lecture 8. choice under uncertainty - stephen...

Lecture 8. Choice Under UncertaintyStephen Kinsella

In previous lectures, we analysed the markets in terms of their inherent uncertainty. Now it is time to show you somebasic models for valuing risk and reward. We will also show you procedures for valuing risky assets. These methods arenot without their shortcomings, however, and you should be aware of these when using the methods in the real world.

In the real world, investors tend to spread their wealth around rather than confine it all to one risky asset. Each investorwill hold a bundle of differentially risky assets, called a portfolio. We speak of the risk and return to this asset portfoliowhen talking about portfolio diversification in the next few sub sections. We want to minimise the riskiness of eachportfolio while maximising its return, or reward. The portfolio is a means to pull out the differences betweensystematic, or diversifiable, risk, and unsystematic (or market) risk.

Pricing a Risky Asset: IssuesEach asset we hold is a claim on future cash not yet generated. The idea behind pricing any risky asset is it's priceshould reflect the present value of its expected cash flow. We want to know this as accurately as possible to gain ameasure of the 'true' value of the asset, to ensure we are not being ripped off when buying, selling, or holding the asset.

Because the future is not known with any certainty, as we have seen in previous lectures, the trader of these assets mustgauge the expected return on the asset using a combination of technical analysis, macroeconomic forecasting, intuition,and blind luck. With each asset, different pricing rules apply. Shares in a company like Apple Computer, can vary inprice by a large amount depending on the company's performance in its industry. Debt-based securities like companypaper have more or less certain cash flow properties, depending on the company. Treasury bonds issued by governmentsare as close to riskless as one gets in real life. Investors can always face exchange rate risk if they buy risky assetsdenominated in a foreign currency, as we can see from the figure below showing the movements of the Euro/Dollarexchange rate from October to December 2007.

DateListPlot@FinancialData@"EURêUSD", AllDD

Oct Nov Dec1.40

1.42

1.44

1.46

1.48

Figure X. Euro-Dollar exchange rate from October 2007 to December 2007.

Pricing a Risky Asset: Return CalculationWe calculate the return, R, on an asset as the sum of the income stream associated with the asset, and any changes in thevalue of that asset over time. Following equation X below, we have

(1)R1 =P1 - P0

P0+

C1P0

=P1 - P0 + C1

P0.

Where R1 is th return on the financial asset at the end of period 1, P1is the price of the asset at the end of period 1, C1isthe income received at the end of period 1.

Printed by Mathematica for Students

Where R1 is th return on the financial asset at the end of period 1, P1is the price of the asset at the end of period 1, C1isthe income received at the end of period 1.

ü ExampleAn asset has a value of 500 euros at the beginning of the year. We expect the market value on the asset to be 600 byyear end; the income of the security is 50 euros. What is the return for the year?

R1 = H600 - 500 + 50L ê 500 = 150 ê 500 = 0.3.

ü ExerciseAn asset has a value of 200 euros at the beginning of the year. The market value of the asset should be 210 euros by yearend; the income of the security is 10 euros. What is the return for the year?

So, we have a 30% return on a per annum basis. In this example, we knew the return at the end of the year exactly. In allreal cases, we don't know this, so we must take an expectation over the years we hold the asset. For some asset i (wherei can be shares in Apple Computer, a US T-bill, company paper, or any other security), we can say the expected return,EHRiL, is given by

(2)EHRiL = P1 R1 + P2 R2 + ... + Pn Rn.

Where in equation now the P1, 2, ... n's represent probabilities of outcomes. These probabilities are subjective, somethingwe have seen in earlier classes. The R1,2,...,n ' s still represent rates of return, but in each period 1, 2,..., n. This equation isa finite arithmetic series, which means the right hand side can be written as

(3)EHRiL = ‚i=1

n

Pi Ri.

ü ExampleTake a look at the table below. Outcome Return H%L Probability Hê1L

1 20 0.05

2 10 0.25

3 5 0.40

4 30 0.10

56

520

0.100.10

Table 1. Table X. Example outcomes, probabilities, and returns.

We know from the equation above that to calculate expected return, we combine each return with the probability of itoccuring over the range of periods or outcomes following

EHRiL = P1 R1 + P2 R2 + ... + Pn Rn.

So we have

EHRiL = .05 H20L + 0.25 H10L + 0.40 H5L + 0.10 H30L + 0.10 H5L + 0.10 H20L = 11 %.

