1. Decisions Under Risk and Uncertainty

2. Risk vs. Uncertainty Risk Uncertainty
Must make a decision for which the outcome is not known with certainty
Can list all possible outcomes & assign probabilities to the outcomes
Uncertainty
Cannot list all possible outcomes
Cannot assign probabilities to the outcomes

3. Measuring Risk Probability Distributions
Chance that an event will occur
Probability Distribution
List of all possible events and the probability that each will occur
probability distribution is essential in evaluating and comparing investment projects
Expected Value or Expected Profit

4. Probability Distribution for Sales

5. Expected Value
Expected value (or mean) of a probability distribution is:
Where Xi is the ith outcome of a decision, pi is the probability of the ith outcome, and n is the total number of possible outcomes

6. Measuring Risk Probability Distributions
Calculation of Expected Profit
Expected profit of an investment is the weighted average of all possible profit levels that can result from the investment under the various state of the economy, with the probability of those outcomes or profits used as weights.

7. Expected Value Does not give actual value of the random outcome
Indicates “average” value of the outcomes if the risky decision were to be repeated a large number of times

8. Variance
Variance is a measure of absolute risk
Measures dispersion of the outcomes about the mean or expected outcome
The higher the variance, the greater the risk associated with a probability distribution

9. Measuring Risk Probability Distributions
Calculation of the Standard Deviation Project A

10. Measuring Risk Probability Distributions
Calculation of the Standard Deviation Project B

11. Identical Means but Different Variances

12. Standard Deviation
Standard deviation is the square root of the variance
The higher the standard deviation, the greater the risk

13. Probability Distributions with Different Variances

14. Coefficient of Variation
When expected values of outcomes differ substantially, managers should measure riskiness of a decision relative to its expected value using the coefficient of variation
A measure of relative risk

15. Decisions Under Risk
No single decision rule guarantees profits will actually be maximized
Decision rules do not eliminate risk
Provide a method to systematically include risk in the decision making process

16. Which Rule is Best?
For a repeated decision, with identical probabilities each time
Expected value rule is most reliable to maximizing (expected) profit
Average return of a given risky course of action repeated many times approaches the expected value of that action

17. Expected Utility Theory
Actual decisions made depend on the willingness to accept risk
Expected utility theory allows for different attitudes toward risk-taking in decision making
Managers are assumed to derive utility from earning profits

18. Expected Utility Theory
Managers make risky decisions in a way that maximizes expected utility of the profit outcomes
Utility function measures utility associated with a particular level of profit
Index to measure level of utility received for a given amount of earned profit

19. Manager’s Attitude Toward Risk
Determined by the manager’s marginal utility of profit:
Marginal utility (slope of utility curve) determines attitude toward risk

20. Manager’s Attitude Toward Risk
Risk averse
If faced with two risky decisions with equal expected profits, the less risky decision is chosen
Risk loving
Expected profits are equal & the more risky decision is chosen
Risk neutral
Indifferent between risky decisions that have equal expected profit

21. Manager’s Attitude Toward Risk
Can relate to marginal utility of profit
Diminishing MUprofit
Risk averse
Increasing MUprofit
Risk loving
Constant MUprofit
Risk neutral

22. Manager’s Attitude Toward Risk

23. Manager’s Attitude Toward Risk

24. Manager’s Attitude Toward Risk

25. Adjusting Value for Risk
Value of the Firm = Net Present Value
r is appropriate discount rate
We will extend this model to deal with an investment project subject to risk.
Two commonly used methods . 1 Risk-Adjusted Discount Rate
2 Certainty Equivalent Approach

26. Risk-Adjusted Discount Rate
This reflects the managers /investors trade off between risk and return. Risk measured by the SD of profits/returns The risk return trade off function of indifference curve ( R) shows that the manager is indifferent among a 10 % rate of return on a riskless asset with SD = 0 ( point A) 20% rate of return with SD =1 (Point C) 32% with SD 1.5 (point D) Difference between the expected rate of return on risky project and riskless project is called risk premium on risky investment Risk –return trade off function ( R) shows that risk premium of 4% is required to compensate for the level of risk given by SD = 0.5 Similarly 10% premium required for an investment with SD =1.0

