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Uncertain Outcomes

Here we study projects that have uncertain outcomes and we view

various ways people may deal with the uncertain situations.

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Overview

Up to this time we have operated as if outcomes occurred with certainty. This may not be the case. We deal with these situations by

1) calculating the mean and variance of each option, and

2) considering the disposition people have for these situations.

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the mean

Say I flip a coin and give you a dollar if it comes up heads, but you give me a dollar if it comes up tails. From your perspective, the mean, or expected value, is

Ex = .5(1) + .5(-1) = 0.

The expected value is a summation problem, where each term in the sum is the probability of an outcome times the value of the outcome. As another example, say a homeowner has no insurance on a $100,000 house with a 2 % chance of a fire doing a total loss. The homeowner has an expected value of:

Ex = .02(0) + .98(100,000) = 98,000

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the variance

Say I flip a coin, but this time give you $10 if it comes up heads and you give me $10 if it comes up tails. From you perspective, the mean, or expected value, is

Ex = .5(10) + .5(-10) = 0.

Notice this second coin example has the same expected value as the first, but the payoff amounts are more extreme. When values are more extreme they are said to have more variation.

A measure called the variance is calculated to give a number to represent the variation. The standard deviation, the square root of the variance, is also commonly calculated. A bigger number by either measure indicates more variation.

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variation in the coin examples

Var1 = .5(1 - 0)2 + .5(-1 - 0)2 = 1.

Var1 = .5(10 - 0)2 + .5(-10 - 0)2 = 100.

Notice the variance is also a summation, but each term is the probability of the outcome times the square of the deviation of the outcome from its expected value.

Deviation = outcome minus expected value.

The standard deviation is the square root of the variance:

sd1 = 1 and sd2 = 10. The $10 gamble has more variation. Options with more variation are said to have more risk.

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Types of people

As we define types of people we consider options A and B. Both have the same expected value and A has zero variation, a sure bet, while B has some variation.

A risk averse person prefers A to B.

A risk loving person prefers B to A.

A risk neutral person is indifferent between A and B.

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applications

Travelers have the option of eating at local restaurants or the national chains as they drive through unfamiliar towns. The national chains are a “sure bet” in that the quality of food is a know value. Risk averse people go to the chains, risk lovers go to the local store and the risk neutral folks are indifferent between the two.

When it comes to our homes, many people are risk averse because we pay to make our wealth a sure bet - the sure bet is the value of the home minus the premium. Notice the premium can not be outrageous.

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applications

Managers of a firm may have any of the attitudes about risk we mentioned before. But, in the modern corporation the manager may be given the incentive to be risk neutral. Stockholders want the highest return, and although with high return there is often high risk, stockholders diversify their portfolios to even out the risk

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St. Petersburg Paradox

Say you are offered the following gamble: A coin will be flipped until heads comes up and the payout is two cents raised to a power where the power is the number of flips it took for the first head to come up. For example, if head comes up on the first flip you get 2 cents, or if it comes up on the third flip you get 8 cents. Experts have shown the expected value of the game is:

Ex = (1/2)2 + (1/2)222 + (1/2)323 + ... = a really huge amount.

How much would you pay to play the game?

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St. Petersburg Paradox

You may often hear that firms will be purchased for the expected value (in present value terms) of the future profits. But just like the example of the gamble in the St. Petersburg Paradox, the risk may be so large that the purchase price is much less than the expected value. The price is discounted to account for the risk.

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Decisions under Risk

A Different look at Utility Theory

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Let’s consider an example.

Say one option for you is to take a bet that pays $5,000,000 if a coin flipped comes up tails and you get $0 if the coin comes up heads.

The other option is that you will get $2,000,000 with certainty. (Say your grandmother will give you $2,000,000 if you do not bet.)

EMV of the bet = .5(5,000,000) + .5(0) = 2,500,000

EMV sure deal = 1(2,000,000) = 2,000,000

Choosing the option with the highest EMV, expected monetary value is a popular decision rule. But, now with a sure bet we may decide to avoid the risky alternative. Would you take a sure $2,000,000 over a risky $5,000,000? Is that your final answer?

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Utility Theory is a methodology that incorporates our attitude toward risk (risk is a situation of uncertain outcomes, but probabilities are known) into the decision making process.