The expected rate of return is then 11%.

ü ExerciseCalculate the expected percentage return on the asset with the following outcomes, probabilities, and returns. Outcome Return H%L Probability Hê1L

1 10 0.15

2 40 0.15

3 15 0.20

4 60 0.20

56

-60-12

0.050.25

Table 2. Exercise table. Calculate the expected return.

Further Notes on Subjective Probability

2 choice_lecture8.nb


Further Notes on Subjective ProbabilitySubjective probability is a tricky concept, and it is worth spending some time discussing it properly. A subjectiveprobability describes an individual's personal judgement about how likely a particular event is to occur. It is not basedon any precise computation but is often a reasonable assessment by a knowledgeable person. The canonical study in thetheory of subjective probability is John Maynard Keynes's (rhymes with 'brains') Treatise on Probability (1921). Hisassessment of decision making under uncertainty is given in his 1937 QJE article:

"[Under uncertainty] there is no scientific basis on which to form any calculable probability whatever. We simply do notknow. Nevertheless, the necessity for action and for decision compels us as practical men to do our best to overlook thisawkward fact and to behave exactly as we should if we had behind us a good Benthamite calculation of a series ofprospective advantages and disadvantages, each multiplied by its appropriate probability waiting to be summed."

From this point of view, probability is a mode of judgment. From this point of view probabilities are “in the mind”~thesubject’s, say, yours. If you say the probability of rain is 70% you are reporting that, all things considered, you wouldbet on rain at odds of 7:3, thinking of longer or shorter odds as giving an unmerited advantage to one side or the other. Amore familiar mode of judgment is flat, “dogmatic” assertion or denial, as in ‘It will rain’ or ‘It will not rain’. In place ofthis “dogmatism”, the probabilistic mode of judgment offers a richer palate for depicting your state of mind, in whichthe colours are all the real numbers from 0 to 1.

Subjective probability theorists assume that there is no outside 'truth', that in fact, we are constantly making up our own'truths' as we go. We assign probabilities to outcomes in an uncertain and personally contingent way, and, in doing so,we create our own distributions. The subjective nature of probability assignments is can be made clearer by thinking ofsituations like a horse race. In this case, most spectators face more or less the same lack of knowledge about the horses,the track, the jockeys, etc. Yet, while sharing the same 'knowledge' (or lack thereof), different people place different betson the winning horse. The basic idea behind this derivation is that by observing the bets people make, one can presumethis reflects their personal beliefs on the outcome of the race. Thus subjective probabilities can be inferred fromobservation of people's actions. This is a deep and important issue in economics, and two paragraphs will not even beginto shed light on the subject. However, important references are Shackle, (1949, 1961), de Finetti (1931, 1937), Popper(1959), and Keynes (1921). Nobellist Robert Aumann (1969) has also made important contributions in this area.

Variance and Expected ReturnWe saw how to derive the variance in abstract statistical terms in tutorials. We are interested in the variability of rates ofreturn because, frankly, they vary. Sometimes by a lot, for example, the stock price of Apple computer varied by anenormous amount from the year 2000 to 2005, as we can see from the figure below:

DateListPlot@MovingAverage@FinancialData@"AAPL", "Jan. 1, 2000", "Value"D,100D, 8"Jan. 1, 2000", Automatic, "Day"<, Joined Ø TrueD

2000 2001 2002 2003 2004 20050

50

100

150

Figure X. 100 day Moving Average of Apple's stock price, Jan 2000- Jan 1 2005.

We want to know the distribution of returns about the average, or median, return. We want to know this because wewant to know how to price the stock to buy, sell, or hold it. We would like a measure of the variability of the stock overtime, and the variance, and it's square root, the standard deviation, give us just that.

The variance is the sum of squared deviations about the mean. We square the deviations so negative rates of return aren'tlost in the summation. The equation for the variance of asset i, s2i, is given by

choice_lecture8.nb 3


The variance is the sum of squared deviations about the mean. We square the deviations so negative rates of return aren'tlost in the summation. The equation for the variance of asset i, s2i, is given by

(4)si2 = P1HR1 - EHRiLL2 + P2HR2 - EHRiLL2 + ... + PnHRn - EHRiLL2,

Where the probabilities of each outcome are given by P1, P2, etc., and the rate of return on each outcome is given byR1,2, etc. The equation above is also a series, so we can summarise it for n outcomes as

(5)si2 = ‚

i=1

n

Pi@Ri - EHRiLD2.

