Managerial Economics Uncertainty Aalto University School of Science Department of Industrial Engineering and Management January 12 – 28, 2016 Dr. Arto Kovanen, Ph.D. Visiting Lecturer

 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

 Example:  Let x(i) be the value of the i-th outcome (i = 1, 2, …, k)  Let p(i) be the probability (or likelihood) that the i-th outcome will take place  The E(x) = x(1)*p(1) + … + x(k)*p(k) where the sum of p(i)’s equals one  Variance = V(x) = σ 2 measure how much each outcome deviates from the expected value (or mean = μ)  Example:  For the above sample variance is given by (x(1) – μ) 2 p(1) + … + (x(k) – μ) 2 p(k) = σ 2  Large variance means that there is a lot of dispersion 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? Uncertainty– general

 Importance of risk-free 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

 Because of diminishing marginal utility of wealth, individuals will pay to avoid risk and will need to be compensated for taking 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 considered “risk (default) free”  But even fixed-income securities have market risk (that is, their prices are not fully predictable)  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 Return to risk (cont.)

 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 Insurance risk

 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 Investment risks

 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 - example

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 Oil drilling (cont.)