Applied Quantitative Analysis for Finance
Applied Quantitative Analysis for Finance FIN 203
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This 8 page Class Notes was uploaded by Rose Harvey I on Wednesday October 28, 2015. The Class Notes belongs to FIN 203 at Wake Forest University taught by Umit Akinc in Fall. Since its upload, it has received 11 views. For similar materials see /class/230707/fin-203-wake-forest-university in Finance at Wake Forest University.
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Date Created: 10/28/15
Risk Preference and Utility Financial decisions very often if not always involve risky choices Therefore valuing risky opportunities is a major challenge for nancial managers If several alternative actions were comparable in the amount of risk they involved the choice among them would have been easy Most rational decision makers would agree on the correct choice the one with the largest expected return EMV However when the alternatives involve dissimilar risks as they often do the correct choice is not as straight forward The maximum EMV criterion does not always describe the behavior of rational individuals because EMV is completely blind to risk and people pay attention to risk in making nancial choices To illustrate how EMV leads to uncharacteristic decision behavior consider a risky situation where you are asked to invest 10000 in a venture where there is one percent probability for a pro t of 1000000 and 99 chance of a complete bust The expected value of this investment is 100 But no sane individual would invest in this venture where losing 10000 is a near certainty Another example is a proposal to ip a fair coin for winning 100010r losing 10000 The expected value ofthis stake is 50 But again most rational people would not risk their hardeamed 10000 for an even chance of a 1 gain Without loss of generality since we are talking prudent business decision making we will assume that risk is undesirable and that unless they are somehow compensated for it most decision makers would try to avoid risk as much as possible In both examples above despite positive EMV the disutility of the inherent risk sways most rational people away from the choice based on the EMV criterion To further illustrate the conceptual dif culty in the evaluation of risky decision alternatives stemming from the subjective nature of risk avoidance consider an example where you put 1 to win 2 or nothing with even probabilities EMV 1 If risk is indeed undesirable we should expect rational people to be unwilling to pay the 1 for this lottery for they prefer their 1 to stay in their pocket rather than risking it on the lottery which has a risky expected payoff of 1 nothing or 2 However many people will as some of you might take this gamble Should we consider such people irrational Not at all those willing to take the gamble are being adequately compensated by the enjoyment and the thrill of the gamble Thus the risk behavior of sane individuals is a bit more complex than simple risk avoidanceiif the stakes are not too high they may even be risk seeking The whole multibillion dollar gambling industry is founded on marketing this enjoyment As the stakes increases however the disutility of the risk begins to exceed the utility of the enjoyment and thrill of the gamble We would nd very few buyers for a 50000 ticket for stakes of 100000 or nothing with equal probabilities The disutility of coming up empty and losing the ticket price of 50000 would be too much for the thrill of the gamble to compensate for most people A decision maker in making his choice has to account and make provisions for the risks underlying the alternatives There are two implicit components in his task to quantify risk and to establish a mechanism by which the measured risk can be traded against expected return Conceptually the later is the more dif cult step since the suf cient compensation for a given amount of risk depends on the risktaking behavior of the decision makeriinherently a subjective matter There are at least two alternative approaches one can take in measuring and trading risk the well known meanvariance approach and utility theory The rest of this note will brie y review the meanvariance concept but more fully develop the utility approach Mean Vanance Aggroach forRlsk Analysrs F u dumbhmons The nsk ofloslng one39s home to ahumeane ean be modeled by atwo valued Thu tl Mum rv vananee W assumes that every nskraverse declslon makerhas an rndllferenee euryequot where lncreaslng nsk w ofretum n kls 0 thus the value39 ofthe opportunrty ls EMV e A a where 7t ean be rnterpreted as the unlt eost ofnsk39 graph deprets a nskraverse deersron makerwho eonsrders porntA nsk free A F th declslon maker L A nsk ofC Furthermore for or tor certamty eqmvalent Uulrty Aggroach to Rlsk Analysrs ln monetary terms thrs approaeh assoerates a cardlnal quantr able uulrty functlon for varlous amounts of money The seale usedto measure uulrty ls arbltxary andls set by eonvenuon commonlyth 39 39 39 quot f Lquot L ith 1any other 39 39 39 A 39 L autili and 1 39 rational 39 39 39 39 39 39 1 mo is anon decreasing function ofamount ofmoney 22 no rational decision maker would prefer less money to more A 39 39 39 39 39 while a linear one models indifference to risk To see this consider the example you are otfered for a price a lottery ticket which will pay 2500 or 12500 with equal probabilities The expected return EMV for the ticket is 7500 05L2500 05012500 Would you pay 7500 to own this ticket 1fyou are risk averse ou would notifor ariskaverse person the additional utility ofmaking aprofit of 5000 12500 a 7500 does not quite balance the disutility oflosing5000 7500 2500 However there is aprice less than 7500 