Total Transactions (millions) (b) Year of Operation ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0) BMMR ARIMA (1, 0, 1) 5% Bound 25% Bound 75% Bound 95% Bound (c) FIGURE 1 (continued) Toll transaction forecast streams with various time series model options: (b) empirical validation, mean absolute prediction errors, and (c) long-term forecasts, expected values and confidence intervals. portfolio, categorized as conservative-, neutral-, or aggressive-level forecasts. Confidence intervals for only the preferred model specification are illustrated in Figure 1c. Expected value forecast paths from all other models lie within 5% to 75% bands of the preferred specification. These confidence intervals can qualitatively help visualize a best-case–worst-case forecast outlook. A wide range of possible future deviations from the expected values (resulting from uncertainty and unknown future market factors) are expected to lie within these upper and lower confidence bounds. The selected time series models each predict a unique set of forecasts. The ARIMA (0, 1, 1) specification and the BMMR model each have a good empirical fit along with the smallest range of prediction errors (≤1%). The other first-order specifications, such as ARIMA (1, 0, 1) and ARIMA (1, 1, 0), have slightly higher but acceptable MAPE ranges; however the second-order specification, ARIMA (2, 1, 0), appears to have the highest MAPE range of the candidate models, and forecasts during the validation period also appear to be higher and thus more aggressive. The forecast portfolio composition for the case study is quite diverse and contains a good mix of aggressive and conservative model assets. Forecast Portfolios Forecast Portfolio Risk and Return Monte Carlo simulation of transaction forecasts projected by the time series models provides a range of forecast options and helps quantify expected returns and risks for each. The situation also helps in identifying individual options contributing most to the overall portfolio risks. As with the time series models, transaction forecasts are treated as random variables, and expected forecast returns for a Shah and Jammalamadaka 87 particular year are computed with the expected value of the probability distribution of forecasts. Forecast risk is ultimately measured by the estimated variance, or standard deviation around the expected value of the distribution. More specifically, Gaussian transaction distributions are assumed in this study since they are a function of the distribution error terms in the underlying time series models. In essence, transaction forecasts are treated as random variables with an expected value and variance of projections in every year along the future planning horizon. In line with MPT assumptions, it is assumed that planners and agencies are generally risk averse concerning forecasts, meaning that given two portfolios offering matching forecast returns (or expected values of forecast projections), the less risky one (lesser variance) is more likely to be preferred. Investors are sometimes willing to take additional risks if compensated by higher expected returns, and conversely, investors targeting higher expected returns must accept more risk. Similarly, more optimistic forecast expectations or more aggressive revenue projections come with higher risks or portfolio variances. The resulting trade-off between forecasting returns and risks achieved from portfolio optimization is the same for all project stakeholders. However, different parties often evaluate trade-offs differently according to respective risk preferences. For a general case of n individual forecast options or models i, let ri denote the forecast returns projections from each option, and E(ri) its expected value. Given this set of risky options, a set of weights W describing the optimal distribution of a forecast portfolio of size n is defined. It is the proportion in which the overall portfolio is split among the individual model options. In addition, let wi be the optimal weight for an option i, which indicates the proportion of the overall forecast returns assigned to model i. All weights are also assumed to be nonnegative. The returns from the overall forecast portfolio containing the above forecast assets are denoted by rp, and the portfolio return’s expected value is computed by the following vector combination of individual options: n E (rp ) = ∑ wi E (ri ) (2) i =1 The sum of optimal weights across individual options is equal to one, shown by (i, j ) and standard deviations of individual assets σi and σj as follows: cov (ri , rj ) = ρij σi σj (5) The portfolio variance can also be expressed in relation to the correlation coefficient as follows: n n var (rp ) = ∑ ∑ wi wj ρij σi σj (6) i =1 j =1 A weight vector w is an n-dimensional vector that represents the weight assignment across individual forecast options for each portfolio containing n assets. Portfolio variance can be represented in the vector notation [with the transpose of the weight vector (wT)] as follows: var (rp ) = w T i ∑ i w (7) p In summary, the estimate of the expected return for each forecast option or model is its average value across the range of forecasts in any planning year. The estimate of variance is the