ü Example 4Take a look at the table below.

Outcome Return H%L Probability Hê1L1 20 0.05

2 10 0.25

3 5 0.40

4 30 0.10

56

520

0.100.10

Table 3. Example outcomes, probabilities, and returns.

We calculated the value of EHRiL as 11% above. Here the variance is given by

si2= 0.05 H20 - 10L2 + 0.25 H10 - 10L2 + 0.40 H5 - 10L2 + 0.10 H30 - 10L2 + 0.10 H5 - 10L2 + 0.10 H20 - 10L2

= 67.5. The standard deviation of this variance is the square root, and the the square root of 67.5 is 8.2.

ü ExerciseCalculate the variance of the table below.

Outcome Return H%L Probability Hê1L1 10 0.15

2 40 0.15

3 15 0.20

4 60 0.20

56

-60-12

0.050.25

Table 4. Exercise Table. Calculate the Variance.

We can use programs like Mathematica to estimate the variance of a stock, say, Apple Computer (for a change) from 1January, 2000, to the present. You can see at a glance that the returns on a stock are not normally distributed over time.If they were, the standard deviation would be a lot smaller than 1629.39 , or 40.36.

Variance@FinancialData@"AAPL", "Jan. 1, 2000", "Value"DD1629.29

What does this mean?

Let's assume for a moment that returns to stocks are normally distributed. What would they look like? These returnswould follow the distribution below

Plot@PDF@NormalDistribution@0, 1D, xD, 8x, -4, 4<,AxesLabel -> 8"Return", "Probability"<D



-s2 +s2

-4 -2 2 4Return

0.1

0.2

0.3

0.4Probability

Now, variations in stock prices are generally positive, so let's only look at the positive part of the distribution. Of course,returns can be negative, but we'll leave that aside for a moment and return to it.

Plot@PDF@NormalDistribution@0, 1D, xD, 8x, 0, 4<,AxesLabel -> 8"Return", "Probability"<D

1 2 3 4Return

0.1

0.2

0.3

0.4Probability

Now lets look at the distribution of Apple's stock price over the last year, say, from 1 January, 2006 to 1 January, 2007.We have

<< Histograms`x = FinancialData@"AAPL", "Jan. 1, 2006", "Value"D;Histogram@x, AxesLabel Ø 8"Volume", "Returns"<D

75 100 125 150 175Volume

10

20

30

40

50

Returns

Now look at the distribution of the stock over a longer period---from 1 January 2000 to 1 January 2007.



Now look at the distribution of the stock over a longer period---from 1 January 2000 to 1 January 2007.

x = FinancialData@"AAPL", "Jan. 1, 2000", "Value"D;Histogram@x, AxesLabel -> 8"Volume", "Returns"<D

50 100 150Volume

100

200

300

400

Returns

A completely different picture emerges, much less like the Gaussian distribution most models claim for their bases.

More importantly, when we compare these distributions in three dimensions, a completely different picture emerges.First, let's look at the Gaussian distribution in 2 dimensions. You can see most of the values are clustered in the centre,like in the 2 dimensional case.

Graphics@Point@RandomReal@NormalDistribution@D, 82000, 2<DDD

Now look at the same data, but visualised in 3 dimensions.



Graphics3D@Point@RandomReal@NormalDistribution@D, 82000, 3<DDD

What does our Apple computer data look like? It is essentially Pareto distributed, so it looks something like this in 3dimensions.



Graphics3D@Point@RandomReal@ParetoDistribution@1, 2000D, 82000, 3<DDD

You can see the returns are clustered in one corner with a few wild, wild outliers. Those points far off in the edges of thebox represent very odd days of trading, when gains and losses were being recieved at high volumes. This is the characterof the stock market as we see it today. Those far away dots represent the outliers we can see more clearly in a 70 yearretrospective of the Dow Ryan Industrial Index, as we see below.

DateListPlot@8Ò1, Log@10, Ò2D< & üüü FinancialData@"^DJI", AllD,Joined Ø True, Filling Ø BottomD

Figure X. Dow Ryan Industrial Index volatility over 70+ years.

How much variance is involved in calculating this index? Something on the order of 10,000. Markets are turbulent overlong stretches, as we said in week 1.



