Evaluating Alternative Operations Strategies to Improve Travel Time Reliability (2013)

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108 Appendix B presents an approach for determining the eco- nomic benefits of improving travel time reliability under recurring-event scenarios. In this approach, uncertainty aris- ing from recurring events is converted to a certainty-equivalent measure so that conventional evaluation methods can be used to place a value on the cost of unreliability. The certainty- equivalent measure is a method that allows the value of reli- ability to be expressed in terms of the increase in the average travel time a person would accept to eliminate uncertainty. For example, suppose that Mr. A is told to draw a card from a full deck. If he draws a red card he wins $100, and if he draws a black card he wins nothing. Given the likelihood of either winning or losing, if Mr. A could be paid $50 to not play the game, then $50 would be his certainty-equivalence. Thus, Mr. A has placed a dollar value on removing unreliability. He has forgone the chance at $100 but has also avoided the equal possibility of receiving no payment. In this way, a vari- able that can be characterized by its probability distribution can be converted to a certainty-equivalent measure. The approach presented for determining the value of reli- ability associated with recurring events is extended to rare events in Appendix C. Recurring and rare events (and the unreliability produced by them) differ in the probability dis- tributions that characterize them. Unreliability produced by recurring events is assumed to display statistical behavior that is best represented by normal distributions (often, a lognormal distribution). Rare or “extreme” events are better represented by a class of distributions known as extreme value (EV) distri- butions. It is often possible to observe the effect of recurring events on network performance directly by observing the vari- ability in speeds or other network performance metrics. In the case of unreliability caused by rare events, it may not be pos- sible to observe the statistical nature of the unreliability in real- world data, because the number of such incidents in a given region may be so low that coincident performance measure- ments are lacking. The rare-event approach is briefly described in this appendix and is presented in full in Appendix C. Once the certainty-equivalent measure has been computed, valuation can proceed as if the values involved were determin- istic. The value of reliability can be derived for multiple user groups or market segments by applying a separate value of time that corresponds to each user group along with the observed average volume for each user group on the roadway segment. The total value of reliability is then computed as the sum of the reliability values for each user group on the highway segment. To illustrate this concept, let us consider a single link of a roadway network that experiences considerable variation in the average travel time because of recurring congestion. Imag- ine that a commuter would be willing to spend an additional minute per mile if the uncertainty in travel time were elimi- nated by a technical or policy action. This additional minute per mile is the certainty-equivalent of the unreliability caused by a recurring source of unreliability. The cost of unreliability to the commuter, converted to the certainty-equivalent mea- sure, can be monetized by the value that he or she places on each minute of additional delay. Most studies report the value of time as being between 40% and 50% of the wage rate for average trips and a much higher rate (around 85% of the wage rate) for commuting and intercity trips. The value of time for freight movement depends on the size of the truck and the value of the commodity hauled, typically with a value ranging between $30 and $60 per hour. In the following example, the certainty-equivalent value of 1 minute/mile is used. It is assumed that the average value of time of users of the roadway is $20/hour (33.3 cents/minute), the roadway segment is 10 miles long, and it is used by 6,000 users/hour during the time of day when unreliability in travel times is experienced. The cost of unreliability to users on the facility over a 1-hour period would be approximately $20,000 (1 minute/mile × 33.33 cents/minute × 10 miles × 6,000 = 2,000,000 cents or $20,000). This concept for calculating the value of travel time reliability is illustrated in Figure B.1. The concepts for determining the economic benefits for improving travel time reliability are presented in detail in the A p p e n d i x B Determining Economic Benefits of Improving Travel Time Reliability

109 sections following. These include measuring the value of reli- ability, valuing reliability for recurring events, measuring net- workwide impacts, quantifying the value of time, and applying the methodology. A discussion follows on conclusions for the options theoretic approach and a recurring-event reliability valuation example. Measuring the Value of Reliability Evaluating the economic cost of unreliability on the highway system requires a method for valuing reliability. This, in turn, requires measuring unreliability in a consistent manner and a method of placing a value on the costs of unreliability. The methodology outlined in this section provides a consistent means for characterizing reliability and for valuing reliability by using the value of time measures described above. This methodology is adaptable to both recurring reliability issues and the degradation of network performance caused by rare events (such as bridge failures, flooding, and other natural disasters presented in Appendix C). Although recurring and rare phenomena arise out of pro- cesses with very different stochastic properties, the challenge of analyzing these different sources of reliability is similar, and the solutions have a common approach. In both cases, the uncertainty that these two processes impose upon the cost of travel may be converted to a certainty-equivalent mea- sure to enable the use of conventional evaluation methods for determining the value of reliability. In the case of rare events, the facts are different, but the virtue of the certainty-equivalence notion is the same. For example, if knowledge of the processes that cause a bridge failure can be characterized by an assignment of a probability distribution to the bridge-failure event, then the uncertainty about the occurrence of a bridge failure can be used to derive a certainty-equivalent measure of the number of such events. This quantity can then be associated with the resulting impacts on network performance and valued using conventional tech- niques for measuring network performance. These techniques involve measuring the incremental delay associated with the bridge failure and the time period over which the delay persists. The ultimate goal of valuing network unreliability is to guide investments into policies or technical solutions that reduce the economic burden of unreliability. Thus, the methodology for valuing unreliability naturally incorporates considerations of present valuation (the time value of money) and the discount rates associated with considering event timing. Measuring the value of reliability involves five consider- ations: (a) use of options theory, (b) a real option for travel time, (c) measuring the economic cost of unreliability, (d) eval- uating reliability management policies, and (e) characterizing reliability for recurring events. Figure B.1. Concept for valuing travel time reliability.

110 Practitioners and experts in the real-options branch of finan- cial analysis have a very specific meaning for “real option.” Usu- ally, the term refers to real assets in the capital budget context. Because it is not known that anyone else has studied travel time reliability by using the options approach, it is not clear what is the proper terminology to use. However, Professor Lenos Trigeorgis, a well-known expert in the field of real options, has applied the term “real options” in an analogous setting, that is, a decision to increase flexibility of production processes as defense against exchange rate variability. Therefore, the term “real option” is used in a liberal sense for the options theoretic approach for the valuation of travel time reliability because the certainty-equivalent measure is easily converted to minutes (time). Application of options formulas of the Black–Scholes type to nonfinancial options has to respect certain underlying assumptions of the Black–Scholes model. Analysts should apply Black–Scholes to real options in circumstances in which they make the most logical sense. A full discussion of the under lying assumptions of option-pricing models of the Black–Scholes type and their implications in the development of real options, such as those presented here, can be found in Kodukula and Papudesu (2006). Because real options are not traded in a formal market, the so-called arbitrage assumption is commonly highlighted as a limitation on the applicability of Black–Scholes to real options. However, as Kodukula and Papudesu (2006, p. 84) point out: We believe that a categorical denial of the validity of the models to real option problems is inappropriate and that a “no arbi- trage” condition is only a limitation of the model and can be overcome easily by proper adjustment. Practitioners have used three different types of adjustment: 1. Use an interest rate that is slightly higher than the riskless rate in the option pricing model. 2. Use a higher discount rate in calculating the discounted cash flow (DCF) value of the underlying asset. 3. Apply an “illiquidity” discount factor to the final option value. Basically, the objective of these adjustments is to account for any overvaluation caused by not meeting the “no arbitrage” condition. All three methods, therefore, decrease the option value, making it more conservative. Hence, the development of real options and proper adjust- ment make development of a real option for travel time pos- sible, despite the fact that the commodity is not traded in a market in which the arbitrage condition is necessarily assured. Damodaran (2005), in The Promise and Peril of Real Options, provides a relatively accessible treatise of the Black–Scholes formula and real options. In addition to providing an intro- duction to the use of options, the paper includes discussion Options Theory The field of economics that provides a mechanism for con- verting uncertainty into a certainty-equivalent value is options theory. Options theory provides insights into the value of cur- rent and future opportunities whose value is not known with certainty but whose opportunities can be characterized proba- bilistically. The classic application of options theory in a finan- cial context is to answer such questions as, “How much should I be willing to pay to be able to buy (or sell) a security at a given point in the future at a specific, prearranged price?” An option to buy at a fixed price is a “call option,” and an option to sell at a fixed price is a “put option.” Economists have determined that many financial arrange- ments that involve valuing uncertainty (such as insurance con- tracts, prepayment penalties on mortgages, or car lease terms) can be analyzed as financial options of one sort or another. In insurance, for example, buyers might ask how large an insur- ance premium they would be willing to pay to avoid the risk associated with the loss from a fire. The option can provide an opportunity to buy or sell (“exercise the option”) at a specific point in time or any time up to a specific time. The opportunity to buy or sell at a specific point in time is a “European option,” and the opportunity to buy or sell at any time up to a specific time is an “American option.” Simple examples of the application of a European and an American option are described below: • An example of an application of a European option is in commodity trading. Here, individuals are buying the right to purchase or sell an asset at a contracted price at the end of the contract period. Trading in commodity futures is of the European option type, because people buy the right to purchase or sell wheat, corn, or other products at a speci- fied (strike) price in the future. • Buying the right to purchase or sell an asset at a contracted price at any time during the contract period is an American option. Insurance policies are a type of American option, because the commodity being traded (money value of insured object) can be called or put at any time during the policy coverage. Economists have expanded the notion of options beyond financial instruments to so-called real options. Real options involve the analysis of quantities such as commodities and time. Formulas have been developed to compute the value of various types of real options under a wide range of conditions. Many real-world insurance policies insure such real outcomes as a policy that ensures that a communications satellite will function at a specified level of performance or for a specified number of years. Application of options theory to travel time reliability constitutes one formulation of a real option.

111 uncertainty can be abstracted from to facilitate valuing the unreliability of a road system. Specifically, the real option for- mulation allows for answering the question posed above with a certainty-equivalent delay, which can be converted to addi- tional travel time per mile. Measuring the Economic Cost of Unreliability Once the certainty-equivalent value of a stochastic incidence of a performance metric or the underlying events has been deter- mined, valuation can proceed as if the values involved were deterministic. To continue the example of recurring unreliabil- ity phenomena, the certainty-equivalent value of the uncertain travel time performance greatly facilitates monetization of the road’s unreliability. Specifically, assume that the option value of the travel time unreliability is 1 minute/mile. That is, the traveler would be willing to spend an additional minute per mile if the down- side uncertain risk of slower speeds or longer travel times was eliminated by some technical or policy action. This implies that the unreliability cost to the traveler is equal to whatever value he or she places on each minute of additional delay. Using available evidence on the value of time in a given travel setting allows the traveler to place a dollar value on a road’s unreliability. To repeat the example that was previously provided in this appendix: if the average value of time of users of the roadway is $20/hour, then the value of time is 33.3 cents per minute, per mile, per user. If the certainty-equivalent value of delay is 1 minute/mile, the road segment displaying this unreliability is 10 miles long, and there are 6,000 users per hour at the time of day that this unreliability is displayed, the cost of unreliability is approximately $20,000/hour or $7 million/year for each hour of the day that this level of unreliability occurs. If we use the same example as above, the dollar value of the reliability can be easily calculated to account for multiple road-user groups or trip types. Assume that there is a 10-mile road segment with an equivalent option value for the travel time unreliability equal to 1 minute per mile. On the basis of the a.m. peak values of time developed by the Puget Sound Regional Council, the dollar value associated with the unreli- ability on this road segment is shown in Table B.1. The dollar value of reliability is calculated as the certainty- equivalent of delay (1 minute), multiplied by the user group volume, multiplied by the value of time per minute per hour (60 minutes) × facility length (10 miles). Evaluating Reliability Management Policies Once the cost of unreliability is established, this information can be used to evaluate the cost-effectiveness of strategies to manage unreliability. To do this, it is necessary to determine the effect of the strategy on improving reliability and then of the conditions for formulating real options and a com- parison of a real options approach to decision making under uncertainty in contrast to the more traditional discounted cash flow models. A Real Option for Travel Time To illustrate the applicability of options theory to the issue of travel time, let us examine the phenomenon of unreliability, which occurs on a relatively frequent basis over the course of a year, and how it might be analyzed by using options theory. Imagine that a single link of a network is involved and that we have observed the speeds and resulting travel times on this link for many days. Assume that those observations have led to the conclusion that the travel time has an average value, but there is considerable variation in value around that aver- age. That is, travel times on any specific day are unlikely to be the average, but rather something above or below the mean. (Put differently, the link does not provide reliable service.) Further, over the period of observation, the travel times expe- rienced are distributed lognormally. Of interest is devising a succinct measure of the unreliabil- ity of this link. One way to do that is to ask the question, “How much longer is the travel time I would accept in return for no uncertainty about the travel time?” This question sounds very much like the questions that arise in deciding how much one might be willing to pay to insure property. Given that insur- ance contracts can be represented by options, the question per- taining to travel time unreliability can be answered with the right formulation and parameterization of an options formula. This option formulation for travel time reliability is derived from options representations of insurance. In other words, the basic insight of the approach is that one can think of unreliabil- ity as analogous to the occurrence of an undesirable outcome in some random-event context (e.g., an accident that impairs the value of a car). In an auto insurance context, one can think of the insurance policy as a mechanism for compensating the driver for any lost value due to an accident during the life of the contract. Carrying this notion over to travel time reliabil- ity, one can imagine that an insurance policy could be crafted that compensated the driver for the unexpected occurrence of speeds below the expected (average) speed. Such a policy does not exist for daily vehicle travel, although such policies do exist for long trips (e.g., overseas travel insurance). So, if one accepts that the concept of speed insurance makes sense, then the Black–Scholes formulation makes sense, and one can calculate the speed-equivalent “premium” to be ensured compensation for encountering speeds less than the mean (expected) speed. Thus, the premium of our insurance contract is the excess delay we are willing to pay to be guaranteed a travel time equal to today’s average. The specific mathematics of this are presented later, but this example illustrates how travel time

