1. Managerial Economics Uncertainty
Aalto University
School of Science
Department of Industrial Engineering and Management
January 10 – 26, 2017
Dr. Arto Kovanen, Ph.D.
Visiting Lecturer

2. Uncertainty– general
Thus far we have examined economic decisions under the assumption that there is no uncertainty
However, we cannot fully predict what will happen in the future, hence there is always some uncertainty
How does uncertainty affect firm’s decisions?
First, some concepts:
How do we measure uncertainty (risk or randomness)?
Expected value = E(x) = μ = the sum of likely outcomes weighted by their individual probabilities
The weights (probabilities) should sum up to unity

3. Uncertainty– general
A decision problem under uncertainty is characterized by:
A set of alternatives or options
The possible states of the world
The possible consequences of each action
All uncertainties in the standard problem are captured by known probabilities of each event
Rationality requires that a person maximizes his/her expected utility subject to one’s uncertainties and one’s preferences
Is uncertainty adequately represented by probabilities?

4. Uncertainty– general
Standard deviation = σ = √(σ2) is another way of measuring the randomness of the data in a sample
With large numbers, the metrics (μ and σ) will also be large – scaling required
Coefficient of variation = CV = σ/μ (normalizes the unit)
This is more useful because we know that in a sample 1 with σ1 > σ2 of sample two, but the same mean values, μ1 = μ2, the date is more dispersed
Are you risk averse, risk neutral or risk lover?
What does it mean in practice?

5. Keynes on Uncertainty
John Maynard Keynes was a famous British economist
He not only made a distinction between what is certain from what is only probable (i.e., we can determine to what extent the event is likely to take place)
According to Keynes, game of roulette is not subject to uncertainty – it is subject to probability
Or the weather is only modestly uncertain
For Keynes, the prospect of war in Europe (in the 30s) or the price of copper were examples of uncertainty
This is because for these events, “there is no scientific basis to form any calculated probabilities”. We simply do not know!

6. Uncertainty-example Which option would you prefer? Example 1:
A sure gain of $240
A 25% chance of winning $1,000 and a 75% chance of winning nothing
Example 2:
A certain loss of $750
A 75% chance of losing $1,000 and a 25% chance of losing nothing
Example 3:
How much would you pay to play for flipping a coin? If it is heads up, you receive $2. If it is tails, we flip again and if it is heads up, you receive $4. If it is tails, we flip again and so on.

7. Uncertainty-example Which option would you prefer?
Example 1: most prefer sure gain over uncertainty, even though expected utility is $250 > $240. That is, people are usually risk averse
Example 2: most people hate to accept a certain loss and therefore like to take a chance with probabilities
Example 3: this is called St. Petersburg’s paradox. The game has an expected value of infinity (i.e., …), but most people want to pay much less for this game. This points to risk aversion that makes the expected utility associated with the game finite even when the game is played infinite number of times.

8. Uncertainty-used car
Suppose you can buy a used car with $1,000 and sell it for $1,100, making $100 in profit
However, the car may be a lemon and would cost $200 to repair. If it is a peach, it would cost $40 to repair
Probability of a lemon is 20%
If you don’t buy a used car, your cost is $0
Assume you are risk neutral
Draw a decision tree for this problem
Expected value is $28 = 0.2*(-$100) + 0.8*($60)
How much would you pay for perfect information to know if the car is a lemon or a peach?

9. Uncertainty-used car (cont.)
How do we figure this out?
We would not buy a lemon, but it would cost $X to have that information. The expected value of that information is 0.2*$X.
We would buy a peach, but it would cost $X, which then reduces the profit of $60. So the expected value of that transaction is 0.8*($60 - $X)
The overall expected value is then $48 - $X, which we need to compare with the outcome without perfect information ($28)
Hence the maximum amount you want to pay for the perfect information is $48 - $X = $28, which equals $20
If you pay < $20, you would gain over expected value

10. Return to risk
Importance of risk-free (is there such thing) and risky return
An individual is risk averse if he/she prefers a sure amount to a risky payoff with the same expected value
Example: Individual’s utility is given by U = 100M0.5. The individual is offered $1,000 if he flips “heads” but will lose $1,000 if he flips “tails”. Initial M = $50,000
The expected value of the game is zero since both options have the same probability and equal payoffs (with opposite signs) = called “a fair gamble”, which risk-neutral individual would accept
Because of the change is utility due to loss is larger than the change in utility due to winning (non-linear), a risk-averse person is not going to accept the gamble

