1.3 Uncertainty

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dice question mark1.3. Uncertainty

+ define risk and uncertainty
+ define risk as used in financial analysis
+ introduction to simple expected utility theory
+ introduction to simple prospect theory

Lecture  What is uncertainty, and how do we deal with it?

Part I – Risk and Uncertainty
There’s a small cottage industry built on the distinction between risk and uncertainty. One of the founders of this industry was Frank Knight, a University of Chicago economics professor who taught there between the late ’20s and the mid-’50s. Although even Knight seems confused, the basic distinction is that risk is something you can insure against–which means the probability of specific losses (events) is known or can be calculated, and uncertainty is when you can’t assign probability to outcomes.

I generally find most conversations about risk and uncertainty unenlightening. Many get bogged down in semantics, many are just confused or confusing, and rarely do they increase my understanding. You should read the paper “What is Risk?” even though I don’t agree with everything that’s written there. I think there IS a valid (philosophical) discussion that we could have about uncertainty, risk, and related topics. I personally like the idea of using risk as a term to describe a action with a known probability distribution over outcomes. I also think there’s ignorance–where because of lack of experience or knowledge, we just don’t have a complete probability distribution or description of the outcomes and likelihood of each, but we know a little bit about it. Pure Knightian uncertainty either describes everything (we never know the complete probability distribution of most things, except maybe all the games offered at a casino) or it describes nothing (other than some truly obscure or irrelevant things.) I tend to use the term UNCERTAINTY as a general word to describe that which is unknown in advance. I’m uncertain how each of you will do in this class, but come December 20, I’ll have a complete and perfect idea!

As for risk, it’s just not a particularly helpful term because of how it’s used in everyday speech. According to Merriam-Webster, risk is “possibility of loss or injury” (definition 1) or “the chance that an investment (as a stock or commodity) will lose value” (definition 4). Nearly all uses of the term risk highlight the negative aspect of loss, or danger. We wouldn’t call a lottery risky–there’s a certain loss (the cost of the ticket) and only money to be won, not lost. The Russian roulette scene in the Deer Hunter is clearly one of risk!
If you look up a finance or investment book, you’ll find that risk has a far less negative definition. The NASDAQ website defines risky asset as “an asset whose future return is uncertain.”  Sounds like uncertainty to me–or at least “unknown in advance.” Yes, you can own an asset that falls in value over time (bad,) or you can own one that increases in value over time (good.) But the disciplines of economics and financial economics use the term risk to mean uncertainty, and it is neither good nor bad.

So we are faced with a dilemma. Two terms, many definitions, no clarity. I DO NOT use the term risk to mean “bad outcome.” To me, there is nothing negative about risk, other than the very general thought that I’d rather know, in advance, what is going to happen than not (but that’s called risk aversion–more later.) I use risk and uncertainty interchangeably. Not because I want to confuse you, but because I want to de-emphasize the loss/negative-only interpretation of risk.

From a statistical view point, uncertainty is the basis for probability theory. There’s an experiment in which some process (rolling a dice) generates specific outcomes (the number of dots on the upper face). If the die is fair, then we can specify the probability distribution over all outcomes: Pr(N=i, i=1…6) = 1/6. This distribution satisfies all the rules of a probability distribution. It DESCRIBES our uncertainty. We KNOW one of six faces will appear, but WHICH face is unknown in advance.

Note that uncertainty is a consequence of how we experience time. (This is deep…) Because time seems linear (past, present, future) then before something happens we are uncertain as to the outcome, so we must form expectations as to what will happen. Once it happens, there is no more uncertainty–the outcome is realized and we can observe it. As good Bayesians, we can use the observed outcomes to revise our understanding of the probability distribution, and make better predictions of the future.

In finance, asset risk is measured by the variation in returns from that asset. As we will see in the next topic (2.1), we can calculate the expected return and risk measure for every asset, and every portfolio of assets. This is a fundamental statistic when choosing and managing portfolios.

