Portfolio Theory

An Investor’s Preference

1

2 3

4

Return

Risk

• 2 dominates 1 - same risk, but higher return• 2 dominates 3 - same return, but lower risk• 4 dominates 3 – same risk, but higher return

Indifference Curves

Return

Risk

Increasing Expected Utility

Indifference Curve

- Represents individual’s willingness to trade-off return and risk

- Assumptions:

1) 5 Axioms

2) Prefer more to less (Greedy) => Max E[U(W)]

3) Risk aversion

Expected Return E(R)

Standard Deviation σR

Indifference Curve


- Assumptions:

1) 5 Axioms


3) Risk aversion

4) Assets jointly normally distributed

Indifference Curves

Increasing Expected Utility

Indifference Curve


- Assumptions:

1) 5 Axioms


3) Risk aversion


The transitionFrom U(RETURN, RISK)

To U(expected return, standard deviation of return)

• This transition means the objects of choice for an investor is normally distributed.

• Because only if a distribution is normally distributed can it be described only by its mean and standard deviation.

• This poses a question:“Does that mean the risky assets investors invest in have to be always normally distributed in terms of their returns?”

The answer: NO! Individual asset’s return does not have normal distribution.

So why should we assume normally distributed returns for the objects of choice?

Indifference Curve


- Assumptions:

1) 5 Axioms


3) Risk aversion


The transitionFrom U(RETURN, RISK)

To U(expected return, standard deviation of return)

• This transition means the objects of choice for an investor is normally distributed.

• Because only if a distribution is normally distributed can it be described only by its mean and standard deviation.

• This poses a question:“Does that mean the risky assets investors invest in have to be always normally distributed in terms of their returns?”

The answer: NO! Individual asset’s return does not have normal distribution.

So why should we assume normally distributed returns for the objects of choice?

BECAUSE RATIONAL INDIVIDUAL INVESTORS CHOOSE TO INVEST IN A PORTFOLIO BUT NOT A SINGLE ASSET!

Jointly normally distributed?

Individual stock return may not be normally distributed, but a portfolio consists of more and more stocks would have its return increasingly close to being normally distributed. NOTE: (N = No. of stocks in the portfolio)

Jointly normally distributed?1st moment: Mean = Expected return of portfolio

2nd moment: Variance = Variance of the return of portfolio (RISKNESS)

Mean and Var as sole choice variables => Distribution of return can be adequately described by mean and variance only

That means, distribution has to be normally distributed.

3 reasons to support mean-variance criteria

1) As the table shows, a portfolio consisting of large number of risky assets tend have a return that is very close to normally distributed.

2) The fact that investors rebalance their own portfolios frequently will act so as to make higher moments (3rd, 4th, etc) unimportant (Samuelson 1970).

3) Cost-effective consideration: data required to compute 3rd and 4th moments are very demanding. Both computational and cost inefficient.

Conclude• Thus we are convinced that expected returns and variances are good

enough to describe the objects of choice.

• From the investor’s point of view, it is cost-effective, and more intuitive.

• From the objects of choice, if the unit of choice is an investment portfolio with large number of risky assets, it is close to normally distributed.

• So we are cool with the indifference curves.

• Now, we’ll work on the computations of expected return and standard deviation (i.e., the square root of variance) of an investment portfolio.

• We make our lives easy by assuming individual asset’s return is normally distributed to illustrate. We don’t buy this assumption, but for the purpose of math, we’ll stick with it.

Math Review I

• Asset j’s return in State s:

Rjs = (Ws – W0) / W0

• Expected return on asset j:

E(Rj) = ∑sProb(State=s)xRjs

• Asset j’s variance:

σ2j = ∑sProb(State=s)x[Rjs- E(Rj)]2

• Asset j’s standard deviation:

σj = √σ2j

• Thus, Rj ~ N(E(Rj), σ2j)

Math Review I• Covariance of asset i’s return & j’s return:

Cov(Ri, Rj)= E[(Ris- E(Ri))x(Rjs- E(Rj))] =∑sProb(State=s)x[Ris- E(Ri)]x[Rjs- E(Rj)]

• Correlation of asset i’s return & j’s return:ρij = Cov(Ri, Rj) / (σiσj)

-1 ≤ ρij ≤ 1When ρij = 1 => i and j are perfectly positively

correlated. They move together all the time.When ρij = -1 => i and j are perfectly negatively

correlated. They move opposite to each other all the time.

