FIN 400 Notes 2 - Utility Theory PDF

FIN 400:Notes 2 UTILITY THEORY  Introduction Utility theory provides a model of how individuals make a choice amongst competing alternatives Such choice drives demand for goods and services Which in turn determines the true mkt price/value of assets It also gives an idea of...

FIN 400:Notes 2 UTILITY THEORY  Introduction Utility theory provides a model of how individuals make a choice amongst competing alternatives Such choice drives demand for goods and services Which in turn determines the true mkt price/value of assets It also gives an idea of how individuals behave in the face of uncertainty which is fundamental to understanding how financial mkts operates The theory is the building block of most financial decision making processes. It assumes a single person economy with a bundle of resources The individual has to decide upon current consumption and saving some of the resources for the future  Its therefore a current consumption vs future consumption decision The decision depends on the perceived utility the individual gets from each action His is basically a consumption vs investment decision that establishes an exchange rate This rate is the additional units at which he will be indifferent between current and future consumption A collection of these exchange rates combine to produce a single interest rate also known as the price of future consumption for the whole  At this market determined or established rate no one is worse off, in fact almost everybody is better off As such an exchange economy that uses mkt determined prices/interest rates to allocate resources across time is superior to an economy without the price mechanism. The price mechanism is the best system of allocating resources to their most productive use  And firms that make use of mkt determined interest rates in selecting projects creates wealth in the economy Consumption vs Utility The total utility function  Individuals prefer more consumption to less - Principle of Non-satiation  Marginal utility will always be positive  That is, as more and more units of consumption are added there will always be an increase in satisfaction  There is a diminishing marginal utility returns to scale as consumption increases  This means that at higher consumption levels the additional satisfaction that one gets from each added unit decreases  Current and future consumption combinations  There is a combination of current (C0) and Future (C1) consumption that an individual regard as having equal total utility and as such he is indifferent btn the two  Therefore these form an indifference curve and all combinations of (C0) and (C1) on the same indifference plane/surface have the same total utility  The decision marker is indifferent btn point A with consumptions (CoA,C1A) and  Indifference Curves Point A has more future consumption but less current consumption than Point B yet both has the same total utility Indifference Curves  Indifference Curves Note: The slope at point A is greater than at B on the same indifference curve indicating that individuals at lower current consumption level require more additional future units in order to give up current consumption Conclusion: The discussion so far explains the consumption side of the utility theory, next we shall discuss the production /technological limitations or constraints In an economy, individuals may desire to consume or invest certain levels of resources which may not be sustained by the production capacity. We therefore have to consider both the demand and supply side  Productive Opportunities These are investment opportunities that allow individuals to turn a unit of current consumption/resources[savings/investments] into more than one unit of future consumption  Each individual in the economy has a schedule of investment opportunities that can be arranged from the highest rate of return down to the lowest rate of return This rate is called the marginal rate of transformation (MRT) Although depicted as a linear function any decreasing function would do This implies diminishing marginal returns because the more the individual invests the lower the marginal rate of return Individuals will make all investments whose marginal return is higher than their rate of time pref for current consumption i. e keep investing as long as MRS Y , the individual should divest to get to 0 0 point B  Here at point B the investor’s required marginal rate of substitution (MRS) equals the marginal rate offered by the production opportunity set(MRT) MRS = MRT Also, the individual’s consumption in each time period is exactly equal to the output from production (P0 = C0* and P1 = C 1* ) Thus signifying equilibrium points  Summary of the production Opportunity schedule The existence of investment opportunity in the economy allows consumers to attain higher levels of utility Without investment opportunities consumers would be restricted to a utility level that is provided by their basic endowment In the above graph the consumer was able to move from point (Yo,Y1) to point B An individual should continue to invest as long as the return from investing (MRT) is greater than the consumer’s required return(MRS) The optimal point is when MRT = MRS If MRT < MRS the consumer should divest to get to the optimal MRT = MRS The Theory of Choice & Decisions under uncertainty 18  Theory of Choice This theory is often referred to as rational choice theory. It is a framework for understanding social and economic behaviour The basic notion of the theory being that aggregate social behaviour results from the behaviour of individual actors, each of whom is a making their individual decision The question of choice stems from the basic principle of economics that, in any situation there is limited resources that has to satisfy unlimited needs and wants 19  Cont’n As an economic principle the theory assumes that