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Chapter 6. Risk and Return: Past and Prologue. Rates of Return: Single Period. HPR = Holding Period Return P 1 = Ending price P 0 = Beginning price D 1 = Dividend during period one. Rates of Return: Single Period Example. Ending Price = 24 Beginning Price = 20 Dividend = 1

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## Chapter 6

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Chapter6

Risk and Return: Past and Prologue

Rates of Return: Single Period

HPR = Holding Period Return

P1 = Ending price

P0 = Beginning price

D1 = Dividend during period one

Rates of Return: Single Period Example

Ending Price = 24

Beginning Price = 20

Dividend = 1

HPR = ( 24 - 20 + 1 )/ ( 20) = 25%

Data from Text Example p. 154

1 2 3 4

Assets(Beg.) 1.0 1.2 2.0 .8

HPR .10 .25 (.20) .25

TA (Before

Net Flows 1.1 1.5 1.6 1.0

Net Flows 0.1 0.5 (0.8) 0.0

End Assets 1.2 2.0 .8 1.0

Returns Using Arithmetic and Geometric Averaging

Arithmetic

ra = (r1 + r2 + r3 + ... rn) / n

ra = (.10 + .25 - .20 + .25) / 4

= .10 or 10%

Geometric

rg = {[(1+r1) (1+r2) .... (1+rn)]} 1/n - 1

rg = {[(1.1) (1.25) (.8) (1.25)]} 1/4 - 1

= (1.5150) 1/4 -1 = .0829 = 8.29%

Dollar Weighted Returns

Internal Rate of Return (IRR) - the discount rate that results present value of the future cash flows being equal to the investment amount

• Considers changes in investment
• Initial Investment is an outflow
• Ending value is considered as an inflow
• Additional investment is a negative flow
• Reduced investment is a positive flow
Dollar Weighted Average Using Text Example

Net CFs 1 2 3 4

\$ (mil) - .1 - .5 .8 1.0

Solving for IRR

1.0 = -.1/(1+r)1 + -.5/(1+r)2 + .8/(1+r)3 +

1.0/(1+r)4

r = .0417 or 4.17%

Quoting Conventions

APR = annual percentage rate

(periods in year) X (rate for period)

EAR = effective annual rate

( 1+ rate for period)Periods per yr - 1

Example: monthly return of 1%

APR = 1% X 12 = 12%

EAR = (1.01)12 - 1 = 12.68%

Characteristics of Probability Distributions

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

Normal Distribution

s.d.

s.d.

r

Symmetric distribution

S

E

(

r

)

=

p

(

s

)

r

(

s

)

s

Measuring Mean: Scenario or Subjective Returns

Subjective returns

p(s) = probability of a state

r(s) = return if a state occurs

1 to s states

Numerical Example: Subjective or Scenario Distributions

StateProb. of State rin State

1 .1 -.05

2 .2 .05

3 .4 .15

4 .2 .25

5 .1 .35

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

E(r) = .15

S

2

Variance

=

p

(

s

)

[

r

-

E

(

r

)]

s

s

Measuring Variance or Dispersion of Returns

Subjective or Scenario

Standard deviation = [variance]1/2

Using Our Example:

Var =[(.1)(-.05-.15)2+(.2)(.05- .15)2...+ .1(.35-.15)2]

Var= .01199

S.D.= [ .01199] 1/2 = .1095

Real vs. Nominal Rates

Fisher effect: Approximation

nominal rate = real rate + inflation premium

R = r + i or r = R - i

Example r = 3%, i = 6%

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

Fisher effect: Exact

r = (R - i) / (1 + i)

2.83% = (9%-6%) / (1.06)

Annual Holding Period ReturnsFrom Figure 6.1 of Text

Geom Arith Stan.

Series Mean% Mean% Dev.%

Lg Stk 11.01 13.00 20.33

Sm Stk 12.46 18.77 39.95

LT Gov 5.26 5.54 7.99

T-Bills 3.75 3.80 3.31

Inflation 3.08 3.18 4.49

Annual Holding Period Risk Premiums and Real Returns

Risk Real

Lg Stk 9.2 9.82

Sm Stk 14.97 15.59

LT Gov 1.74 2.36

T-Bills --- 0.62

Inflation --- ---

Allocating Capital Between Risky & Risk-Free Assets
• 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 (cont.)
• Issues
• Demonstrate how different degrees of risk aversion will affect allocations between risky and risk free assets

rf = 7%

srf = 0%

E(rp) = 15%

sp = 22%

y = % in p

(1-y) = % in rf

Example Using the Numbers in Chapter 6 (pp 171-173)

E(rc) = yE(rp) + (1 - y)rf

rc = complete or combined portfolio

For example, y = .75

E(rc) = .75(.15) + .25(.07)

= .13 or 13%

Expected Returns for Combinations

Possible Combinations

E(r)