at which you mig t consider buying the ticket and assuming the risk How much at most would you pay for this lottery ticket For the sake ofdiscussion let s say m would pay 6000 some one else might pay more or less and no more This means that you are indifferent between 6000 for sure and the risky lottery 22 h L i uu As before CE ofthe lottery and the difference between the EMV and the certainty equivalent or 1500 as the nskpnzmmm RP As can be seen in the following graph apositive risk premium implies riskaversion and thus a concave utility function Such autility function has the property that the marginal utility of additional wealth decreases as wealth increases For someone with 20000 annual income a 1000 bump in salary would be vigorously celebrated while the same increase to quot mun b l 39 Ineresti ly quot 39 39 39 utility formoney is the same phenomenon as risk avoidance 10 1 3 0m 1 00 I 5000 7500 10000 12500 15000 X mnmmqnvbtm quotL r r quot ik outcomea probabilityquot quot 39 39 39 r39 quot 39 39 39 39 L r39 Suppose L fL A quot L A L o 394 39 twoofthe outcomes and replace them with their certainty equivalent and thus end up with a binary lottery This approach can be applied as many times as necessary to break any number of risky outcomes into a series of binary lotteries which are easier to analyze Before we go any further let us reiterate the purpose of trying to measure and trade risk and return The object in both approachesimeanvariance and utility is to incorporate the risk as a factor in choosing among alternatives This could be done either by substituting certainty equivalent risk adjusted return for the riskblind EM V as the decision criterion or equivalently using the criterion of maximum expected utility To illustrate the later consider a decision alternative with returns ole X2X3 XN with probabilities of p1 p2 p3 pN While the expected monetary value of this alternative is EM V pi the expected utility is Z Pi 39 Determining the utility function There are two main approaches for constructing the utility function of a decision maker The first as you saw in the text is to elicit it from the decisionmaker by offering him lotteries consisting of the best and the worst outcomes To brie y illustrate this process in a particular decision situation where the best outcome is 12500 and worst 0 suppose we arbitrarily set U0 0 and U12500 1 namely we will assign utility values between 0 and 1 to outcomes in the range from 0 to 12500 Say we want to find for some decision maker U5000 First we find the breakeven probability p0 of winning in the lottery such that expected monetary value ofthe lottery of 0 and 12500 is exactly 5000 Namely we solve 5000 0 1p0 12500 p0 forpo This yields p0 04 Ifthe decision maker is risk indifferent he should regard the lottery 0 12500 with a winning probability of 04 as desirable as a sure 5000 since by construction the lottery s EMVis 5000 If however he is risk averse he should prefer the sure 5000 to a risky 5000 EM V of the lottery The lottery however becomes more attractive if the winning probability increases gradually beyond 04 closing the utility gap between the sure 5000 and the lottery Thus for some probability of winning better than 04 the decision maker should be indifferent between the lottery and the sure 5000 Say this happens for a specific decision maker for a probability of winning of 055 i e the decision maker is as happy with the lottery if the probability of winning is 055 as he is with 5000 for sure This means that the expected utility of the lottery is equal to the utility of 5000 or U5000 1055 U0 055 U12500 By convention U0 0 and U12500 1 Substituting these in the equation we get U5000 055 This process can be repeated a number of times to obtain the utility of the decision maker for various amounts between 0 and 12500 to approximate his entire utility function Notice that the way the indifference lottery is set up implies that the certainty equivalent is 5000 and since the EM V 1 0550 055 12500 6875 the risk premium is 1875 The second approach is to use predetermined mathematical functions having properties appropriate to the decision maker in order to approximate his utility function For instance UX R has most of the properties of utility function of rational individuals U0 0 Uoo 0039 it is non decreasing and concave exhibiting the property of diminishing marginal utility of money but undefined for negative monetary results Two more commonly used mathematical functions for this purpose are the negative exponential utility function and the logarithmic utility function These represent different risk behaviors and thus may be appropriate in different situations for different decision makers 1 As you will see this convention simplifies the process of assigning utilities to outcomes averslon behavlor If a person Wants the same nsk premlum for a blnary lottery regardless of ms o after line lottery lfthe declslon maker starts wth varlous levels of wealth 39 Thls declslon vve nsk averse The neganve exponentlal unler functlon UX 2 W has constant 3750 3250 250 0 or 7500 5500 5000 2500 or 10000 500 5000 or 125 TV only parameter Tu W A 39 n r nl ThevalueofR esMz wth equal probabllmes 2 g 5 and 7025 10 and 705 1000 and 70500 etc makerfor the loss ofMZ Appropriate Rfordne deelslon maker ls thathorWhlch he ls N w RM n larger than M the unllty ofwlnnlng wlll be less than the dlsuullty ofloslng anol vrse versa 4i ullllwmwmmnu agnnvmlnsmu rm u M x R nzhgexlx 20000 flndthe cenamty equlvalent andthe nsk premlum N V applylng me followlng formula CE Example If a declslon maker39s c c o than this is a fairly risk tolerant decision maker willing to risk losing 10000 for fty percent chance of making 20000 EMV 042500 02 0 0410000 