Expected Profit Under Maximin and Minimax Rules
Chin Hon Tan, Guanghua Han
Department of Industrial and Systems Engineering, National University of Singapore
1 Engineering Drive 2, Singapore 117576
Joseph C. Hartman
Department of Industrial and Systems Engineering, University of Florida
303 Weil Hall, P.O. Box 116595, Gainesville, FL 32611-6595
Abstract
The attractiveness of a project is often determined by computing the expected net present value of cash flows across the
project horizon. However, expected net present value does not measure risk. In this paper, we study the performance
of two decision rules that consider risk, maximin net present value and minimax regret, within a project investment
setting through numerical experiments. Despite its risk averse nature, the minimax regret rule results in solutions with
expected net present values that deviate modestly from the maximum expected net present value under many situations. In addition, the minimax regret rule outperforms the maximin net present value rule over repeated experiments,
highlighting that the former is less conservative.
Keywords
Project evaluation, expected present value, maximin, minimax regret.
1. Introduction
When cash flow and interest rates are deterministic, the net present value (PV) of a project indicates its economic
appeal. When there is uncertainty in the cash flow, the expected net present value (EPV) of a project can be used to
evaluate its expected profitability. Hence, EPV can be used to rank projects when there are multiple projects to select
from. In particular, it follows from the Law of Large Numbers [1] that a decision maker is almost surely better off in
the long run by selecting projects with the highest EPV.
However, managers rarely select projects based on EPV alone. It can be shown that, under certain axioms (completeness, transitivity, continuity and independence), individuals seek projects with the highest expected utility [2]. Hence,
the expected utility of projects can, in theory, be used to rank and select projects. However, the utility function of the
decision maker may not be available in practice. In addition, preferences that violate expected utility theory have been
observed experimentally [3].
Besides expected utility, researchers have also proposed evaluating uncertain prospects based on expected gains and
associated risks. For example, Markowitz [4] associated the risks of a project with its volatility and proposed a meanvariance framework which seeks to maximize expected gains and minimize variance. However, it has been highlighted
that variance, which penalizes gains and losses equally, may not be an appropriate representation of risk. Various
asymmetric risk measures, including semivariance, maximum loss and Value-at-Risk (VaR), have subsequently been
proposed [5].
In the expected utility and mean-risk approaches, it is often assumed that uncertainties in cash flows are represented
by known probability distributions. However, this may not be true in practice, especially for projects without precedent. Under these situations (i.e., random state of nature is not completely understood), researchers propose a robust
approach, including the maximin criterion and the minimax regret criterion, which seeks to maximize the smallest
possible outcome and minimize the largest regret that may be experienced, respectively [6].
The minimax regret rule, which evaluates missed opportunities rather than the worst possible outcome, is generally
considered to be less conservative than the maximin rule [6, 7]. Perakis and Roels [8] highlight that in the classical
newsvendor problem, the profits achieved by ordering the quantity that minimizes maximum regret is generally higher
Tan, Han and Hartman
than the profits achieved by ordering the quantity that maximizes the smallest profit under various maximum entropy
distribution assumptions. In addition, they also find that the minimax regret rule performs well, on average, under a
revenue management context [9].
In this paper, we study the performance of the maximin and the minimax regret rule in selecting projects with uncertain
cash flows. In our experiments, the EPV obtained under the minimax regret rule deviates modestly from the maximum
EPV. In addition, the minimax regret rule outperforms the maximin rule in the long run, indicating that the former is
less conservative than the latter under a project investment setting.
2. Model description
Consider an investor who has to select exactly one project from a set of N possible projects, including a “do-nothing”
project, when appropriate. Projects are evaluated across the horizon T . Let Ait denote the cash flow of project i in
period t. Let PVi denote the net present value of project i:
T
PVi = ∑ αt Ait ,
(1)
t=0
where α denotes the periodic discount factor. Suppose that Ait is dependent on a set of K uncertain parameters
X = {X1 , X2 , ..., XK } such that:
K
Ait =
∑ mikt Xk ,
(2)
k=1
where mikt are finite known constants. We assume that all Xk are defined on the same probability space (ΩX , FX , PX ),
where ΩX is the K-dimension sample space, FX is the set of events and PX is the probability measure associated with
X. We say that X is rectangular if ΩX = [a1 , b1 ] × [a2 , b2 ] × · · · × [aK , bK ]. For a rectangular multivariate random
variable X, it is possible for any element of X to take the value of its upper support when another element of X takes
the value of its lower support. We note that independence among the elements of X is sufficient, but not necessary, for
X to be rectangular.
