Bm410 Investments

Theory 2: Modern Portfolio Theory (MPT) and Efficient

Frontiers

How to reduce the risk of your portfolio

Real Objectives

Remember “Portfolio Theory” is a difficult subject to understand. It is in essence the attempt to answer the two critical questions:

• 1. How do you build an optimal portfolio?

• 2. How do you price assets?• The next few class periods will be

devoted to answering those two questions! Hang in there!!!

Other Objectives

A. Show how covariance and correlation affect the power of diversification to reduce portfolio risk (to give the “optimal” portfolio)

B. Construct efficient portfoliosC. Understand the theory behind factor

models

E(r)E(r)

E(rE(rpp) = 15%) = 15%

rrff = 7% = 7%

= 22%= 22%00

PP

FF

PP

S = 8/22S = 8/22

E(rE(rpp)) - - rrff = 8% = 8%

CAL: (Capital

AllocationLine)

Review of the CAL

Slope: Reward to variability ratio: ratio of risk premium to std. dev.

Risk premium

This graph is the risk return combination available by choosing different values of y. Note we have E(r) and variance on the axis.

A Portfolio Theory Timeline

CAL MPT CAPM SML APT

Note: Dates are approximate

1920s 1950s 1960s 1960s 1970s

Risk Aversion and Allocation

Key concepts regarding risk:• Greater levels of risk aversion lead to larger

proportions of the risk free rate

• Lower levels of risk aversion lead to larger proportions of the portfolio of risky assets

• Willingness to accept high levels of risk for high levels of returns would result in leveraged combinations

A. Show how Covariance and Correlation affect Diversification

• Previously, investors assumed that there was no risk in investing in government securities

• While there was no “default” risk, other types of risk became apparent

• There came a change in this “risk” view • People started realizing that there was risk to the

so-called risk-free asset—other risks• The risk-free asset wasn’t so risk-free

anymore• Analysts started using variance (standard

deviation) as a measure of risk or volatility• This opened up a world of opportunities

Harry Markowitz—the father of Modern Portfolio Theory (MPT)

Harry Markowitz came up with Modern Portfolio Theory in 1952• Key was his point of view regarding portfolios

• He looked at the assets and their contribution to the overall portfolio risk, not just individual risk!

• He noted that investors must be compensated for risk that cannot be diversified away, but not for risk that was diversifiable!

• And he did it all without computers or calculators He was later awarded the Nobel Prize in Economics for

this important work. This chapter is based on his work

MPT Notations I – (Just memorize them)

1. The return (or expected return E(rp)) of a portfolio is the weighted return of each of the assets.

Return: Expected Return:

rp = w1r1 +w2r2 E(rp) = w1E(r1) +w2E(r2)

Note 1: w1 +w2 = 1

w1=weight asset1, r1 = return asset1,

E(r) = Expected Return

• Note 1 means that the portfolio is subject to a “no short-selling” constraint that says the weights of all assets must equal 100%

MPT Notations II (continued)

2. The risk (σp2

or variance) of a portfolio is the squared weighted risk (variance) of each asset plus the cross-product

σp2

= w12σ1

2 + w22σ2

2 + 2w1w2 (ρ1,2σ1σ2 ) cross-term

• Note: It is not just the weighted risk. It is the squared weighted risk plus the cross-product

(This is a two security portfolio)

σ1 is the standard deviation (σ12 variance) of asset 1

ρ1,2 is the correlation coefficient between assets 1 and 2

Problem #1

What is the relationship of the portfolio standard deviation to the weighted average of the standard deviations of the component assets?

Remember:σp

2 = w1

2σ12 + w2

2σ22 + 2w1w2 (ρ1,2σ1σ2 )

Answer #1

In the special case that all assets are perfectly correlated, the portfolio standard deviation will be equal to the weighted average of the component standard deviations. Otherwise, the portfolio standard deviations will be less than the weighted average of the component standard deviations.

MPT Notations III (continued)

3. The measure of the way the assets in the portfolio move together is given by its covariance.

Covariance = ( ρ1,2 ) * σ1σ2

• By itself, the covariance is just a number. However, if we divide the covariance by the standard deviation of each asset, we get the traditional correlation coefficient

Correlation Coefficient (ρ1,2 ) = covariance / σ1σ2

The goal for an MPT “optimal” portfolio:

• To maximize return for a given level of risk

• To minimize risk for a given level of return

MPT Notations (continued)

The Key to diversification is the correlation = rho)

p2

= w121

2 + w222

2 + 2w1w2 (12 )

This 2w1w2 (12 ) is called the cross product What happens to the cross produce when the = x?