3000 CE 20000ln 04e 250020900 02e 0 04e 39100002 v 200001n 04 11331 02 1 04 06065 200001n8959 219969 RP EMVi CE 3000 219969 80031 The logarithmic utility function models decreasing riskaversion behavior Most people tend to become less riskaverse as they become wealthier Insurance buying behavior of many people illustrates this behavior Younger and thus poorer people tend to opt for small deductibles in their auto insurance because it may be a real hardship to cover the deductible in case of an accident However as the same people become more wealthy their ability to cover the same deductible increases making them more willing to take the risk of having to cover the deductible In fact when they become so wealthy that the potential loss from an accident is miniscule compared to their wealth they may not even buy insurance We see that many very large companies with large net worth opt for self insurance whereas smaller companies transfer many risks they may face to insurance companies To illustrate this behavior let us look at the previous example lottery with various initial wealth positions As the initial position wealth increases the risk premium for the lottery decreases Although there are many mathematical 2500 or 1 functions exhibiting this property the logarithmic function U X ln X A is frequently used in modeling the diminishing riskaverse behavior in financial markets The function is defined forX gt A where A is interpreted as the decision maker s entire net worth It is undefined for losses exceedingA since such losses are tantamount to complete ruin The risk aversion decreases ie decision maker s risk premium decreases as the he becomes richer For this function the certainty equivalent of n discrete uncertain returns X with probability p uncertain returns is given by CEX1AF XZ A X3 14 X 21 A Example if decision maker s A 20000 find the certainty equivalent and risk premium of the returns in the above example CE 2500 20000 4 0 20000 2 10000 2000004 7 20000 CE 497984 72478 617801 7 20000 2298196 EMV 3000 as before RP 3000 7 229817 70183 As we pointed out earlier in the meanvariance approach a risky investment is valued by adjusting the mean returns for the risk or as EM V A 02 Notice that a constant 1 implies constant riskaversion behavior and A 0392 is the risk premium that does not depend on EM V It is interesting to note that the exponential utility function approximates 7 by 05R Also the approximation becomes better as the returns distribution approaches to a normal distribution Thus the certainty equivalent of a normally distributed returns can easily be calculated as CE EMV 05R 0392 A complete example Consider the following three investment options real estate indexed fund and a new venture in nanotechnology The returns to the investments are estimated depending on the state of the economy and are given as in the following payoff table The investor has 1M to invest and does not wish to invest in more than one option The probabilities of Bearish Neutral and bullish conditions in the economy are estimated to be 025 035 040 New 25 2 30 a Calculate the EM V of all three and make a recommendation if the investor is risk neutral Converting the returns to monetary payoffs in 000 EMVRE 2025 3035 9040 415 EMVUN 5025 035 15040 475 EMVNV 25025 2035 30040 505 Based on the max EM V appropriate decision criterion for riskindifference N V is the best investment alternative b Suppose the decision maker us risk averse whose utility for money is U X R where X is a sum ofmoney and UX is his utilityfor this sum Does the utilityfunction imply decreasing constant or decreasing risk averse behavior To answer this question we need to see if the utility function implies increasing constant of decreasing risk premiums as the starting monetary position wealth increases Let s define the convenient lottery of 0 and 1000 with even probabilities We will calculate the risk premium for two arbitrary starting positions 0 and 1000 and see what happens to the risk premium If the starting position is 0 the lottery will result in 0 or 1000 EM V 500 Certainty equivalent is calculated by answering the question what is the sure amount whose utility is equal to the expected utility ofthe lottery Expected utility ofthe lottery is 5 0 1000 1581 thus the sure value whose utility equals 1581 is 15812 250 which leads to a risk premium of RP EllV7 CE 500 7 250 250 Assume now the starting point is now 1000 thus the lottery will end in 1000 or 2000 with expected utility of 051 1000 12000 3817 which corresponds to a CE of 38172 145711 since the EMV ofthe lottery is 1500 the risk premium is 4289 This analysis shows that as the individual becomes richer his riskaversion decreases c The decision maker s utility for money is negative exponential with R 5 000000 Which investment is the best We can calculate the CE of each investment and choose the best All amounts are in thousands CERE 5000 ln251e205000 35e39305000 40e39905000 4131 CEIN 5000 ln25e505000 35e390 40 elm5000 4676 CENV 5000 ln25e2505000 35e205000 40 e393005000 4557 Thus the Index fund is the best investment Notice that this investor has a fairly large capacity for risk R 5M means he is willing to risk 25M for an even chance for a gain of 5M and thus a fairly risky investment is chosen d Assume the decision maker now has a logarithmic utilityfunction withl 10M Which investment is best CERE 10000 20251000030 3510000904 7 10000 4140 CEIN 10000 502510000035100001504 7 10000 4713 CENV 1000025025100002035100003004 7 10000 4805 New venture is the selected investment for this investor Further exercise Verify the choices made in parts c and d by calculating the expected utility of each investment
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