portfolio selection problem modified

CHAPTER SEVEN

THE PORTFOLIO SELECTION PROBLEM

04/08/2023 Syed Karim Bux Shah 2

INTRODUCTION

• Portfolio is a collection of securities.• With a given amount of wealth and

securities, an investor can design innumerable portfolios.

• THE BASIC PROBLEM:– given uncertain outcomes, what risky securities

should an investor own?


INTRODUCTION

• THE BASIC PROBLEM:– The Harry Markowitz Approach

• assume an initial wealth• a specific holding period (one period)• a terminal wealth• diversify


INTRODUCTION

• Initial and Terminal Wealth• recall one period rate of return

where rt = the one period rate of return

wb = the beginning of period wealth

we= the end of period wealth

b

bet w

wwr


INITIAL AND TERMINAL WEALTH

• DETERMINING THE PORTFOLIO RATE OF RETURN– similar to calculating the return on a security– FORMULA

0

01

w

wwrp



• DETERMINING THE PORTFOLIO RATE OF RETURNFormula:

where w0 = the aggregate purchase price at time t=0

w1 = aggregate market value at time t=1

0

01

w

wwrp



• OR USING INITIAL AND TERMINAL WEALTH

where

w0 =the initial wealth

w1 =the terminal wealth

01 1 wrw p


THE MARKOWITZ APPROACH

• MARKOWITZ PORTFOLIO RETURN

– portfolio return (rp) is a random variable

– defined by the first and second moments of the distribution• expected return• standard deviation



• MARKOWITZ PORTFOLIO RETURN– defined by the first and second moments of the

distribution• expected return (mean returns)• standard deviation (dispersion of returns about

mean)



• MARKOWITZ PORTFOLIO RETURN– First Assumption:

• Non-satiation: investor always prefers a higher rate of portfolio return/higher terminal wealth.

• This leads to a conclusion “Given two portfolios with similar risk, investor would prefer the portfolio with higher returns.

Portfolio Returns Risk

A 12% 10%

B 8% 10%

Preferable Portfolio



• MARKOWITZ PORTFOLIO RETURN– Second Assumption

• Risk aversion: assume a risk-averse investor will choose a portfolio with a smaller standard deviation

• in other words, these investors when given a fair bet (odds 50:50) will not take the bet, i.e. $5 if head, and $-5 if tail. Note expected return on this is 0=(5*0.5)+(-5*0.5).

Portfolio Returns Risk

A 12% 10%

B 12% 08%

Preferable Portfolio



• MARKOWITZ PORTFOLIO RETURN– INVESTOR UTILITY– DEFINITION: is the relative satisfaction derived by

the investor from the economic activity- work, consumption, investment.

– It depends upon individual tastes and preferences-One individual may not seek same satisfaction/utility from same activity.

– It assumes rationality, i.e. people will seek to maximize their utility

– Utility wealth function: shows relationship between utility and wealth.



• MARGINAL UTILITY– each investor has a unique utility-of-wealth

function– incremental or marginal utility differs by

individual investor and depends upon the amount of wealth one already possesses.

– Richer investor value marginal $ less than a poor investor does.



• MARGINAL UTILITY– Assumes

• diminishing characteristic: As one has more of wealth, additional/marginal unit of wealth will add positive utility but on decreasing rate i.e. utility derived from marginal unit will keep on decreasing with successive units.

• An investor with diminishing marginal utility is risk averse and such an investor rate certain investment higher than riskier one.

• nonsatiation• Concave utility-of-wealth function



UTILITY OF WEALTH FUNCTION

Wealth

Utility Utility of Wealth

110105100

UcUr

103

Certainty equivalent

Risk premium


Conclusions

• Uc=Utility from certain investment• Ur=Utility from risky investment• Uc > Ur• The amount of positive utility derived from an

additional $1 < the amount of negative utility (disutility) resulted from loss of $1.

• Note: This is evident from the slope of utility wealth function which is increasing on decreasing rate. At any point on curve slope towards right is lower than the slope to left (Concavity).


Understanding Certainty Equivalents and Risk Premiums

• Suppose you are given two options A and B for investing $100.A: that you will earn Rs.105 with certainty.B: that you will earn either Rs.110 or nothing, probability of both events is 50:50.

Note: Both options have same expected pay off i.e. Rs.105. • Which option would you choose?• Your decision depends upon your attitude to risk. You are:

Risk indifferent, if both options are equally attractive to you.Risk averse: if you choose option A, preferring safe $ to risky $.Risk taker: if you choose plan B.


Understanding Certainty Equivalents and Risk Premiums

• A risk averse investor will choose option B only if:~ ceteris paribus, he receives lesser pay off in riskless investment e.g. Rs.101~ ceteris paribus, he receives even higher pay off in risky investment (Option B) e.g. Rs.120.

Note there must be an amount, where the investor regard both investments equally. For example in the Option A, if instead of certain $105, you are offered $103 and as a result you now regard both certain and risky investments equal, i.e. you derive same level of expected utility from both options. We call $103 Certainty Equivalent (CE). And the difference between expected payoff and CE is called Risk Premium (RP), a compensation to investor for additional risk taking.

The more risk averse you are the higher risk premium you demand and hence the lower CE, you have.

Risk averse have positive RP, risk neutral have zero risk premium, and risk takers have negative risk premium.

