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Portfolio Mathematics:Risk with Risk-Free Asset Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviatio

Fisher effect: Approximation

R = r + i or r = R - i

Example: r = 3%, i = 6%

R = 9% = 3%+6% or r = 3% = 9%-6%

Fisher effect: Exact

Real vs Nominal Rates

i 1

R

1 r

0 1

06

0 09

.

0

% 83

P D

P P

HPR

0

1 0

2 40

48

1) Mean: most likely value

2) Variance or standard deviation

3) Skewness

* If a distribution is approximately normal, the distribution is described by characteristics 1 and 2

Characteristics of Probability Distributions

Symmetric distribution

r s.d s.d.

Normal Distribution

Subjective returns

‘s’ = number of scenarios considered

p i = probability that scenario ‘i’ will occur

r i = return if scenario ‘i’ occurs

Measuring Mean: Scenario

) r ( E

E(r) = (.1)(-.05)+(.2)(.05) +(.1)(.35) E(r) = 15 = 15%

Using Our Example:

σ 2 =[(.1)(-.05-.15) 2 +(.2)(.05- 15) 2 +…] =.01199

σ = [ 01199] 1/2 = 1095 = 10.95%

Subjective or Scenario Distributions

Measuring Variance or Dispersion of Returns

2 s

1 i

=

Standard deviation = [variance] 1/2 = σ

Risk Free T-bills Profit = 5 Risk Premium = 22-5 = 17

Risky Investments with Risk-Free Investment

 Investor’s view of risk

 A measures the degree of risk aversion

Risk Aversion & Utility

Risk Aversion and Value: The Sample Investment

Variance or Standard Deviation

• 2 dominates 1; has a higher return

• 2 dominates 3; has a lower risk

• 4 dominates 3; has a higher return

Utility and Indifference

Curves

Example (for an investor with A=4):

Indifference Curves

Expected Return

Standard Deviation Increasing Utility

Portfolio Mathematics: Assets’ Expected Return

Rule 1 : The return for an asset is the probability weighted average return in all scenarios

) r ( E

Portfolio Mathematics:

Assets’ Variance of Return

Rule 2: The variance of an asset’s return is the expected value of the squared deviations from the expected return

2 s

Portfolio Mathematics:

Risk with Risk-Free Asset

Rule 4: When a risky asset is combined with a risk-free asset, the portfolio standard deviation equals the risky asset’s standard deviation multiplied

by the portfolio proportion invested in the risky asset

σ

σ p = w risky asset × risky asset

Rule 5: When two risky assets with variances σ12 and σ22 respectively, are combined into a portfolio with portfolio weights w1 and w2,

respectively, the portfolio variance is given by:

Portfolio Mathematics:

Risk with two Risky Assets

) r , r ( Cov w

w 2 w

2

σ

 Possible to split investment funds between safe and risky assets

 Risk free asset: proxy; T-bills

 Risky asset: stock (or a portfolio)

Allocating Capital Between Risky & Risk Free Assets

 Examine risk/return tradeoff

 Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

Allocating Capital Between Risky & Risk Free Assets

The Risk-Free Asset

 Perfectly price-indexed bond – the only risk free asset in real terms;

 T-bills are commonly viewed as “the” risk-free asset;

 Money market funds - the most accessible risk-free asset for most investors

Portfolios of One Risky Asset

and One Risk-Free Asset

 Assume a risky portfolio P defined by :

E(r c ) = yE(r p ) + (1 - y)r f

r c = complete or combined portfolio

If, for example, y = 75

E(r c ) = 75(.15) + 25(.07)

= 13 or 13%

Expected Returns for

Combinations

* Rule 4 in Chapter 5

*

Variance on the Possible

Combined Portfolios

C

CAL (Capital Allocation

 Borrow at the Risk-Free Rate and invest in stock

Indifference Curves and

 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

Risk Aversion and

Allocation

CAL with Risk Preferences

CAL with Higher Borrowing Rate

σ p = 22%

Risk Reduction with

Diversification

Number of Securities

St Deviation

Market Risk

Unique Risk

w 1 = proportion of funds in Security 1

w 2 = proportion of funds in Security 2

r 1 = expected return on Security 1

r 2 = expected return on Security 2

1 w

n

1 i

1 1

σ 1 2 = variance of Security 1

σ 2 2 = variance of Security 2

Cov(r 1 ,r 2 ) = covariance of returns for

Security 1 and Security 2

Two-Security Portfolio:

