FNCE 625 – Investment Analysis and Management

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Investments: Analysis and Management

Fourteenth Edition

Gerald R. Jensen and Charles P. Jones

Chapter 7

Portfolio Theory

Investment Decisions

Involve uncertainty

Focus on expected returns

Estimates of future returns need to consider and manage risk

Investors often overly optimistic about expected returns

Goal is to reduce risk without affecting returns

Accomplished by building a portfolio

Diversification is key

Risk and Return Measures 1

Ex post Calculations

Mean (Average) Return

Variance of Return

Risk and Return Measures 2

Ex ante Calculations

Expected return:

Variance of Returns

Where, Ps equals probability of state s and Rs equals return in state s.

Dealing With Uncertainty

Risk – the fact that an expected return may not be realized

Investors must think about return distributions

Probabilities weight outcomes

Assigned to each possible outcome to create a distribution

History provides guide but must be modified for expected future changes

Distributions can be discrete or continuous

Calculating Expected Return

Expected return for asset “i” E(Ri)

Weighted average of all possible returns (Ri,s) included in the probability distribution

Each outcome weighted by probability of occurrence (Ps)

Referred to as expected return

Calculating Risk

Variance and standard deviation used to quantify and measure risk

Measure spread (dispersion) around the mean

Variance of returns is in percent squared

Standard deviation of returns (σ) is the square root of variance and is measured in percent

Modern Portfolio Theory

Framework for selection of portfolios based on risk and expected return

Used, to varying degrees, by financial managers

Quantifies benefits of diversification

Security correlations are crucial in determining portfolio risk

An asset with high volatility may have low risk

Portfolio Expected Return

Weighted average of the individual security expected returns

Each asset “i” has a weight, w, which represents the asset’s value as a percent of the portfolio value

Portfolio Risk 1

Portfolio risk is measured by the variance or standard deviation of portfolio returns

Portfolio variance is impacted by two characteristics:

The variance in returns for the individual assets included in the portfolio

The co-movement of returns for the individual assets included in the portfolio

Portfolio Risk 2

Portfolio risk is “not” the weighted average of individual security risks

The risk of individual securities is “not” the crucial consideration

Diversification almost always lowers risk

An asset with high σ may add little to portfolio risk

Portfolio Risk 3

Variance of a Portfolio

σij = covariance of asset i and asset j

Risk Reduction in Portfolios 1

Market risk affects all firms, cannot be diversified away

It is systematic i.e., part of the system

The larger the number of securities, the smaller the exposure to any particular risk

“Insurance principle”

Only issue is how many securities to hold

Risk Reduction in Portfolios 2

Random (or naïve) diversification

Diversifying without looking at how security returns are related to each other

Marginal risk reduction gets smaller as securities are added

Random diversification is beneficial but not optimal

Risk reduction kicks in as securities added

Research suggests it takes a large number of securities to eliminate majority of risk

Security Co-movement

Correlation (ρij) and covariance (σij) measure the tendency for security returns to move in the same or opposite directions

Correlation (ρij)

ρij > 0	securities move together
ρij < 0	securities move apart
ρij = 0	no tendency one way or the other
ρij = −1	perfect negative correlation
ρij = +1	perfect positive correlation

Correlation and Portfolio Risk

Returns to H-Tech

Returns to Giffen

Portfolio: 50% Giffen & 50% H-Tech

Correlation Coefficient

When does diversification pay?

With perfect positive correlation, risk is a weighted average, therefore, no diversification benefit

With perfect negative correlation, expected return can be assured

With zero correlation, significant risk reduction can be achieved

Cannot eliminate risk

Negative correlation or low positive correlation is ideal, but unlikely

Calculating Portfolio Risk 1

Three inputs to calculate portfolio risk

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

Calculating Portfolio Risk 2

Generalizations

The lower the correlation/covariance between securities, the better

As the number of securities increases:

Number of covariances grows quickly

The importance of covariance relationships increases

The importance of each individual security’s risk decreases

Simplifying Markowitz Calculations

Markowitz full-covariance model

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

set of unique covariances for n securities

Markowitz suggests using an index to which all securities are related

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FNCE 625 – Investment Analysis and Management

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