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Portfolio Management Lecture 1: A simple model of portfolio management

Rules and other stuff

› Examination and grading

› The examination for this part of the course will be on xxxx, 2015.

› Check the Exam Schedule on the Faculty Web Page for more information and/or possible changes.

› There is a resit at the end of block 1.2

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Grading

› The grade of Portfolio Management is a weighted average of the exam, and ....

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Other stuff:

› There is a set of questions for each week. Answers will be available after the class through Nestor. You prepare these questions on an individual basis.

› We will use laptops during class for practicing with the models presented in the book/literature.

› Please try to organize the groups in such a way, that each group has it least one laptop during a class. Make sure that your battery is charged.

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Utility functions & Risk tolerance

› Risk is probably the most ambiguous element of modern portfolio theory and asset choice.

› People tend to confuse their risk expectations with their risk attitude.

› Risk attitude can be modelled using utility functions.

› Risk tolerance is the willingness of people to bear risk in exchange for return (risk premium)

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A simple utility function

T

REf2

where T represents the risk tolerance, R is the expected return on a portfolio and σ is standard deviation

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A simple portfolio problem

› We like to choose a portfolio of stocks and riskless bonds that maximizes our utility.

› Rs is the return on stocks, Rf is the return on bonds, and σs is the standard deviation of stock returns.

› Let x be the fraction of assets in stocks.

• Rp = x Rs + (1-x) Rf

• σp = x σs

› We want to choose x in such a way that we maximize our utility f:

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T

xRxxRf

ss

fs

22

1max

5

For example:

Rs 12%

Rf 8%

Sigma 20%

T = 1

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The solution to the problem

2

2

2

02

s

ss

fs

s

fs

RRTx

xT

RRdx

df

x=50%

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Allocation to risky assets

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5

Risky tolerance

Ris

ky a

sset

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How to measure risk tolerance

METHOD 1:

Design questionnaires and interview individuals (expected)

METHOD 2:

› Implicitly derive it from portfolio data (realized)

› We need to know the

1. Risk level (standard deviation) of the risky assets

2. The risk premium

3. The allocation to risky assets

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What is a reasonable value for risk tolerance?

› Following the implicit method (METHOD 2)

› According to Dimson et al, the equity risk premium is somewhere between 3.5% and 5.25%.

› Risk estimates for the stock market index are between 18% and 25%.

TRR

xfs

s

s

2

2 Allocation 1.000 1.000

Rs-Rf 0.053 0.035

Sigma 0.180 0.250

T 1.234 3.571

For a fully invested investor

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Expected risk: implied volatility

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Different beliefs

› Even persons with the same risk tolerance could hold very different portfolios if their expectations are different.

T 1 1 1 1

Rs-Rf 4% 8% 2% 4%

Sigma 15% 20% 25% 25%

Risky asset 89% 100% 16% 32%

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The impact of liabilities

› An important factor is leverage (L/S).

› Investors with fixed liabilities L, should have different optimal portfolios.

› Leverage magnifies risks in the asset portfolio.

› Given a certain level of risk tolerance, an investor without liabilities has a desire for more risky assets than an investor with fixed liabilities.

› NB: Fixed liabilities have zero correlation with the risky assets.

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Example

› Consider an investor who is interested in her wealth S at the investment horizon.

› Wealth 𝑆 = 𝐴 − 𝐿. (Surplus). We deduct liabilities from the assets, because the investor will not be able to spend the entire value of the asset portfolio. Examples of liabilities are consumer loans, pension obligations, mortgage debts, and the need to cover children’s future study expenses.

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Expression of surplus return and risk

› In the presence of liabilities, the return on an investor’s wealth becomes

› And the risk becomes:

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laas rrS

Lrr

,

Example:

› An investor with A=100 and L=50.

› The investor has a risk tolerance of 2. Consistent with the previous example, we assume that the expected return on stocks is 8% and a risk-free rate of 5%.