average value of the squared deviations around the forecast mean, and the estimate of covariance is the average value of the cross product of deviations across the different forecast options under consideration. When options are combined into a forecast portfolio, the expected value and variance of the portfolio accommodates an individual option’s properties and the covariance of the available options. Mean-Variance Optimization Strategies This subsection discusses the model formulations for a few standard classes of portfolio optimization methods. For the current study, five optimization models—each with a different policy goal—are formulated and are developed for the general n-option (or n-forecast model) portfolio. All models are part of the broader mean-variance optimization class of models that aim to jointly optimize the expected values and variance of a variable. n ∑w =1 (3) i i =1 The portfolio return computation involves finding the weighted average return of the individual components. Portfolio variance σ 2p, however, is defined in relation to the individual variance of options and the expected variability of all component asset pairs (i, j ) or the forecast covariance. The estimate of the covariance matrix of individual options is represented by Σp. The overall risks of a portfolio of forecast options is equal to the weighted average covariance of the individual forecast returns, denoted as follows in the scalar notation: Global Minimum Forecast Portfolio Variance Minimum forecast portfolio variance (MFPV) is a constrained minimization problem that seeks to minimize the overall forecast portfolio risk subject to basic constraints related to the weight on an individual forecast option wi. This specific formulation is called “global” MFPV because no conditions or expectations are tied to the portfolio’s overall forecast returns as long as the portfolio’s variance or risks are minimized. The formulation for this case is as follows: min σp w ∈W n (8) n var (rp ) = σ 2p = ∑ ∑ wi wj cov (ri , rj ) (4) i =1 j =1 Covariance can also be expressed in relation to the correlation coefficient ρij between the returns from a pair of forecast options subject to n ∑w =1 i i =1 (9) 88 Transportation Research Record 2670 wi ≥ 0 ∀i ∈(1, n ) (10) The first constraint indicates that the individual weights must sum to one, and the second is the nonnegativity constraint that indicates all weights must be nonnegative. The solution to this model is trivial since the selection involves merely choosing the model with the least observed variance. Minimum Forecast Portfolio Variance with Target Forecast Returns The MFPR with target forecast variance model is similar to the global MFPR model and attempts to maximize the forecast portfolio returns, but with concern about additional risks. Hence, it includes an additional constraint related to the forecast portfolio variance, trying to limit it at the lowest possible variance across individual options (or target variance). The formulation for this case is as follows: max E (rp ) (18) w ∈W The minimum forecast portfolio variance with target forecast returns formulation is very similar to the global MFPV formulation, but includes additional expectations concerning forecast returns, which helps address the dual objective of maximum forecast returns and minimum forecast risks. The formulation for this case is as follows: min σp MFPR with Target Forecast Variance subject to n ∑w =1 (19) i i =1 wi ≥ 0 ∀i ∈(1, n ) (20) (11) w ∈W σp ≤ min (σ i ) ∀i ∈(1, n ) (21) subject to n ∑w =1 (12) i The last expression is the target minimum portfolio variance constraint. i =1 wi ≥ 0 ∀i ∈(1, n ) E (rp ) ≥ max ( E (ri )) (13) ∀i ∈(1, n ) (14) As can be seen from the model formulation, this model has an additional third constraint requiring the forecast portfolio returns to be greater than or equal to the best possible returns from all individual model assets. Global Maximum Forecast Portfolio Returns The global maximum forecast portfolio returns (MFPR) model is also a single-objective optimization problem, aiming to maximize forecast portfolio returns without limitations on the additional risks accumulated alongside. The formulation for this case is as follows: Global Maximum Forecast Portfolio Efficiency The global maximum forecast portfolio efficiency (MFPE) formulation jointly optimizes risk and return by developing a new portfolio performance metric called the “portfolio-efficiency ratio,” which is the ratio of forecast return to risk, along the lines of the Sharpe ratio for risky assets (19). The Sharpe ratio is similar, but it includes the difference between the returns from risky assets and risk-free assets in the numerator. There is no risk-free forecast option considered in this study as all the time series models considered have stochasticity. The objective in this optimization strategy is to maximize the portfolio-efficiency ratio, while keeping the weights within the boundaries. No constraints are assigned either for target maximum value of returns or target minimum value of risks; thus, it is referred to as “global MFPE.” The formulation for this case is as follows: max max E (rp ) (15) w∈W w ∈W E (rp ) σp subject to subject to n (16) ∑w =1 (17) wi ≥ 0 The solution