Combining Equations (1) and (2), we get:
T
PVi
=
∑ αt Ait
t=0
T K
=
∑ ∑ αt mikt Xk
t=0 k=1
K T
=
∑ ∑ αt mikt Xk
k=1 t=0
K
=
∑ cik Xk ,
(3)
k=1
T
where cik = ∑t=0
αt mikt describes the impact of Xk on the net present value of project i. Note that PVi is a random
variable. It follows from Equation (3) that the expected
net present
value of project i, denoted by EPVi , is:
"
#
K
EPVi = E
K
∑ cik Xk
=
∑ cik E[Xk ].
k=1
k=1
In this paper, we assume that each Xk is bounded from below and above by ak and bk , respectively. Under this condition,
it follows from the Principle of Maximum Entropy that the uniform distribution with lower and upper supports ak and
bk , respectively, best describes Xk [10]. Hence:
K
1 K
EPVi = ∑ cik E[Xk ] = ∑ cik (ak + bk ).
2 k=1
k=1
A risk neutral investor will select the project with the highest EPV. However, a risk averse investor may prefer another
project if the project with the highest EPV is risky. Next, we consider two criteria, minimum PV and maximum regret,
that reflect the risk associated with a project. Let MPVi denote the minimum PV of project i:
MPVi = min{PVi }.
X
The following proposition highlights how the minimum PV of a project can be computed.
Tan, Han and Hartman
Proposition 1. If X is rectangular:
K
∑ YikMPV ,
MPVi =
k=1
where:
YikMPV
(
cik ak
=
cik bk
if cik ≥ 0
otherwise.
Proof of Proposition 1. From Equation (3):
K
K
MPVi = min ∑ cik Xk =
X
k=1
{cik Xk } .
∑ min
Xk
k=1
The second equality follows from the rectangular property
of X. The theorem follows from the fact that:
(
cik ak if cik ≥ 0
min {cik Xk } =
Xk
cik bk otherwise.
Proposition 1 states that if X is rectangular, the minimum PV of a project can be determined by letting each Xk take
the value of its lower and upper support when cik is non-negative and negative, respectively.
Next, we determine the regret that each project may experience. Let Ri be a random variable that denotes the regret of
project i:
Ri = max{PV j } − PVi .
(4)
j
The regret of project i is defined as the difference between the PV of the best project under the realized scenario and
the PV of project i under the same scenario. Suppose the investor selects project i. At the end of the horizon, it is
determined that project i has a net present worth of $1M. In retrospect, the investor realizes that project j, with a net
present worth of $1.3M, has the highest net present worth under the realized scenario. In this example, the regret of
project i is $1.3M − $1M = $0.3M. Let MRi denote the maximum regret that may be experienced by a investor that
selects project i:
MRi = max{Ri }.
X
The following proposition highlights how the largest possible regret associated with a project can be computed.
Proposition 2. If X is rectangular:
K
MRi = max ∑ YiMR
jk ,
j
where:
YiMR
jk
k=1
(
(c jk − cik )bk
=
(c jk − cik )ak
if c jk − cik ≥ 0
otherwise.
Proof of Proposition 2. From Equation (4):
MRi
= max max{PV j } − PVi
j
X
)
(
(
K
= max max
j
X
(
= max
j
∑ c jk Xk
− ∑ cik Xk
k=1
k=1
K
∑ max
Xk
)
K
c jk − cik Xk
)
k=1
The second equality follows from Equation (3) and the third equality follows from the rectangular property of X. The
theorem follows from the fact that:
(
(c jk − cik )bk if c jk − cik ≥ 0
max c jk − cik Xk =
Xk
(c jk − cik )ak otherwise.
Proposition 2 states that if X is rectangular, the maximum
regret of a project can be determined by letting each Xk take
the value of its upper and lower support when c jk − cik is non-negative and negative, respectively.
In this paper, we are interested in the EPV of projects that are selected using the following risk averse decision rules:
Tan, Han and Hartman
1. MMPV rule: Select the project with the largest MPVi . We refer to this as the maximin PV rule. When X is
rectangular, Proposition 1 can be used to determine the minimum PV of each project and the project with the
largest minimum PV can subsequently be identified by comparing the MPVi of the projects.
2. MMR rule: Select the project with the smallest MRi . We refer to this as the minimax regret rule. When X is
rectangular, Proposition 2 can be used to determine the maximum regret of each project and the project with the
smallest maximum regret can subsequently be identified by comparing the MRi of the projects.
In particular, we are interested in the difference between the EPV of these projects and the optimal EPV. Let ∆PV and
∆R denote these differences:
EPVR − EPVmin
EPVPV − EPVmin
and
∆R =
,
∆PV =
EPVmax − EPVmin
EPVmax − EPVmin
where EPVmax = maxi {EPVi }, EPVmin = mini {EPVi }, EPVPV is the expected net present value of the project with the
largest MPVi and EPVR is the expected net present value of the project with the smallest MRi .