11. Return to risk (cont.)
Because of diminishing marginal utility of wealth, individuals will pay to avoid risk and will need to be compensated for taking additional risk
Assume that individuals care about average wealth and dislike variance of outcomes
Risk cannot be easily avoided by means of, for instance, diversification, albeit it can be reduced
Can we measure risk? Beta coefficient for stocks (beta equals = 1 for broad market index
Small capitalization stocks, commodities are more risky due to their volatility

12. Return to risk (cont.)
A risk-neutral person would ignore the riskiness of the gamble and focus only on the expected payoff
Risk-return trade-off important for companies as well
When σ > 0, i.e., there is risk, the entrepreneur needs to receive a higher payoff compared to a certain payoff
Example: there are two projects, one is risk-free and the other one has an uncertain return
Expected return can be written as follows:
E(x) = θx(rf) + (1 – θ)x(r) where θ = share of risk-free investment, x(rf) = risk-free return and x(r) = uncertain return (applies to risk neutral person)

13. Return to risk (cont.)
Can we price risk (x(r) – x(rf))? This compares return of a risky asset (e.g., stock) to a risk-free asset (e.g., government bond)
E.g., U.S. government securities have very low default risk and hence are often called “risk (default) free”
But even fixed-income securities have market risk (that is, their prices are not fully predictable and can change quickly and by significant amount)
Is bank deposit a risk free asset?
Nominal return may be, but real, inflation-adjusted not
Banks can become insolvent, so focus on those with a high credit rating (e.g., AA)

14. Return to risk (cont.) How to account for risk?
Risk-adjusted return can be written as a combination of the risk-free rate and market risk premium
E(x) = x(rf) + β*RP* (Capital-Asset Pricing Model)
RP = (x(r) – x(rf)) is called the “risk premium”
β = σ(r)/σ(p) is the slope called the “risk premium”; the steeper the slope, the greater the additional expected return from higher share of risky assets (volatile assets have high beta)
σ(p) = std. of the portfolio and σ(r) = std. of the risky asset

15. Insurance risk Assume that the initial wealth is $300,000
There is 10% probability that wealth will fall in value to $60,000
Possible loss is $240,000 with probability of 0.1
How much would you pay for insurance to cover the risk of loss in wealth?
The expected value of wealth is $276,000 (E(x))
But same utility is given by $267,800 (E(U(x)) because E(U(x)) > U(E(x)) due to concavity of the utility curve
Draw a curve to see this

16. Investment risks
For a company making an investment into a machinery, the cost of the machine is up-front and known
If the investment is financed with borrowed money, the cost of the loan may also be known (if at fixed rate)
What is not know is the cash flow resulting from output produced with the new machine
Recall the expression for net present value (NPV) that sums the firm’s future discounted cash flows
In reality, each period’s cash flow is expected, subject to uncertainty (this is equal to the “profit guidance” of companies)

17. Investment risks
Example: company’s projected net revenues are for the next five years $100,000 per year, subject to the signed probabilities by the management: 0.9, 0.8, 0.7, 0.6, and 0.5 (i.e., on the fifth year, the likelihood to receive the projected revenue is 50-50). The risk-free discount rate is 10%.
The risk-adjusted NPV is then $272,600 (calculate); the risk-unadjusted return is $379,100 (calculate)
This takes into account uncertainty related to future returns – makes a big difference for decision-making
Actual, ex post, returns can differ from the projected, but the calculation helps to gauge the return to firm’s investment

18. Oil drilling - example
A company involved in oil drilling must decide whether to drill at the given site before the option period expires
The cost of drilling is $200,000, which will be lost if the drilling site is “dry”, but the company will earn a profit over the life of the well equal to $600,000
The probability of striking oil is 0.6 while the likelihood of hitting a dry well is 0.6
If the company does not drill, then it will have no costs and earn no profits

19. Oil drilling (cont.)

20. Oil drilling (cont.)
What is the expected profit from drilling (shown in the circle)?
In reality, decisions are much more complex
There may be multiple wells with different outcomes and probabilities
How does one undertake decision-making in such a situation?
Let’s look at the following example

22. Oil drilling (cont.)
The values in the circles indicate the expected outcome from a particular well
Note the last line on the bottom for no oil, which as a high probability (0.66) and will incur a cost of $400,000
Given the probabilities, drilling is not profitable
Hence the company should allow the option to expire