to this model is trivial, as one can simply select the forecast asset projecting the maximum forecast returns while assigning the maximum weight (or rather the entire portfolio weight) to it. However, high returns also contain high risks, and under most practical applications the objective may be to minimize overall variance as opposed to singularly maximizing returns. σp > 0 n ∑ wi = 1 i =1 wi ≥ 0 (22) ∀i ∈(1, n ) (23) i i =1 ∀i ∈(1, n ) (24) (25) Unlike the earlier global MFPV and MFPR models that place emphasis on stand-alone forecast risks and returns, the MFPE model considers the ratio or forecast efficiency, which jointly places emphasis on both. Shah and Jammalamadaka 89 Global Equally Weighted Forecast Portfolio also computed. The current set of optimal portfolios is developed for a future planning year, 2020, and is shown only for demonstration. The framework is generic and can be applied for any future year of interest. The effectiveness of the diversification strategy during selection of stand-alone forecast options can also be quantified. For example, risks and returns yielded by the several combinations of optimal portfolios (including the passive portfolio) can be compared with those from the individual model options. Essential characteristics of individual forecast options are shown in the Figure 2, which shows the performance metrics for individual forecast options and combined forecast portfolios. As the name suggests, the global equally weighted forecast port folio (EWFP) achieves simple diversification by spreading out weights equally across the individual forecast assets. This strategy is not necessarily optimal (and the weight values are not necessarily optimal), but practically, it is used as a fair starting point for more detailed optimal portfolio strategies, such as those described in this section. It can also be a diversification strategy in the absence of any additional tools such as optimization; thus, it is sometimes referred to as a “benchmark” portfolio. Although equally weighting a broad spectrum of forecast assets may not be the most optimal strategy, it can often be a useful guideline for analyzing and developing a basic understanding of the forecast landscape because it serves as a neutral measurement of all available forecast options. The egalitarian approach does not prefer any specific asset over another. In addition, since no additional strategies are used to further diversify on the basis of any planning or investment goals, the global EWFP port folio is referred to as a “passive” portfolio. Notwithstanding the passive strategy, the degree of risk diversification in this case may still be a better option when stand-alone forecast assets are considered. Forecast Portfolio Versus Individual Portfolios Forecast variance for returns projected by individual model assets, their expected returns, and their efficiency ratios are plotted in Figure 2, a–c, respectively. Forecast returns from the global MFPR and MFPV with target forecast returns strategies correspond with the highest possible returns projected by individual forecast assets. However, forecast risks, or variance, under the risk-averse strategies, including global MFPV, MFPR with target forecast variance, and the efficiency-oriented global MFPE strategy, are found with lower forecast risks than those carried by all individual forecast assets. In addition, forecast-efficiency ratios achieved under these three strategies have been found to be higher than efficiencies under all individual forecast assets. Apart from the optimal portfolio strategies, the global EWFP strategy suggests a passive portfolio with equal weight allocation and underscores the merits of diversification. The portfolio variance for the passive portfolio is lower than almost all individual assets in both case studies, and forecast-efficiency ratios Optimization Model Solutions and Optimal Forecast Portfolios Solutions for optimization problems described in the section on mean-variance optimization strategies include optimal weights across different forecast options. Values for objective functions under different optimization strategies include forecast portfolio returns, portfolio risks, and forecast portfolio-efficiency ratios, which are 18 Individual forecast assets Forecast portfolio combinations Forecast Variance or Risks (millions) 16 14 12 10 8 6 4 2 0 ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0) BMMR ARIMA (1, 0, 1) Global MFPV MFPV with Target Forecast Returns Global MFPR MFPR with Target Forecast Variance Global MFPE (tangency portfolio) Individual Forecast Assets and Optimal Portfolios (a) FIGURE 2 Performance metrics for individual forecast options and combined forecast portfolios: (a) forecast variance or risks. (continued on next page) Global EWFP (passive portfolio) 90 Transportation Research Record 2670 82 Individual forecast assets Forecast portfolio combinations 80 78 Forecast Returns (millions) 76 74 72 70 68 66 64 62 60 ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0) BMMR ARIMA (1, 0, 1) Global MFPV MFPV with Target Forecast Returns Global MFPR MFPR with Target Forecast Variance Global MFPE (tangency portfolio) Global EWFP (passive portfolio) MFPV with Global MFPR MFPR with Global MFPE