› An investor without liabilities chooses 2

2

0.03

0.32=

1

3.

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10

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An investor with leverage 50/50=1 chooses a fraction of 16.67% in stocks

Chapter 2: Risk Profiling

› Risk tolerance is the willingness of an individual to take risk.

› Determining someone’s risk appetite is more than finding out how much he or she wants to invest in stocks, bonds, hedge funds, and whatever assets you may consider

› Risk appetite may change due to someone’s fall or rise in wealth level, income, age, personal psychological make-up, etc..

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The neo-classical view

› Investors are rational and maximize their utility

› An individuals’ risk tolerance is constant over time

› Risk tolerance is the same for all financial decisions.

› The shape of the utility function determines the risk preferences of the investor

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Expected utility theorem

U(W) utility function representing the utility of wealth W

Expected Utility Theorem:

E(U) = Σ U(W) P(W)

A utility function is nothing more than a weighting function or a scoring system

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| 23 The efficient frontier

Risk

Ret

urn

The efficient frontier is the

Opportunity set of investment

Alternatives that present

Rational choices for risk

Averse investors

Utility function

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Example: Soccer scores

› In the past, the following was used in counting soccer scores in a tournament:

Outcome of game Points

Win 2

Tie 1

Loss 0

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Example: Soccer scores

› A common way of counting soccer scores in a tournament is by using the following system:

Outcome of game Points

Win 3

Tie 1

Loss 0

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Example: Soccer scores (2)

› Consider two equally strong teams using different strategies:

1. The first team plays a risky strategy involving an offensive playing style

2. This team plays a conservative strategy involving a defensive playing style.

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Example: Soccer scores (3)

› Suppose that team 1 has a 50% probability of winning and a 50% probability of losing

› Suppose that team 2 has a 10% probability of winning, an 80% probability of an equal outcome, and a 10% probability of loosing.

› On average, after 10 games:

› Team 1: has 5*3 = 15 points (old score: 10)

› Team 2: has 1*3 + 8*1 = 11 points (old score: 10)

An example

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The investor has the following utility function:

U(w)= 0.5×w+w2

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Properties of utility functions

1. non-satiation:

economic subjects prefer more over less

U'(W) > 0

2. Risk preference

› U"(W) < 0: risk-aversion

› U"(W) = 0: risk-neutrality

› U"(W) > 0: risk-loving.

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Risk aversion

› Risk aversion implies that I reject a fair gamble.

› A fair gamble is one that sacrificies 1 unit of wealth in return for a lottery with an expected wealth of 1

› I will only expect lotteries whose expected value is larger than its investment value.

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0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20

Uti

lity

of

we

alt

h

Wealth

Utility functions

Risk loving

Risk neutral

Risk averse

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0

5

10

15

20

25

0 2 4 6 8 10 12 14 16 18 20

Uti

lity

of

we

alt

h

Wealth

Utility functions

Risk loving

Risk neutral

Risk averse

1. U’(w)>0: Upward sloping

2. U”(w)>0: increasing upward slope … Risk

loving

3. U”(w)=0: Constant slope …. Risk neutral

4. U”(w)<0: Decreasing upward slope…. Risk

aversion

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| 33 An example

Suppose that the investor uses a logarithmic utility

function: U(W) = ln(W)

Alternatief 1

Alternatief 2

W

1 = 5

P(W

1) = 1/3

W

2 = 10

P(W

2) = 1/3

W = 10

P(W) = 1

W

3 = 15

P(w

3) = 1/3

So the investor prefers alternative 2

W U P W U P

5 1.61 0.33

10 2.30 0.33 10 2.30 1.00

15 2.71 0.33

E[U] 2.21 E[U] 2.30

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E[W] = 10

With risk

E[W] = 10

Without risk

The utility function LN(W) exhibits risk aversion!

