Financial Services [Applications]

Transcription

1 Financial Services [Applications] Tomáš Sedliačik Institute o Finance University o Vienna tomas.sedliacik@univie.ac.at 1

2 Organization Overall there will be 14 units (12 regular units + 2 exams) Course process (place, time, program) Mid-term exam is scheduled or March 30 th 2012 Final exam is preliminarily scheduled or May 11 th 2012 During the course there will be 6 problem sets to be solved at home. For this purpose you will always have at least one week time. You may then present your solutions in class in the ollowing session. 2

3 Organization Evaluation: A midterm and a inal exam, each weighted with 40 percent o the overall score. The remaining 20 percent are awarded or solving the problem sets (15 percent; 6 x 2.5 or each problem set) and or class participation (5 percent; 2 x 2.5 or each presentation). A considerable part o the course material is covered by the presentation slides which I will use in class. Large part o the course material is based on the book Risk Evaluation, Management and Sharing, Harvester Wheatshea, by Louis Eeckhoudt and Christian Gollier. In particular, I recommend to read chapters 2 5, 9 and 10! 3

4 Organization All relevant parts o the course material can be downloaded rom the course homepage: Also, during the term every important inormation about the course will be announced on this homepage. In case you have any questions do not hesitate to ask! You can also write me an on tomas.sedliacik@univie.ac.at. You can also contact our secretary, Ms. Neumeyer, on +43-(0) i you have questions o organizational nature. 4

5 Overview 1. Risk evaluation [Decisions under Risk] i. Decision theory and Risk theory (brie overview) ii. Expected Value and Mean-Variance criterion iii. Expected Utility criterion a. Certainty equivalent, asking price und bid price b. Risk premium c. Degree o risk aversion 2. Applications i. Portolio optimization ii. Optimal insurance coverage 5

6 Decision Theory Making investment or any other decisions which do not only inluence the present but also the uture one automatically deals with uncertainty, since anything that is going to be realized in the uture cannot be predicted with certainty! In the terminology o the decision theory one speaks about certainty when only one state o nature can occur! When two or more states o nature (outcomes) are possible one distinguishes the ollowing two cases: 1) i [objective] probabilities can be attributed to all the possible states o nature we call it a risky situation i.e. decision under risk! 2) I no [objective] probabilities are known we call it an uncertain situation i.e. decision under uncertainty! 6

7 Risk Theory [Decision Under Risk: Lotteries] What is a lottery? To be precise, each lottery is itsel a random payo with known probability distribution! The so called payos are just realizations o the lottery in particular states o nature! In general, any set o payos with known probabilities may be called a lottery! Each lottery can be represented by a random variable with payos x 1, x 2 x n and the correspoding probabilities p 1, p 2 p n (when discrete) or x~ ~ (x) on a positive interval [a, b] (when continuous); where (x) is the probability distribution unction (density unction) o x. Note that a sure payo is also a lottery with the only possible payo having a probability o 1. x~ 7

8 Risk Theory [Decision Under Risk: Lotteries] Where do we encounter lotteries? We permanently encounter lotteries in our every day lie (e.g. travel by the underground, cheating during an exam or even acing more important decisions such as the establishment o a company, marriage, investment in the inancial market etc.) Thereore we must [and we also do] permanently evaluate lotteries in order to make decisions between them! Any subject (e.g. a person) which has to make a decision [between lotteries] we will call a decision maker (DM). 8

9 Risk Evaluation We will denote the initial wealth which the DM is endowed with when making a decision between lotteries by w 0. For simplicity, we will assume that the only payos which the DM takes into consideration are represented by his inal wealth, denoted by w. Also, as long as it is reasonable we will deine lotteries such that they are additive to wealth, i.e. w~ = w ~ x In general, o course, the relation between wealth and an arbitrary lottery may have dierent orms! 0 + 9

10 Risk Evaluation In general, to make a decision a DM ollows a particular objective (criterion), which ully represents his or her personal preerences. He or she then evaluates each possible lottery with respect to that particular criterion and inally chooses the one with the highest score. In coherence, one can assume that there is a particular value unction, denoted by V(w ) which the DM seeks to maximize. Accordingly when deciding among two lotteries he or she chooses the one with the higher value o the value unction, i.e. V(x) > V(y) x y. In other words, he or she chooses the lottery which best its his or her objective (criterion)! 10

11 Expected Value Criterion (EV) In this case we assume that the DM makes his decision based purely on the expected value o his inal wealth. Accordingly the value unction has the ollowing orm: V ( w~ ) =Ε( w~ ) =Ε( w + ~ x) = w + ( ~ x) 0 0 Ε where [as you certainly know] Ε n ( ~ x) = p( ~ x = x ) x or in the continuous case Ε( ~ x) = x( x) i= 1 i i b a dx 11

12 Expected Value Criterion (EV) Obviously, the risk involved in a lottery has no eect on the expected value o the lottery s payos! Thus we can call a DM ollowing the EV criterion to be risk neutral! Example: The ollowing two lotteries perorm equally w.r.t. the EV: x i p(x i ) y i p(y i ) 50 0, ,5 although each o them is characterized with dierent risk! 12