If1,2 is +1 = The cross product is addedIf1,2 is 0 = The cross product is droppedIf1,2 is –1 = The cross product is subtracted

r = 0r = 0

E(r)

r = 1r = 1r = -1r = -1

r = -1r = -1

= .3= .3

13%

8%

12% 20% St. Dev

Two security portfolios with different correlations

The Impact of Correlation

Problem #2

An investor is consider adding another investment to a portfolio. To achieve the maximum diversification benefits, the investor should add, if possible, an investment that has which of the following correlation coefficients with the other investments in the portfolio? • a. -1.0 b. -0.5 c. 0.0 d. +1.0

Answer #2

  a. –1.0. This is perfectly negative correlation, which helps achieve maximum diversification. However, adding any company will a correlation less than 1 will help reduce portfolio risk.

Questions

Do you understand the importance of the correlation coefficient in determining portfolio risk?

B. Efficient Frontiers

What is the efficient frontier?• It is a graphical representation of a combination of

all assets that will give either the highest return for a given level of risk, or the lowest risk for a given level of return

Why do we care?

• If our portfolio is on the efficient frontier, we will have the highest return for our risk level (or lowest risk for our return level) and is our “optimal portfolio”

• Being on the optimal frontier is a goal to strive for

We have calculated a two security portfolio. What about a three security portfolio?

• The return is: rp = w1r1 + w2r2 + w3r3

• The risk is: p2

= w121

2 + w222

2 + w3232 +

2w1w2 Cov(r1r2) + 2w1w3 Cov(r1r3) +

2w2w3 Cov(r2r3)

• Note with 3 variables you have 3 cross terms

Remember, Cov(r1r2) = (12 )

We have calculated a two security portfolio. What about a three security portfolio?

• The return is: rp = w1r1 + w2r2 + w3r3

• The risk is: p2

= w121

2 + w222

2 + w3232 +

2w1w2 Cov(r1r2) + 2w1w3 Cov(r1r3) +

2w2w3 Cov(r2r3)

• Note with 3 variables you have 3 cross terms

Remember, Cov(r1r2) = (12 )

Efficient Frontiers (continued)

Efficient Frontiers (continued)

• In General, For an “n” Security Portfolio:

rp = Weighted average of the “n” securities

You must have “n” forecasts for expected returns

p2 = Standard deviation of the portfolio

You must consider all pair wise covariance measures, or (n*(n-1))/2 calculations

• If you have 50 stocks, you will have 50 expected returns and (50*50-1)/2) or:

• 1,225 correlation coefficients (or covariance) measures.

• That is a lot to calculate.

Efficient Frontiers (continued)

What is the Mean Variance Criterion?• Mean variance criterion (the criteria of return and risk)

states:

• Portfolio A dominates Portfolio B if:

• E(rA) >= E(rB)

• [the expected return of asset A is greater or equal to the expected return of asset B]

and

• σA <= σB

• [the variance of A is less than or equal to the variance of B]

Efficient Frontiers (continued)

What is the minimum variance portfolio?

• It is the combination of assets in a portfolio which gives the lowest available risk

Why is it useful?

• It gives a starting point of minimum portfolio risk.

• From this point you can take on additional risk, but only if you want and are compensated for this risk

What is the formula for calculating the MVP?

W1 = [ σ22

- ρ1,2(σ1 * σ2 )] / [σ12 + σ2

2 - 2ρ1,2(σ1 * σ2 )]

W2 = (1-W1)

Problem #3

Suppose we had two assets with ρ1,2 = .2:• Asset 1: E(r1) = .10 σ1 = .15• Asset 2: E(r2) = .14 σ2 = .20

Calculate the minimum variance portfolio, i.e. the point of lowest risk. What are the: • a. Weights of the assets in the minimum variance

portfolio (MVP), • b. The expected return of that portfolio, and • c. The standard deviation of that portfolio?

Remember:

W1 = [ σ22

- ρ1,2(σ1 * σ2 )] / [σ12 + σ2

2 - 2ρ1,2(σ1 * σ2 )]

W2 = (1-W1)

Answer #3

a. The minimum variance portfolio weight is at W1 = [ σ2

2 - ρ1,2(σ1 * σ2 )] / [σ1

2 + σ22 - 2ρ1,2(σ1 *

σ2 )]:

• W1 = [ .22 - .2(.15*.2 )] / [.152+.22 -2*.2(.15*.2 )] or

• W1 = .673 or 67.3%

• W2 = (1-.673) or

• W2 = .327 or 32.7%

b. The expected return of the MVP is

• rp = w1r1 +w2r2 :