Risk

) r , r ( Cov w

w 2 w

w 1 2 1 2 2 2 2 2 1 2 1 2

2

σ

ρ 1,2 = Correlation coefficient of returns

σ 1 = Standard deviation of returns for Security 1

σ 2 = Standard deviation of returns for Security 2

Covariance

2 1

2 , 1 2

1 , r ) r

(

Range of values for ρ 1,2

Three-Security Portfolio

3 3

2 2

1 1

p w r w r w r

) r , r ( Cov w

w 2

) r , r ( Cov w

w 2

) r , r ( Cov w

w 2

w w

w

3 2

3 2

3 1

3 1

2 1

2 1

2 3

2 3

2 2

2 2

2 1

2 1

2

p

+

+ +

+ +

+ σ

+ σ

+ σ

=

σ

Generally, for an n-Security Portfolio:

i i

k j

n

1 i

2 i

2 i 2

Returning to the Two-Security Portfolio

2 2 1

w 2 w

w 2 w

Two-Security Portfolios with Different Correlations

ρ = 3

ρ = -1

ρ = -1

 Relationship depends on correlation coefficient

 -1.0 < ρ < +1.0

 The smaller the correlation, the greater the risk reduction potential

 If ρ = +1.0, no risk reduction is possible

Portfolio of Two Securities:

Correlation Effects

Minimum-Variance

Combination

 Suppose our investment universe

comprises the following two securities:

Minimum-Variance Combination: ρ = 2

) r , r ( Cov 2

) r , r (

Cov w

B A

2 B

2 A

B A

2 B A

− σ

+ σ

0 )

2 0 )(

20 )(

15 ( 2 )

15 ( )

20 (

) 2 0 )(

20 )(

15 ( )

20

(

− +

−

=

3267

0 w

1

w B = − A =

Minimum -Variance:

Return and Risk with ρ = 2

 Using the weights w A and w B we determine minimum-variance portfolio’s

characteristics:

% 31

11

%) 14

)(

3267

0 (

%) 10

)(

6733

0 (

09

171 )

2 0 )(

15 )(

20 )(

3267

0 )(

6733

0

(

2

) 20 (

) 3267

0 ( )

15 (

) 6733

+ +

=

σ

% 08

13 09

.

171

σ

Minimum -Variance Combination: ρ = -.3

 Using the same mathematics we obtain:

w A = 0.6087

w B = 0.3913

 While the corresponding

minimum-variance portfolio’s characteristics are:

r P = 11.57% and

s P = 10.09%

Summary Reminder

 Objective:To present the basics of modern portfolio selection process

 Capital allocation decision

 Two-security portfolios and extensions

 The Markowitz portfolio selection model

 The optimal combinations result in lowest level of risk for a given return

 The optimal trade-off is described as the efficient frontier

 These portfolios are dominant

Extending Concepts to

All Securities

The Minimum-Variance

Frontier of Risky Assets

E(r)

Efficient frontier

Individual assets

σ

 The set of opportunities again described by the CAL

 The choice of the optimal portfolio depends on the client’s risk aversion

 A single combination of risky and riskless assets will dominate

Extending to Include A

Riskless Asset

Alternative CALs

M

E(r)

CAL (Global minimum variance)

CAL (A) CAL (P)

P

M

σ

Portfolio Selection &

Risk Aversion

E(r)

Efficient frontier of risky assets

More risk-averse investor

U’’’ U’’ U’

Q

σ

Less risk-averse investor

Efficient Frontier with Lending & Borrowing

F

P E(r)

 Equilibrium model that underlies all modern financial theory

 Derived using principles of diversification with simplified assumptions

 Markowitz, Sharpe, Lintner and Mossin are researchers credited with its development

Capital Asset Pricing

Model (CAPM)

 Individual investors are price takers

 Single-period investment horizon

 Investments are limited to traded financial assets

 No taxes, and transaction costs

Assumptions

 Information is costless and available to all investors

 Investors are rational mean-variance optimizers

 There are homogeneous expectations

Assumptions (cont’d)

 All investors will hold the same portfolio of risky assets – market portfolio

 Market portfolio contains all securities and the proportion of each security is its market value as a percentage of total market value

 The market portfolio is on the efficient frontier and, moreover, it is the tangency portfolio

Resulting Equilibrium

Conditions

 Risk premium on the market depends on the average risk aversion of all market participants

 Risk premium on an individual security is a function of its covariance with the market

Resulting Equilibrium Conditions (cont’d)

Capital Market Line

M = The market portfolio

 The risk premium on individual securities is a function of the individual security’s contribution to the risk of the market portfolio

 Individual security’s risk premium is a function of the covariance of returns with the assets that make up the market portfolio

Expected Return and Risk

on Individual Securities

Security Market Line

1.25 y

ß 6

.08

Disequilibrium Example

E(r) 15%