Target Forecast Target Forecast (tangency Returns Variance portfolio) Global EWFP (passive portfolio) Individual Forecast Assets and Optimal Portfolios (b) 25 Individual forecast assets Forecast portfolio combinations Forecast Eﬃciency Ratio 20 15 10 5 0 ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0) BMMR ARIMA (1, 0, 1) Global MFPV Individual Forecast Assets and Optimal Portfolios (c) FIGURE 2 (continued) Performance metrics for individual forecast options and combined forecast portfolios: (b) forecast returns and (c) forecast-efficiency ratio. Shah and Jammalamadaka 91 were higher. Thus, overall portfolio optimization strategies and diversifications generally outperform the individual forecast assets in regard to optimizing risk and achieving improved efficiency ratios. Optimal Weight Assignment Results for the optimal weight assignments across the different forecast options are shown in Figure 3. These weights hold under current estimates of individual forecast returns, variance, and covariance. Results for all five mean-variance optimization classes of strategies are shown, but results for the passive portfolio strategy are not shown because it is simply an equal weight assignment. In the first case study application, the global MFPV, MFPR with target forecast variance, and global MFPE models suggest diversification across individual forecast assets; however, some models, such as the global MFPR and MFPV with target forecast returns strategies, do not. The latter models are inclined to maximize forecast returns, and the optimization models find that assigning maximum (or rather the entire) weightage on the asset providing the highest returns in both cases, or the ARIMA (2, 1, 0) model, provides the highest overall portfolio returns. This is a trivial solution from the global MFPR strategy and does not result in a diversified portfolio. As previously mentioned, this approach generally is the riskiest proposition and, in simple terms, amounts to “putting all one’s eggs into one basket.” Other strategies, however, provide relatively more diversified portfolios. For example, the global MFPV strategy, as expected, suggests the comparatively highest allocation on a model with the lowest variance, which is ARIMA (1, 0, 1). In addition, the optimal weight for this model is almost three times that of the second preferred model, which is BMMR. For the MFPR with target forecast variance strategy, a higher weight is assigned to the BMMR model than the ARIMA (1, 0, 1) model, though the corresponding weights are not much different. The global MFPE strategy results in the ARIMA (1, 0, 1) model receiving the highest weightage, followed by the BMMR. Asset weights in optimal portfolios are to be driven by the respective individual asset characteristics, such as their expected returns, variance, and forecast efficiency. These optimal weights, naturally, are sensitive to the current optimization model formulations and constraints. Thus, relaxing any constraint on the decision variables, particularly the nonnegativity, may result in different optimal combinations, but that aspect has not been explored in the current study. Recommendations for Risk Management This section develops policy and practice takeaways from the current study approach and the findings from the case study application. Efficient Portfolios In an environment of future uncertainty and concerns about its effect on future facility performance and revenue potential, an analytical risk-management approach for forecasts using optimal portfolio strategies is created. The merits of diversifying across individual risky forecast assets, models, or both and combining into optimal portfolios are also presented. These optimal portfolios can be combined further to develop “efficient” portfolios to achieve improved degrees of risk–return trade-off. These are essentially convex sets of portfolios that are generated by the linear combination of any pair of two optimal portfolios (20, 21). The combination of pairs of optimal forecast portfolios can be chosen with the use of the weights of the form α and (1 − α). Thus, given any two optimal portfolios p1, p2 with weight vectors w1, w2, respectively, the following convex combination is an efficient portfolio pe: pe = α i w1 + (1 − α ) i w2 (26) These assigned weights are directly proportional to the fraction of the total toll asset capital invested in each portfolio. Qualitatively, these values reflect the emphasis placed on a specific portfolio strategy. For example, when two mutually inverse objectives, such as Optimal Forecast Asset Model Weights 1.2 1 0.8 0.6 0.4 0.2 0 Global MFPV ARIMA (0, 1, 1) MFPV with Target Global MFPR MFPR with Target Forecast Returns Forecast Variance Forecast Portfolio Optimization Strategy ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0) FIGURE 3 Weight allocations across forecast options under optimal portfolio strategies. BMMR Global MFPE ARIMA (1, 0, 1) 92 global MFPV and global MFPR portfolios, are chosen to combine and develop an efficient portfolio, they can represent two extreme objectives of a planning and financing spectrum. One end of the spectrum aspires for highest returns and optimistic projections without concern for the high risks, while the