MANAGERIAL ECONOMICS 02

DECISION MAKING UNDER RISK AND UNCERTAINTY (13/10/2011)

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Prof. Dr. Hasan ALKAS

RISK VS. UNCERTAINTY

Risk Possible outcomes are known Probailities can be assigned to outcomes Objective vs. Subjective probabilities

Uncertainty Possible outcomes are (partly) unknown and/or Probabilities cannot be assigned to outcomes

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AXIOMS OF PREFERENCE RELATIONS

Axiom 1 – Completeness For any 2 bundles, A and B, either A B, or B A, or A~B.

Axiom 2 – Transitivity Consider any 3 bundles A, B and C. If A B and B C, then

A C. Similarly, if A~B and B ~C, then A ~C.

Axioms 1 and 2 imply Rational Preference Relation

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ADDITIONAL PROPERTIES OF PREFERENCE RELATIONS Additional Property 1 – More is preferred to less “Non-satiation” Higher indifference curve.

Additional Property 2 – Diminishing Marginal Rate of Substitutions (MRS) As more of a good, say apples, is obtained, the rate at

which she is willing to substitute, say, apples for bananas, decreases.

“Convexity” Marginal utility positive but falling as consumption of any good rises

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REVEALED PREFERENCES

Paul Samuelson (1938) Addresses question of how to obtain information about preferences. Underlying reasoning: Individuals reveal their preferences when choosing between alternatives

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WEAK AND STRONG AXIOM OF REVEALED PREFERENCES Requirement: Consistent choices

Weak Axiom of Revealed Preferences

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Whenever bundle A is revealed preferred to bundle B, bundle B is

never revealed preferred to bundle A.

Strong Axiom of Revealed Preferences Given we have bundles B1, B2, …, Bn. Suppose B1 is revealed

preferred to B2, B2 is revealed preferred to B3, …, Bn-1 is revealed

preferred to Bn. Then B1 is revealed preferred to Bn.

With more then two bundles we might end up with a circle relationship Circle: A revealed preferred to B, B revealed preferred to C, C revealed preferred to A

EXPECTED UTILITY I

Assume: Individual can choose between a number of risky

alternatives (e.g. different lottery tickets) Each risky alternative may result in one of a number of

possible outcomes (outcome is not known at the time of decision making) (e.g. possible lottery prizes)

Probabilities of outcomes known. (probabilities of prizes of one lottery sum up to one)

Preference relation is continuous and satisfies independence axiom.

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EXPECTED UTILITY II

Discrete case: 𝑈 𝑊� = 𝐸 𝑢(𝑊� ) = ∑ 𝑝 𝑤𝑖 𝑢(𝑤𝑖)𝑛𝑖=1

𝑈 : expected utility 𝑊� : risky prospect 𝑤𝑖, i = 1…n: n possible outcomes 𝑝 𝑤𝑖 : probability of outcome 𝑤𝑖 𝑢(𝑤𝑖): utility of outcome 𝑤𝑖 (Bernoulli utility function)

Example

Assume: 𝑢 𝑤𝑖 = 𝑤𝑖

Flip coin twice

For two heads or two tails you get 5€

For one head and one tail you have to pay 2€

𝑈 𝑊� = ∑ 𝑝 𝑤𝑖 𝑢(𝑤𝑖)𝑛𝑖=1 = 2(0.5)25 + 2 0.5 2 −2 = 1.5

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ATTITUDE TOWARDS RISK I

Risk averse: Preferring a sure amount of X€ to a risky prospect with an expected value of X€.

𝑢 𝐸[𝑊� ] ≥ 𝐸 𝑢 𝑊� Jensen‘s inequality concave function

Risk neutral: Indifferent between risky prospect with an expected value of X€ and a sure amount of X€.

𝑢 𝐸[𝑊� ] = 𝐸 𝑢(𝑊� )

Risk loving: Preferring a risky prospect with an expected value of X€ to a sure amount of X€.

𝑢 𝐸[𝑊� ] ≤ 𝐸 𝑢(𝑊� )

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ATTITUDE TOWARDS RISK II

RISK AVERSION

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HOW TO MEASURE RISK AVERSION?

Arrow-Pratt Measures

Arrow-Pratt Measure of Absolute Risk Aversion (ARA)

𝐴 𝑤 = −𝑢𝑢𝑢(𝑤)𝑢𝑢(𝑤)

Arrow-Pratt Measure of Relative Risk Aversion (RRA)

𝑅 𝑤 = −𝑢′′ 𝑤𝑢′ 𝑤

𝑤 = 𝐴 𝑤 𝑤

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ARROW-PRATT MEASURES

Degree of risk aversion is related to the curvature of the utility function

Curvature can be represented by second derivative BUT: The second derivative is not invariant to positive

linear transformations, but our understanding of utility functions requires that

Hence, we need to normalise the second derivative Normalising with respect to the first derivative gives a

measure invariant to positive linear transformations

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ARROW-PRATT MEASURES

Decreasing relative risk aversion: �As wealth increases, the individual becomes less risk averse with respect to gambles that are the same in proportion to his wealth level.

Decreasing relative risk aversion implies decreasing �absolute risk aversion, i.e. as wealth increases, the individual becomes less risk averse with respect to gambles that are the same in absolute terms.

Finance theory often assumes constant relative risk �aversion.

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SYSTEMATIC VS. UNSYSTEMATIC RISK

• Risk

• Systematic Risk (Market Risk) • Risk factors that affect a large number of assets, such as changes in GDP, inflation, interest rates, etc.

• Unsystematic (Business Specific) • Risk factors that affect a limited number of assets, such as labor strikes, part shortages, etc.

• Portfolio with different assets can eliminate diversifiable risk for the most part.

• As more and more assets are added to a portfolio, risk measured by σ decreases, but still some risk remains.

• The relevant risk measure is Beta β-factor, which measures the riskiness of an individual asset in relation to the market portfolio (see CAPM or SML).

• β = 1.0 : same risk as the market

• β < 1.0 : less risky than the market

• β > 1.0 : more risky than the market

• As most investors are Risk Averse they don’t like risk and demand a higher return for bearing more risk.

• The standard deviation of returns is a measure of total risk

• Total risk = systematic risk + unsystematic risk.

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DECISIONS UNDER UNCERTAINTY

Maximax Criterion Identify best outcome for each possible decision and choose the

decision with the maximum payoff thereof Maximin Criterion Identify the worst outcomes for each decision and choose the

decision with the maximum payoff thereof Minimax Regret Criterion Determine the worst potential regret associated with each

decision and choose the decision with the minimum worst potential regret

Equal Probability Criterion Assume each state of nature is equally likely to occur, compute

the average payoff for each equally likely possible state of nature and choose the decision with the highest average payoff

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THE REAL OPTION APPROACH

Uncertainty creates opportunities.

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Valu

e

Uncertainty

Traditional View

Real Options View

Managerial Options increase Value

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