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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,
Transportation Research Record: Journal of the Transportation Research Board,
No. 2670, 2017, pp. 83–94.
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 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
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).
1  N xi − xˆι
N  i =1 xi
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
Annual Total Transactions (millions)
Forecast Model Validation Year
ARIMA (0, 1, 1)
ARIMA (1, 1, 1)
ARIMA (2, 1, 0)
ARIMA (1, 0, 1)
ARIMA (1, 1, 0)
FIGURE 1 Toll transaction forecast streams with various time series model options: (a) short-term forecasts,
comparison with actuals.
(continued on next page)
Transportation Research Record 2670
MAPE (%)
ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0)
Forecast Assets Models
ARIMA (1, 0, 1)
Annual Total Transactions (millions)
Year of Operation
ARIMA (0, 1, 1)
ARIMA (1, 1, 1)
ARIMA (1, 1, 0)
ARIMA (2, 1, 0)
ARIMA (1, 0, 1)
5% Bound
25% Bound
75% Bound
95% Bound
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
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:
E (rp ) = ∑ wi E (ri )
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
cov (ri , rj ) = ρij σi σj
The portfolio variance can also be expressed in relation to the
correlation coefficient as follows:
var (rp ) = ∑ ∑ wi wj ρij σi σj
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
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.
∑w =1
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
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
var (rp ) = σ 2p = ∑ ∑ wi wj cov (ri , rj )
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
∑w =1
i =1
Transportation Research Record 2670
wi ≥ 0
∀i ∈(1, n )
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 )
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
∑w =1
i =1
wi ≥ 0
∀i ∈(1, n )
w ∈W
σp ≤ min (σ i )
∀i ∈(1, n )
subject to
∑w =1
The last expression is the target minimum portfolio variance
i =1
wi ≥ 0
∀i ∈(1, n )
E (rp ) ≥ max ( E (ri ))
∀i ∈(1, n )
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 E (rp )
w ∈W
E (rp )
subject to
subject to
∑w =1
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
∑ wi = 1
i =1
wi ≥ 0
∀i ∈(1, n )
i =1
∀i ∈(1, n )
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
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
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
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
Individual forecast assets
Forecast portfolio combinations
Forecast Variance or Risks (millions)
ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0)
ARIMA (1, 0, 1) Global MFPV
MFPV with
Global MFPR
MFPR with
Global MFPE
Individual Forecast Assets and Optimal Portfolios
FIGURE 2 Performance metrics for individual forecast options and combined forecast portfolios: (a) forecast variance or risks.
(continued on next page)
Global EWFP
Transportation Research Record 2670
Individual forecast assets
Forecast portfolio combinations
Forecast Returns (millions)
ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0)
ARIMA (1, 0, 1) Global MFPV
MFPV with
Global MFPR
MFPR with
Global MFPE
Global EWFP
MFPV with
Global MFPR
MFPR with
Global MFPE
Target Forecast
Target Forecast (tangency
Global EWFP
Individual Forecast Assets and Optimal Portfolios
Individual forecast assets
Forecast portfolio combinations
Forecast Efficiency Ratio
ARIMA (0, 1, 1) ARIMA (1, 1, 1) ARIMA (1, 1, 0) ARIMA (2, 1, 0)
ARIMA (1, 0, 1) Global MFPV
Individual Forecast Assets and Optimal Portfolios
FIGURE 2 (continued) Performance metrics for individual forecast options and combined forecast portfolios: (b) forecast returns
and (c) forecast-efficiency ratio.
Shah and Jammalamadaka
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
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
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.
Global MFPE
ARIMA (1, 0, 1)
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
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.
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
Forecast Returns (millions)
Forecast Variance or Risks (millions)
Forecast Conservativeness Index (`)
Forecast Conservativeness Index (`)
Forecast Portfolio Efficiency Ratio
Forecast Conservativeness Index (`)
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
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,
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).
The authors thank several anonymous reviewers for their thorough
reviews and comments on the initial manuscript.
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The Standing Committee on Revenue and Finance peer-reviewed this paper.
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