other end has the singular objective of minimizing risks while adopting conservative forecast projections. Forecast Conservativeness Index As per Equation 26, let α be the weight assigned to the global MFPV portfolio, the remainder (1 − α) the weight on the global MFPR portfolio. Under such a combination, alpha values can be assigned according to the risk averseness levels of individuals or agencies. They are referred to in this study as the “forecast conservativeness index” (FCI). From the perspective of agencies or investors, FCI can represent a wide range of possible objectives trying to diversify forecast assets while distributing risks. Also, the global MFPR strategy does not necessarily suggest diversification, whereas the global MFPV does. Under this convex combination, FCI can reflect the weight given to a diversification strategy as opposed to a concentrated single-asset strategy. Global MFPV places emphasis solely on forecast risks, while global MFPR places emphasis only on returns. Thus, choosing a higher alpha or a larger FCI indicates a higher weight on MFPV and, conversely, more of an inclination toward minimizing forecast risks rather than gaining high returns from the portfolio. As a result, higher alpha indexes may be more preferable for more conservative agencies or investors, and the corresponding portfolios can also be classified as risk averse. However, a lower alpha value or a lower FCI indicates a lower emphasis on minimizing risks but higher emphasis on maximizing returns. Such “aggressive” or optimistic portfolios indicate higher risk proclivity. FCI cannot be generalized as a metric of risk aversion for a convex combination of any two general optimal portfolios. It holds only under the choices of global MFPV and global MFPR portfolios to develop the efficient portfolio combination, which is a result of two inverse portfolios with one focusing unequivocally on risks and the other focusing only on returns. This combination represents polarized objectives for minimizing risks and maximizing returns, along with the relative degree of each captured through FCI. The effects of selecting a range of FCI values along with the forecast-efficiency ratio of the forecast portfolio demonstrated in Figure 4, highlight the forecast conservativeness indexes and portfolio characteristics. Trends show that increasing FCI indicates a higher degree of forecast risk averseness. Also consistent with objectives, forecast variance or risk levels are found to decrease. In line with the basic characteristics of any common financial portfolio produced with MPT, forecast portfolios developed in this study indicate that forecast returns at lower risk levels are also low in proportion. This trend can once again be observed in the trend of decreasing forecast returns with increasing conservativeness or FCI. The third set of plots indicates that the forecast portfolio-efficiency ratios improve with higher values of FCI; this improvement can be attributed to a decline in forecast variance. Role of System-Specific Characteristics Apart from risk-averseness preferences, system characteristics such as operations may also play a role in the selection of FCI ranges. Transportation Research Record 2670 For the purposes of long-range planning and toll infrastructure programming, toll network planners and bidders in coordination with local agencies can adopt FCI ranges according to their understanding of what fits the respective system best. In addition, governmental support or assurance on investment returns may promote optimism in bidders and contractors, which could include a guaranteed level of minimum revenue, which is designed to alleviate concerns related to demand risk (22). Toll facilities in developing areas that have modest near-term projections of traffic and demographic growth might prompt the selection of more conservative forecasts (or higher FCI values). Inversely, low-variance forecasts or high-growth facilities may create additional room for risks. Such facilities may prompt adoption of relatively aggressive forecast portfolios (or lower ranges of FCI values). More often though, or in the absence of any conspicuous system characteristics, planning decisions tend to gravitate toward a lowto moderate-risk category and, as found here, can help improve forecast–portfolio efficiency by trying to achieve an optimal risk–return trade-off. Conclusions Summary of Research and Findings The study leveraged several analytical strategies, including stochastic time series models, Monte Carlo simulations, and MPT-based optimal and efficient portfolios. The goal was to develop a risk management framework for toll road traffic forecasting and capital investments. The framework was demonstrated with a case study and included two classes of time series models: ARIMA and BMMR. These two models are considered available forecast options but can be replaced by any other available forecasting engines, if desired. In the postvalidation forecasting regime, recognizing the added magnitudes of uncertainty given “randomness” in future projections and market volatility, an MPT-based portfolio optimization approach was developed to jointly optimize forecast returns and risks. Five different optimal portfolio