112 a lognormal distribution, while the stochastic nature of rare events necessitates adapting the more traditional options for- mula. For rare events, an application of stochastic variables displaying a generalized extreme value distribution has been adopted and is presented in Appendix C. Both recurring and rare events cause unreliability, so recur- ring and rare events are distinguished only by differences in the frequency distributions that characterize them. Unreli- ability produced by recurring events is assumed to display statistical behavior best represented by normal distributions (often, lognormal distributions). In the case of unreliability caused by recurring events, it often is possible to observe the effect of these events on network performance by directly observing the variability of speeds or travel time. This is because normal distributions often best describe high-frequency phenomena, so the chances of having useful network performance data improves. In the case of unreliability caused by rare events, variability in speeds or travel time may not be measured directly. For exam- ple, the number of bridge failures may be so low that changes in performance measures may be difficult to study directly. (In cases such as major accidents, for example, there may be a way to directly measure the influence of these rare events on speed variability.) The next section describes the options theoretic approach for valuing reliability for recurring events. Valuing Reliability for Recurring events Unreliability produced by recurring events is defined as the variability in travel time that occurs as a result of events such as accidents, incidents, and poor traffic-signal timing. Over the course of a year, travel times may display high volatility at the same time of day, on individual roadway segments, and on specific paths or routes. On a somewhat less frequent, but nonetheless recurring basis, weather and other random, natural events can impair network performance. Normal rain events, for example, impair network capacity and performance in a transitory fashion. perform the same options theoretic computation under the improved conditions. The policy may, of course, affect both the average travel time and its volatility. For simplicity, assume that the policy does not affect average travel times, but rather, reduces the variance (volatility) of travel times. For example, a traveler information system, incident management system, or some other treatment may reduce the travel time variance such that the options value of unreliability is now only 30 seconds/mile on the roadway segment described above. There is now a sav- ings of approximately $3.5 million/year for each hour of the day that is affected by the treatment. Thus, a treatment that costs less than $3.5 million/year would be worthwhile (cost-effective) to implement because of its impact on improving reliability. In a real-world appli- cation, an improvement to traffic information systems might generate improvements in reliability over many years. As traf- fic grows, values of time evolve, and the computation of the options value of reduced volatility is repeated for each period of time during the life of the improvement. These future sav- ings can be reduced to a present value in the planning year by discounting the stream of annual options values. The advan- tage of the certainty-equivalent approach is that user benefits from improvements in travel time reliability can be treated deterministically, just as they are in other traditional user benefit categories in standard transportation investment or policy evaluations. Characterizing Reliability for Recurring Events Options formulations for travel time reliability have been developed to correspond to the valuation of reliability related to recurring and rare events. The stochastic nature of rare events (such as bridge failures, road closures due to flood- ing, and other events) is quite different from the uncertainty that characterizes recurring events. Recurring events (such as crashes, weather-related events, and other common sources of travel time variation) can generally be characterized with Table B.1. Example Dollar Value of Reliability for Multiple Road User Groups Road User Group Share of Volume Volume Value of Time/Hour Value of Reliability/ Hour Single-occupancy vehicle 70% 4,200 $26 $18,200 High-occupancy vehicle, 2 passengers 15% 900 $30 $4,500 High-occupancy vehicle, ≥3 passengers 8% 480 $38 $3,040 Vanpool 2% 120 $102 $2,040 Heavy trucks 5% 300 $50 $2,500 Total 6,000 $30,280

113 I = the guaranteed speed, in mph r = the annualized, risk-free continuously com- pounded interest rate s = variability of V; square root of the log-value varia- tion process of V T - t = option length in years, where T is the expiration date of the option Equation B.1 illustrates how the travel time reliability option can be formulated by following standard insurance options formulations, that is, recasting an insurance option as a “speed guarantee insurance” policy. From the mathematics of the underlying options theory, we know that (all other things being equal) the options value of a speed guarantee does the following (and, hence, affects the cost of unreliability of speeds): • Increases with the variability of speeds, • Increases with the guaranteed speed, • Decreases with the length of the contract, and • Decreases with the average speed. Absent any quantification of the options value, these insights are helpful in understanding how the characteristics of un - reliability, or the benefits of remediation of unreliability, are affected by statistical properties of unreliability. Two of the elements in the travel time reliability option are the interest rate and the option length. The value of the interest rate used in the formulation should be the real, annual riskless rate of return. This rate varies somewhat with macro economic conditions, but it should reflect the real dis- count rate the market is applying to value funds received in the future versus today. This is also called the “time value of money” in finance parlance. The interest rate in the imple- mentation of the Black–Scholes formula should be a low single-digit annual rate in the vast majority of macro economic settings. The option survives for a fixed amount of time, calculated from the lowest 5% speed implicit in the speed distribution. This could be smaller or larger. It is done to avoid the complexity of valuing serial options when multiple road segments, each with a different speed distribution, are involved. In this sense, it is akin to what is called a “capped” option, and it imparts a conservative value (lower) to the option value. This value (and hence the assumed life of the put option [insurance contract]) can be changed by the user. To implement the option, we need to provide the inputs listed in Equation B.1. In the examples presented below, the following features of the option are used: • The log mean and log standard deviations of speed are derived from data on segment speeds for a 5-minute time- of-day interval, using a year of daily speed observations. • The interest rate is set to a riskless, short-term interest rate. The certainty value of unreliability associated with recur- ring events can be derived by using options valuation tech- niques that employ a lognormal frequency distribution. This is achieved by recasting the question of reliability in terms of a speed insurance problem, which in turn, can be addressed by options theory. The options theoretic approach answers the hypothetical question: “How large a reduction in average speed should a traveler be willing to accept in return for a guaranteed mini- mum travel speed?” The options formula determines the speed reduction premium a traveler would be willing to pay for a minimum-speed-guarantee insurance policy. A simple speed insurance case is one in which the “cover- age” of the insurance is relatively short and the option can be invoked at the end of the life of the insurance period. Recast as a speed-reliability problem, this makes sense because it is of interest to know the burden placed on the traveler, who finds that the speed has been impaired by the volatility created by recurring events. A short-lived option that pays off at the end of its life is a European put option—that is, an option of finite life that can be exercised at the end of the option’s life. Such an option compensates the holder for any losses incurred if actual per- formance is poorer than the contracted performance guaran- tee. If performance is greater than the expected performance guarantee, then the option has no value. A travel time option, which is expressed as a certainty- equivalent of delay measured by speed or additional travel time, can be monetized by using the dollar value of travel time. However, to preserve the underlying distributional assump- tion of lognormality, it is better to compute the real option in terms of speed, as shown in Equation B.1, which is speed guarantee for recurring events. , 2 1P V t Ie N d V N dT r T t T( ) ( ) ( )= −( )− − where ln 2 (B.1) 1 2 2 1 d V I r T t T t d d T t T ( )( ) ( ) ( ) = + + σ − σ − = − σ − and where P(VT, t) = value of a European put option in mph, as a function of link speed and option length N(x) = the cumulative standard normal evaluated at x VT = the (unknown) speed experienced traversing the link, in mph ≈ a random variable, distributed lognormally

114 average speed, log average speed, and the standard deviation are calculated from observed speed data. The option value (mph) is calculated from the average log speed and log stan- dard deviation calculated from the data and other inputs to the options formulation previously described by using Equation B.1. When Equation B.1 is used, the option value will be expressed in the speed reduction that drivers would be willing to accept to obtain their speed guarantee. The option value, expressed in mph per hour can be converted to minutes by subtracting the option value in speed from the speed guarantee and determin- ing the additional average travel time the road user is willing to accept in exchange for travel time reliability. With the option values shown in Table B.2, the option val- ues (in mph or minutes) can be plotted against the average annual speed for each 5-minute period. From Figures B.3 and B.4, we can conclude that the value of unreliability at an average speed of 20 mph is approximately 0.8 minutes/ mile, whereas the value of unreliability at a speed of 60 mph is only about 0.1 minutes/mile. These measures may be idio- syncratic to the facility studied. However, the calculations illustrate that by converting speed volatility to delay equiva- lents, we can consistently measure the cost of unreliability and the benefits of eliminating unreliability. When one wishes to value unreliability over an arbitrarily long evaluation horizon, the fixed-life feature of the Euro- pean put option is inappropriate. First, there is no assumed fixed life, so a perpetual option formulation is required. In addition, the American put option, allowing exercise of the option any time during the (perpetual) life of the option, is the only exercise feature that makes sense in the context of a perpetual valuation horizon. The value of an American put option with perpetual life can be calculated from Equation B.2, which has been adapted from McDonald (2002). The option in Equation B.2 • The guaranteed or “insured” speed is set to the average his- torical speed. • The option length is set to the time (in years) that it takes to traverse the segment at the lowest speed observed in the historical data. Figure B.2 illustrates the lognormal distribution of speed for a 5-minute interval in the a.m. peak period with data from the Puget Sound Region for a 1-year period. Table B.2 illustrates the value of the implied put option at various average annual speeds and log variability. This exhibit displays the certainty-equivalent value both in mph and in minutes of delay. The data suggest that speed variability imposes a burden on highway users. A rational and risk-neutral user would be willing to sacrifice speed to avoid traveling considerably below the average speed. This is especially the case when the log standard deviation in speeds is large relative to the aver- age speed. Although intended only to be illustrative, the data in Table B.2 reveal that the volatility of speeds relative to aver- age speeds is not monotonically related to average speed in the real world. The relative volatility and unreliability (and, hence, the value of speed guarantees in mph) is the largest in the speed range of 45 mph. However, as Table B.2 illustrates, the value of the speed-guarantee option (when expressed in minutes) declines monotonically with speed. Thus, a policy that improves average speeds will decrease total delay, inclu- sive of the certainty-equivalent volatility burden. Policies that reduce the volatility of speed directly (without necessarily improving average speed) can also be shown to decrease total delay inclusive of unreliability. As previously stated, the data in Table B.2 are illustrative. They are provided to show the option values associated with varying speeds and log variability on a facility. In this table, the Figure B.2. Illustration of a lognormal distribution of speed.