13 Expected Value Criterion (EV) Which one o the two lotteries wouldyou choose? Are you also indierent between them? The EV criterion is not always plausible! It is suitable i there are many independent random variables involved in a lottery (e.g. an average damage o multiple insurance units), as the overall risk may be negligible! However, i a lottery consists o only a ew or even just one independent random variable the EV criterion may be less suitable, as the overall risk may be substantial! 13

14 Expected Value Criterion (EV) Example: Diversiication Consider a person which aces a risk represented by the ollowing lottery: x i p(x i ) 0,99 0,01 This could or example be the potential loss (damage) resulting o that person s car robbery. 14

15 Expected Value Criterion (EV) What is the expected value and the variance resp. standard deviation o the potential loss aced by that person? Now suppose that two persons [each threatened by such risk] come together and agree on bearing airly the costs o the aggregate loss to be realized, i.e. each person covers (shares) exactly one hal o the aggregate loss. Thus, instead o considering her own loss, each person considers the average loss to be realized in the group. What is now the expected value and the variance resp. standard deviation o the potential loss aced by each person? 15

16 Expected Value Criterion (EV) Let us now assume that not just 2 but persons do share their losses as suggested above (e.g policyholders insured by a particular insurance company). What is the variance resp. the standard deviation o the average loss or the underlying portolio o policyholders? In general: Var n ~ xj n ~ nvar ( ) ( ) ( ~ j x) Var( ~ = 1 1 x) x Var = Var ~ x = = = n 2 n j= 1 ~ σ ~ σ x = n ( ) ( x) j n 2 n Note that the random variables need to be linearly independent! 16

17 Expected Value Criterion (EV) Obviously the variance decreases in proportion to the number o independent lotteries included in the average payo (loss). ~ Var( ~ x ) = and thereore Var( x) = = The standard deviation o the average loss amounts to less than 10. In case o 1 mil. policyholders it would even lie below 1. In case the risk [here represented by the variance resp. standard deviation] is that small, the evaluation o a lottery based on the expected value alone is quite reasonable! 17

18 Expected Value Criterion (EV) Note that the expected loss per policyholder is 100, thus the expected total loss amounts to 100 * = 1 M. I this loss is airly distributed among all policyholders each o them has to cover an expected loss o 100. Surely there will be policyholders who actually suer a loss (their car being stolen) and others with no loss at all. In the aggregate, however, with a large number o policyholders on board with minor variations one in one hundred cars will be stolen. In consequence, the loss rom an individual car robbery varies more strongly than the average loss rom all car robberies in the group. Hence, [simply due to a lower risk involved] in the latter case the EV criterion is more suitable than in the ormer. 18

19 Saint Petersburg Paradox A well known example to show that the EV criterion may not be suicient to evaluate a lottery is the so called Saint Petersburg Paradox! In a game a coin is tossed until a tail irst time appears and the game ends. At the end o the game the player receives 2 n monetary units as a reward, where n is the number o heads appeared during the game. Obviously the expected value o this game (lottery) is ininity although you will scarcely ind a person willing to pay an ininite amount o money to participate on this game. 19

20 Mean-Variance Criterion (MV) In general, when people make decisions they also tend to take risk into account, not just the expected value o the payos. The MV criterion is one (among many others) which allows risk to be considered in the evaluation o lotteries. However, it is important to note that MV reers to risk only in orm o the variance or standard deviation o a lottery s payos. In general, the corresponding value unction is a unction o E(x) and Var(x). Formally: V ( w~ ) ( ~ ( ~ = { Ε w ), Var w )} 20

21 Mean-Variance Criterion (MV) It is plausible to assume that V ( w ~ Ε( w ~ The sign o the irst derivative w.r.t. Var(w ) is rather ambiguous and it does in act oer propositions about the risk attitude o the DM: V ( w ~ ) 0... Risk Seeking ( ~ > Var w ) V ( w ~ ) Var( w ~ ) V ( w ~ ) ( ~ Var w ) < = ) ) > 0 Risk Aversion Risk Neutrality Note that here risk is considered only in orm o variance! 21

22 Mean-Variance Criterion (MV) In practice, a special orm o the MV criterion is preerably used, the value unction being linear in both E(w ) and Var(w ) where k is a constant. V ~ ~ ~ ( w ) =Ε( w ) kvar( w ) It is obvious that i k > 0 (k < 0) the variance has a negative (positive) eect on the value o the lottery. I k = 0 the risk [measured only in orm o variance] has no inluence on the value o the lottery, thus the EV criterion is in act a special case o the MV criterion! 22

23 Mean-Variance Criterion (MV) Let us now compare the two lotteries rom slide 12 or a DM A with k = 2 and alternatively an other DM B with k = -1! V ( ~ x) V ( ~ x) A B = 100 2* 2500< V ( ~ y) = 100 ( 1)*2500> V ( ~ y) A = 100 2*0 B = 100 ( 1)*0 preerence o A ~ x p ~ y preerence o B ~ x ~ y Obviously a DM with k = 0 would be indierent between the two lotteries! 23

24 Mean-Variance Criterion (MV) In this particular example, one can classiy A and B as risk averse and risk seeking respectively. However, one should keep in mind that the risk is represented here exclusively by the variance! I risk is measured in this way one can make the ollowing statement: Out o two lotteries with equal expected value and dierent variance a risk averse (seeking) DM will preer the lottery with lower (higher) variance! Note that a simpliication o this kind can only be applied when the DM ollows the MV criterion, but not in general! 24