Evaluating Alternative Operations Strategies to Improve Travel Time ...

SHRP 2 L11: Final Appendices
Equation 4 - The Option Value of a Rare Event Process Distributed EV (Gumbel)
−(
x−µ
)
β
−(
x−µ
)
⎛
⎞
β
− σ
⎜ σ e
V
⎟
e e
*
F(
V;
K,
x,
π , s)
⎝
⎠
Gumbel
=
if V < V or
β −1
⎛ Kβ
⎞
β⎜
⎟
⎝ β −1⎠
−(
x−µ
) −(
x−µ
)
⎛
⎞
− σ
*
= ⎜ σ e
V
⎟
e e
− K if V ≥ V
⎝
⎠
where
β =
1
2
π
− +
2
s
⎛ π 1 ⎞
⎜ −
2
⎟
⎝ s 2 ⎠
2r
+
2
s
and where
F() = the present discounted value function of the option
V = the present discounted value of the event when it occurs
K = the present discounted value of the cost investing to mitigate impacts
x = the number of events, x > μ
π = the mean of the process that generatesV, π > 0
s = the standard deviation of the process that generatesV, s > 0
r = risk − free interest rate
*
V = the value of the event below which it is worth continuing to wait to invest
µ = location parameter for the Gumbel distributuion
σ = scale parameter for the Gumbel distributuion
The formulation in Equation 4 can be used to evaluate the cost of rare events (and their
mitigation). For example, suppose that a transportation authority is considering investing in a
project that would protect a segment of highway from the effects of avalanches in the segment
right-of-way. The agency wishes to know whether it is worthwhile to do so. Specifically, the
agency wants to know how the value of mitigation (V) and the cost of mitigation (K) associated
with the project compare.
Put in more formal terms, the agency wishes to know what the certainty-equivalent benefit would
be under various scenarios of the value, V, of the avalanche closures, and the cost, K, of mitigation.
The agency would proceed as follows:
• Historical information on the frequency of avalanche events would be used to derive the
values of the scale and location parameters of the Gumbel EV.
• Any uncertainty about the present value of the event would be addressed by providing
information about π and s.
• The number of simultaneous events sought by the strategy is entered as x.
2
> 1
K
VALUATION OF TRAVEL-TIME RELIABILITY FOR RARE EVENTS Page C-9