Evaluating Alternative Operations Strategies to Improve Travel Time ...
Evaluating Alternative Operations Strategies to Improve Travel Time ...
Evaluating Alternative Operations Strategies to Improve Travel Time ...
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SHRP 2 L11: Final Appendices<br />
Step 2 – Calculate the Certainty-Equivalent Value of Reliability<br />
A. Choose the appropriate option formulation (European Put option or American<br />
Put option).<br />
B. Determine the risk-free interest rate <strong>to</strong> be used in the analysis.<br />
C. Calculate the contract length, based on the 95 th percentile speed.<br />
D. Calculate the certainty-equivalent value of reliability for the roadway using the<br />
options formula.<br />
E. Convert the certainty-equivalent value from miles per hour <strong>to</strong> minutes per<br />
mile.<br />
A. Choose the appropriate option formulation (European Put option or American Put option).<br />
For this example, the European Put option was employed because:<br />
• The European Put option gives us the traveler’s value of reliability for each trip<br />
made, given the observed or expected speed variability. The value of unreliability<br />
can be multiplied by the number of commuters and work days <strong>to</strong> tell us the<br />
commuter value of unreliability for a period of time, appropriate for the evaluation<br />
of a strategy.<br />
• The European put option is selected since the option is formulated using a finite<br />
time horizon, where the option is exercised at the end of the option life (i.e., once<br />
the traveler has traversed the bridge segment.)<br />
B. Determine the risk-free interest rate <strong>to</strong> be used in the analysis. (A risk-free interest rate<br />
identifies the rate at which one is guaranteed a return on investment.)<br />
A value of 5.00% risk-free interest rate was assumed.<br />
Thus, r = 0.05<br />
C. Calculate the contract length, based on the 95 th percentile speed.<br />
The contract length (T-t), is calculated as the travel time <strong>to</strong> cover the segment at the 95 th<br />
percentile speed, determined using the mean log-speed and the standard deviation of the<br />
log-speed. For this example the lowest 5% speed is 23.65 mph and the segment length is 4<br />
miles. The contract length is expressed in years since the interest rate is an annual rate.<br />
D. Calculate the certainty-equivalent value of reliability for the roadway using the options<br />
formula.<br />
⎛ 60 /13.22 ⎞<br />
The certainty-equivalent value of reliability T − t = ⎜<br />
⎟×<br />
10 is calculated by first calculating three<br />
⎝365× 24×<br />
60 ⎠<br />
parameters: PV (<br />
T<br />
,) t , d1 and d2. PV (<br />
T<br />
,) t is calculated using the formula for volatility in finance, which<br />
represents the standard deviation of the log speed, divided by the square root of the contract<br />
length. The variables d1 and d2 are two points on the cumulative log-normal distribution,<br />
N(d), N(d1), and N(d2) are thus probabilities that the cumulative probability will be less than<br />
d1 or d2, respectively. In the Black Scholes formula, N(d2) is the probability the option will be<br />
exercised and N(d1) is called the option delta. It measures how the option value changes with<br />
volatility.<br />
SAMPLE PROBLEM-QUANTIFYING THE ECONOMIC BENEFIT OF IMPROVING TRAVEL-TIME RELIABILITY Page D-7
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