115 and where P(I, ∞) = the value of the perpetual American put option in mph, as a function of the speed guarantee V = the (unknown) speed experienced traversing the link, in mph I = the guaranteed speed, in mph r = the annualized, risk-free continuously com- pounded interest rate s = variability of V; the square root of the log-value variation process of V The perpetual American put option is useful in valuing a policy intended to control the speed variability of the morning com- muting period over a long period of time. The parameters of can be valued in a case in which the lognormal speed vari- ability can be used to parameterize the option. The valua- tion would yield the certainty-equivalent value of various speed guarantees, I, associated with various average speed measures, V. P I I m m m V I m , • •∞( ) = − −   1 1 where ( )= − σ − σ − + σm r r r12 12 2 (B.2)2 2 2 2 Table B.2. Illustrative Option Calculation for an Urban Freeway Time of Day Average Annual Speed (mph) Log Average Annual Speed SD Log Speed Option Value Miles per Hour Minutes per Mile 14:00 57.11 4.02 0.25 5.64 0.115 14:05 56.56 4.01 0.29 6.49 0.137 14:10 55.16 3.95 0.42 9.25 0.219 14:15 53.78 3.92 0.43 9.14 0.228 14:20 52.52 3.89 0.45 9.34 0.247 14:25 51.08 3.84 0.50 10.09 0.289 14:30 50.81 3.84 0.48 9.69 0.278 14:35 50.01 3.81 0.51 10.16 0.306 14:40 48.02 3.76 0.53 10.10 0.333 14:45 46.31 3.71 0.57 10.37 0.374 14:50 45.03 3.67 0.59 10.47 0.404 14:55 43.64 3.63 0.60 10.24 0.422 15:00 41.52 3.57 0.61 9.92 0.454 15:05 41.11 3.55 0.63 10.19 0.481 15:10 39.47 3.49 0.66 10.21 0.530 15:15 37.48 3.42 0.68 10.00 0.582 15:20 35.79 3.38 0.67 9.39 0.596 15:25 34.15 3.32 0.67 8.97 0.626 15:30 33.69 3.31 0.67 8.80 0.630 15:35 33.93 3.32 0.66 8.75 0.615 15:40 32.68 3.28 0.67 8.54 0.650 15:45 29.00 3.15 0.67 7.59 0.734 15:50 26.30 3.05 0.64 6.64 0.771 15:55 24.67 3.00 0.63 6.06 0.791 16:00 23.16 2.94 0.61 5.53 0.812 16:05 23.10 2.94 0.60 5.42 0.796 Source: Counter data at I-405/I-90, southbound, 2007. Note: The calculations apply a guarantee equal to the average speed for each time of day. SD = standard deviation.

116 the lognormal speed distribution must then be estimated in a manner consistent with this long-lived option (i.e., using long histories of the morning commuting speeds). Because the reli- ability measure applies to a long time interval, the certainty- equivalent value of unreliability is higher than in the finite European put option. Parameterizing the Options Model The examples provided above illustrate how options theory can be used in a setting in which the unreliability problem is caused by recurring events and the speed performance metric is distributed lognormally. The same method can be applied to circumstances other than the volatility of speed measured in 5-minute intervals. The parameters of the options formulation should be mea- sured to be consistent with the network performance along the facility under study. For example, performance measures can be derived that are specific to a longer period, such as an entire weekday, the a.m. peak hour, weekend travel, and so forth. In all cases, the log mean and log standard deviations need to be estimated from available data or extrapolated from other studies. In the case of an a.m. peak-hour study, travel times or speed data could be assembled for that time of day Average Annual Speed Th e O pt io n Va lu e of A vo id in g Sp ee d Va ria bi lity , M ea su re d as th e Av e ra ge S pe ed R ed uc tio n W illi ng to B e Pa id (in M ile s p er Ho ur) Figure B.3. Illustrative option values (in mph) versus average speed. Average Annual Speed Th e Op tio n V al ue o f A vo idi ng S pe ed V ar ia bil ity , M ea su re d as th e In cr ea se d Av er ag e T ra ve l T im e W illi ng to B e Pa id (in M inu tes pe r M ile ) Figure B.4. Illustrative option values (in minutes) versus average speed.

117 its surrounding economic conditions. Hence, for short-life options, short-term interest rates (expressed on a per annum basis) should be used to represent these opportunity costs. The approach developed for recurring events can be gen- eralized, such that the value of unreliability is calculated on the basis of the speeds and travel times experienced on the facility. This approach does not necessarily rely on specific information about the source of the unreliability, as long as the source and the speeds are lognormally distributed. Thus, system performance data determine the underlying stochas- tic nature of unreliability on a facility. Table B.3 summarizes the disruption data identified in SHRP 2 Reliability Project L03 for use by agencies in reporting reliability performance measures (Cambridge Systematics, Inc. et al. 2013). Reliance on the normal or lognormal distribution is stan- dard in options theory and a requirement for using the Black– Scholes formulation. The Black–Scholes model is based on the normal (or lognormal) distribution of the underlying asset. for a year of data. The log mean and log standard deviation would then be computed from this data sample. In the case in which the average travel time for the entire weekday is of interest, daily average travel times could be constructed for each of 250 weekdays in a year. The life of the option is determined differently in each of these two cases. In the case of the a.m. peak-hour study, the life of the option would be set to 1 hour, and the time period would be examined. In the case of the study of weekday per- formance, a 24-hour life would be assumed. In all cases, these lives would be expressed in years for consistency with the standard interest rate term. The interest-rate parameter in the calculation is provided to respect the yield on risk-free alternative uses of the trav- elers’ resources. In short-life options, such as those used to represent recurring unreliability problems, the effect of dif- ferent interest-rate assumptions is not particularly material. Nevertheless, it is important to place the analysis properly in Table B.3. Summary of Disruption Data Characteristics by Type of Disruption Disruption Type Data Collected Source Availability Location Traffic incidents Accident data DOTs State database Archives Private sector Major urban or tourist areas Statistics on nature (size, severity, duration) of accident Disruption of roadway operations (number of lanes closed) Response after accident Data on traffic incidents (stalled or disabled vehicles, debris) Traffic.com TMC operators and freeway service patrols Traditional crash data Specific incident-response programs Weather Basic weather data National Climatic Data Center of the National Oceanic and Atmospheric Administration Areawide Work zones and construction Planned construction activity (number of lanes closed, period of time) DOTs State database Urban and rural Fluctuations in demand and special events For large events: time and date, nature of changes in traffic demand For other events: no data collected Public- or private-sector traffic management plans Urban areas Signal timing Traffic control plans (signal cycles, phase length, order, signal offsets, and base ramp metering) Operating traffic agencies Usually available Specific timing plans implemented Operating traffic agencies Rarely available Phase length actually operated Operating traffic agencies Almost never implemented Bottlenecks and inadequate base capacity Change in geometry (lane drops) All roadway agencies Good data Traffic patterns (weave and merge sections) Visual disruptions (sightseeing, rubbernecking) No record Minor changes in functional capacity Potentially available Note: DOTs = departments of transportation; TMC = transportation management center.

118 help to characterize the stochastic nature of future system performance. Valuing Reliability for Rare Events Although a sufficiently long time series of high-resolution traffic performance data (e.g., highway speeds) tends to dis- play lognormal distribution, one can imagine many situa- tions in which a transportation agency is concerned about events or sources of variability that are extremely rare. Some examples include physical phenomena such as earthquakes, avalanches, and particularly severe or rare flooding. Examples could also include bridge failures from various causes and terrorist acts that disrupt transportation links or networks. Ignoring the prospect of rare events and using pure Gauss- ian assumptions instead is at the heart of many financial and engineering catastrophes, which include long-term capital and elements of the financial crisis that it precipitated. One of the reasons to distinguish between recurring and rare events in this discussion and the development of the options theo- retic approaches is to draw attention to the rare-event issue. Unfortunately, implementation of strategies to protect against rare events in a cost-effective way is very difficult because of the problem of characterizing the event distribution and the complexity of mathematically representing the proper invest- ment strategy. This is especially challenging in the setting of highway infrastructure development and operation. The options theoretic approach to the valuation of travel time reliability is extended to rare events by using new options formulations aimed at addressing rare events, specifically events that follow extreme value distributions. Research on options theory using extreme value distributions is relatively new, and has not been applied, to our knowledge, to settings other than an example application for research and devel- opment in the pharmaceutical industry. Therefore, the rare- event methodology is presented in Appendix C, and future research is recommended to address uncertainty arising from rare events in investment decision making. Measuring networkwide impacts Policies and strategies that are intended to improve road- way reliability may affect only certain segments or an entire regional network. Similarly, the adverse impacts of phenom- ena such as flooding, bridge failures, or accidents may occur on just a few segments or over large portion of a regional net- work. Thus, the final step in mitigating system unreliability is to consider methods for addressing the scale of the impact. The certainty-equivalent options perspective can be applied to an individual segment or to an entire network as long as the appropriate data exist to provide the necessary parameters. At times, the analysts may be asked to extrapolate the effects of In the real option for travel time reliability, the assumed asset is speed, and so the assumption of the lognormality of speed (e.g., to compute the certainty-equivalent measure in min- utes of travel time) is an assumption for the options theoretic approach. The lognormal has been shown to be the most-appropriate distribution for high-frequency speed data collected from roadways. Traffic engineering research has confirmed the validity of the use of the lognormal distribution for travel time and speed. The lower bound of zero and longer right tail of the distribution make the lognormal particularly appro- priate for the typically skewed speed data. SHRP 2 L03 cites Rakha et al. (2006), which confirms the use of the log normal assumption for speeds and travel times in the context of travel time reliability. Other recent papers include El Faouzi and Maurin (2007), Emam and Al-Deek (2006), Leurent et al. (2004), and Kaparias et al. (2008). Strategies for improving travel time reliability may focus on a specific cause or address multiple sources of unreliability. Determining the various sources of unreliability by using dis- ruption data may then aid in determining the potential strate- gies to be implemented. Monetizing the Certainty-Equivalent Values Because the certainty-equivalent values of an option can be expressed in minutes or hours of delay, unreliability can be monetized by applying the appropriate values of time to these delay measures. Measuring unreliability in future periods permits the analyst to determine the present dis- counted value of unreliability and to evaluate long-term net- work improvements or management policies that are directed at addressing unreliability. Similar to other parameters of the options approach, certainty-equivalent values should be selected to measure what is important to the reliability issue being addressed. For example, if a study is focused on improving the reliabil- ity of freight movement, the analyst may only be interested in a strategy’s effect on the certainty-equivalent delay experi- enced by trucks. In this case, only the value of time for truck travel might be used to monetize the value of unreliability. In general, most strategies that are used to reduce unreliabil- ity will benefit both freight and passenger travel. In general, the options value of unreliability should be monetized by using a traffic-weighted average value of time by vehicle class and by trip purpose. To evaluate long-term strategies and treatments to improve system reliability, it is necessary to understand how traffic growth affects both average speeds and the volatility of those speeds. Average speed relationships that are derived from historic data may help determine the future levels of unreliability. In addition, microsimulation models may also