E(rp) = 15%

P

rf = 7%

F

0

s

22%

If y = .75, then

= .75(.22) = .165 or 16.5%

c

If y = 1

= 1(.22) = .22 or 22%

c

If y = 0

= 0(.22) = .00 or 0%

c

Combinations Without Leverage

s

s

s

Using Leverage with Capital Allocation Line

Borrow at the Risk-Free Rate and invest in stock

Using 50% Leverage

rc = (-.5) (.07) + (1.5) (.15) = .19

sc = (1.5) (.22) = .33

CAL

(Capital

Allocation

Line)

E(r)

P

E(rp) = 15%

E(rp) -

rf = 8%

) S = 8/22

rf = 7%

F

s

0

P

= 22%

Risk Aversion and Allocation
• 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
Quantifying Risk Aversion

E(rp) = Expected return on portfolio p

rf = the risk free rate

.005 = Scale factor

A x sp = Proportional risk premium

The larger A is, the larger will be the addedreturn required for risk

Quantifying Risk Aversion

Rearranging the equation and solving for A

Many studies have concluded that investors’ average risk aversion is between 2 and 4

Chapter7

Efficient Diversification

n

S

W

=

1

i

i

=1

Two-Security Portfolio: Return

rp = W1r1 +W2r2

W1 = Proportion of funds in Security 1

W2 = Proportion of funds in Security 2

r1 = Expected return on Security 1

r2 = Expected return on Security 2

s12 = Variance of Security 1

s22 = Variance of Security 2

Cov(r1r2) = Covariance of returns for

Security 1 and Security 2

Two-Security Portfolio: Risk

sp2= w12s12 + w22s22 + 2W1W2 Cov(r1r2)

Covariance

Cov(r1r2) = r1,2s1s2

r1,2 = Correlation coefficient of

returns

s1 = Standard deviation of

returns for Security 1

s2 = Standard deviation of

returns for Security 2

Correlation Coefficients: Possible Values

Range of values for r1,2

-1.0 <r < 1.0

If r = 1.0, the securities would be perfectly positively correlated

If r = - 1.0, the securities would be perfectly negatively correlated

Three-Security Portfolio

rp = W1r1 +W2r2 + W3r3

s2p = W12s12

+ W22s22

+ W32s32

+ 2W1W2

Cov(r1r2)

+ 2W1W3

Cov(r1r3)

+ 2W2W3

Cov(r2r3)

In General, For an n-Security Portfolio:

rp = Weighted average of the

n securities

sp2 = (Consider all pair-wise

covariance measures)

Two-Security Portfolio

E(rp) = W1r1 +W2r2

sp2= w12s12 + w22s22 + 2W1W2 Cov(r1r2)

sp= [w12s12 + w22s22 + 2W1W2 Cov(r1r2)]1/2

E(r)

13%

r = -1

r = .3

r = -1

8%

r = 1

St. Dev

12%

20%

TWO-SECURITY PORTFOLIOS WITH DIFFERENT CORRELATIONS

r = 0

Portfolio Risk/Return Two Securities: Correlation Effects
• Relationship depends on correlation coefficient
• -1.0 <r< +1.0
• The smaller the correlation, the greater the risk reduction potential
• If r = +1.0, no risk reduction is possible

s

Sec 1

E(r1) = .10

= .15

r

= .2

12

s

Sec 2

E(r2) = .14

= .20

2

Minimum Variance Combination

1

s 2

- Cov(r1r2)

2

=

W1

s 2

s 2

- 2Cov(r1r2)

+

2

1

= (1 - W1)

W2

Minimum Variance Combination: r = .2

(.2)2 - (.2)(.15)(.2)

=

W1

(.15)2 + (.2)2 - 2(.2)(.15)(.2)

W1

= .6733

W2

= (1 - .6733) = .3267

Minimum Variance: Return and Risk with r = .2

rp = .6733(.10) + .3267(.14) = .1131

s

= [(.6733)2(.15)2 + (.3267)2(.2)2 +

p

1/2

2(.6733)(.3267)(.2)(.15)(.2)]

1/2

= [.0171]

= .1308

s

p

Minimum Variance Combination: r = -.3

(.2)2 - (.2)(.15)(.2)

=

W1

(.15)2 + (.2)2 - 2(.2)(.15)(-.3)

W1

= .6087

W2

= (1 - .6087) = .3913

Minimum Variance: Return and Risk with r = -.3

rp = .6087(.10) + .3913(.14) = .1157

s

= [(.6087)2(.15)2 + (.3913)2(.2)2 +

p

1/2

2(.6087)(.3913)(.2)(.15)(-.3)]

1/2

= [.0102]

= .1009

s

p

Extending Concepts to All Securities
• 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

The minimum-variance frontier of risky assets

E(r)

Efficient

frontier

Individual

assets

Global

minimum

variance

portfolio

Minimum

variance

frontier

St. Dev.