A simple example: Asset j

60%

40%

$150,000 Good State: rgood = ($150,000 – $100,000) / $100,000 = 50%

$80,000 Bad State: rbad = ($80,000 – $100,000) / $100,000 = -20%

$100,000

Expected Return: (Notation: Let αs denote the Prob(State=s)

E(rj) = ∑sαsrjs = 60%(50%) + 40%(-20%) = 22%

Variance:

σ2j = ∑sαs[rjs- E(rj)]2 = 60%(50%-22%)2 + 40%(-20%-22%)2 = 11.76%

Standard Deviation:σj = √σ2

j = √11.76% = 34.293%

Math Review II

• 4 properties concerning Mean and Var

Let ũ be random variable, a be a constant

1) E(ũ+a) = a + E(ũ)

2) E(aũ) = aE(ũ)

3) Var(ũ+a) = Var(ũ)

4) Var(aũ) = a2Var(ũ)

Portfolio Theory – a bit of history

• Modern portfolio theory (MPT)—or portfolio theory—was introduced by Harry Markowitz with his paper "Portfolio Selection," which appeared in the 1952 Journal of Finance. 38 years later, he shared a Nobel Prize with Merton Miller and William Sharpe for what has become a broad theory for portfolio selection.

• Prior to Markowitz's work, investors focused on assessing the risks and rewards of individual securities in constructing their portfolios. Standard investment advice was to identify those securities that offered the best opportunities for gain with the least risk and then construct a portfolio from these. Following this advice, an investor might conclude that railroad stocks all offered good risk-reward characteristics and compile a portfolio entirely from these. Intuitively, this would be foolish. Markowitz formalized this intuition. Detailing a mathematics of diversification, he proposed that investors focus on selecting portfolios based on their overall risk-reward characteristics instead of merely compiling portfolios from securities that each individually have attractive risk-reward characteristics. In a nutshell, inventors should select portfolios not individual securities. (Source: riskglossary.com)

Link to his Nobel Prize lecture if you are interested:• http://nobelprize.org/economics/laureates/1990/markowitz-lecture.pdf

Illustration: A case of 2 risky assets• Assume you have 2 risky assets (x & y) to

invest, both are normally distributed.

Rx ~ N(E(Rx), σ2x) & Ry ~ N(E(Ry), σ2

y)

In your investment portfolio, you put a in x, b in y.• a + b = 1 (a and b in %)

Your Portfolio’s Expected Return E(Rp) is:

E(Rp) = E[aRx + bRy]

=aE(Rx)+ bE(Ry)

Illustration: A case of 2 risky assets• Rx ~ N(E(Rx), σ2

x) & Ry ~ N(E(Ry), σ2y)

To calculate your Portfolio Variance:

σ2p = E[Rp - E(Rp)]2

= E[(aRx + bRy)-E[aRx + bRy]]2

= E[(aRx - aE[Rx])+(bRy - bE[bRy])]2

= E[a2(Rx - E[Rx])2 + b2(Ry - E[Ry])2 + 2ab(Rx- E[Rx])(Ry - E[Ry])]

= a2 σ2x + b2 σ2

y + 2abCov(Rx, Ry)

= a2 σ2x + b2 σ2

y + 2abCov(Rx, Ry)

Var σ2p = a2 σ2

x + b2 σ2y + 2abσxσyρxy

s.d. σp = √(a2 σ2x + b2 σ2

y + 2abσxσyρxy)

• Thus, the portfolio’s return is normally distributed too Rp ~ N(E(Rp), σ2

p)

σp = √(a2 σ2x + b2 σ2

y + 2abσxσyρxy)σp increases as ρxy increase.