• rp = .673* .10 + .327 * .14 = 11.3%

Answer #3

c. The minimum variance portfolio risk or standard deviation is: p

2 = w1

212 + w2

222 + 2w1w2 (12 )

p = [.6732 2 + .3272 2 + 2 * .673 *.327 *

p = .1308 or 13.08%

Extending These Concepts to All Securities

Key Points:The optimal combinations result in lowest level of

risk for a given returnThe optimal combinations dominant, i.e., are mean

variance efficient and pass the mean variance criterion

The optimal combinations is described as the efficient frontier

The goal for an MPT “optimal” portfolio:To maximize return for a given level of riskTo minimize risk for a given level of return

E(r)

Efficientfrontier

Globalminimumvarianceportfolio MinimumMinimum

variancevariancefrontierfrontier

Individualassets

Standard Deviation

The Efficient Frontier and Minimum Variance Portfolio

Extending the Efficient Frontier to Include Riskless Asset

What happens if we include a riskless asset in the portfolio?• The optimal combination goes from

curvilinear to linear• A single combination of risky and riskless

assets will dominate

E(r)

CAL (Globalminimum variance)

CAL (A)CAL (P)

M

P

A

F

P P&F A&FM

A

G

P

M

s

Efficient Frontier with CALs

Dominant CAL with a Risk-Free Investment (F)

What happens when you combine a risk-free asset with a dominant CAL?• CAL(FP) dominates other lines -- it has the

best risk/return or the largest slope

• Slope = (E(R) - Rf) /

E(RP) - Rf) / PE(RA) - Rf) / • Regardless of risk preferences,

combinations of P & F dominate

Implications for Portfolio Construction

Determine your opportunity set• Determine your preferred set of assets, i.e., which

assets are potential candidates for the portfolio• Calculate your efficient frontier (and discard any

portfolios below the minimum variance portfolio Choose the optimal risky portfolio (ORP)

• This dominates all alternative feasible lines Choose your appropriate mix between the risky (ORP)

and the risk free (T-bills) assets• The result is called the separation property:

portfolio choice is two independent decisions: the determination of the ORP and the personal choice of the best mix of the risky and risk free asset.

Choosing a Single Portfolio

How do you choose a single portfolio among all the options on the efficient frontier?• One way is to use iso-utility curves, two-

dimensional graphs of investors preferences

More risk averse

Moderately risk averse

Less risk averse

Risk Loving

CAL and Utility CurvesE(r)

CAL (Globalminimum variance)

CAL (P)

P

F

P P&F A&FM

P

s

Implementation of MPT

What has happened since the development of Modern Portfolio Theory in 1956?• It has not caught on as quickly as would have been

expected.

• This was due to two problems:

• 1. Too much information was needed

• 2. Too much computational power was required to calculate optimal portfolios

Implementation of MPT (continued)

What was needed:• Simplification of the amount and type of

informationFor one portfolio (i.e. one point on the frontier)

of 100 securities, you need:100 expected returns and 4,950 covariance

calculations (n * (n-1) / 2 ) (or 100 standard deviations and 4,950 correlation coefficients)

• Simplification of computational proceduresPerhaps there are other ways to forecast that

covariance matrix, i.e. single and multi index-models, grouping techniques, etc.

Questions

Do we understand the concept of efficient frontiers?

C. Understand the Theory behind Factor Models

What about if, instead of doing all the calculations for your portfolio as suggested by MPT, you used instead an optimal proxy for all the stocks in the market, at P (kind of like an optimal index fund)?You could compare your stock to this proxy, and

your portfolio risk standard would be the weighted risk standard for all your stocksIt would simplify dramatically the amount of

work to be done by analysts and portfolio managers (analysts like that!)

You would create, in essence, a single- (or multi-) factor model, depending on your assumptions

Prelude to Factor ModelsE(r)

CAL (P)

P

F

P

P

M

Standard deviation

P becomes our optimal proxy for the market, similar to an index fund

Factor Models

What are factor models?Statistical models designed to estimate both the

systematic (marketed related) and firm-specific (individual firm related) risk with the goal of relating risk (both types) to investment returns on assets.

What are single factor models?Factor models which assume that stock prices

move together because of a common movement, which is assumed to be the market

What are multi-factor models?Factor models which assume that stock prices

move based on more factors than the common movement, the market, alone.