ü Covariance and Correlation of Rates of ReturnOutcome Probability Return on RA Return on RB

1 0.1 20 12

2 0.2 10 -10

3 0.1 11 11

4 0.2 13 20

5 0.1 14 72

6 0.3 50 10

Table 5. Table of returns on two assets, RA and RB.

The expected rates of return for A and B are EHRAL = 24%, EHRBL = 14.5 %. We determine the coveriance of A and B byapplying the following formula:

(6)sAB = ‚i=1

n

PiHRAi - EHRALL HRBi - EHRBLL.

This works out as

0.1(20-24)(12-14.5)+0.2(10-24)(-10-14.5)+0.1(11-24)(11-14.5)+0.2(13-24)(20-14.5)+0.1(14-24)(72-14.5)+0.3(50-24)(10-14.5)

=-30.55.

A negative correlation implies the the rates of returns on A and B tend to move in opposite direction, while a positivecorrelation indicates they move in the same direction.

ü ExerciseWhat does this mean for portfolio selection? Can you give examples where a negative correlation between assets is agood thing for a risk-averse investor?

Look at the table below. Calculate the covariance and the correlation coefficient. Outcome Probability Return on RA Return on RB

1 0.1 20 12

2 0.2 10 -10

3 0.1 11 11

4 0.2 13 20

5 0.1 14 72

6 0.3 50 10

Table 6. Table of returns on two assets, RA and RB.

A more useful and tractable way to consider the relationship between two assets is the correlation coeffcient between Aand B, rAB, defined as

(7)rAB =sAB

sA sB.

Here sAis just the standard deviation of A, which we work out using the formula given above:Subscript@s, AD^2 = ⁄i=1

n Pi@Ri - EHRiLD2 =

0.1 H20 - 24L^2 + 0.2 H10 - 24L^2 + 0.1 H11 - 24L^2 + 0.2 H13 - 24L^2 +

0.1 H14 - 24L^2 + 0.3 H50 - 24L2 = 17.17

By similar reasoning, sB =27.6, so we can calculate rAB = -30.55 ê H17.17 * 27.6L= -0.064.



What does this mean? Correlation coefficients can take values between -1 and +1. They are measures of co-movement,with values closer to -1 or +1 indicating stronger positive or negative relationships between the two stocks. If the twostocks were positively correlated, this means if the price of Asset A were to rise, so would the price of asset B. Thereverse is also true.

ü ExerciseCalculate the correlation coefficient for Table 6 above.

Risk Aversion and Attitudes to RiskPeople have different preferences when it comes to risk. The idea behind defining a concept like risk aversion is todifferentiate between individuals. Imagine two assets, X and Y, with the same associated level of risk, but with differingstandard deviations. The risk-averse person will want to choose the level of risk that is lower. A risk-neutral person willbe indifferent between the two choices. Risk-loving people will go for the higher level of risk, at least in this scenario. Ingeneral, we tend to think of most people as risk-averse (Samuelson, 1963).

In the risk-return space, as graphed below, we can see the differences between investors. When investors are risk averse,they won't take on increased risk unless the are compensated by an increase in expected return. The curves you seebelow encapsulate the idea that different people have different preferences when it comes to risk.

E(R)

Risk (Std Dev)

IC1IC2

A

C

B

σ1 σ2

Indifference curves in this space become steeper as we increase the level of risk (i.e., the standard deviation) associatedwith holding that much risk. As we increase the amount of risk a person must hold, they become less likely to want tohold that level of risk. The coefficient of absolute risk aversion for some bundle x, given by

(8)ru = -U '' HxL êU ' HxL,determines the slope of each indifference curve.

„ Teasing this idea out a bit more

Suppose you face a risky choide: with some probability p you will recieve wealth W1, and with probability H1 - pL you'llreceive wealth W2.

It can be shown that, under some strong assumptions, the utility function



(9)EHUHWLL = p.UHW1L + H1 - pL UHW2LDescribes the expected utility the person can expect to receive from the bet. This formulation of the utility function iscalled the von Neumann-Morgenstern utility functional form.

Risk aversion in this formulation looks like the example below.

Example

Given a choice between 50 euros for certain, and an 50:50 straight bet on 100 or nothing, they'll go for 50 for certain.