119 has incorporated link-level augmentations to allow mea- surement of unreliability effects. Although it is applied only to freeway links (with a representative stochastic rendering for all links), it is a convenient way to automatically consider the benefits of improving reliability when evaluating various strategies and policies (not just policies designed to address unreliability issues). This regional-model approach also has the potential to help extrapolate events that occur on only a few links in the network to determine the impact on the network as a whole. Xie and Levinson (2008) used this general method to evaluate the effect of the failure of the I-35W bridge in the Minneapo- lis, Minnesota, region. In this case, total delay was addressed, but reliability was not. The accuracy of this method is limited only by the capa- bilities of the model. Obviously, a model that simulates the dynamic behavior of traffic (and the volatility of travel speeds throughout the network) provides a better way to represent unreliability with and without the application of various strategies or policies. The evaluation exercise then proceeds to capture the performance data (link by link or in the aggre- gate across the network) and apply the valuation methods described above to the reliability metrics. Mathematical Representation of Network Reliability A third approach is to employ mathematical models of net- work reliability to simulate the impact of a strategy or policy on network performance. These models differ from regional models used by metropolitan planning organizations (MPOs) and other agencies to evaluate network improvements because they are pure mathematical constructs rather than simulation models per se. Mathematical models can be considered to be sketch models as opposed to regional-model implementations. In this regard, these mathematical models may not be fully faithful to real-world networks, but they offer a way to evaluate generic policies in abstract network representations. Examples of these models include those by Clark and Watling (2005), Kaparias et al. (2008), Iida (1999), and Bell and Iida (2003). Unlike many regional models implemented for project evalu- ation, these mathematical models may capture system reliabil- ity issues in a way that can guide policy. Depending upon the context, these models can be used to evaluate either reliability issues associated with recurring events or with rare events. The impacts on reliability can be measured on a few links or across the network. None of these three methods directly measures impacts beyond those that occur on the network itself. Broader eco- nomic impacts must be represented, if appropriate, by other tools that link transportation infrastructure to regional eco- nomic viability. unreliability measured on one link to the entire network. For example, a bridge failure or avalanche may be confined to a single segment or group of segments, but deterioration of reli- ability in that area may propagate elsewhere in the network. There are three alternative approaches to addressing the aggregation of reliability considerations on the network level: (a) direct measurement at the network level, (b) region- specific travel models, and (c) mathematical representation. Each of these approaches has advantages and disadvantages. Direct Measurement of Unreliability at the Network Level This method involves measuring unreliability separately for each roadway segment in the roadway network. If each segment can be measured, the network-level effects can be aggregated from the individual link effects. Alternatively, unreliability could be measured at the network level. For example, one might measure speed volatility using network- wide vehicle miles traveled (VMT) and vehicle hours trav- eled (VHT) data. The mean and standard deviation of the log mean and standard deviation of the ratio of regional VHT and VMT could be computed by vehicle class and by trip type from daily observations over some period of time. The certainty-equivalent value of unreliability then can be calcu- lated directly with the European put option approach described earlier in this appendix. The certainty-equivalent delay associated with each category of travel could then be monetized by applying the appropri- ate value of time. Strategies or policies that mitigate unreliable travel could then be evaluated with conventional benefit-cost techniques, so long as the strategy or policy can be character- ized by a change in the reliability performance measures. Measurement with Region-Specific Travel Models Simulation of the effect of strategies or policies that are intended to improve reliability at the network level can be measured with modeling techniques for the regional net- work that are sensitive to the impact of these strategies (such as dynamic tolling) that affect system reliability. In this approach, a model of the regional network is used to exam- ine the effect of any changes in reliability that occur on the affected segments or on the network as a whole. Certainty- equivalent option models can be used in conjunction with microsimulation, dynamic traffic assignment, or other model platforms that measure how unreliability is affected by a pricing-, or operational-, or capacity-improvement policy. This approach has the potential to be both comprehensive and respectful of regional network idiosyncrasies. The Puget Sound Regional Council’s four-step travel demand model

120 RP data from HOT-lane facilities have also been used to improve SP survey instruments and vice versa. As a result, researchers can create scenarios reflecting realistic trip alter- natives, thereby reducing measurement errors from respon- dent perceptions of scenarios that are familiar to them. Passenger-Travel Value of Time Since the initial conduct of studies for value of time, the relationship between the value of time and user income has been well established. Waters (1992) conducted a literature review, summarizing the research from the 1960s, 1970s, and 1980s with regard to the value of time as a percentage of the wage rate. Most of the studies reported the value of time to be between 40% and 50% of the wage rate for commuting trips and a much higher percentage (around 85% of the wage rate) for intercity trips. Fewer value-of-time estimates were specific to leisure trips. These estimates were highly variable—ranging from 35% to more than 200% of the wage rate. Consistent with the find- ings published by Waters (1992), Small (1992) also observed that the value of time with respect to user income varied from 20% to 100% of the wage rate across various industrialized cities. Small found that a good estimate for the value of time is roughly half of the road user’s hourly wage rate. Miller (1989) also reviewed the value-of-time literature and suggested that the driver’s value of time is 60% of the wage rate and the passenger’s value of time is 40% of the wage rate. Another finding reported by Miller is that the value of time is approximately 30 percentage points higher in congested condi- tions as compared with free-flow conditions. From this find- ing, Miller suggested that the driver value of time in congested conditions is approximately 90% of the wage rate and that the passenger value of time in congested conditions is 60% of the wage rate. Although it is unpleasant to drive in congested condi- tions, travelers may be choosing to travel during peak peri- ods because they place a higher value on their travel time or on the schedule delay associated with departing at a less- congested time, or both. If this were not the case, travelers could adjust their departure time so as to travel during the shoulder times adjacent to the peak when congestion levels are not as severe. In observing travelers on the New Jersey Turnpike, Ozbay and Yanmaz-Tuzel (2008) found that the value of time is higher during the peak period than during the prepeak and postpeak periods for both commuting and for leisure trips. The peak period value of time was found to be $19.72 for commut- ing trips and $17.16 for leisure trips. This value of time was approximately $2.00 higher (about 10% higher) in the peak than in the prepeak and postpeak periods for both commuting trips and leisure trips, with the exception of prepeak leisure Quantifying the Value of Time The dollar value of reliability is determined by multiplying the certainty-equivalent penalty (measured in minutes per mile) by the value of time. The value of reliability can be derived for multiple roadway user groups by applying a separate value of time that corresponds to each user group, along with the observed average volume for each user group on the road- way segment. The total value of reliability is then computed as the sum of the reliability values for each user group on the roadway segment. A literature review on the value of time was conducted to determine the value of time for passenger travelers and for freight movers. Literature related to the value of time has been a productive topic for research. The more-useful mea- sures of time value have been developed from mode-choice or path-choice (route-choice) studies based on household and shipper surveys or from studies of traveler behavior on facilities such as high-occupancy toll (HOT) lanes. Although the value of travel time may be referred to as an average value, the value of travel time is recognized to take on a range of values that depend on user income and other demographic characteristics (in the case of passen- ger travel) or on operations and shipment characteristics (in the case of freight travel). The value of time may also depend on factors such as trip purpose, time of day, or travel conditions. Value of Time Methodology and Data Value of time modeling and estimation dates back to at least the 1960s. At that time, early empirical research focused on traveler mode choice to determine the value of time by look- ing at the marginal rate of substitution between travel time and cost across mode alternatives. In addition to mode choice, route choice and, to a lesser extent, housing choice, models have also been developed to help practitioners understand traveler behavior and decisions with respect to travel time, travel cost, and residence and work location (Small 1992). When revealed-preference (RP) data have been difficult to obtain, the alternative for researchers has been to conduct stated-preference (SP) surveys. SP surveys ask respondents to make travel decisions for hypothetical scenarios in which the alternatives in each scenario have different travel times, travel costs, or other trip attributes. Brownstone and Small (2005) note the difficulty in creating realistic scenarios and accu- rately presenting the scenarios such that the key variables of interest are properly understood and measured. When com- paring results from studies based on RP data and SP surveys for HOT-lane facilities in Southern California, Brownstone and Small (2005) observed that the SP surveys tend to yield much lower results than do the RP data.

121 et al. (2007) also used data from the SR-91 HOT lanes in a mixed-logic model to detect time-dependent heterogeneity in the value of time. Estimates of the value of time during the morning commuting period started out relatively high at $16.50 at 5:00 a.m., built throughout the morning, and dropped sharply by 9:30 a.m. Table B.4 lists some of the value-of-time estimates from recent SP surveys and HOT-lane facility RP studies for pas- senger travelers. trips, which exhibited nearly the same value of time as peak leisure trips. The morning peak period has been found to be the time of day with the highest observed value of time, consistent with the findings that travelers with inflexible schedules (such as work schedules) have a higher value for travel time reliability. Ghosh (2001) found higher values of time of $22.00/hour (75% of the wage rate) for the morning commute period along the I-15 HOT-lane facility in San Diego, California. Liu Table B.4. Passenger Value of Time from SP and RP Studies Travel Type or Model Type Value of Time per Hour (% wage rate) Data or Study Year Study Type, Location Reference Prepeak commute $16.72 2005 RP, N.J. Turnpike Ozbay and Yanmaz-Tuzel (2008) Peak commute $19.72 2005 RP, N.J. Turnpike Ozbay and Yanmaz-Tuzel (2008) Postpeak commute $17.35 2005 RP, N.J. Turnpike Ozbay and Yanmaz-Tuzel (2008) Prepeak leisure $17.03 2005 RP, N.J. Turnpike Ozbay and Yanmaz-Tuzel (2008) Peak leisure $17.16 2005 RP, N.J. Turnpike Ozbay and Yanmaz-Tuzel (2008) Postpeak leisure $15.33 2005 RP, N.J. Turnpike Ozbay and Yanmaz-Tuzel (2008) MnPASS subscribers that were early or on time, p.m. peak $10.62 2007 SP, I-394 MnPASS Tilahun and Levinson (2009) MnPASS subscribers that were late $25.42 2007 SP, I-394 MnPASS Tilahun and Levinson (2009) MnPASS non-subscribers who were early or on time $13.63 2007 SP, I-394 MnPASS Tilahun and Levinson (2009) MnPASS nonsubscribers who were late, a.m. peak $10.10 2007 SP, I-394 MnPASS Tilahun and Levinson (2009) Median value of time per hour $21.46 (93%) 2000 RP, SR-91 and I-15 Small et al. (2006) Median value of time per hour $11.92 (52%) 2000 SP, SR-91 and I-15 Small et al. (2006) Route-choice model $19.22 1998 RP and SP, SR-91 Lam and Small (2001) Route- and time-of-day choice models $4.74 1998 RP and SP, SR-91 Lam and Small (2001) Route- and mode-choice models $24.52 1998 RP and SP, SR-91 Lam and Small (2001) Transponder, route-choice model $18.40 1998 RP and SP, SR-91 Lam and Small (2001) Transponder, mode- and route-choice models $22.87 (72%) 1998 RP and SP, SR-91 Lam and Small (2001) Passenger travel—$125,000–$175,000 annual income $7.11 1998 SP Calfee and Winston (1998) Passenger travel—$7,500–$12,500 annual income $3.06 1998 SP Calfee and Winston (1998) Major U.S. metro areas, median $3.88 (19%) 1998 SP Calfee and Winston (1998) Peak period (a.m.) $30 1998 RP, I-15 Brownstone et al. (2003) Passenger vehicles $8–$16 1999–2000 RP, SR-91 Sullivan (2000) Peak period (commuter) $13–$16 1999–2000 RP, SR-91 Yan et al. (2002) Peak period (a.m.) $22 (75%) 1998 RP, I-15 Ghosh (2001) Median $15 (52%) 1998 RP, I-15 Ghosh (2001) Afternoon commute $9–$22 (30%–75%) 1998 RP, I-15 Ghosh (2001) Note: MnPASS = Minnesota PASS express lanes.

