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
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
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?
Uncertainty general Standard deviation = σ = (σ 2 ) 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?
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!
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: Uncertainty-example 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.
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., 1 + 1 + 1 + ), 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.
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?
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
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 = 100M 0.5. 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
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
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)
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)
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
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
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)
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
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
Oil drilling (cont.)
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
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