Single Factor Model

The Single Factor Asset Model:Ri = E(ri) + ßiM + ei

What is it saying?• The excess return on a security (over cash) is equal

to the expected excess holding period return plus the impact of:

• a. systematic factors or market related surprises (zero assumes there are “no surprises”), and

• b. firm-specific factors, or firm-specific surprises

Note that both M and ei have zero expected values as each represent the impact of unanticipated events

Single Factor Model (continued)

Single Factor Model notation and terms:• Notation: Ri = E(ri) + ßiM + ei

• ßi = index of a securities’ sensitivity to the supposed market factor

• M= some macro factor, which in this case, is the unanticipated movement of some macro factor

Key assumption:• The common factor is a broad market index, such as

the S&P500, and is at P on the efficient frontier.• Is it really a broad index? Is it at P?

• This is a major simplification of reality

Single Factor Model:Specification

Is this a good model?• A model is of little use unless we can test it. It

is not testable in its above form. To make it testable, we:

• 1. Use the rate of return on a broad index of securities (i.e., the S&P 500) for a proxy. So now ßiM becomes ßiRm, or M is equal to Rm,

the return on the market

• 2. The expected holding period return E(Ri) becomes ai, because that is the stock’s excess return assuming the market’s excess return is zero

Single Factor Model:Specification (continued)

So the new model, which is now testable, becomes:

• Ri = ai + ßiRm + ei

or substituting in the risk free rate becomes

(ri-rf) = ai + ßi(rm - rf) + ei

(ri - rf) = i + ßi(rm - rf) + ei(ri - rf) = i + ßi(rm - rf) + ei

Single Factor Model:Specification (continued)

Risk Premium Market Risk Premium

or Index Risk Premium

i= the stock’s expected return if the market’s excess return is zero

ßi(rm - rf) = the component of return due to

movements in the market index

(rm - rf) = 0

ei = firm specific component, not due to market movements

Firm specific Risk

Single Factor Model:Specification (continued)

Using the Risk Premium format• Let:

• Ri = (ri - rf)

• Rm = (rm - rf)

• The equation become

• Ri = ai + ßi(Rm) + ei

Single Factor Model:Fitting the Data: Regress Ri with Rm

Excess Returns (i)Or ri - rf Security

CharacteristicLine or best fit

.. ...... ..

..

.. ..

.. ....

.. ....

.. ..

.. ....

....

..

.. ..

.. ....

.. ....

.. ..

.. ....

.. ....

.. ..

..

.. ...... .... .... ..

Excess returnson market indexOr rm - rf

Algebraic Representation of the reg. line = E(Ri|Rm) = a i + ßiRm. The slope is also called the slope coefficient or simply beta.

Note that this line is average tendencies, not actual. It shows the effect of the index return on our expectations of Ri

Single Factor Model:Key Components of Risk

What are the key components of risk?• Market or systematic risk

• Risk related to the macro economic factor or market index

• Unsystematic or firm specific risk• Risk not related to the macro factor or

market indexTotal risk = Systematic + Unsystematic

Single Factor Model:Advantages and Disadvantages

AdvantagesReduces the number of inputs for diversificationEasier for security analysts to specializeFewer calculations

DisadvantagesMay be too much of a simplificationMay be a far cry from realityThere may be more than a single factor to explain

market returns

Problem #4

Investors expect the market rate of return to be 10%. The expected rate of return on the stock with a beta of 1.2 is currently 12%. If the market return this year turns out to be 8%, how would you review your expectations of the rate of return on the stock?

Answer #4

The expected return on the stock would be your beta (1.2) times the market return or:

1.2 * 8% = 9.6%

Likewise, you could also determine how much the return would decrease by multiplying the beta times the change in the market return or:

1.2 * (8%-10%) = -2.4% + 12% = 9.6%

Questions

Do we understand how factor models came about?

Review of Objectives

A. Can you see how covariance and correlation affect the power of diversification to reduce portfolio risk?

B. Can you construct efficient portfolios?

C. Do you understand the theory (and history) behind factor models?

Question

A three-asset portfolio has the following characteristics:

• Asset E(r) s.d. Weight

A 15% 22% .5

B 10% 8% .4

C 6% 3% .1

What is the expected return on this 3 asset portfolio?

Answer

E(r) = waE(r)a + wb E(r)b + wcE(r)c

= .5 (.15) + .4(.10) + .1(.06)

= .075 + .04 + .006 = .121 or 12.1%

Problem

Consistent with capital markets theory, systematic risk:

i. Refers to the variability in all risky assets caused by macroeconomic and other aggregate market-related variables

ii. Is measured by the coefficient of variation of returns on the market portfolio

iii. Refers to non-diversifiable riska. i only b. ii only c. i and iii only d. ii and iii only

Answer

c. i and iii only. Systematic risk is non diversifiable, market-related risk.