122 the value of time.) The cost-based approach produces more- conservative estimates because the value of time includes only the costs of driver time and inventory (also sometimes the time-based vehicle depreciation cost) and does not take into account the value of time from the shipper’s perspective. Cost-based estimates should, however, include the inventory cost, which is calculated by using an hourly discount rate, and the value of the shipment. The inventory costs are generally a very small portion of the value of time and do not reflect damage or the perishability of the shipment. The revenue approach calculates the value of time as the net increase in profit from the reduction in travel time by making assumptions about the level of utilization for the time sav- ings. Although the revenue method seems to be rarely used, compared to the cost-based and stated preference methods, two studies—Hanning and McFarland (1963) and Waters et al. (1995)—report ranges for the value of time by using the revenue approach. Hanning and McFarland (1963) estimated the value of truck time as $17.40 to $22.60 (1998 dollars), while Waters et al. (1995) find a wide range for the value of truck time, from $6.10 to $34.60 per hour. Kawamura (2000) notes that using the revenue-based approach is potentially problematic because “actual behavioral changes under a policy or program are determined by the perceived value of time, the benefit-loss calculations based on this method will be inaccurate except in the cases in which truck operators possess a perfect knowledge of the marginal profit.” With data collected from an SP survey on congestion pric- ing, Kawamura (2000) found the mean value of truck time to be approximately $30.00 (converted to 2008 dollars). The value of time for California operators varied by carrier and operation types, but not necessarily with respect to shipment weight. Additional findings from Kawamura (2000) are shown in Table B.5. This table shows that carriers whose drivers are paid by the hour have a higher value of time than do carriers who have fixed-salary drivers. In addition, for-hire carriers have a higher value of time than do private operators. In summary, there is a strong consensus in the literature that the value of time for passenger travel depends on the driver wage rate. This is consistent with economic theory that the opportunity cost of not working (driving or enjoying lei- sure activities) is equal to the workplace value of time. It has also been found that trips during peak periods and during congested travel conditions tend to have a higher value of time. The higher value of time in peak periods also reflects the higher value of time for commute trips, in particular for morning commute trips, which tend to be less flexible. The value of time during the morning commuting period has been observed to be 75% of the wage rate. Freight-Mover Value of Time Few studies are devoted to the value of time for freight movers (trucks) beyond the relationship between the value of time and driver pay and inventory costs. Kawamura (2000) noted the shortage of research, particularly with respect to the lack of research on the variation of truck value of time with respect to carrier and shipper operating characteristics. (Shippers generate the need to move freight, and carriers accommo- date that need.) Just as passenger-vehicle value of time varies by user characteristics and trip purpose, the value of time for freight movers also seems to depend on carrier characteristics and shipment attributes. Factors such as the carrier operations (for hire or private, truckload or less than truckload), driver pay type, and shipment characteristics (such as commodity value or shipper characteristics) are recognized as potential factors influencing the freight mover value of time. Kawamura (2000) lists three methods used in studies to determine freight mover value of time. The first method is a cost-based approach (or factor cost) that divides the value of time into its constituent cost elements. The largest portion of the hourly cost is the truck driver wage and compensation, with vehicle depreciation and inventory costs representing a much smaller share of the hourly cost. (The vehicle operating costs should be accounted for separately and not included in the value of time.) The second method is the revenue-based approach in which the freight mover value of time is esti- mated as the increase in carrier revenues derived from 1 hour of travel time savings. The third method is the SP survey, which uses data from motor carrier managers and shippers to estimate the value of freight-mover time based on their choice among hypothetical travel scenarios. Freight value-of-time estimates from the cost-based approach is based on the hourly opportunity (or direct) cost of the truck driver and inventory, excluding vehicle operating costs. (Vehicle operating costs such as fuel, tires, and mainte- nance should be accounted for separately in benefit-cost and other analyses, though some studies do report vehicle-related expenses or the total marginal costs of truck operation as Table B.5. Findings from SP Survey Truck Operator Type Value of Timea All trucks $30.91 Private carrier $23.25 For-hire carrier $36.98 Truckload operations $33.02 Less-than-truckload operations $29.85 Hourly paid drivers $33.55 Other pay type $19.95 Source: Kawamura 2000. a Converted to 2008 dollars.

123 Guidance and Recommended Rates This section describes the U.S. DOT Departmental Guidance on the Valuation of Travel Time in Economic Analysis (1997b). The U.S. DOT guidance is the basis for the FHWA Highway Economic Requirements System (HERS) value-of-time esti- mates. This guidance is also used in the FHWA Highway Freight Logistics Reorganization Benefits Estimation Tool (2008). The empirical basis for the U.S. DOT Guidance on the Valuation of Travel Time in Economic Analysis (1997b) is the clustering of estimates for passenger travelers at around 50% of the hourly wage rate. The U.S. DOT guidance pro- vides the recommended values for the value of time as a percentage of the hourly wage, differentiating values for local passenger personal travel, local passenger business travel, intercity passenger personal travel, intercity passen- ger business travel, and truck-driver travel. Acknowledg- ing the variation in the value of time estimates found in the literature, the U.S. DOT provides plausible ranges for con- ducting sensitivity analyses. The values of travel time recom- mended by U.S. DOT are shown in Table B.6. The value of travel time as a percentage of the wage rate is converted to a dollar value by using wage data from the Bureau of Labor Statistics for business travel and truck-driver wage rates. The median household income from the U.S. Census Bureau is used for personal travel. The U.S. DOT guidance for truck travel time is similar to the cost-based approach, with 100% of the full driver compensation recommended for the truck- driver value of time. HERS uses the value of time for seven vehicle classes: two passenger-vehicle classes and five truck configurations. The value of time is computed by using the U.S. DOT guidance for driver and occupant time plus estimates on the vehicle depre- ciation cost per hour and the freight inventory cost per hour. Variation in the truck value of time for the different vehicle classes is primarily due to the average vehicle occupancy assumption applied to the driver compensation for each vehi- cle class. These values are shown in Table B.7. Smalkoski and Levinson (2005) implemented an adaptive SP survey for carriers and shippers in Minnesota and analyzed their willingness to pay for operations permits during the spring load-restriction period. The mean value of time was $49.42, with a 95% confidence interval of $40.45 to $58.39. The authors were unable to produce value-of-time estimates for different carrier groups, but like Kawamura, they observed that for-hire firms seem to have a “considerably” higher value of time than do other groups. The FHWA Freight Management and Operations publica- tions have reported the value of time for freight as a range between $25 and $200 per hour. The high end of the FHWA range is based on the Small et al. (1999) truck value-of-time estimated range of $144 to $192 per hour. Given the likely idiosyncrasy of the estimates from the SP data collected by Small et al. (1999), the upper end on the range for freight mover value of time may be closer to between $75 and $100 (with the overall range of truck value of time between $25 to $100), and likely not as high as $200, on average. The Southern California Association of Governments adopted a value of $73/hour for use in freight studies based on the FHWA publications. Similarly, after reviewing the lit- erature on the truck value of time, the Puget Sound Regional Council, together with the freight working group at the Wash- ington State Department of Transportation (DOT), adopted truck value-of-time estimates of $40/hour for light trucks, $45/hour for medium trucks, and $50/hour for heavy trucks (Outwater and Kitchen 2008). Value-of-time estimates derived with a cost-based approach have consistently placed the value of time for freight at around $25 to $30/hour, which is primarily based on driver compen- sation and benefits, with a small fraction of the hourly cost from vehicle depreciation and inventory costs. If one were to include additional shipper value of time costs (in addition to the direct driver time and time-related operating costs), the freight-mover value of time may be closer to $40 or $50 per hour and up to $75 per hour for some freight corridors. Table B.6. U.S. DOT Recommended Values of Time in 2008 Dollars Type of Travel/Traveler Value of Time (per person-hour as a percentage of wage rate) National Hourly Wage (or Hourly Median Household Income) Value of Time (dollars per person-hour) Passenger travel Personal (local) 50% (35%–60%) $25.12 $12.55 ($8.79–$15.07) Personal (intercity) 70% (60%–90%) $25.12 $17.60 ($15.07–$22.61) Business 100% (80%–120%) $29.20 $29.20 ($23.28–$34.92) Freight movers Truck drivers (local and intercity) 100% $23.30 $23.30

124 The U.S. DOT guidance on the value of time is given as a percentage of the wage rate per person-hour. The dollar value per person-hour can be converted to value per vehi- cle hour by multiplying the value per person-hour by the average vehicle occupancy (AVO) for each user group. Data from the 1995 National Household Travel Survey is used for the passenger travel AVO assumption used in the HERS value-of-time estimates. The AVO for the four-axle and five- or-more-axle combination trucks is based on a study of team drivers. The six-tire truck and three- to four-axle truck AVOs are assumed values. Although the value-of-time estimates found in the liter- ature take on a range of values, there is a clear relationship between the passenger value of time and wage rate. Several studies also find that the value of time during peak periods is higher than during off-peak periods. The trip purpose, particularly during the morning commuting period, influ- ences the value of time. The recommended values shown in Table B.8 are based on the strongest relationships empiri- cally demonstrated in the literature. These values largely fol- low the current U.S. DOT guidance. Many regional agencies or state DOTs may have their own values of time that reflect the local wage rates. With the same hourly rates as in Table B.8 for calculat- ing the hourly value of time from the U.S. DOT guidance, the dollar value of time per person-hour is presented in Table B.9. The dollar value of time per person-hour can be converted to HERS computes the inventory costs for four- and five-axle combination trucks by applying an hourly discount rate to the average payload value. The payload value is estimated by the average payload weight multiplied by the average shipment value—expressed in dollars per pound for truck shipments. Although the payload for a four-axle combination truck would be lower than the payload for a five-axle combination truck, the value of the commodity shipped may be higher. Thus, the inventory costs for these two truck configuration classes are set to the same value. The inventory costs for lighter trucks (three- to four-axle and six-tire trucks) are assumed to be negligible, and personal vehicles are assumed to not be trans- porting goods. The HERS inventory costs do not include spoilage costs, damage, or inventory depreciation—only the cost of holding the inventory during transport (HERS 2005, p. 5.5.5). Vehicle depreciation costs are the average dollar value that a vehicle’s value declines for each hour of use, adjusted so that mileage-based usage is accounted for sepa- rately in the vehicle operating costs. The HERS value-of-time method for trucks has been adopted for use in the FHWA Highway Freight Logistics Reorganiza- tion Benefits Estimation Tool, with modifications made to the inventory-cost method. In the freight logistics tool, the analyst can specify up to five commodities and their value and relative share of freight moving through the corridor. Total inventory cost is then the weighted average reflecting the mix of com- modities shipped on the particular roadway. Table B.7. Value of Time from HERS Criteria Automobile Truck Small Medium 4-Tire 6-Tire 3- to 4-Axle 4-Axle Combination 5-Axle Combination Business travel Value per person $29.20 $29.20 $29.20 $23.30 $23.30 $23.30 $23.30 Average vehicle occupancy 1.43 1.43 1.43 1.05 1.00 1.12 1.12 Vehicle depreciation $1.40 $1.87 $2.45 $3.41 $9.22 $8.25 $7.93 Inventory $0.79 $0.79 Business travel value per vehicle $43.16 $43.62 $44.20 $27.88 $32.52 $34.35 $34.02 Personal travel Value per person $12.55 $12.55 $12.55 Average occupancy 1.43 1.43 1.43 Percentage personal 89% 89% 75% Personal-travel value per vehicle hour $17.95 $17.95 $17.95 Average value per vehicle hour $20.72 $20.77 $24.51 $27.88 $32.52 $35.14 $34.82 Source: HERS-ST 2005 Technical Report. Note: Costs were converted to 2008 dollars using U.S. DOT guidance for value per person and HERS cost indexes for other cost items.