individuals always make prudent and logical decisions that provide them with the highest amount of personal utility. This therefore requires that the individual is able to determine with a certain level of certainty and consistence the level of reward to be received from each decision This basically means that the individual should be able to measure and compare different decision scenarios in order to make a rational or value maximising decision This is achieved through the achievement of the 20 following axioms:  Axioms of Choice under Uncertainty To develop the theory of rational decision making in the face of uncertainty, it is necessary to make some precise assumptions about an individual’s behaviour. Known as the axioms of cardinal utility, these assumptions provides the minimum set of conditions for consistent and rational behaviour We present these five axioms of choice next 21  Axiom 1: Comparability (completeness) for a complete set (U) of choices, an individual is able to say whether they a) Prefers X over Y → X > Y or b) Prefers Y over X → Y >X or c) Is indifferent between the two → Y~X 22  Axiom 2: Consistence (Transitivity) a) if an individual prefers X to Y , and prefers Y to Z, then X is preferred to Z at all times b) if an individual is indifferent as to X and Y , and is also indifferent as to Y and Z, then s/he is indifferent between X and Z 23  Axiom 3: Strongly Independence If there is a gamble where an individual has a probability of a of receiving outcome X and a probability of (1- a) of receiving outcome z The gamble is written G(x, z: a ) Strong independence says if the individual is indifferent as to x and y then the individual will be indifferent between the first gamble G(x, z: a ) and another gamble G(y, z: a ) 24  Axiom 4: Measurability If outcome Y is preferred less than X but more than Z, then there exist a unique probability a such that the individual will be indifferent between y and a gamble between x with a probability a and Z with a probability of (1- a) If x> y≥ z or x ≥ y> z, then there exist a unique a such y ̴ G(x, z: a ) 25  Axiom 5: Ranking If alternatives y and u both lie somewhere between x and z and we can establish gambles such that an individual is indifferent between x(with probability a1) and z, while also indifferent between u and a second gamble, this time between x (with probability a2) and z, then if a1 is greater than a2 , y is preferred to u. 26 Utility Theory Under Uncertainty 27  Overview Economic decisions made under uncertainty are essentially gambles. Let’s first look at some gambles, and then come back to decisions under uncertainty. Initially, our gamble will be to flip a fair coin (one where the probability of a head is 0.5 and the probability of a tail is 0.5). The payout will depend on which side of the coin is showing when the coin lands at rest. 28  Examples Facts about several gambles with a fair coin: Gamble 1 – heads means you win $100 and tails means you lose $0.50 Gamble 2 – heads means you win $200 and tails means you lose $100 Gamble 3 – heads means you win $20000 and tails means you lose $10000 Note what a person would lose on each gamble. Many people would say the loses in gambles 2 and 3 make them uneasy and they wouldn’t take those gambles. But, some folks out there might take gambles 2 and 3. 29  Digress – the mean What is the mean or the average of the numbers 4 and 6? You probably said 5 and you are right. This could be written (4+6)/2 = (4/2) + (6/2) = (1/2)4 + (1/2)6, where in this last form you see each number multiplied by ½. In this context the mean is said to be a simple weighted average, with each value weighted by ½. What would the weights be if we wanted the average of 4, 5, and 6? 1/3! In general with n numbers the weight is 1/n. In other situations we may look at a weighted average (not simple), though the weights are found in a different way. 30  Back to example The expected value of a gamble is a weighted average of the possible payout values and the weights are the probabilities of occurrence of each payout. We talk about EVi as the expected value of gamble i. EV1 =.5(100) +.5(-0.50) = 50 – 0.25 = 49.75. (Notice when you lose the loss is subtracted out.) EV2 =.5(200) +.5(-100) = 100 – 50 = 50 EV3 =.5(20000) +.5(-10000) = 10000 – 5000 = 5000 31  Example In our example the EV for each gamble is positive. The EV is the highest for gamble 3. But, remember we said not many folks would probably like it because of the uneasiness they would feel by losing the 10000. Von Neumann and Morgenstern (1947) created a model we now call the expected utility model to deal with situations like this. They indicated folks make decisions based not on monetary values, but based on utility values. Of course the utility values are based on the monetary values, but the utility values also depend on how people view the world. 32  Expected Utility 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. 33  Expected 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 It is useful to employ a graph like this in value 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. Monetary value 34 In general we say people have one of three attitudes toward risk. People can be risk avoiders, risk seekers , or indifferent toward risk (risk neutral). Utility Utility values are assigned value Risk avoider to monetary values and the general shape for each type of person is shown at the Risk indifferent left. Note that for equal increments in dollar value Risk seeker the utility either rises at a decreasing rate(avoider), constant rate or increasing Monetary rate. value 35  Utility Here we show a generic example with a risk avoider. Two monetary values of interest are, U(X2) say, X1 and X2 and U(X1) those values have utility U(X1) and U(X2), respectively $ X1 X2 36  Expected Utility In expected utility theory we want to focus on wealth values and utility values. Gambles will lead to adjustments in wealth. Let’s call W the initial wealth which can always be retained if no gamble is taken, and call the wealth at a loss X1 = W – loss, and the wealth at a win X2 = W + win. The expected value of wealth from a gamble is then (p1 is the probability of a loss and p2 is the probability of a win) EV = p1X1 + p2X2 Note we called the expected value of a gamble EV and I now have the expected value of wealth with a gamble being EV. EV will mostly stand for expected value of wealth, unless37otherwise stated.  