strategies were evaluated, and a convex combination of two of those (global MFPV and global MFPR) was used to develop an efficient forecast portfolio. A global MFPE strategy aiming to forecast the portfolio-efficiency ratio along the lines of a Sharpe return-to-risk ratio is also implemented. In addition, an egalitarian global EWFP strategy with an equal weight assignment leading to a passive portfolio is developed, which achieves lower forecast risks and better forecast efficiency than most individual forecast options. These findings highlight the forecast risk mitigation merits of portfolio strategies and diversification. Some specifications may achieve an efficient trade-off, but may also carry higher forecast variances and ultimately increase the risks associated with the overall forecast portfolio. Some ARIMA specifications appear aggressive in projections compared with other low-variance model options such as BMMR. Thus, there is no single preferred model, and the choice can vary depending on individual risk proclivities. To articulate such challenges, a discrete metric is developed called FCI, which can serve a wide range of risk-averseness levels (given an efficient portfolio combination consisting of global MFPV and global MFPR portfolios). Higher FCI values lead to less-risky forecast portfolios, as well as portfolios with lower expected forecast returns. However, portfolio-efficiency ratios are found to improve with higher FCIs. Agencies and financiers, however, are generally Shah and Jammalamadaka 93 80 14 Forecast Returns (millions) Forecast Variance or Risks (millions) 16 12 10 8 6 4 2 0 0 .1 .2 .3 .4 .5 .6 .7 .8 Forecast Conservativeness Index (`) .9 78 76 74 72 70 68 1 0 .1 .2 .3 .4 .5 .6 .7 .8 Forecast Conservativeness Index (`) (a) .9 1 (b) Forecast Portfolio Eﬃciency Ratio 25 20 15 10 5 0 0 .1 .2 .3 .4 .5 .6 .7 .8 Forecast Conservativeness Index (`) .9 1 (c) FIGURE 4 Forecast conservativeness indexes and portfolio characteristics: (a) forecast portfolio variance or risks, (b) forecast portfolio returns, and (c) forecast portfolio-efficiency ratios. more conservative in practice, with less predilection for risks by opting for higher ranges of FCI values at least in the early stages of planning. Study Limitations and Future Research Avenues Certain practical limitations of the proposed methods are highlighted. First, the dependence on time series methods can limit cases in which historical data might not be available for estimating models and conducting simulations. Thus, the current framework’s applicability may be limited to existing toll facilities or brownfield projects, rather than newly planned greenfield corridors. Second, from a modeling standpoint, an apparent limitation of a univariate time series forecasting approach is that it does not consider any additional variables other than its own past values and prediction errors. It is a high-level assessment based exclusively on toll transactions or observed traffic volumes. Typical comprehensive T&R studies for forecasting generally involve an interactive demand model and toll diversion routines that include a host of additional demographics, demand-side and supply-side (network) variables, and constraints. Third, the procedure is demonstrated on systemwide toll data, which may be too macroscopic. Further research is needed to validate the process with more microscopic gantry-level transaction data, for which operations may be more controlled and local-level effects more pronounced. The current method is thus not designed to fully substitute or even serve as a proxy for the relatively more comprehensive T&R study approach. However, it can complement those studies by helping cross validate their model projections, mimic historical variations, and quantify potential forecast risks. In combination with conventional four-step demand models, these methods can help to improve forecast reliability and ultimately improve planning and investment decisions. These limitations open avenues for several continuing and future research possibilities. Multivariate time series models that consider additional supply-side variables or historical demographic trends, 94 such as generalized autoregressive conditional heteroskedasticity, are worth exploring. Given that the project finance landscape changes quite dynamically and as new capital and risk-sharing instruments emerge in urban toll road projects, planners and investors may not always conform to the rationality assumptions of the MPT, such as inherent risk averseness. Additional insights may be borrowed from behavioral finance to design assorted portfolios applicable in different contexts, along with more innovative portfolio optimization strategies for toll road assets (23). Acknowledgments The authors thank several anonymous reviewers for their thorough reviews and comments on the initial manuscript. References 1.Jammalamadaka, P., Y. Jarmarwala, W. Hirunyanitiwattana, and N. Mokkapati. Comparative Analysis of Traffic and Revenue Risks Associated with Priced Facilities. Presented at 14th TRB National Transportation Planning Applications Conference, May 2013. 2. Bain, R. Error and Optimism Bias in Toll Road Traffic Forecasts. 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