125 state DOTs, regional MPOs, or the 2008 National Household Travel Survey are other sources for determining the volume by user group or by trip purpose. Applying the Methodology With the options theoretic approach, the costs of unreliability or the benefits of potential reliability improvement strategies can be assessed. By converting uncertain performance metrics to certainty-equivalent values, valuation of reliability improve- ments can be achieved by simply applying appropriate values of time to the certainty-equivalent delay measure. This can be done for multiple user groups and trip purposes to the extent that the composition of the traffic stream can be determined. To implement this approach, some judgments must be made: 1. The statistical nature of unreliability must be known or assumed. It must be determined whether the unreliabil- ity occurs because of events that cause speeds to be dis- tributed lognormally, or by extreme value type, or other distributions. In some cases, it is easier to represent the distribution of the events rather than the highway perfor- mance metric (such as speeds or travel times). This is the case in situations in which highway performance metrics are not collected comprehensively over a long period of time, but in which data are available on the events that cause unreliable travel conditions. Formal statistical tests for the goodness of fit, such as the Kolmogorov–Smirnov test, can be used to verify the distribution of the data, in addition to the use of descriptive statistics and visual inspection (e.g., histograms). 2. A time horizon for the evaluation effort must be defined that is appropriate to the reliability measurement pro- cess. If a finite horizon to the evaluation is appropri- ate, then an options formulation can be selected that has a finite life. If a long time horizon characterizes the process that generates unreliability, a perpetual options value per vehicle by using average vehicle occupancy assump- tions to scale the dollar value per person-hour. The freight-mover value of time, shown in Table B.10, should be based on 100% of the truck-driver wage and ben- efits, plus the hourly inventory costs. Hourly inventory costs can be calculated as the value of commodities shipped in a corridor divided by the hourly discount rate. Once the value of time for each user group is developed, the vehicular volume for each user group may be identified so that the value of an increase in reliability can be deter- mined for each group. Volume counts by vehicle class (or truck share of the total volume) are commonly reported and tabulated from traffic recorder data. Truck share of the total volume is often captured from loop-detector data, particularly in freight corridors. Truck volume data can also be derived from Highway Performance Monitoring System data for sample segments. The 2003 U.S. DOT guidance suggests creating a weighted average for automobile travel by assuming that 94.4% of vehicle miles are personal trips and 5.6% of vehicle miles are business related. Household travel surveys conducted by Table B.8. Passenger Traveler Value of Time as Percentage of Average Wage Rate Trip Purpose, Time of Day Value of Time (as percentage of wage rate) Range of Values for Sensitivity Analysis Peak period (morning commute) 75% 50%–90% Peak period (p.m.) 60% 40%–70% Leisure, other trip purpose 50% 30%–60% Intercity travel 70% 60%–90% Business travel 100% 80%–120% Table B.9. Passenger–Traveler Value of Time Trip Purpose, Time of Day Value of Time (dollars per person-hour) Range of Values for Sensitivity Analysis Peak period (morning commute) $18.80 $12.60–$22.60 Peak period (p.m.) $15.00 $10.00–$17.60 Leisure, other trip purpose $12.60 $7.50–$15.10 Intercity travel $17.60 $15.10–$22.60 Business travel $29.20 $23.40–$35.00 Note: Business travel hourly rate is based on wage and compensation; other trip purposes are based on national median household income following U.S. DOT guidance. Table B.10. Freight-Mover Value of Time Freight-Mover Group Value of Time (dollars per hour) Range of Values for Sensitivity Analysis Light truck (Single unit or 6-tire truck) $30 $30–$55 Medium truck (3- to 4-axle truck) $50 $40–$75 Heavy truck (4-axle and 5-or-more-axle combination) $60 $40–$80

126 Comparison of Methodologies for Valuation of Travel Time Reliability The options theoretic approach is a new approach for the valuation of travel time reliability. The typical approaches to the valuation of travel time reliability are based on discrete choice models that use SP or RP data. Given the limited num- ber of natural experiments that allow for the collection of RP data, the majority of this work has tended to focus on SP survey data. Comparing the options theoretic method to RP and SP approaches is useful, given that most of the research in this area has been conducted using the RP-SP empirical approaches. This section presents the following: • A discussion of some conceptual and practical consider- ations of relying on the RP-SP approach. Some of these considerations call into question the practical utility of the RP-SP approach. • A numerical comparison of the value of reliability of the options approach and that estimated by the RP-SP approach. For the latter, the estimates produced by Small et al. (2006) are used. With their data, the necessary parameters to implement the options theoretic approach are computed with the same speed distribution and values of time in their paper. Conceptual Considerations in Relying on RP or SP Methods Numerous aspects of the RP-SP approach make reliance on this method problematic in practice. Some of the contrasts between the options approach and RP and SP approaches are the following: • The RP and SP approaches are not economical to apply, requiring costly studies for application. Putting aside any general skepticism about the reliability of SP procedures in particular, even this method is relatively costly to imple- ment and subject to the same statistical issues and biases that creep into interview-based contingent valuation, con- joint, and similar analyses. • The RP and SP approaches are not agnostic procedures free of functional form assumptions akin to those of the options approach. In particular, these approaches implic- itly adopt utility function specifications that are usually asserted, rather than demonstrated. The function form is usually linear in its arguments (or some nonlinear speci- fication to introduce risk aversion). This is analogous, in a mathematical sense, to the Black–Scholes approach, which employs assumptions about risk, too. In its simplest expres- sion, Black–Scholes assumes risk neutrality, but has been shown to be robust to the assumption of risk aversion. formulation may be more appropriate. Thus, one might evaluate recurring congestion phenomena by using short-life options. Rare events (such as bridge failures, flood events, etc.) may require a longer or even perpetual option life assumption. 3. It must be determined whether the unreliability is to be valued only at the end of an assumed time horizon or whether it is to be valued whenever (during the option’s life) the unreliability is to occur. European option for- mulations are appropriate for the former, and American options formulations are appropriate for the latter. 4. After the appropriate framework for valuation is deter- mined, data must be assembled. These data should identify the necessary probability distributions. In summary, the options formulation will be determined from the frequency of the phenomena addressed by the strategy, the data availability for the roadway and event type being evaluated, and the statistical distribution displayed by the data. For typical recurring events and for very rare events, the choice of options formulation may be obvious. In the case of rare events, speed data will often not exist, and the options formulations for recurring events will not be appropriate. Given an identified treatment or policy to improve reliabil- ity and the necessary options formulas and data, the unreli- ability processes can be converted into certainty-equivalent measures and treated as deterministic, rather than proba- bilistic. Application of the appropriate values of time then provides a method for comparing the value of unreliability with and without implementation of the treatment or policy to improve reliability. The reliability benefit from a particu- lar treatment or policy is then determined by comparing the value of unreliability without the treatment to the value of unreliability with the treatment. The change in the total value of unreliability is determined by calculating the present value of the benefit from imple- menting the treatment. To conduct a benefit-cost analysis of the treatment or policy, the benefit from the strategy is then compared to the present value of the cost of implementa- tion and the annual maintenance or operations costs of the treatment or policy. The annual outlays associated with the treatment or policy must be discounted to their present value, just as the annual benefits from improvement reli- ability are discounted to their present value. Strategies with the highest benefit-to-cost ratio will provide the greatest level of benefits when compared to the strategy implemen- tation and maintenance costs for the lifetime of the project. When comparing alternative treatments and policies, the evaluation should use the same discount rate for comput- ing the present value of the treatment benefits and costs and use similar time frames.

127 Small et al. model yielded a value of unreliability of $19.56 per hour. When applied to the 10-mile segment of SR-91, the value of unreliability was then measured to be $0.52 (for the 10-mile segment). The Small et al. (2006) paper provides enough information to enable measuring the volatility of speeds in a manner com- patible with the options theoretic approach. Specifically, the median speed on the general-purpose (free) lanes is reported to be 53 mph. The corresponding 20th percentile speed (imputed from the 80th percentile travel time, in minutes) is 46 mph. These speed data, combined with the assumption of a lognormal distribution of speeds, yield a log mean speed of 3.9703 and the log standard deviation of speed of 0.1683. With use of the log standard deviation and the assumption of a 5% discount rate (to which the calculations are very insen- sitive), the put option value can be calculated for an option whose life lasts long enough for the traveler to traverse the 10-mile facility. All that remains to be done is to hypothesize the “speed guarantee” appropriate to this setting. In this report’s char- acterization of unreliability, the average speed experience of travelers, measured over time, is taken as the speed against which they measure unreliability and so is the desired “guar- anteed” speed. Unfortunately, Small et al. (2006) did not know the average speed experience at the various times their RP sample traveled. What they knew was the median speed over the entire 4-hour a.m. peak (53 mph). However, vari- ous speed guarantees in the options formula can be tested to see what speed guarantee corresponds to Small et al.’s revealed $0.52 value of reliability over the 10-mile facility. That speed is 57 mph. Alternatively, by using the 53-mph peak-period median value as the guaranteed speed (though not the correct datum for the options approach), the value of reliability cal- culated by the options approach is $0.29. Measured either way, it is clear that the options approach and the RP-SP approach yield quite consistent values. Indeed, Small et al. (2006) also estimated an SP model, which produced a value of reliability much smaller than the estimate based on RP. Given the computational cost and flexibility of the options approach, it is clear that it is a valuable method of evaluating travel unreliability. Conclusions for the Options Theoretic Approach This appendix presents an options theoretic approach for the valuation of travel time reliability. Options theory is a well-established methodology from the field of financial economics used for the valuation of assets in the presence of uncertainty. Although options theory in finance deals with the dollar value of financial instruments, real options deal • The RP and SP analyses usually also postulate a fairly spe- cific characterization of the context of unreliability—for example, that it arises out of a particular manifestation of a scheduling–cost problem, and so forth. The options theoretic approach is no more restrictive; it simply postu- lates that there is a willingness to pay for insurance (hypo- thetically) that compensates drivers for not experiencing below-average speeds that are, in turn, drawn from a log- normally distributed delay process. • A major difference is that the options theoretic approach allows separation of the value-of-time issue from the “real” unreliability issue. Because the existing travel models carry values of time internally for other purposes (mode choice and traffic assignment), the RP and SP approaches (typi- cally confounding time–time savings valuation and traffic variability) are harder to integrate into the modeling suite. In contrast, the options approach allows unreliability to be introduced directly into traditional, volume-delay speci- fications used in travel model platforms. Given that the primary empirical input of the options approach is speed- distributional information, it imposes light additional bur- dens on the modeler. The required data on speed variations are plentiful, easily calculated from loop-detector histories, and can be made idiosyncratic to individual network links. For the same reason, the options approach is friendlier in microsimulation model settings. Numerical Comparison A numerical comparison demonstrates that the options the- oretic approach yields values of unreliability similar to the RP-derived estimates of Small et al. (2006). The additional complexity and cost of measuring and implementing measures derived from RP and SP would be worthwhile, perhaps, if the simpler and less-demanding options approach yielded vastly different measures. In fact, however, the measures are virtually indistinguishable. This is demonstrated below by comparing an options measure to that derived in Small et al. (2006). This work is a highly regarded implementation of the RP-SP approach. Small et al. (2006) used a mixed logit model with both RP and SP data from the unique setting of California’s SR-91 express lanes. This setting is unique because a driver using the congested general-purpose lanes can pay a toll and enjoy both a faster speed of travel and (an assumed) zero variabil- ity in travel speed. This allowed the authors to estimate both the value of travel time savings and the value of unreliability. Using the RP data, they measured the value of travel time at $21.46/hour (93% of the average wage rate). The study was conducted under 4-hour a.m. peak conditions, which expose general-purpose lane users to considerable volatility in speeds and travel times. Under the conditions in place, the

