1

Estimating Return and Risk

Estimating Return and Risk

Chapter 7Jones, Investments:

Analysis and Management

2

Investment Decisions

Investment Decisions

Involve uncertainty Focus on expected returns

– Estimates of future returns needed to consider and manage risk

Goal is to reduce risk without affecting returns– Accomplished by building a portfolio– Diversification is key

3

Dealing With Uncertainty

Dealing With Uncertainty

Risk that an expected return will not be realized

Investors must think about return distributions, not just a single return

Use probability distributions– A probability should be assigned to

each possible outcome to create a distribution

– Can be discrete or continuous

4

Calculating Expected Return

Calculating Expected Return

Expected value – The single most likely outcome

from a particular probability distribution

– The weighted average of all possible return outcomes

– Referred to as an ex ante or expected return

i

m

1iiprR)R(E

5

Calculating RiskCalculating Risk

Variance and standard deviation used to quantify and measure risk– Measures the spread in the probability

distribution

– Variance of returns: 2 = (Ri - E(R))2pri

– Standard deviation of returns: =(2)1/2

– Ex ante rather than ex post relevant

6

Portfolio Expected Return

Portfolio Expected Return

Weighted average of the individual security expected returns– Each portfolio asset has a weight,

w, which represents the percent of the total portfolio value

– The expected return on any portfolio can be calculated as:

n

1iiip )R(Ew)R(E

7

Portfolio RiskPortfolio Risk

Portfolio risk not simply the sum of individual security risks

Emphasis on the risk of the entire portfolio and not on risk of individual securities in the portfolio

Individual stocks are risky only if they add risk to the total portfolio

8

Portfolio RiskPortfolio Risk

Measured by the variance or standard deviation of the portfolio’s return– Portfolio risk is not a weighted

average of the risk of the individual securities in the portfolio 2

i

2

p

n

1i iw

9

Risk Reduction in Portfolios

Risk Reduction in Portfolios

Assume all risk sources for a portfolio of securities are independent

The larger the number of securities the smaller the exposure to any particular risk– “Insurance principle”

Only issue is how many securities to hold

10

Risk Reduction in Portfolios

Risk Reduction in Portfolios

Random diversification– Diversifying without looking at relevant

investment characteristics– Marginal risk reduction gets smaller and

smaller as more securities are added A large number of securities is not

required for significant risk reduction International diversification is

beneficial

Portfolio Risk and Diversification

p %

35

20

0

Number of securities in portfolio10 20 30 40 ...... 100+

Portfolio risk

Market Risk

12

Markowitz Diversification

Markowitz Diversification

Non-random diversification– Active measurement and

management of portfolio risk– Investigate relationships between

portfolio securities before making a decision to invest

– Takes advantage of expected return and risk for individual securities and how security returns move together

13

Measuring Comovements in Security Returns

Measuring Comovements in Security Returns

Needed to calculate risk of a portfolio:– Weighted individual security risks

»Calculated by a weighted variance using the proportion of funds in each security

»For security i: (wi i)2

– Weighted comovements between returns»Return covariances are weighted using the

proportion of funds in each security

»For securities i, j: 2wiwj ij

14

Correlation CoefficientCorrelation Coefficient

Statistical measure of relative co-movements between security returns

mn = correlation coefficient between securities m and n mn =+1.0 = perfect positive correlation

mn =-1.0 = perfect negative (inverse) correlation

mn =0.0 = zero correlation

15

When does diversification pay?– Combining securities with perfect positive

correlation provides no reduction in risk»Risk is simply a weighted average of the

individual risks of securities

– Combining securities with zero correlation reduces the risk of the portfolio

– Combining securities with negative correlation can eliminate risk altogether

Correlation CoefficientCorrelation Coefficient

16

CovarianceCovariance Absolute measure of association

– Not limited to values between -1 and +1

– Sign interpreted the same as correlation

– The formulas for calculating covariance and the relationship between the covariance and the correlation coefficient are:

BAABAB

iBi,BA

m

1ii,AAB

pr)]R(ER)][R(ER[

17

Calculating Portfolio Risk

Calculating Portfolio Risk

Encompasses three factors– Variance (risk) of each security– Covariance between each pair of

securities– Portfolio weights for each security

Goal: select weights to determine the minimum variance combination for a given level of expected return

18

Generalizations – The smaller the positive

correlation between securities, the better

– As the number of securities increases:»The importance of covariance

relationships increases»The importance of each individual

security’s risk decreases

Calculating Portfolio Risk

Calculating Portfolio Risk

19

Simplifying Markowitz

Calculations

Simplifying Markowitz

Calculations Markowitz full-covariance model

– Requires a covariance between the returns of all securities in order to calculate portfolio variance

– Full-covariance model becomes burdensome as the number of securites in a portfolio grows»n(n-1)/2 unique covariances for n securities

Therefore, Markowitz suggests using an index to simplify calculations

Efficient PortfoliosEfficient Portfolios

An efficient portfolio has the smallest portfolio risk for a given level of expected return

Alternatively, an efficient portfolio maximizes the expected return for a given level of portfolio risk

Porfolios located on efficient frontier dominate all other portfolios 20

Diversifiable vs. Nondiversifiable

Risk

Diversifiable vs. Nondiversifiable

Risk Diversifiable, or nonsystematic, risks

are those risks which are unique to individual companies– Adding nonperfectly correlated stocks to

a portfolio reduces its nonsystematic risk Nondiversifiable risks are called

systematic risks– systematic risks include interest rate

risk, market risk and inflation risk 21

Capital Asset Pricing Model (CAPM)

Capital Asset Pricing Model (CAPM)

Beta– Beta is a measure of the systematic risk

of a security that cannot be avoided through diversification`

– The overall market has a beta of 1»Riskier stocks, those which are more volatile

than the market have Betas greater than 1»Less risky stocks have Betas less than 1

CAPMCAPM

Required rate of return– ki = Risk free rate + Risk

premium»=RF + ßi[E(Rm) - RF]

Security Market Line (SML) is the linear relationship between an asset’s risk and its required rate of return