Expected Utility The following is what needs to be considered to get the expected utility of a gamble: 1) Start with a person’s initial wealth W, 2) If the gamble is taken identify X1 and X2, 3) Assign to each wealth value in 2) the respective utility value U1, and U2 and in a graph connect the U1 and U2 values with a chord. 4) Calculate the expected value of wealth with the gamble EV. 5) Calculate the expected utility of the gamble as EU where EU = p1U1 + p2U2, and find the value on the chord above the EV. 38  Utility U2 U1 EU $ X1 EV X2 39  Example continued Say utility is assigned by the function U = sqrt(wealth) and a person has initial wealth 10000. Then for the three gambles we had before the EU’s would be EU1 =.5sqrt(9999.5) +.5sqrt(10100) = 100.248 EU2 =.5sqrt(9900) +.5sqrt(10200) = 100.247 EU3 =.5sqrt(0) +.5sqrt(30000) = 86.603 So in terms of EU’s the preferred order of gambles for this person is gamble 1, then 2, and the 3. When we looked at EV’s the order was 3, 2, and 1. So expected utilities of gambles may have a different rank ordering than when looking at the EV’s. 40  Fair gambles A fair gamble is one where the expected value of the gamble is zero, i.e., p1(-loss) + p2(win) = -p1(loss) + p2(win) = 0. This implies that the expected value of wealth with the gamble is equal to the value of wealth when not gambling at all, which you might call your certain wealth. For fair gambles EV = p1(W – loss) + p2(W + win) = p1W +p2W – p1(loss) + p2(win) = (p1+p2)W + 0 = W, since p1+p2=1. 41  Risk averse fair gamble In this graph I have the generic Utility view of a risk avoider. With the fair gamble we have the EV and the EU is on the chord above the EV. Uw If the person does not gamble EU wealth will be W and the utility there is just read off the utility function here as Uw (note a risk averter has diminishing marginal X1 EV X2 wealth utility of wealth.) =W 42  Risk Averse fair gamble For the fair gamble we again know EV = W, but for a risk averse person Uw > EU. Thus we can conclude risk averse folks will not accept fair gambles. On the next slide you can see I thickened part of the horizontal axis and the chord connecting the two points on the utility function associated with the wealth values under the gamble. The probabilities of the gamble could be changed (and the gamble would no longer be fair) and the only way the person would accept the gamble over having the certain wealth W is if the EV was greater than W*. So, a risk averse person may gamble, but it has to be at favorable odds. 43  Risk averse fair gamble If the EV of a gamble is above Utility W* (and is no longer a fair gamble, but a favorable one), then the person will end up on the chord segment that has not been Uw thickened and thus only then have EU EU>Uw. X1 EV X2 wealth =W W* 44  Risk Seeker fair gamble In this graph I have the generic view of a risk seeker. U With the fair gamble we have the EV and the EU is on the chord above the EV. EU If the person does not gamble wealth will be W and the Uw utility there is just read off the utility function here as Uw (note a risk seeker has X1 EV X2 W increasing marginal =W utility of wealth. 45  Risk Seeker fair gamble For the fair gamble we again know EV = W, but for a risk seeker person Uw < EU. Thus we can conclude risk seeker folks will always accept fair gambles. On the next slide you can see I thickened part of the horizontal axis and the chord connecting the two points on the utility function associated with the wealth values under the gamble. The probabilities of the gamble could be changed (and the gamble would no longer be fair) and the only way the person would NOT accept the gamble over having the certain wealth W is if the EV was less than W*. So, risk seeker may NOT gamble, but it has to be at 46 unfavorable odds.  Risk Seeker fair gamble If the EV of a gamble is below W* (and is no longer a U fair gamble, but an unfavorable one), then the person will end up on the EU chord segment that has not been thickened and thus only Uw then have EU 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. 58 Back on the slides 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) = EU. The payoff C is called the certainty equivalent of the gamble. 59  Say we have a risk avoider and the gamble G leaves Y1 p1 % of the time and Y2 p2% of the time. U The risk premium, rp, of a U(Y2) gamble is the EV of the EU gamble minus the certainty equivalent of the gamble. U(Y1) rp = EV - 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. Y Y1 C EV Y2 60  A Gamble of no fire insurance Say Y2 is value of property if no fire and Y1 is the value of U the property with a fire. The EV = p1Y1 + p2Y2. EU = p1u(Y1) + p2U(Y2) U(Y2) EU C is the certainty U(Y1) equivalent of the gamble. Y Y1 C EV Y2 61 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. Y2 – C is called the reservation price for insurance. 62 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 - EV = Y2 - (p1Y1 + p2Y2) = Y2(1 - p2) -p1Y1 = p1(Y2 - Y1), so Y2 - EV is the actuarially fair premium 63 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. 64 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. 65 U situation without information Y1 C EMV Y2 Y U situation with information Y Y1’ c’ EMV’ Y2 66 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. 67 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. 68