Implication: given a (and thus b, because they add up to 1), if ρxy is smaller, the portfolio’s variance is smaller (i.e., the risk is lower)

Diversification: you want to maintain the expected return at a definite level but lower the risk you expose. Ideally, you hedge by including another asset of similar expected return but highly negatively correlated with your original asset.

Qs.: Can you link this concept to the reason why we focus on expected return and s.d. of return as the objects of choice?

Illustration: A case of 2 risky assets

DiversificationProposition: portfolio of less than perfectly correlated assets always offer better risk-return opportunities than the individual component assets on their own.

Proof:

If ρxy = 1 (perfectly positively correlated)

then, σp = a σx + b σy

=> portfolio s.d. is merely the weighted avg. s.d.If < 1 (less than perfectly correlated)

then, σp < a σx + b σy

=> portfolio s.d. is less than the weighted avg. s.d.

Varying the portfolio weight

13%

%8E(Rp)

a0% 100%

Suppose:

Rx ~ N(13%, (20%)2) & Ry ~ N(8%, (12%)2)

E(Rp) = E[aRx + bRy]=aE(Rx)+ bE(Ry)

a = portfolio weight on Asset x (in %), a+b=100%

20%

12%

σp

a0% 100%

Suppose:

rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)

σp = √(a2 σ2x + b2 σ2

y + 2abσxσyρxy)

ρxy=1



Min-Variance opportunity set with the 2 risky assets

= 1

13%

%8

12% 20%

σp

E(Rp)

20%

12%

σp

a0% 100%

Suppose:

rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)

σp = √(a2 σ2x + b2 σ2

y + 2abσxσyρxy)

ρxy=0.3




13%

%8

12% 20%

= .3

σp

E(Rp)

20%

12%

σp

a0% 100%

Suppose:

rx ~ N(13%, (20%)2) & ry ~ N(8%, (12%)2)

σp = √(a2 σ2x + b2 σ2

y + 2abσxσyρxy)

ρxy=-1




13%

%8

12% 20%

= -1

= -1

σp

E(Rp)

Min-Variance opportunity set with the Many risky assets

σp

Efficientfrontier

Min-variance opp. Set- Given a level of expected return, the lowest possible risky is?

Individual risky assets

E(Rp)

Min-Variance opportunity setMin-Variance Opportunity set – the locus of risk & return combinations offered by portfolios of risky assets that yields the minimum variance for a given rate of returnE(Rp)

σp

Efficient setEfficient set – the set of mean-variance choices from the investment opportunity set where for a given variance (or standard deviation) no other investment opportunity offers a higher mean return.

E(Rp)

σp

Investors’ choices with many risky assets, no risk-free asset

Efficient set

Morerisk-averseinvestor

U’’’ U’’ U’

Q

PS

Lessrisk-averseinvestor

E(Rp)

σp

Introducing risk-free assets

• Assume borrowing rate = lending rate• Then the investment opp. set will involve any

straight line from the point of risk-free assets to any risky portfolio on the min-variance opp. set

• However, only one line will be chosen because it dominates all the other possible lines.

• The dominating line = linear efficient set• Which is the line through risk-free asset point

tangent to the min-variance opp. set.• The tangency point = portfolio M (the market)

Capital market line = the linear efficient set

E(RM)

5%=Rf

σM

M

E(Rp)

σp

rf

A

M

Q

B

CMLE(Rp)

σp


rf

A

M

Q

B

CMLE(Rp)

σp


This guy’s investment portfolio consist of around 50% in risk-free asset, 50% on the market portfolio

This guy’s investment portfolio consist of around -50% in risk-free asset, 150% on the market portfolio

2-Fund Separation

• All an investor needs to know is the combination of assets that makes up portfolio M as well as risk-free asset. This is true for any investor, regardless of his degree of risk aversion.

2-Fund Separation:• “Each investor will have a utility-maximizing

portfolio that is a combination of the risk-free asset and a portfolio (or fund) of risky assets that is determined by the line drawn from the risk-free rate of return tangent to the investor’s efficient set of risky assets.”

Portfolio Theory

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