Slides_Fundamentals of Investments - Chapter 17 doc

Š The expected return on a portfolio is a linear combination of the expected returns on the assets in that portfolio... Š Note that portfolio variance is not generally a simple combinat

Don’t Put All Your Eggs in One Basket

Diversification and Asset Allocation

Our goal in this chapter is to examine the role of diversification and asset allocation in investing

Goal

Š The role and impact of diversification were

first formally explained in the early 1950s by Harry Markowitz

Š Based on his work, we will look at how

diversification works, and how we can be sure

we have an efficiently diversified portfolio

p return return

expected

Expected Returns

Calculating the Variance

Š Variance is calculated as the sum of the

squared deviations from the expected return multiplied by their probabilities

p return expected return 2variance

Š The standard deviation is the square root of the variance Standard deviation = σ = √variance

Calculating the Variance

Š One convenient way of describing a portfolio

is to list the percentages of the portfolio’s total value that are invested in each portfolio asset

We call these percentages the

portfolio weights.

Portfolios

Group of assets such as stocks and bonds held by an investor

Š The expected return on a portfolio is a linear

combination of the expected returns on the assets in that portfolio

where E(R P ) = expected portfolio return

w i = portfolio weight of portfolio asset i

E(R i ) = expected return on portfolio asset i

( ) = ∑ [ × ( ) ]

i

i i

R E

Š Note that portfolio variance is not generally a simple

combination of the variances of the portfolio assets.

Š Moreover, it may be possible to construct a portfolio

of risky assets with zero portfolio variance!

where VAR(R P ) = variance of portfolio return

p s = probability of state of economy s

E(R s ) = expected portfolio return given state s

( ) = ∑ [ × { ( ) ( ) − } ]

s

P s

s

R

Diversification and Portfolio Risk

The Principle of Diversification

Why Diversification Works

Š Positively correlated assets tend to move up

and down together, while negatively correlated

assets tend to move in opposite directions

Correlation

The tendency of the returns on two assets to move together Imperfect correlation is the key reason why diversification reduces

portfolio risk as measured by the portfolio standard deviation

Why Diversification Works

Š The correlation coefficient is denoted by

Corr(R A , R B) or ρ It measures correlation and ranges from -1 (perfect negative correlation) to

0 (uncorrelated) to +1 (perfect positive correlation)

Why Diversification Works

Why Diversification Works

Why Diversification Works

Calculating Portfolio Risk

Š For a portfolio of two assets, A and B, the

variance of the return on the portfolio is:

( A B )

B A

B A

B B

A A

p w σ w σ 2 w w σ σ Corr R R

where w A = portfolio weight of asset A

w B = portfolio weight of asset B

such that w A + w B = 1

Correlation and Diversification

Š Suppose that as a very conservative,

risk-averse investor, you decide to invest all of your money in a bond mutual fund Is this decision a wise one?

Correlation and Diversification

Correlation and Diversification

Correlation and Diversification

Š The various combinations of risk and return

available all fall on a smooth curve

Š This curve is called an investment opportunity

set because it shows the possible combinations

of risk and return available from portfolios of these two assets

Š A portfolio that offers the highest return for its

level of risk is said to be an efficient portfolio.

Š The undesirable portfolios are said to be

dominated or inefficient.

More on Correlation & the Risk-Return Trade-Off

Risk and Return with Multiple Assets

The Markowitz Efficient Frontier

Š For the plot, the upper left-hand boundary is

the Markowitz efficient frontier All the other possible combinations are inefficient

Š Note that Markowitz analysis is not usually

extended to large collections of individual assets because of the data requirements

Markowitz efficient frontier

The set of portfolios with the maximum return for a given standard deviation

Work the Web

 Perform a Markowitz-type analysis at:

Chapter Review

Š Diversification and Portfolio Risk

Î The Effect of Diversification: Another Lesson from Market History

Î The Principle of Diversification

Š Correlation and Diversification

Î Why Diversification Works

Î Calculating Portfolio Risk

Î More on Correlation and the Risk-Return Off

Trade-Chapter Review

Š The Markowitz Efficient Frontier

Î Risk and Return with Multiple Assets