Expected payoff (EP)= Risk Premium (RP) + Certainty Equivalent (CE)CE = EP-RPRP = EP-CE


INDIFFERENCE CURVE ANALYSIS

• INDIFFERENCE CURVE ANALYSIS– DEFINITION OF INDIFFERENCE CURVES:

• a graphical representation of a set of various risk and expected return combinations that provide the same level of utility


INDIFFERENCE CURVE ANALYSIS

• INDIFFERENCE CURVE ANALYSIS– Features of Indifference Curves:

• no intersection by another curve• “further northwest” is more desirable giving greater

utility• investors possess infinite numbers of indifference

curves• the slope of the curve is the marginal rate of

substitution which represents the nonsatiation and risk averse Markowitz assumptions


Indifference Curves Analysis

Return

Risk

A

B C

further northwest

A risk averse investor will choose Portfolio A, which offers highest returns, with relatively lower risk.


PORTFOLIO RETURN

• CALCULATING PORTFOLIO RETURN– Expected returns

• Markowitz Approach focuses on terminal wealth (W1), that is, the effect various portfolios have on W1

• measured by expected returns and standard deviation


PORTFOLIO RETURN

• CALCULATING PORTFOLIO RETURN– Expected returns:

• Method One:

rP = w1 - w0/ w0


PORTFOLIO RETURN– Expected returns:

• Method Two:

where rP = the expected return of the portfolio

Xi = the proportion of the portfolio’s initial value invested in security i

ri = the expected return of security i

N = the number of securities in the portfolio

N

tiip rXr

1


Expected returns

• Portfolio expected return is a weighted average of expected returns of its constituents securities, i.e. each security contributes to portfolio by its expected return and its proportion in portfolio.


PORTFOLIO RISK

• CALCULATING PORTFOLIO RISK– Portfolio Risk:

• DEFINITION: a measure that estimates the extent to which the actual outcome is likely to diverge from the expected outcome


PORTFOLIO RISK


where sij = the covariance of returns

between security i and security j

2/1

1 1

N

i

N

jijjiP XX


PORTFOLIO RISK


• COVARIANCE– DEFINITION: a measure of the relationship between two

random variables– possible values:

» positive: variables move together» zero: no relationship» negative: variables move in opposite directions


Note: -1 ≤ ρ i j ≤ +1

PORTFOLIO RISK

CORRELATION COEFFICIENT– rescales covariance to a range of +1 to -1

where

jiijij

jiijij /

Note: Covariance between two securities i and j = correlation between i and j x Standard deviation of I x Standard deviation of j.

ρ i j = +1: denotes perfectly positive relationship between i and j’s returns, implying that as returns of i increase so does j’s.ρ i j = -1: denotes perfectly negative relationship.ρ i j = 0: indicate no identifiable relationship.


Graphical representation of correlation

a) Perfectively Positively correlated returns

b) Perfectively negatively correlated returns

A’s return

B’s returnB’s return

A’s return


Calculating Portfolio Risk

Exp: Given the following variance-covariance matrix for three securities A, B, and C, as well as the percentage of the portfolio for each security, calculate the portfolio’s risk (standard deviation σp.

Variance-covariance Matrix

Security A(50%)

Security B(30%)

Security C(20%)

Security A 459 -211 112

Security B -211 312 215

Security C 112 215 179


Calculating Portfolio Risk

(.5x.5x459) =114.75

(.5x.3x-211)=-31.65

(.5x.2x112)=11.2

(.3x.5x-211)=-31.65

(.3x.3x312)=28.08

(.3x.2x215)=12.9

(.2x.5x112)=11.2

(.2x.3x215)=12.9

(.2x.2x179)=7.16

2/1

1 1

N

i

N

jijjiP XX Solution: We know PF risk equals

Note: This reduces to((.5x.5x459) + (.3x.3x312) + (.2x.2x179) +

2 (.5x.3x-211) + 2 (.5x.2x112) + 2 (.3x.2x215)) ½ = (134.89) ½ = 11.61%


Calculating Portfolio Risk% of PF in each stockSec A 0.5Sec B 0.3

Sec A Sec B Sec C Sec C 0.2Sec A 459 -211 112Sec B -211 312 215Sec C 112 215 179

Solution:114.75 -31.65 11.2-31.65 28.08 12.9

11.2 12.9 7.16

Portfolio Variance = 134.9

Portfolio SD = 11.61 %

Variance-covariance Matrix

Some important points about Variance-covariance Matrix:

1. It is Square Matrix, having N2

elements for N securities.2. Variance appear on the

diagonal of matrix.3. The matrix is symmetric.


Risk-seeking Investor

• Risk seeking investor will prefer: – a gamble when presented a choice.– Large gambles over small gambles, because utility

gained from winning is greater for him than disutility gained from loosing.

– on indifference curve position of Farthest northeast• Risk seeking investors utility functions will be

convex and their indifference curves will be negatively sloped.


Risk-neutral investors

Risk neutral investors:• are indifferent to risk.• Have horizontal indifference curves (IC).• Will prefer farthest north position on IC.

Risk

ReturnIC Preferable

Portfolio15%

14%

AB

10% 20%

Note that as risk-neutral investor just for 1% additional expected returns (from portfolio A compared to B) is willing to take 10% additional risk. Such an investor consider the return factor only, ignoring risk altogether.

portfolio selection problem modified

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