Managerial EconomicsUncertainty

Aalto UniversitySchool of Science

Department of Industrial Engineering and Management

January 12 – 28, 2016Dr. 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) = σ2 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)2p(1) + … + (x(k) – μ)2p(k) = σ2

Large variance means that there is a lot of dispersion

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?

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 = 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

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.)