“ Mathematicians estimate money in proportion
to its quantity, and men of good sense in
proportion to the usage that they may make of it.”
Gabriel Cramer (1704-1752)
Swiss mathematician
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MBA elective - Models for Strategic Planning - Session 8
Risk, Risk Attitudes, and Certainty Equivalents
Essential Ideas
Expected Values (EV) do not reflect the risks present in decisions
Certainty Equivalent (CE) measures the risk-adjusted value of a
risky project
CE is derived from the utility function, which captures the decision
maker’s valuation of payoffs and risk attitude
Utility can be measured
In most cases, estimating a single parameter, RT (Risk Tolerance)
is enough
Ensures a consistent risk evaluation across all projects
© Ph. Delquié
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Expected Values do not reflect risk
Firms’ and Individuals’ choices often deviate from EV
• Buying insurance
• Diversifying assets, activities
Why?
• Cash constraints
• Value is not linear with payoff
• Large losses may hurt too much
• Simply, a dislike for uncertainty and randomness
Is it right to make decisions not based on EV?
Prudent outlook?... or “Myopic” behavior?
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Choice Under Risk: Example
Project A
Project B
0.25
3 € million
0.75
2 € million
0.75
1 € million
0.25
0 € million
• Expected Value of A and B is the same.
• Variance of A and B is the same.
• Still, we are not indifferent between A and B 9
• How much would you pay/ask for switching from A to B?
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Using a Utility Function to make decisions
1/2
?
1/2
u(Payoff)
10,000
0
u(5000) > u(Gamble)
5,000
u(10000)
u(5000)
u(gamble): EU =
RP
½ u(10000) + ½ u(0)
u(0)
0€
CE € ≤
5000 €
EV €
10000 €
Payoff
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Expected Utility decisions
Gamble X = (p1,x1; p2,x2; 9 ; pn,xn)
xi is a net final payoff (terminal value of the decision tree)
Calculate Expected Utility of gamble EU = Σ pi u(xi)
EU measures the worth of the gamble in utility units;
reflects the risk of the gamble, i.e. dispersion of payoffs
Caution: EU ≠ u(EV) Σ pi u(xi) ≠ u(Σ pi xi) unless u is linear!
Decision Rule Choose gambles with highest EU.
Quite simply: replace x by u(x) in the decision tree.
Advantages
- Reflects risk preferences
- Allows for individual differences
- Provides a rationale for one-time decisions
- Provides consistent valuation of risks
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Certainty Equivalent: a risk-adjusted measure of value
Definition: CE(X) = a sure payoff worth exactly as much as gamble X
i.e., u(CE(X)) = E[u(X)] that is:
CE(X) = u–1(E[u(X)])
Interpretations:
- You are indifferent between receiving CE for sure or taking the gamble
- Your CE is the lowest amount for which you would be willing to sell your
rights to the Gamble
- CE provides a risk-adjusted measure of a gamble’s value,
expressed in the same units as the payoffs.
Risk Premium (RP):
def: RP = EV − CE
thus: CE = EV − RP
RP measures a discount for risk
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3 steps in constructing a gamble’s Certainty Equivalent
Gamble
Utilities
x1
p1
p2
p1
x2
p2
§
pi
pn
u(x1)
pi
xi
§
1. Take utilities
pn
xn
u(x2)
§
u(xi)
§
u(xn)
2. Calculate EU
Certainty Equivalent of
the Gamble
Expected Utility of
the Gamble
CE = u −1 ( EU )
n
EU = ∑ pi u (xi )
n
= u −1 ( ∑ pi u (xi ) )
3. Take u inverse
i =1
i =1
Payoff scale
Utility scale
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Risk Attitudes
Risk Averse
Risk Neutral
Risk Seeking
Downside change in
asset position carries
more weight than
equivalent upside
Downside and upside
changes in asset
position have the same
weight
Upside change in
asset position has
more weight than
equivalent downside
For any gamble:
For any gamble:
For any gamble:
CE < EV
CE = EV
CE > EV
RP > 0
RP = 0
RP < 0
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Common Mathematical Forms for Utility Functions
• Exponential
−x/RT
u(x) = 1 − e
RT = ‘Risk Tolerance’ coefficient
• Power
u(x) = (x + w)α
(for x > −w)
• Logarithm
u(x) = ln(x + w) (for x > −w)
• Other
Quadratic, Linear + Exponential, Hyperbolic, etc9
Inspecting Excel graphs
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The Exponential Utility function: u(x) = 1 − e−x/RT
Built-in TreePlan: automates the EU and CE calculations
Provides good local approximation to other forms
Assumes risk aversion is constant, i.e. independent of wealth
Facts about the ‘Risk Tolerance’ coefficient, RT
RT is in the same units as the payoff x
Higher values of RT mean lower risk aversion
As RT → ∞ : exponential utility → linear, hence CE → EV
Same value of RT should be used to evaluate all prospects
Direct formula for certainty equivalents:
CE = EV – σ2/(2RT) where: EV = Mean, σ2 = Variance of gamble
- Formula is exact if the gamble has a Normal distribution
- Approximates well if distribution bell shaped or risk is small (σ << RT )
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Utility Measurement
Why measure utility?
Prescribe decisions
1) Elicit decision-makers’ preferences for elementary risks,
which may be introspected with lucidity/confidence
2) Use these measures to prescribe choices in complex situations
Verify/ensure consistency in decisions
Compare risk taking of different Business Units Managers,
or same Manager across situations
(e.g. Mineral Resources Exploration Units)
Example: Measuring utility with ASSESS
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Scale for Utility Measurement
Facts about utility scale
- The unit of utility is arbitrary:
set of two points as you please, e.g. u(Worst) = 0 and u(Best) = 1
- Utility can be rescaled by a positive linear transformation
If u(⋅) represents your utility, then so does v(⋅) = a u(⋅) + b (with a > 0)
−x/RT
−x/RT
e.g. u(x) = 1 − e
v(x) = A − Be
(B > 0)
u(⋅) and v(⋅) will lead to the same decisions and certainty equivalents
- Similar to a temperature scale, e.g. changing from °F to °C
- Permissible operations: expectation only !
(cannot add utilities, cannot take ratio of utilities)
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Corporate Risk Aversion
Risk aversion is a concern in few, high-stake decisions
Some rules of thumb for setting Risk Tolerance coefficient: *
- For companies taking moderate risks, set RT ≈ Net Income [1]
- For large, diversified firms, set RT ≈ 1/6 × firm’s equity [2]
- For oil exploration units, set RT ≈ 1/4 × unit’s annual budget [3]
The same Risk Tolerance should be used throughout the company
Use a risk-free discount rate to estimate the CE of a risky NPV,
(otherwise you may double-count risk: once with discounting and again with CE)
* based on empirical studies with senior management & business unit managers
[1] McNamee, P. & Celona, J. (1990). Decision Analysis with Supertree (2nd Ed.), The Scientific Press: San Francisco, p. 122.
[2] Howard, R. A., 1988. Decision Analysis: Practice and Promise. Management Science, 34, 679-695.
[3] Walls, M. R., Morahan, T. & Dyer, J. S., 1995. Decision Analysis of Exploration Opportunities in the Onshore US at
Phillips Petroleum Company. Interfaces, 25, 39-56.
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To do for next session?
Review concepts and Solution Set 8
To be posted on website shortly
Prepare Exercise Set 9
Applications of concepts developed in Session 8
No new theory will be covered
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