„ St. Petersburg ParadoxImagine the following scene: you are asked to bet some amount to enter a game. If your bet is the highest out of all theplayers, you are allowed to play. Let's say you bet 2 euros, and that was the highest. The rules are, on the first toss, if it'sheads, you get 2 euros. On the second, you get 22 = 4, on the nth toss you get 2n, and so on, so your earnings look likethis:

(10)1 ê 2 * 2 + 1 ê 4 * 1 ê 8 + ... += 1 + 1 + 1 + ...

Most people would only pay a finite amount to enter the game, but the potential rewards are infinite!

For most people, then the following relationship holds:

UIpW1 + H1 - pL W2M > pUHW1L + H1 - pL UHW2Lor

UHEHWLL > EHUHWLLWhich, in W - UHWL space, looks like the figure below.

Figure X. The von Neumann-Morgenstern utility functional form in W-U(W) space.



Allais Paradox(ref) The Allais paradox, is a choice problem designed by Maurice Allais to show an inconsistency of actualobserved choices with the predictions of expected utility theory. The problem arises when comparingparticipants' choices in two different experiments, each of which consists of a choice between two gambles, Aand B. The payoffs for each gamble in each experiment are as follows:

Experiment 1 Experiment 2Gamble 1A Gamble 1B Gamble 2A Gamble 2BWinnings ChanceWinnings ChanceWinnings ChanceWinnings Chance$1 million 100% $1 million 89% Nothing 89% Nothing 90%Nothing 1% $1 million 11%$5 million 10% $5 million 10%

Presented with the choice between 1A and 1B, most people choose 1A. Presented with the choice between 2Aand 2B, most people choose 2B. This is inconsistent with expected utility. The point is that both gambles givethe same outcome 89% of the time (the top row; $1 million for Gamble 1, and zero for Gamble 2), so, inexpected utility, these equal outcomes should have no effect on the desirability of the gamble. If the 89%‘common consequence’ is disregarded, both gambles offer the same choice; a 10% chance of getting $5million and 1% chance of getting nothing as against an 11% chance of getting $1 million. (It may help to re-write the payoffs. 1A offers an 89% chance of winning 1 million and a 11% chance of winning 1 million,where the 89% chance is irrelevant. 2B offers an 89% chance of winning nothing, a 1% chance of winningnothing, and a 10% chance of winning 5 million, with the 89% chance of nothing disregarded. Hence, choice1A and 2A should now clearly be seen as the same choice, and 1B and 2B as the same choice).

Ellsberg Paradox(ref) Suppose you have an urn containing 30 red balls and 60 other balls that are either black or yellow. Youdon't know how many black or yellow balls there are, but that the total number of black balls plus the totalnumber of yellow balls equals 60. The balls are well mixed so that each individual ball is as likely to be drawnas any other. You are now given a choice between two gambles:

Gamble A Gamble BYou receive $100 if you draw a red ball You receive $100 if you draw a black ballAlso you are given the choice between these two gambles (about a different draw from the same urn):Gamble C Gamble DYou receive $100 if you draw a red or yellow ball You receive $100 if you draw a black or yellow ball

Since the prizes are exactly the same, it follows that you will prefer Gamble A to Gamble B if, and only if, youbelieve that drawing a red ball is more likely than drawing a black ball (according to expected utility theory).Also, there would be no clear preference between the choices if you thought that a red ball was as likely as ablack ball. Similarly it follows that you will prefer Gamble C to Gamble D if, and only if, you believe thatdrawing a red or yellow ball is more likely than drawing a black or yellow ball. If drawing a red ball is morelikely than drawing a black ball, then drawing a red or yellow ball is also more likely than drawing a black oryellow ball. So, supposing you prefer Gamble A to Gamble B, it follows that you will also prefer Gamble C toGamble D. And, supposing instead that you prefer Gamble D to Gamble C, it follows that you will also preferGamble B to Gamble A.

When surveyed, however, most people strictly prefer Gamble A to Gamble B and Gamble D to Gamble C.Therefore, some assumptions of the expected utility theory are violated.



ü Differing Levels of Risk AversionWe can study the effects of different levels of risk aversion on levels of utility attained with the folowing tool.

HInitialise this cell to use the code, from the Wolfram Demonstrations project .

risk aversion coefficient 1

type of utility function constant absolute risk aversion constant relative risk aversion

0 10 20 30 40 50

-40

-20

0

20

x

utility@xD

constant absolute risk aversion



lecture 8. choice under uncertainty - stephen...

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