27. Adjusting Value for Risk

28. The risk return trade off curve would be steeper (R’) for more risk averse manager and less steep (R’’) for a less risk averse manager.
More risk –averse manager facing R’ would require a premium of 22%(c’) for project with SD=1, while a less risk –averse manager with R’’ would require a risk premium of only 4% for the same investment

29. R= net cash flow or return C= initial cost of investment
Project is undertaken if its NPV greater than or equal to zero, or larger than that for an alternative project

30. There is a project with expected net cash flow /return 45,000 for the next five years and initial cost 100,000. Risk adjusted discount rate is 20% Then , we have If firm perceived this as much risky project and used 32% as k its NPV would be

31. Adjusting Value for Risk
Certainty Equivalent Approach
Certainty Equivalent Coefficient
This modifies the numerator of the valuation model
Here R is risky net cash flow
r is risk free discount rate
α is certainty equivalent coefficient
The investor must specify the certain sum that yields to him same utility or satisfaction of the expected risky sum from the investment. The value of α ranges from 0 to 1.
0 means too risky project
1 means risk free project

32. If the manager regarded the sum of with certainty as equivalent to the expected (risky) net cash flow of per year for the next five years therefore α = 36000/45000= 0.8 Take 10% as risk free discount rate Then NPV would be If firm perceived this as much more risky and applied 0.62 as certainty equivalent coefficient the NPV would be 5, These answers are closed to those under previous method

33. Manager’s Utility Function for Profit

34. Expected Utility of Profits
According to expected utility theory, decisions are made to maximize the manager’s expected utility of profits
Such decisions reflect risk-taking attitude
Generally differ from those reached by decision rules that do not consider risk
For a risk-neutral manager, decisions are identical under maximization of expected utility or maximization of expected profit

35. Decisions Under Uncertainty
With uncertainty, decision science provides little guidance
Four basic decision rules are provided to aid managers in analysis of uncertain situations

36. Summary of Decision Rules Under Conditions of Uncertainty
Maximax rule
Maximin rule
Minimax regret rule
Equal probability rule
Identify best outcome for each possible decision & choose decision with maximum payoff.
Identify worst outcome for each decision & choose decision with maximum worst payoff.
Determine worst potential regret associated with each decision, where potential regret with any decision & state of nature is the improvement in payoff the manager could have received had the decision been the best one when the state of nature actually occurred. Manager chooses decision with minimum worst potential regret.
Assume each state of nature is equally likely to occur & compute average payoff for each. Choose decision with highest average payoff.

37. Decision Trees
Sequence of possible managerial decisions and their expected outcomes
Conditional probabilities

39. Uncertainty Maximin Criterion
Determine worst possible outcome for each strategy
Select the strategy that yields the best of the worst outcomes

40. Uncertainty: Maximin
The payoff matrix below shows the payoffs from two states of nature and two strategies.
No probabilities given

41. Uncertainty: Maximin
The payoff matrix below shows the payoffs from two states of nature and two strategies.
For the strategy “Invest” the worst outcome is a loss of 10,000. For the strategy “Do Not Invest” the worst outcome is 0. The maximin strategy is the best of the two worst outcomes - Do Not Invest.

42. Uncertainty: Minimax Regret
The payoff matrix below shows the payoffs from two states of nature and two strategies.
Maximum regret or opportunity cost of the wrong decision.
Regret is measured by by the difference between the payoff of a given strategy and the pay off of the best strategy.

43. Uncertainty: Minimax Regret
The regret matrix represents the difference between the given strategy and the payoff of the best strategy under the same state of nature.

44. Uncertainty: Minimax Regret
For each strategy, the maximum regret is identified. The minimax regret strategy is the one that results in the minimum value of the maximum regret.

45. Uncertainty: Informal Methods
Gather Additional Information
Request the Opinion of an Authority
Control the Business Environment
Diversification