Utility value

Monetary value

It is useful to employ a graph like this in our analysis. In the graph we will consider a rule or function that translates monetary values into utility values. The utility values are our subjective views of preference for monetary values. Typically we assume higher money values have higher utility.

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Say we observe a person always buying chocolate ice cream over vanilla ice cream when both are available and both cost basically the same, or even when chocolate is more expensive and always when chocolate is the same price or cheaper. So by observing what people do we can get a feel for what is preferred over other options.

When we assign utility numbers to options the only real rule we follow is that higher numbers mean more preference or utility.

Even when we have financial options we can study or observe the past to get a feel for our preferences.

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In general we say people have one of three attitudes toward risk. People can be risk avoiders, risk seekers , or indifferent toward risk.

Monetary value

Utility value

Risk avoider

Risk indifferent

Risk seeker

Utility values are assigned to monetary values and the general shape for each type of person is shown at the left. Note that for equal increments in dollar value the utility either rises at a decreasing rate(avoider), constant rate or increasing rate.

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Utility

$X1 X2

U(X2)

U(X1)

Here we show a generic example with a risk avoider. Two monetary values of interest are, say, X1 and X2 and those values have utility U(X1) and U(X2), respectively

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Utility

$X1 X2

U(X2)

U(X1)

Say the outcome of a risky decision is to have X1 occur q% of the time and X2 occur (1 – q)% . Then the EMV is

q(X1) + (1 – q)(X2). The expected utility of the risky decision is found in a similar way and without proof I tell the expected utility is

EMV

along the straight line connecting the points on the curve directly above the EMV for the decision. We have the expected utility as EU = qU(X1) + (1 – q)U(X2)

EU

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Utility

$X1 X2

U(X2)

U(X1)

The decision maker may have an option that is certain. If so, the EU is simply the utility along the utility curve. So in this diagram we see that any sure bet greater than Y has an expected utility greater than the expected utility of the risky option.EMV

EU

Y

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Utility theory then suggests that the alternative that is chosen is the one that has the highest expected utility.

Example:

Say a risky alternative has 45% chance of getting $10,000 and a 55% chance of getting -$10,000. Say

U(10,000) = .3 and U(-10,000) = .05 and the U(0) = .15 and say a certain alternative has a value of 0.

EU of risky deal = .45(.3) + .55(.05) = .1625

EU of the certain deal = 1(.15) = .15 The person will choose the risky deal.

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Another Example

Say Utility U = square root of X, where X is a dollar amount received by a person.

Then U(4) = 2 and U(16) = 4, for example.

Say a risky option will pay 4 50% of the time and 16 50% of the time. The expected value is 10 because

.5(4) + .5(16) = 10 and the expected utility is 3 because

.5U(4) + .5U(16) = .5(2) + .5(4) = 3.

Now, if there is an option that will pay more than 9 with certainty, than the certain option is better. Let’s see this on the next slide.

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4 9 10 16 x

U(x)

U(16)=4

EU = 3

U(4)=2

U(x)

Any certain option above 9 gives a utility value greater than the expected utility of the uncertain option.

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Reducing Risk

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In a previous section we mentioned that sometimes we face an uncertain situation with regards to monetary values. We saw

1) The expected monetary value, EMV, of the uncertain situation (what I will now call a gamble) is the sum of some numbers, where each number is a monetary value multiplied by its probability of occurring,

2) The expected utility of a gamble (what I will now write as E[U(G)]) is the sum of some numbers, where each number is the utility of a monetary value multiplied by its probability of occurring,

3) The expected utility of a gamble does not occur on the utility function (unless the person is risk neutral), but on the chord or line segment that connects utility values of each part of the gamble and directly above the EMV of the gamble.

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U

YY1 EMV Y2

U(Y1)

U(Y2)

E[U(G)]

Say we have a risk lover and the gamble G pays Y1 p1 % of the time and Y2 p2% of the time.

EMV =p1Y1 + p2Y2

E[U(G)] = p1U(y1) + p2U(Y2)

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U

YY1 EMV Y2

U(Y1)

U(Y2)

E[U(G)]

Say we have a risk avoider and the gamble G pays Y1 p1 % of the time and Y2 p2% of the time.

EMV =p1Y1 + p2Y2

E[U(G)] = p1U(y1) + p2U(Y2)

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On slides 3 and 4 I show you generic cases of a risk lover and a risk avoider. You see a gamble with monetary values Y1 and Y2, with associated probabilities p1 and p2 (where p2 = 1- p1).