Extending to Include Riskless Asset
• The optimal combination becomes linear
• A single combination of risky and riskless assets will dominate

ALTERNATIVE CALS

CAL (P)

CAL (A)

E(r)

M

M

P

P

CAL (Global

minimum variance)

A

A

G

F

s

P

P&F

A&F

M

Dominant CAL with a Risk-Free Investment (F)

CAL(P) dominates other lines -- it has the best risk/return or the largest slope

Slope = (E(R) - Rf) / s

[ E(RP) - Rf) / s P] > [E(RA) - Rf) / sA]

Regardless of risk preferences combinations of P & F dominate

Single Factor Model

ri = E(Ri) + ßiF + e

F= some macro factor; in this case F is unanticipated movement; F is commonly related to security returns

Assumption: a broad market index like the S&P500 is the common factor

Single Index Model

Risk Prem

Market Risk Prem

or Index Risk Prem

a

= the stock’s expected return if the

market’s excess return is zero

i

(rm - rf)= 0

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

movements in the market index

ei = firm specific component, not due to market

movements

Let: Ri = (ri - rf)

format

Rm = (rm - rf)

Ri = ai + ßi(Rm)+ ei

Estimating the Index Model

Excess Returns (i)

.

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Security

Characteristic

Line

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Excess returns

on market index

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Ri = ai + ßiRm + ei

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 index
• Total risk = Systematic + Unsystematic
Measuring Components of Risk

si2 = bi2sm2 + s2(ei)

where;

si2 = total variance

bi2sm2 = systematic variance

s2(ei) = unsystematic variance

Examining Percentage of Variance

Total Risk = Systematic Risk + Unsystematic Risk

Systematic Risk/Total Risk = r2

ßi2 sm2 / s2 = r2

bi2sm2 / bi2sm2 + s2(ei) = r2

Advantages of the Single Index Model
• Reduces the number of inputs for diversification
• Easier for security analysts to specialize
Chapter8

Capital Asset Pricing and Arbitrage Pricing Theory

Capital Asset Pricing Model (CAPM)
• 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
Assumptions
• Individual investors are price takers
• Single-period investment horizon
• Investments are limited to traded financial assets
• No taxes, and transaction costs
Assumptions (cont.)
• Information is costless and available to all investors
• Investors are rational mean-variance optimizers
• Homogeneous expectations
Resulting Equilibrium Conditions
• All investors will hold the same portfolio for 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
Resulting Equilibrium Conditions (cont.)
• 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
Capital Market Line

E(r)

CML

M

E(rM)

rf

s

sm

M = Market portfolio rf = Risk free rate E(rM) - rf = Market risk premium E(rM) - rf = Market price of risk

= Slope of the CAPM

s

M

Expected Return and Risk on Individual Securities
• 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
Security Market Line

E(r)

SML

E(rM)

rf

ß

ß

= 1.0

M

SML Relationships

b = [COV(ri,rm)] / sm2

Slope SML = E(rm) - rf

SML = rf + b[E(rm) - rf]

Sample Calculations for SML

E(rm) - rf = .08 rf = .03

bx = 1.25

E(rx) = .03 + 1.25(.08) = .13 or 13%

by = .6

e(ry) = .03 + .6(.08) = .078 or 7.8%

Graph of Sample Calculations

E(r)

SML

Rx=13%

.08

Rm=11%

Ry=7.8%

3%

ß

.6

1.0

1.25

ß

ß

ß

y

m

x

Disequilibrium Example

E(r)

SML

15%

Rm=11%

rf=3%

ß

1.0

1.25

Disequilibrium Example
• Suppose a security with a b of 1.25 is offering expected return of 15%
• According to SML, it should be 13%
• Underpriced: offering too high of a rate of return for its level of risk
Security Characteristic Line

Excess Returns (i)

SCL

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Excess returns

on market index

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Ri = ai + ßiRm + ei

Using the Text Example p. 245, Table 8.5:

Excess

GM Ret.

Excess

Mkt. Ret.

Jan.

Feb.

.

.

Dec

Mean

Std Dev

5.41

-3.44

.

.

2.43

-.60

4.97

7.24

.93

.

.

3.90

1.75

3.32

Regression Results:

a

rGM - rf = + ß(rm - rf)

a

ß

Estimated coefficient

Std error of estimate

Variance of residuals = 12.601

Std dev of residuals = 3.550

R-SQR = 0.575

-2.590

(1.547)

1.1357

(0.309)

Arbitrage Pricing Theory

Arbitrage - arises if an investor can construct a zero investment portfolio with a sure profit

• Since no investment is required, an investor can create large positions to secure large levels of profit
• In efficient markets, profitable arbitrage opportunities will quickly disappear
Arbitrage Example from Text pp. 255-257

Current Expected Standard

Stock Price\$ Return% Dev.%

A 10 25.0 29.58

B 10 20.0 33.91

C 10 32.5 48.15

D 10 22.5 8.58

Arbitrage Portfolio

Mean Stan. Correlation

Return Dev. Of Returns

Portfolio

A,B,C 25.83 6.40 0.94

D 22.25 8.58

Arbitrage Action and Returns

E. Ret.

* P

* D

St.Dev.

Short 3 shares of D and buy 1 of A, B & C to form P

You earn a higher rate on the investment than you pay on the short sale

APT and CAPM Compared
• APT applies to well diversified portfolios and not necessarily to individual stocks
• With APT it is possible for some individual stocks to be mispriced - not lie on the SML
• APT is more general in that it gets to an expected return and beta relationship without the assumption of the market portfolio
• APT can be extended to multifactor models