In conclusion, the most important thing is this: a risky asset not only offers the chance of losing a lot of value, it also offers the chance of increasing in value a lot. If you want a “ten-bagger” (a term coined by Peter Lynch) you must find an asset with great up-side potential, which will almost surely come with some large down-side potential as well! We can think of asset risk as having two “sides” – there’s up-side or opportunity, and there’s down-side or exposure. They can be acknowledged, and managed, but not separated. Risk is the investor’s friend (and enemy!)

Part II – Risk Preferences
Go to this website and take the risk assessment. This is an academic quiz, so as you complete it you are also contributing to research–thanks. Do you take risks? Do you have health insurance? Would you drive your car home after having two drinks with a friend? Would you buy a lottery ticket? Most people don’t THINK about their attitudes to risk very much, but their choices tell us a lot about how they view risk. Unfortunately it’s not easy to describe a person’s willingness to accept risk. Some people pay health insurance premiums, but text while they drive. Buying insurance says you’re risk averse, while texting while driving is risk seeking behavior.

There’s a lot that economists, psychologists, behavioral economists have written about risk. For a long time, the basic model was Expected Utility Theory.

Expected Utility Theory
Consider a game in which you are faced with two choices:
A: Lose $10 with certainty
B: Toss a fair coin, if Heads you win $20, if Tails, you lose $20.
Would you play this game? Since most people don’t like losing money, they would probably play the game. There’s a 50:50 chance you will win $20. But there’s also the same chance you would lose more than $10–$20 in fact. Is there a way of thinking about this game more systematically?

In statistics there’s a concept called Expected Value, which can be calculated for probability distributions. EV = Pr(L1).L1 + Pr(L2).L2 + … + Pr(Ln).Ln where there are n possible outcomes Li, i=1…n and Pr(Li) is the probability that outcome Li occurs. All the probabilities have to sum to one, to be sure it’s a probability distribution.
Take choice B above: L1 = -$20, L2 = +$20, Pr(L1) = Pr(L2) = 0.5
so EV(game) = -$10 + $10 = $0.
Choice B is “worth” $0. Compared to the certain loss of $10 from choice A, it would seem best to go for the toss of the coin.

Consider another game:
A: Receive $12 for sure, or
B: toss a fair coin: if Heads win $20, if Tails win $4.
The expected value of B is EV(B) = (0.5)$4 + (0.5)$20 = $12. So choice B has the same (expected) payoff as the certain payoff from choice A, but choice B is risky. You can’t actually win $12 from B, only $20 (good) or $4 (not so good.) Many more people would take the certain choice in this game. But why? The explanation lies in expected utility.

Assume there is a utility function that represents the utility/satisfaction/well-being obtained from payoffs in games of chance (the outcomes are uncertain.) We can compare the utility of the expected value of choice B with the expected utility of choice B. That is:
(i) U[EV(B)] = U[(0.5)4 + ((0.5)20)]
(ii) EU(B) = (0.5)U($4) + (0.5)U($20)

Expected Utility Graphic

In the diagram above, the Expected Utility function is concave–it displays diminishing marginal utility as income increases. In this case the utility of the expected value of the outcomes is greater than the expected utility from tossing the coin. In other words, a person with this shaped utility function would prefer a certain $12 than play a game with an expected value of $12. If a person has such a utility function, they are risk averse. If a person has a convex utility function, they would prefer to toss the coin than take the $12, and are thought of as risk-loving, or risk-seekers.

The diagram of the simple model reveals a few other interesting things. Notice that the utility from playing the risky game, B, is EU(B), which is the same level of utility associated with the certain amount C on the horizontal axis. The person is indifferent between tossing the coin and the certain payment C. The expected value of the coin toss is $12, but the equivalent certain payment is less than $12. Maybe it’s $10. The amount C is called the certainty equivalent, and the difference between the EV(B) and C is important. Consider offering this person insurance that would pay them $C rather than face the risky outcome of the coin toss. How much would the person pay to buy the insurance? Up to the difference EV(B)-C. A risk averse person would rather a lower certain payment than have to face a risky choice.

Expected Utility Theory is quite powerful, but has many limitations. There are too many inconsistencies with observed behavior for some people to think it’s a good model. One alternative model has emerged called Prospect Theory.