I now want to show something we saw in the previous section, but I want to be more precise in my language.

Sometimes we may have an opportunity that is known with certainty. The utility of the opportunity will be on the utility function for the individual and will be noted U(C).

The decision rule for choosing between a gamble and a certain payoff is

-choose the certain option when U(C) > E[U(G)], and

-choose the gamble when U(C) < E[U(G)].

Of course, when the two are equal the individual would be indifferent between the two.

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Back on slides 3 and 4 I have some vertical dashed lines. I put them there on purpose. I want you to think of the location as values of a certain payoff, I now call C, and then we can see that

U(C) = E[U(G)]. The payoff C is called the certainty equivalent of the gamble.

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U

YY1 EMV Y2

U(Y1)

U(Y2)

E[U(G)]

Say we have a risk avoider and the gamble G pays Y1 p1 % of the time and Y2 p2% of the time.

The risk premium, rp, of a gamble is the EMV of the gamble minus the certainty equivalent of the gamble.

rp = EMV - C and will always be positive for a risk avoider.

The risk premium for a risk lover will be negative and it will be zero for a risk neutral person.

C

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U

YY1 EMV Y2

U(Y1)

U(Y2)

E[U(G)]

C

A Gamble of no fire insurance Say Y2 is value of property if no fire and Y1 is the value of the property with a fire.

The EMV = p1Y1 + p2Y2.

E[U(G)] = p1u(Y1) + p2U(Y2)

C is the certainty equivalent of the gamble.

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If a person buys insurance it changes the risky situation into a certain situation.

If Y2 - C = fee paid for insurance the individual will have C with certainty. To see this we note

If no fire the individual has Y2 - fee = C, and

If fire the individual gets restored to Y2 and has still paid the fee so the certain property value is C.

SOOOOOO

Y2 - C is really the maximum fee the person would pay for insurance and they would like to pay less.

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Let’s take the point of view of the insurance company - and we do not have to look at the graph here.

They pay claim of Y2 - Y1 p1% of the time and they pay 0 p2% of the time for an expected claim of

p1(Y2 - Y1) + p2(0) = p1(Y2 - Y1)

This is called the actuarially fair insurance premium - meaning this is the minimum they have to charge to be able to pay out all the claims.

Now look in the graph - here is an amazing result:

Y2 - EMV = Y2 - (p1Y1 + p2Y2) = Y2(1 - p2) -p1Y1

= p1(Y2 - Y1), so Y2 - EMV is the actuarially fair premium

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Review

Y2 - EMV = least insurance company will charge,

Y2 - C = Most person will pay,

(Y2 - C) - (Y2 - EMV) = EMV - C is the room the person and the insurance company have to negotiate for the insurance. Before we said EMV - C was the risk premium and now we see it is the most the person would pay over the actuarially fair premium to insure against the gamble.

Now insurance companies pay out claims and pay employees and electricity and other admin. expenses. The company has to get some of EMV - C to pay these expenses.

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People won’t buy the insurance if the insurance company needs more than EMV - C to cover its other expenses because the person would have more utility without it in that case.

Next let’s look at how information can be beneficial in reducing risk.

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Y

U

Y

U

Y1 C EMV Y2

Y1’ c’ EMV’ Y2

situation without information

situation with information

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On the previous slide I show two graphs. Both have the same utility function for an individual. The top graph is a situation where the individual has no information and the bottom graph shows what happens when more information is obtained.

Note more information may not eliminate risk, but it can reduce it. Let’s study an example to show context.

Say an individual can buy a painting and if it is a real master painting the wealth of the individual will be Y2. If the painting is a fake the individual will lose some of his expenditure because the painting is no big deal - wealth is Y1. We see the certainty equivalent of the gamble is C. Presumably the individual will buy the painting if the certainty equivalent of the gamble is better than his wealth by not buying the painting at all.

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Now say the person can hire a painting expert to see if the painting is a fake or not. If the expert says the painting is a fake then you will not buy it and will not lose on the low end. But if it is a real painting you will have the same high end wealth because you will buy the painting.

So information from an expert in this case gives you the same high end value but makes your low end value better than without information.

But, the expert is going to want to charge you for the information. How much should you pay?

Since C’ is the certainty equivalent with information and C is the certainty equivalent without the information, the person would pay up to C’ - C for the information and the utility of the person would be improved.