128 an approach for the valuation of travel time reliability for rare events, and as an approach for optimal investment deci- sion making, given the uncertainty related to low-probability, high-consequence events. Summary of Reviewer Comments and Responses Comment 1 One can create many different classification schemes within the field of options and financial derivatives. However, in view of the need to impute dollar values to improvements in travel time reliability in Project L11, the literature suggests three relevant classes: 1. Financial options: These are concerned with valuing a financial asset given the underlying price, the strike price, the time to expiration, the volatility (variability), and the risk-free interest rate. 2. Real options: These relate to the valuation of one or more contingencies that unfold over time, such as a decision to proceed with Phase 2 of a project after making a positive feasibility determination under Phase 1. 3. Valuing insurance contracts: These refer to what a buyer of insurance is willing to pay for an insurance contract based on the present value of expected loss. Each of these approaches is indicative of how one might value improvements in travel time reliability. None by itself appears to be strictly applicable, and none may be possible without using supplemental techniques to evaluate how road users trade off reductions in the variability in travel time against reductions in average travel time and out-of-pocket costs. Response 1 The comment presents a useful taxonomy for the types of options. Our travel time reliability option formulation is derived from options representations of insurance. In other words, the basic insight of the approach is that one can think of unreliability as analogous to the occurrence of an undesirable outcome in some random event context (e.g., an accident that impairs the value of a car). In an auto insurance context, one can think of the insurance policy as a mechanism for compensating the driver for any lost value due to an accident during the life of the contract. Carry- ing this notion over to travel time reliability, one can imagine that an insurance policy could be crafted that compensated the driver for the unexpected occurrence of speeds below the expected (average) speed. Such a policy does not exist for daily vehicle travel, although such policies do exist for long with stochastic variables when quantities are measured in terms of real units, such as time or actual commodities. Given the novel approach developed in this research, an extensive review of the options theoretic approach was per- formed by a number of expert reviewers. Experts on options theory and finance were consulted; they provided multiple rounds of review of the appropriateness and validity of the approach. Expert reviewer comments largely focused on the mapping of the travel time reliability option formulation to the analogous elements in the Black–Scholes formula. Use of an insurance option for travel time variability is a rather original application of options theory for the valuation of travel time reliability, as suggested by the reviewers. However the use of real options and these techniques is not new in the context of transportation. The reviewer comments and responses are summarized below. Given the unfamiliarity with such methods and questioning of the approach because of its highly mathematical nature, it is suggested that a work- shop or conference highlighting the use of various financial techniques in the transportation field will help to advance the understanding and application of these methods. The options theoretic approach for the valuation of travel time reliability is an option formulation that determines the option value for travel time reliability in terms of the travel time that roadway users would be willing to sacrifice to obtain a speed guarantee. With the certainty-equivalent of delay, practitioners can compute the value of reliability for multiple user groups, by using well-established values of time with the deterministic value associated with travel time reliability. An advantage of the options theoretic approach is that it is a robust and compact method that can be tailored to reli- ability analysis for specific roadways and the observed travel time variability on them. Unlike SP surveys, the options theo- retic approach is not based on fixed idiosyncratic data. The options theoretic approach can be generalized to travel time variability experienced on other roadways or in other regions. It uses readily available traffic data. This method converts a stochastic variable (travel time) into a certainty-equivalent measure that can be treated deterministically in the evalua- tion of a project or operational treatment. A limitation of the options theoretic approach is that the formula is inappropriate for analyses in which travel time variability cannot be characterized by a lognormal distri- bution. Determining which options formulation should be used for specific analyses (particularly for rare events) poses a potential challenge to the implementation of this methodol- ogy by public agencies, particularly if analysts are unfamiliar with travel time distributions and benefit–cost evaluation frameworks. In Appendix C, the rare-event formulation for incorporat- ing the valuation of reliability into investment decisions is pre- sented. The rare-event approach was developed to investigate

129 commonly used when considering money because this range is close to the historical interest rates people could get by putting their money in something like a bank account with guaranteed interest rate. However, it is not clear how to jus- tify the assumption that the travel speed will grow at any rate with certainty over T - t time along the target road segment. If we can properly justify applying a guaranteed interest or growth rate r to travel speed, then the value of r will need to be determined carefully because the result of the model depends heavily on the value of r. Response 4 The value of the interest rate used in the formulation should be the real, annual riskless rate of return. This rate var- ies somewhat with macroeconomic conditions but should reflect the real discount rate the market is applying to value funds received in the future versus today. This is also called the “time value of money” in finance parlance. The reviewer is correct that the interest rate in the implementation of the Black–Scholes formula should be a low, single-digit annual rate in the vast majority of macroeconomic settings. Comment 5 “Real options” is a specific branch of options evaluation. It does not concern financial assets, but rather, other tangible assets in which the value of the option depends upon one or more contingent events. Examples are raising tolls at such time when traffic volumes reach a certain level, investing in the next stage of drug development assuming prelimi- nary drug trials are successful, pursuing a line of research once a positive feasibility determination has been made, and developing the next component of a modular electronics platform once the market for additional modules has been established. In each of these examples, the value is con- ditional upon an event occurring in the future or upon a condition state being realized. The term “real options” for the most part has a specific meaning among those who are expert in this branch of financial analysis. It is possible that the semantic elasticity of the term “real” options allows it to be applied to valuing travel time reliability. However, one of the country’s leading experts on real options whom we con- sulted does not think this approach is applicable to valuing travel time reliability. Response 5 The reviewer is correct that the term “real options” is com- monly used in a capital budgeting context and that this set- ting involves real (versus financial) assets. Others, however, make the distinction between whether the option is purchased trips (e.g., an accident that impairs the value of a car). In an auto insurance context, one can think of the insurance policy as a mechanism for compensating the driver for any lost value due to an accident during the life of the con- tract. Carrying this notion over to travel time reliability, one can imagine that an insurance policy could be crafted that compensated the driver for the unexpected occurrence of speeds below the expected (average) speed. Such a policy does not exist for daily vehicle travel, although such policies do exist for long trips (e.g., overseas travel insurance). So, if one accepts that the concept of speed insurance makes sense, then the Black–Scholes formulation we are using makes sense, and one can calculate the speed-equivalent “premium” to be assured compensation for encountering speeds less than the mean (expected) speed. Comment 2 The question is whether the Black–Scholes option pricing equation can be used to value a decrease in the variability of travel times over a specified road segment. As I read your report I had two questions in mind: First, is the Black–Scholes model applicable? Second, have you applied it correctly? My answers are yes and yes. Response 2 We agree with the reviewer that the Black–Scholes model is applicable to the problem of travel time variability, and we believe that this model has been appropriately applied. Comment 3 How to justify applying any interest rate (growth rate) to travel speed (or time) in the way used in the model and how to select the value of this interest rate for travel speed (or time)? Response 3 A finite and fixed option life (insurance contract life) is nec- essary to derive a value of the insurance premium or option value. The assumption to apply an interest rate (growth rate) is arbitrary (although less so than assumptions of other approaches that assume that all that matters in measurement of reliability is the probability of tail events). Comment 4 It appears that the value of r is arbitrarily set; and there does not seem to be any justification for it. The example sets the value of r to be 5% (or 3%). A 3% to 5% interest rate is

130 security when the option matures. While the two contracts are similar in some aspects, they are priced in different ways using very different paradigms. The insurance contracts are priced by actuarial methods that are based on historical statistics and underwriting. The put option is priced using a Black–Scholes type model. Response 6 Real-world contracts (mortgages, insurance, options on trad- able securities, etc.) have features that do, indeed, complicate the valuation exercise. However, it has long been recognized that an insurance contract is fundamentally a put option. Actuarial complexities arise because of the need to mea- sure such things as the distribution of life expectancy, acci- dent rates, etc., and to consider and contain adverse selection distortions. Our modeling of the value of insurance against traveling slower than the expected speed is focused on deter- mining the underlying value of such insurance if it existed. We are not modeling how rates would be set if I were imag- ining starting a commute-trip insurance business. It is the underlying value of the risk (not the operating challenges of insurers, etc.) that is of interest. Comment 7 Reliance on normal/lognormal distribution—again this is fairly standard in option theory in order for the pricing to be tractable. The appendix would benefit from greater dem- onstration that this assumption is appropriate for the types of data likely to be used. In particular, formal tests (e.g., a Kolmogorov-Smirnov test) could be employed to evaluate the assumption and identify the sensitivity of the results to departures from this assumption. Response 7 The authors agree with and appreciate the comments of this reviewer. The only response we offer concerns the reliance on assumption of lognormality in the simpler (recurring events) setting. Although it is true that little formal testing of the log- normality of speed data goes on, it is a widely appreciated feature in practice. Vehicle counting systems, such as PeMS, TRAK, and others, are installed in thousands of locations on U.S. highways. These systems produce large quantities of high-resolution traffic volume and speed data, and the log- normality of the distribution of speed is accepted as com- monplace. This does not mean that testing for compliance with this assumption should not be done, of course. Analysts can be instructed to use statistical testing methods to confirm the distribution of their data. or arises naturally in the course of decision making under uncertainty. Still others distinguish between whether the option is tradable or not. We know of no one else who has studied travel time reliability using the options approach, so it is not clear what is the proper terminology to use. However, Professor Lenos Trigeorgis, a well-known expert in the field of real options analysis, applied the real options term in an analogous setting, that is, a decision to increase flexibility of production processes as a defense against exchange rate variability. (In this case, value arose from increasing flexibility.) Comment 6 The original option model was developed to determine the price of traded financial options such as call options, which give the right to buy shares of common stock at a fixed price in the future, or put options, which confer the right to sell shares of common stock at a fixed price in the future. The classic derivation depends on the ability to form so-called arbitrage portfolios of traded securities that hedge out the stochastic component of the prices. In the Black–Scholes case, the other key assumption is that the markets are com- plete, which roughly means that traded securities can span the uncertainty. The original option pricing model has been extended along different dimensions in a variety of ways. For example, the original model has been modified to handle different stochastic assumptions regarding the underlying securities and deal with complicated derivative structures. In addition, the range of application has been extended to value real options, such as the option a mining firm has to open or close a mine. It has also been applied to value cer- tain features of insurance contracts. However, in this case, successful applications relate to the valuation of embedded financial options in insurance and annuity contracts rather than the basic insurance per se. For example, a policyholder under an equity-indexed annuity may participate in the upward moves of the S&P 500 Index and also benefit from downside protection in case the market value falls below an initial investment. These features are routinely valued by using a combination of call and put options on the S&P Index. A standard insurance contract, such as a reinsurance policy on a house, has a payout that resembles the payout on a put option. In the case of the fire insurance contract, the premium is paid up front, and the benefit is equal to the fire damage if any. The fire damage can be viewed as the differ- ence between the value of the house before the fire minus the value after the fire. In the case of a European put option on a financial security, the investor pays the option premium at inception and receives a benefit equal to the difference between the option strike price and the market price of the

131 methodology will yield appropriate values for probabilities of travel times will require a lot of empirical research and com- parison with methodologies currently applied elsewhere in the world (Norway, the Netherlands, United Kingdom, etc.). Without such work, it is very risky to incorporate the method- ology as the standard procedure to be applied. Response 9 The thrust of this comment is that it urges us to compare the options theoretic approach with stated- and revealed- preference approaches to “prove” the appropriateness or validity of the results obtained from the options theoretic approach. Comparing the approaches is a useful suggestion because most folks working in this area are toiling away to develop unreliability valuations using the latter two empirical approaches. The contrasts between the two approaches are the following: 1. The RP and SP approaches are probably impractical meth- ods for widespread application of reliability valuation. This is because they are not economical, requiring separate studies for each application. Even putting aside my general skepticism about the SP techniques, even that approach is relatively costly to implement and subject to the same statistical issues and biases that creep into interview-based contingent valuation, conjoint and similar analyses. 2. The RP and SP approaches implicitly adopt utility function specifications that are certainly debatable in their mathe- matical form (usually linear in its arguments or some non- linear specification to introduce risk-aversion). Hence, they are no more agnostic than the Black–Scholes approach which technically assumes risk-neutrality, but have been shown to be robust to the assumption of risk-aversion. 3. The RP and SP analyses usually also postulate a somewhat specific characterization of the context of unreliability— for example, that it arises out of a particular manifestation of a scheduling-cost problem, etc. The options theoretic approach is no more restrictive; it simply postulates that there is a willingness to pay for insurance (hypothetically) that compensates drivers for not experiencing below- average speeds that are, in turn, generated from draws from a lognormally distributed delay process. 4. A major difference (which I see as an advantage of the options theoretic approach) is that the value of the real option allows separation of the value-of-time issue from the real unreliability issue. Since the existing travel models carry values of time internally for other purposes (mode choice and traffic assignment), the RP and SP approaches (having confounded time valuation and traffic variability), are harder to integrate into the modeling suite. In contrast, Additional citations were added to the appendix to sup- port the appropriateness of our method in the transportation context. For example, SHRP 2 L03 cites Rakha et al. (2006), which confirms the use of lognormal distribution for speed in the context of travel time reliability. Comment 8 By assuming that “recurring and rare phenomena arise out of processes with very different stochastic properties,” the authors are proposing using a mixture of distributions to characterize events that affect time travel reliability. One of the difficulties in using mixtures is identifying the point at which the distributions should be joined; there is no dis- cussion of this in the appendix. The justification for using mixtures is that while recurring events, for which there is often ample data for analysis, might be well characterized by a normal distribution, rare events tend to fall into an extreme tail that would result in an overall distribution that has very fat tail. Response 8 The authors appreciate and agree with these comments. Ignoring the prospect of rare events, and using pure Gauss- ian assumptions instead, is at the heart of many financial and engineering catastrophes, including long-term capital, and elements of the current financial crisis. One of the reasons that we distinguish between recurring and rare events in our discussion is to draw attention to the rare-event issue. Unfor- tunately, implementation of strategies to protect against rare events in a cost-effective way is very difficult because of the problem of characterizing the event distribution and the com- plexity of mathematically representing the proper investment strategy. This is especially challenging in the setting of high- way infrastructure development and operation. We feel the best we can do in a paper such as this is to highlight the issue, offer the skeleton of a methodology, and provide citations to (the very few) papers that hint at how to embed an investment strategy in a rare-event setting. Comment 9 My main comment concerns the travel time reliability valua- tion issues. Although the proposed methodology for measur- ing the value of reliability using option values is innovative. The use of this methodology for this issue is new. From a sci- entific point of view this is very interesting. But in my opinion, the research does not fully address the question of whether the approach can be applied for transportation phenomena. Therefore, this approach presents a risk. To “prove” that the