3. Numerical experiments
To study the performance of the MMPV rule and the MMR rule, we generate a series of random problems as follows:
1. Each cik is obtained by sampling independently from a normal distribution with zero mean and a user-defined
variance σ2 for i = 1, 2, ..., N and k = 1, 2, ..., K. In simple linear regression analyses, mikt are assumed to be
independent and normally distributed. Since cik is a linear combination of mikt and the sum of a series of normal
distribution is normally distributed, we believe the normal distribution is appropriate for sampling cik .
2. For each Xk , we obtain two random samples independently from a uniform distribution U(−β, β), where β is a
user-defined parameter. We set ak and bk to be the smaller and larger of the two samples, respectively. This is
repeated for k = 1, 2, ..., K. A uniform distribution is selected so that the two samples that are used to determine
ak and bk are equally likely to take any values between −β and β.
Note that N, K, σ2 and β are user-defined parameters, which are varied in the experiments to study their effects on the
performance of the MMPV rule and the MMR rule.
Figure 1: Performance of MMPV and MMR rules under different N.
To study the effects of N on the EPV of the projects selected under different decision rules, we estimate the EPV
obtained by the two decision rules by generating 100,000 random problems for each N. We arbitrarily set K = 5,
σ2 = 100 and β = 50. The values of ∆PV and ∆R for each problem are computed and the average ∆PV and ∆R for
different N, denoted respectively by ∆¯ PV and ∆¯ R , are illustrated in Figure 1.
Our experiments indicate that the MMR rule performs well under a project investment setting. In particular, ∆R = 1
when N = 2. This highlights that the MMR rule, which minimizes the maximum regret of a decision, also maximizes
EPV. It is possible to show that this is true for all K, σ2 and β. A proof is provided in [11]. Surprisingly, a risk averse
decision rule, is optimal from a risk neutral perspective. We note that there are many situations in practice where
N = 2. For example, a decision maker is presented with a project and has to decide whether to accept (choice 1) or
Tan, Han and Hartman
reject (choice 2) it. Our results verify that a decision maker who minimizes maximum regret also maximizes EPV
under this setting.
The MMR rule also performed well when there are more than two projects to choose from. In our experiments,
∆¯ R > 0.95 for N = 2, 3, ..., 15. Note that ∆R = 1 and ∆R = 0 implies that the project selected by the MMR rule has the
highest and lowest EPV among all possible projects, respectively. Our results suggest that the deviation between the
EPV achieved under the MMR rule and the maximum EPV is likely to be small under many situations.
The MMPV rule appears to perform well under a project investment setting, especially when N is high. In particular,
we note that ∆¯ PV > 0.9 when N ≥ 5. Despite their risk averse nature, both decision rules appear to perform well in the
long run, with the MMR rule outperforming the MMPV rule across all N (i.e., N = 2, 3, ..., 15).
Figure 2: Performance of MMPV and MMR rules under different K.
To study the effects of K, we generate 100,000 random problems at each K for N = 2, 5 and 10 (see, Figure 2). These
experiments highlight that the performance of both decision rules improve as K increases, which indicates that both
rules are better at evaluating projects with multiple uncertain cost components. In particular, the effect of K on ∆¯ PV
appears to be more pronounced when N is high. Hence, the MMPV rule is expected to perform best when both N and
K are large. Generating another 1,000 random problems for N = 600 and K = 600, we obtain ∆¯ PV = 0.925, which is
significantly higher than the ∆¯ PV value of 0.914 that is obtained for N = 10 and K = 15 (see, Figure 2). In contrast, the
average performance of the MMR rule appears to be insensitive to N, with the exception of N = 2, when K is large.
Figure 3: Performance of MMPV and MMR rules under different σ2 .
In our experiments the experimental parameters σ2 and β were arbitrarily set at 100 and 50, respectively. Further
experiments, using different σ2 and β values, suggest that the performance of the two rules are not influenced by these
parameters (see, Figures 3 and 4).
Tan, Han and Hartman
Figure 4: Performance of MMPV and MMR rules under different β.
4. Conclusions
In this paper, we study the expected performance of two risk averse decision rules under a project investment setting
through numerical experiments. It is observed that the MMPV and MMR rules perform well, on average, when
evaluating projects with multiple uncertain cost components. In particular, our experiments highlight that the MMR
rule maximizes EPV when N = 2 and deviates modestly from the maximum EPV when there are more than two
projects to choose from. Despite its risk averse nature, the MMR rule is effective in selecting projects under many
situations. Furthermore, our experiments highlight that the MMR rule outperforms the MMPV rule under various
settings, which suggests that the MMR rule is generally less conservative than the MMPV rule. In this paper, we
focused solely on the PV of a project. An interesting area of further research is to study how the MMR rule can
be used to select projects with uncertain cash flows when there are multiple considerations (e.g., payback period,
maximum loss, etc.).
Acknowledgments: The authors are grateful to three anonymous referees for their comments. The authors also
gratefully acknowledge support from the National University of Singapore and the National Science Foundation under
Grant No. R-266-000-068-133 and CMMI-0813671, respectively.
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