Prospect Theory
Developed by Daniel Kahneman and Amos Tversky, (Kahneman is a winner of the Nobel prize in Economics (Tversky died before the award was made) and has a recent best seller, Thinking Fast and Slow,) Prospect theory tries to describe people’s behavior more closely than EUT. The major insight of Prospect theory is that people view gains and losses relative to a reference point, rather than absolutely as assumed in EUT. People are sensitive to changes in wealth, and more sensitive to losses than gains of the same magnitude. The diagram below represents a value function (equivalent to the utility function) defined over gains and losses relative to a base point of X*.


Another major element of Prospect theory is the weighting function. Rather than weight outcomes by their objective probability as in EUT, people are modeled as using a subjective probability weighting scheme that tends to give low probability events a larger weight (more importance) and higher probability events less weight (relatively less importance.)
Prospect theory has been successfully applied to financial behavior, especially that of investors. Read this paper by Barberis from 2012 for more details (even though the paper is somewhat technical, the main points are clearly stated.) Check out this very informative reference website for all things Prospect Theory.

Matching Risk Attitudes to Risk Opportunities
One of the major services offered by financial advisors is creating an investment portfolio that matches the risk attitudes or tolerance of the client. By many different means, they assess the risk preferences of the client, then construct a matching risk profile for the assets (securities) in the portfolio. The ideas presented above provide the theory behind assessing risk preferences or attitudes. Unfortunately what these advisors do in practice is not real science, and they have varying success with the matches. Fortunately most clients don’t have a clue, and are forgiving.
The average investor only has a very rudimentary understanding of his/her own risk tolerance. But it’s a good idea to reflect on your own choices (not just an abstract reflection) as a guide to assessing your attitudes to risks.
With knowledge of one’s risk tolerance, an investor can venture out into a world full of investment opportunities. Each opportunity offers an expected return, and an associated risk–more on this later.

Take the Quiz on the Learn website.

Aside: Risk Management
Risk reduction may be a goal, and risk avoidance may be possible, but eliminating risk is practically impossible. There are few certainties in the future (even though everything in the past is knowable, except maybe who killed JFK…) If risk really is uncertainty, then we all face uncertain futures. Rather than trying to eliminate risk, people should be trying to manage it. Managing risk requires understanding the nature of uncertainty, and, when possible, taking advantage of it. Successful investing is very much an exercise in managing risk: reducing it where possible, avoiding when appropriate, and using it when it offers advantage. Risk is a rose–the joy of the scent, the pain of the thorns.

1.  What is Risk? The first chapter from a book titled Strategic Risk Taking by Aswath Damodaran, Finance Professor at NYU provides a nice introduction to the concepts of risk and uncertainty.
2. On Knightian Uncertainty:
3. A somewhat odd English psychology professor talking about investor emotions (He’s clearly not an economist!):http://www.youtube.com/watch?v=fel5sRL9azc
4. Search on Youtube for “NBR Your mind and your money” This is a series of short videos made by the PBS program Nightly Business Report. All interesting (and short.)
5. Wacky YouTube video on EUT: http://www.youtube.com/watch?v=IHsODoIYRH8
6. Review of Prospect Theory by Barberis, 2012.
7. A great interview with Daniel Kahneman, about psychology and economics:

Homework (due Monday Sept 19)
Write one-page (typed) on BETA. It’s a measure of relative risk for an individual stock, or a portfolio of securities. Give examples, and argue if beta is a measure investors should pay attention to.

Discuss any, or all, of the following questions in the online Learn discussion forum for this Topic.
1. The Economist had an article indicating that women are better investors than men. It could be due to different risk attitudes. Read this paper about differences in risk attitudes between women and men:
Does your experience (reading, personal) support these research findings? Give examples.
2. The most common way people manage risk is through insurance. You can buy car insurance and health insurance, for example, but there are lots of risks you can’t buy insurance for (even though Lloyds of London might underwrite any policy for a price.) And even if you can buy insurance, it compensates for losses rather than reverse those losses (you get money when injured, you don’t get your health back.) Do you think we should have more insurance for risks like losing money in the stock market?
3. Why do you think Daniel Kahneman was awarded the Nobel prize in economics? He is a psychologist after all.

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