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Some other examples

1. Risk Loving: U=W+.01W2

2. Risk Neutral: U = W

3. Risk Averse: U = 0.8 + W – 0.02W2

Illustrate (3) risk aversion with an example:

› 50% probability of getting 10 U(10)=8.8

› 50% probability of getting 18 U(18)=12.32

› Expected utility of this lottery = 0.5*8.8+0.5*12.32= 10.56

› To check whether the utility function is concave, we can calculate the utility of a certain outcome of 14.

› If the utility of the uncertain bet is lower, than the investor must have risk aversion. The expected utility is U(14)=10.88. … so we conclude risk aversion

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Mathematical proof:

› U’(W) = 1 -0.04W > 0, only if W<25 (So utility as a quadratic function of wealth is only a valid utility function on a particular domain of wealth)

› U’’(W) = -0.04 < 0, which implies risk aversion

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| 37 Absolute risk aversion (A[W])

1. Increasing absolute risk aversion:

If W ↑ then the dollar amount allocated to risky assets is decreasing.

2. Constant absolute risk aversion:

the allocation to risky assets is independent of W.

3. Decreasing absolute risk aversion:

If W ↑ then the dollar amount allocated to risky assets is increasing.

As a result:

› Consider a risky bet with uncertain outcomes Z

› The amount of money an investor is willing to pay in order to avoid the bet is the risk premium in dollar terms:

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WAWU

WUZZ

2

'

"2

2

1

2

1

› σz is a measure of the dollar risk of the investment opportunity

› U”(W)/W’(W) is the measure of absolute risk aversion

2

2

Z

WA

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Relative risk aversion (R[W])

(i) Increasing relative risk aversion:

the relative (%) allocation to risky assets is decreasing in wealth: R'(W) > 0

(ii) Constant relative risk aversion:

the relative weight of risky asset is independent of the level of wealth: R'(W) = 0

(iii) Decreasing relative risk aversion:

Relative weight of risky asset is increasing with the level of wealth: R'(W) < 0

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A measure of relative risk aversion

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)('

)(")(

)(2)('

)("

2

22

WU

WWUWR

WRWU

WWU zz

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Practice of risk aversion:

Widely held beliefs:

› Risk tolerance decreases with age.

› Women are less risk tolerant than men.

› Unmarried individuals are more risk tolerant.

› Professionals are more risk tolerant.

› Highly-educated people are more risk tolerant

› Self-employed people are more risk tolerant.

› High-income people are more risk tolerant.

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Studies of risk tolerance

› Riley and Chow (1992)

• Based on actual behavior.

• What are the cross-sectional determinants of the relative risk aversion coefficient RRAI –defined as [1-(risky assets/wealth)]?

• Use Survey of Income and Program Participation (1984).

• Age, gender, education, race, marital status, income, wealth

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Risk tolerance and age | 43

Risk tolerance increases with age up to a certain level (55-64) in order to decrease again.

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Education and risk tolerance

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The higher the level of education, the higher the risk tolerance

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Wealth/income and risk tolerance | 47

Hallahan, Faff and McKenzie (2004)

• Based on questionnaires.

• Self-assessed risk tolerance vs. (standardized) FinaMetrica’s risk tolerance score (RTS) (25 questions).

• The link with demographics for 20,000 individuals.

• Issue: Does the RTS predict behavior beyond demographics?

• Face validity. Predictive validity. Reliability. Test norms.

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25

RTS and preferred portfolio | 49

Conclusion

› Psychological concepts (personality) add little or no explanatory power to explaining risk tolerance

› Self-assessed risk tolerance is as good as a long list of question (such as the one that you filled in during class).

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Next week › Mean variance analysis: How to build smart

spreadsheet models.

› Take your laptop to class

› Read Plantinga (2015c) and Elton & Gruber, Ch 4-6 before the lecture …

› Practice with the questions: if you want to discuss one of the questions during class, please send me an email and mention the question number as well as what your problem with the question is.

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