Toll Road Forecast Risk Analysis
using Time Series Models and
Portfolio Optimization
Rohan Shah
Phani Jammalamadaka, PE
May 16, 2017
Raleigh, NC
Outline
 Background and Motivation
• Toll Traffic and Revenue (T&R) Estimation Process & Uncertainty
Elements
 Study Method and Model Development
• Time Series Forecasting
• Forecast Portfolios
• Mean-Variance Optimization Strategies
 Numerical Analysis and Case Study Application
 Policy Insights and Applicability
2
Background and
Motivation
T&R Estimation Process & Uncertainty Elements
4
Study Method and
Model Development
Study Overview
6
Time Series Model Forecasts: Estimation
• Leveraging available historical data
• Two stochastic time series models used as forecasting tools
• Autoregressive Integrated Moving Average (ARIMA), and Brownian Motion
Mean Reversion (BMMR)
• Expected value of estimates are taken,
which give deterministic/baseline
forecast estimates
• Error terms get excluded (zero
expected values)
• These are a
fair starting point and useful in
visualizing the growth trends
Source: https://people.duke.edu/~rnau/411georw.htm
7
Time Series Model Forecasts: Validation
• Validation needed to establish predictive accuracy
• Out-of-Sample Testing approach
• A fraction of the original input data (not used in model estimation) is kept
aside for empirical validation of model forecasts
• Mean absolute percentage error (MAPE) is used:
𝟏
𝑀𝐴𝑃𝐸 =
𝑵
𝑵
𝒊=𝟏
𝒙𝒊 − 𝒙𝒊
∗ 𝟏𝟎𝟎
𝒙𝒊
Where, 𝑥𝑖 = actual observations (data); {𝑥𝑖 } = estimated or forecasted time
series; 𝑁 = input time-series length
8
Time Series Model Forecasts: Simulation
• Both models have ‘error terms’ that qualitatively indicate likely deviation
of forecasts from actual/expected values
• They are random variables
•
Even though their ‘expected value’ is statistically zero (deterministic
forecasting), they may take finite values during actual forecasting
• Stochasticity is introduced
using a Monte-Carlo simulation
of error terms where they are
sampled from a Gaussian
distribution
𝑒𝑡 ~𝑖. 𝑖. 𝑑. 𝒩(0, 𝜎 2 )
Simulated Asset Price
Asset prices simulated using Geometric Brownian Motion
Time (t)
Source: http://www.turingfinance.com/random-walks-down-wall-street-stochastic-processes-in-python/
9
Forecast Portfolio: Concept and Design
• Individual forecast assets or models are combined into a ‘portfolio’ of
forecasts
• Planners/decision-makers can choose from a ‘set’ of forecasts rather a
single forecast source/model
• Some defining variables:





𝑛 = individual forecast options or models 𝑖
𝑟𝑖 = the returns or forecast projections from each
𝐸 𝑟𝑖 = expected value of forecast
𝑊 = vector of weights for the optimal distribution of a forecast portfolio
𝑤𝑖 = optimal weight for model option 𝑖
• Optimal weight is the proportion of the overall forecast to be assigned to
model 𝑖
10
Forecast Portfolio: Risk and Return
Return:
• The returns from the overall forecast portfolio is 𝑟𝑝 , and its
expected value is computed by the vector combination of
individual model returns𝒏
𝑬 𝒓𝒑 =
𝒘𝒊 𝑬 𝒓𝒊
𝒊=𝟏
Risk:
• Each forecast has a risk/variance associated with itself
• Portfolio variance 𝜎𝑝2 , (or standard deviation 𝜎𝑝 ) is the weighted
average covariance of the individual forecast returns𝑛
𝑛
𝑉𝑎𝑟 𝑟𝑝 = 𝜎𝑝2 =
𝑤𝑖 𝑤𝑗 𝐶𝑜𝑣 𝑟𝑖 , 𝑟𝑗
𝑖=1 𝑗=1
11
Forecast Portfolio: Mean-Variance Optimization (MVO)
12
Numerical Analysis and
Case Study Application
Setup and Empirical Validation
Area: Massachusetts Turnpike corridor is used a case-study application, and
historical system-wide toll transactions data dating back to 1950s are acquired
Assumptions/Features: Steady-state operations, Brownfield corridor
Validation: Five ARIMA specifications and one BMMR specification are fit to the
historical time series
14
Long Range Forecasts and Confidence Intervals
• Confidence intervals are plotted only for the preferred model
specification (one with the lowest MAPE)
• Long-range forecast streams are a reflection of the validation
period, indicating a mix of forecast outlooks, ranging from
conservative, neutral, to aggressive forecasts
15
Optimization Solutions and Diversification Merits
• Portfolio-efficiency ratio jointly optimizes risk and return, and
captures the tradeoff offered by a portfolio
• Ratio of expected return of the portfolio to the portfolio
variance
16
Efficient Portfolios: Convex Combination
• Optimal portfolios can further be combined to develop ‘efficient’
portfolios
• Goal is to achieve an even better degree of risk-return tradeoff
• These are convex sets of portfolios (linear combinations of a pair
of optimal portfolios)
• The combination is chosen using weights 𝛼 and (1 − 𝛼)
• Given any two optimal portfolios 𝑝1 , 𝑝2 with weight vectors
𝒘𝟏 , 𝒘𝟐 , their convex combination is an efficient portfolio 𝑝𝑒 :
𝑝𝑒 = 𝛼. 𝒘𝟏 + 1 − 𝛼 . 𝒘𝟐
17
Risk Management using Efficient Portfolios
• Effects of 𝛼 on Portfolio Efficiency and Variance
Max return
zone
Min variance
zone
Max return
zone
Min variance
zone
• Weights (α) reflect the emphasis laid on a policy objective
• When the two mutually inverse objectives Global MFPV and
Global MFPR portfolios are chosen in an efficient portfolio, they
represent two extreme objectives on the risk spectrum
18
Policy Insights and
Applicability
Risk-Return Tradeoff: High Risk-High Return
• Numerical findings highlight the merits of portfolio strategies and
diversification in risk-mitigation
• Some specifications may achieve an efficient tradeoff,
but increase the overall
portfolio variance
• ARIMA: Aggressive
BMMR: Moderate
• There is no single
preferred model, and
the choice depends on individual risk proclivities
20
Risk Preferences and Forecast Model Selection
• In long-range planning, agencies take risks based on systemspecific characteristics
• Toll facilities in developing areas/modest near-term growth
projections => conservative forecasts
• Low-variance forecasts/high-growth facilities => aggressive
forecast portfolios
• Generally, planning decisions may gravitate toward low- to
moderate-risk category
21
Model Limitations
• Historical data availability
• Applicability limited to existing toll facilities or brownfield
projects
• High-level assessment based exclusively on observed data versus
comprehensive T&R studies
• Variance in systemwide/macroscopic toll data versus more
controlled microscopic gantry-level transaction data
22
Applicability
• Not a proxy for more conventional/comprehensive T&R study
approach
• Complement those studies
• Cross-validate their model projections, mimic historical
variations, and quantify potential forecast risks
• Improve forecast reliability, when applied in combination with
TDM-based approaches
• Aid planning and financial investment decisions
23
Q&A
Thank You
Rohan Shah
[email protected]
Phani Jammalamadaka, PE
[email protected]