132 Recurring-event Reliability Valuation example Issue The transportation agency oversees a stretch of highway that experiences significant and variable congestion in the a.m. peak period (6:00 a.m.–10:00 a.m.). This facility is 10 miles long, running north and south, with the central business district at the north end of the facility. Graphically, the speed and variability characteristics of this facility are akin to those depicted in Figure B.5. The large dip in speed at around 8:00 a.m. reflects the slower commuting times dur- ing the peak, relative to the other time blocks on the facility. The variability or unreliability in speed (as measured by the standard deviation of speeds over the course of a year) also seems greater during the peaks. As the a.m. peak-period speed and variability suggest, users on the facility face additional costs in the form of extra time lost while traversing the facility because of travel time variability. The agency is interested in knowing the value of the time that would be saved if a strategy that would improve travel time reli- ability were implemented. This would help the agency to per- form a benefit-cost analysis for the strategy and decide whether it is worth implementing. Solution The variability in the a.m. peak period in the northbound direction on the facility generates costs that are borne by the users of the facility. The cost of the travel time variability can be converted to a certainty-equivalent value, which indicates the additional time motorists are willing to spend on the facil- ity if only the variability in travel times could be eliminated. This certainty-equivalent value of unreliability can then be con- verted to real-dollar values by applying the user’s value of time. • Data needs 44 High-resolution speed data; and 44 Volume data by vehicle type. • Cautions 44 Applicable to data for which travel time variation can be suitably characterized by the lognormal distribution. The steps involved in the calculation are outlined below. Step 1: Characterize the Recurring Congestion Problem A. Obtain speed data for each 5-minute interval (or other appropriate interval) (Table B.11). B. Calculate the average speed and the standard deviation of the log of speed. the options approach allows unreliability to be introduced directly into traditional, volume-delay specifications used in travel model platforms. Since its primary empirical input is speed-distributional information, it imposes light addi- tional burdens on the modeler. The required data on speed variations is plentiful, easily calculated from loop-detector histories, and can be made idiosyncratic to individual net- work links. For the same reason, the approach is friendlier in micro simulation model settings. Comment 10 Because the experts we consulted differ on the validity of applying Black–Scholes to valuing reductions in travel time variability, SHRP 2 Reliability staff held a number of supple- mental conversations to determine under what circumstances Black–Scholes might continue to be useful. Both a member of SHRP 2 staff and one of the reviewers suggested that the options theoretic approach might be useful for ranking dif- ferent ways to improve travel time reliability even though one cannot be confident that calculated values from Black–Scholes have absolute meaning. Response 10 We believe that the measurement technique proposed is much more consistent and transparent than one-off SP or RP findings. The value of travel time is already a necessary input to the travel demand modeling process; our work simply extends the application of those time values to time- certain equivalents of variables measures. Empirically, expe- rienced travel modelers such as Blaine have observed that traffic assignment became more realistic when our measure of the “impedance” of volatility was included in link imped- ance specifications (for freeways). Comment 11 Reliability Project L11 brings to the attention of practitioners and decision makers an analytic method that often is supe- rior to traditional discounted cash flow or discounted present value analysis. This is a valuable silver lining of this research and represents a contribution to the field even if a consen- sus cannot be reached on the validity and applicability of the options theoretic approach to imputing the economic value of improvements in travel time reliability. Response 11 This method is a contribution to the field, and we also feel that the approach provides advantages over the SP and RP methods, as detailed in this appendix.

133 Note: NB = northbound. Figure B.5. Diurnal speed and speed variability. C. Construct the lognormal distribution with the log mean and log standard deviation of the speed to confirm that the speed data are lognormally distributed. Step 1A Table B.11. Speed Data at 5-Minute Intervals Date Time (a.m.) 6:00 6:05 6:10 . . . . . . . . 9:50 9:55 1/2/2007 1/3/2007 . . . . . . . . . . . . . . . Step 2. Calculate the Certainty-Equivalent Value of Unreliability A. Choose the appropriate option formulation (European put or American put option). B. Determine the risk-free interest rate to be used in the analysis. C. Calculate the contract length, based on the lowest 1% speed. D. Calculate the certainty-equivalent value of reliability for the roadway by using the options formula. E. Convert the certainty-equivalent value from mph to min- utes per mile. Step 2A For this example, the European put option is employed for the following reasons: • The European put option gives us the traveler’s value of unreliability for each trip made, given the observed or expected speed variability. The value of unreliability can be multiplied by the number of commuters and work days to give the commuter value of unreliability for a period of time appropriate for the evaluation of a strategy. • The European put option is based on values of variables that are distributed lognormally, as is the speed data on a facility leading to a fitting application. Step 2B A value of 5.00% risk-free interest rate is chosen. Thus, r = 0.05.

134 Step 2e The new average speed at which the commuter is willing to travel if unreliability is eliminated = 32.67 - 4.71 = 27.96 mph. • Time to cover 1 mile at old average speed 1 32.67 = × 60 minutes. • Time to cover 1 mile at new average speed 1 27.96 = × 60 minutes. • [ The certainty-equivalent time per mile that the user is willing to “pay” to eliminate unreliability is given by P(VT, t) = 1 27.96 1 32.67( )− × 60 = 0.31 minutes per mile. Step 3. Evaluate the Change that Affects Unreliability A. Estimate the expected reduction in speed variability achieved through the implementation of the strategy. B. Calculate the certainty-equivalent value for the new scenario. Step 3A “Smart” systems, including collision-warning systems and systems that automatically adjust cruise-control speed from the relative distance of the car ahead, can reduce speed vari- ability and travel time unreliability as they become widely adapted in vehicles. It is estimated that the new system would reduce the speed variability, a, by 50%. Assumptions for the new scenario: • Vehicle volume remains constant over time. • Unknown speed (VT) = 32.67 mph. • Guaranteed speed (I) = 32.67 mph. • a = 0.5 × 0.3635 = 0.1817. Step 3B The European put option and a 5.00% risk-free interest rate are chosen. Thus, r = 0.05. The lowest 1% speed is now 20.18 mph because of the change in the travel time variability. Therefore, the new contract length is given by 60 20.18 365 24 60 10 0.0000566 yearsT t ( )− = × × × = Sigma is calculated with the formula for volatility in finance, which is the standard deviation of the log speed divided by the square root of the contract length: 0.1817 0.0000566 24.17 T t σ = σ − = = Step 2C The contract length (T - t) is calculated as the travel time to cover the segment under consideration at the lowest 1% speed, which is determined by using the mean log speed and the standard deviation of the log speed. For this example, the lowest 1% speed is 13.22 mph, and the segment is 10 miles long. The contract length is expressed in years because the interest rate is an annual rate. 60 13.22 365 24 60 10 0.0000863 yearsT t ( )− = × × × = Step 2D The certainty-equivalent value of reliability, P(VT, t), is cal- culated by first calculating s, d1, and d2. s is calculated with the formula for volatility in finance, which is the standard deviation of the log speed divided by the square root of the contract length: 0.3635 0.0000863 39.12 T t( )σ = α − = = ln 2 1 2 d V I r T t T t T ( )( ) ( ) ( )= + + σ − σ − where VT = the desired speed = 32.67 mph, and I = the guaranteed speed = X = 32.67 mph. Thus, ln 32.67 32.67 0.05 39.12 2 0.0000863 39.12 0.0000863 0.181761 2 d ( ) = + +         × × = 0.18176 39.12 0.0000863 0.18173 2 1d d T t( )= − σ − = − = − Evaluate the standard normal distribution at d1 and d2: 0.18175 0.4279 0.18174 0.57211 1 2 N d N N d N ( ) ( ) ( ) ( ) = = = − = , 32.67 0.57211 32.67 0.4279 4.71 mph 2 1 0.05 0.000086 P V t Ie N d V N d e T r T t T( ) ( ) ( ) ( ) ( ) = − = − = ( ) ( ) − − − A commuter is willing to accept a reduction of 4.71 mph in his or her average speed to eliminate the travel time variability.

135 Value of reliability improvement • Single-occupancy vehicles = 11,484 × 0.30 × (0.31 - 0.14) = $585.68 per mile. • High-occupancy vehicles = 2,871 × 0.60 × (0.31 - 0.14) = $292.84 per mile. • Trucks = 1,595 × 0.83 × (0.31 - 0.14) = $225.05 per mile. Step 4B • Total length of the highway = 10 miles. • [ Total value of reliability improvement for the average a.m. peak hour = ($585.68 + $292.84 + $225.05) × 10 (miles) ≈ $11,036. Step 4C • Total annual value of reliability improvement = 252 week- days/year. • [ Total value of reliability improvement per year (average a.m. peak hour) = $11,036 × 252 days ≈ $2,781,000. The annual value of reliability improvement could be com- pared to the annual costs of the strategy or improvement in a benefit-cost analysis. It is important to note that these savings are for the average a.m. peak hour only and not for the entire day. This example could be repeated for other periods of the day when the speed and variability can be appropriately aggregated, as was done for the a.m. peak period in this example. Perform intermediate and final option value calculation: ln 2 ln 32.67 32.67 0.05 24.17 2 0.0000566 24.17 0.0000566 0.09089 1 2 2 d V I r T t T t T ( ) ( )( ) ( ) ( )= + + σ − σ − = + +         × × = 0.09089 24.12 0.0000566 0.09086 2 1d d T t( )= − σ − = − = − ( ) ( ) ( ) ( ) = = = − = 0.09089 0.4638 0.09086 0.5362 1 2 N d N N d N , 32.67 0.5362 32.67 0.4638 2.37 mph 2 1 0.05 0.0000057 P V t Ie N d V N d e T r T t T( ) ( ) ( ) ( ) ( ) = − = − = ( ) ( ) − − − Thus, the new certainty-equivalent value in minutes per mile is , 1 30.3 1 32.67 60 0.14 minutes per mile.P V tT ( )( ) = − × = Step 4. Calculate the Road-User Value of Reliability Change A. Determine the value of time for different vehicle classes. B. Calculate the value of the reliability improvement for each vehicle class. C. Calculate the value of reliability improvement for the average a.m. peak hour. D. Calculate the total annual value of the reliability improve- ment over the length of the highway for all user groups only for the average a.m. peak hour. Step 4A In Table B.12, volume is equal to the average a.m. hourly vol- ume (over 4 hours for a three-lane facility). Table B.12. Value of Time for Different Vehicle Classes Vehicle Class Volume Value of Time ($/minute) Single occupancy 11,484 0.30 High occupancy 2,871 0.60 Truck 1,595 0.83