Foundations of Finance
Lecture 7
Diversification: Portfolio Theory
Lecture Readings: Foundations of Finance, pp. 259-263, 299-305, 337-349
1. Lecture Overview
During this lecture, we will discuss the concepts of portfolio theory and diversification. Diversification allows an individual to reduce the risk of their investment without sacrificing any expected return simply by spreading their wealth over a portfolio comprising a number of assets in an appropriate way. During the course of the lecture we will discuss:
What diversification is;
How including multiple assets in a portfolio can achieve diversification;
Which assets to include in order to achieve the greatest level of diversification; and,
How to construct a diversified portfolio in practice.
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2. Review of Statistics
Before we start discussing portfolio theory and the concept of diversification, we will briefly review the statistical terms discussed last lecture. These terms are important to understand as they are central to portfolio theory. More specifically, we will revisit the definitions of:
Random variables;
Expected values; and,
Standard deviation and variance.
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2. Review of Statistics
Random Variables:
A random variable is one that can take on any number of different values. Each value has an associated probability of occurring. The uncertainty associated with the outcome of a random variable is described by a probability distribution, with the most commonly used distribution being the normal distribution, see below:
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2. Review of Statistics
Expected Values:
The value we expect a random variable (X) to take is known as its expected value or mean
Standard Deviation:
Standard deviation, s, is a measure of spread. It is based on how far each value of X varies from the mean. A small standard deviation means that the data doesn’t vary a lot from the mean. A large standard deviation means that the data is more spread out.
Variance:
The variance, s2, is simply the standard deviation squared.
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3. Diversification
Recall from last week the assumption that investors are risk averse and therefore prefer less risk to more. Diversification provides a means of reducing risk faced by investors without sacrificing expected return by combining assets that don’t move perfectly together in a portfolio.
Note that the ideas we are about to discuss relate to a 2-asset portfolio case but can be extended to consider more than 2 assets (these extensions are considered in FINM2003 Investments).
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Total risk decreases as the number of risky assets increases
Due to diversification
Decreasing unsystematic risk
Decreases to average covariance
Systematic risk always present
Economic factors
Influence all risky assets
Cannot diversify away
Portfolio total risk
2p
n
Number of risky assets in portfolio
Systematic risk
Unsystematic risk
(= average of all covariances
between assets in portfolio)
3. Diversification
Systematic
Risk ()
Un-systematic
Risk
Total
Risk
()
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Corporate Finance
Risk and return 1
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Measuring How Assets Move Together
Correlation and Covariance:
The covariance between variables X and Y, sXY, is a measure of association between the two variables. For example, we may be interested in whether there is an association between the return on a company’s stock and the return on the stock market in general:
If the stock market always went up at the same time company’s stock went up, the covariance would be positive;
If the return on the stock and the return on the market were not associated in any way, the covariance would be near zero; and,
If when the market went up, the stock went down, the covariance would be negative.
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Measuring How Assets Move Together
Correlation and Covariance (Continued):
However, covariance is sensitive to the scale of measurement of X and Y and therefore the degree of association (as opposed to the sign) is difficult to interpret. Conversely, the correlation coefficient is a standardised measure of association between two variables. It is standardized as correlation measures must lie between negative one and positive one. This makes it easy to gauge the extent to which two variables are associated. The correlation coefficient, xy, is calculated as:
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Measuring How Assets Move Together
Correlation:
We now consider the following list of correlation values between two assets, Asset 1 and Asset 2, 12:
Case 1: Perfect positive correlation (12=+1);
Case 2: Perfect negative correlation (12=-1); and.
Case 3: Non-perfect correlation (-1< 12<+1).
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Measuring How Assets Move Together
Case 1: Perfect positive correlation (12=+1)
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Measuring How Assets Move Together
Case 2: Perfect negative correlation (12=-1)
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Measuring How Assets Move Together
Case 3: Non-perfect correlation (-1< 12<+1)
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3. Diversification
The optimal investment strategy in terms of constructing a portfolio to reduce risk depends on the properties of the two assets and how they are related to one another.
Before we discuss the “ideal” diversification properties, we will go through how to calculate the expected return, standard deviation and variance on a 2-asset portfolio. We will then use these concepts to prove that diversification allows us to reduce risk without sacrificing expected return.
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4. Diversification
Consider the two assets below:
Two important characteristics of any asset are the:
1. Expected return; and,
2. Standard deviation around that expected return.
We know that investors prefer higher expected returns and lower standard deviations (lower risk). Neither of the two assets in
the diagram above is superior in both measures.
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Portfolio Expected Return
Now consider forming a portfolio that includes both Asset 1 and Asset 2 in some proportion. From the properties of random variables reviewed earlier in the lecture, the expected return of this portfolio, E(Rp), is given by:
Where:
w1 =Proportion invested in Asset 1;
w2 =Proportion invested in Asset 2;
E(R1) =Expected return on Asset 1; and,
E(R2) =Expected return on Asset 2.
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Portfolio Expected Return
Example: Expected Return of a Portfolio
If a portfolio comprises 50% of Asset A and 50% of Asset B, and the expected returns on these assets are 10% each, the expected return on the portfolio is calculated as:
The fact that the expected return on the portfolio is 10% is unsurprising given the return on both assets included in the portfolio is also 10%.
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Discussion
Portfolio return is just the weighted average of the individual stock returns.
Shouldn’t the portfolio standard deviation also be the weighted average of the individual stock standard deviations?
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NO! The key element is the degree of correlation (and hence covariance) among the assets in the portfolio. The lower the correlation, the greater the gains from diversification. The standard deviation of a portfolio should be less than the weighted average of the individual stock standard deviations (exception when ρ = 1). Assets do not move perfectly together. Hence, there is a benefit of diversifying across 2+ assets.
Corporate Finance
RIsk and return 1
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Portfolio Variance and Standard Deviation
The variance of the expected return on a portfolio is the weighted sum of the variances of the individual assets plus the weighted sum of the covariances of the individual assets, or:
Substituting in the formula for covariance, we can also write the portfolio variance equation as:
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Portfolio Variance and Standard Deviation
The standard deviation of the portfolio, p which provides us with a measure of its risk, is simply the square root of its variance, or:
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Portfolio Variance and Standard Deviation
Example: Variance and Standard Deviation of a Portfolio
Return to the earlier example of a portfolio comprising 50% of Asset A and 50% of Asset B, where the standard deviations of these assets are both 12%. If the correlation between these assets is , the variance and standard deviation of the portfolio are calculated as:
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Portfolio Variance and Standard Deviation
The previous example illustrates a very important idea: Combining assets that do not move perfectly together in a portfolio reduces variance and, therefore, standard deviation without lowering expected return.
The question is how do we reduce risk by the greatest amount?
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Portfolio Variance and Standard Deviation
The ideal situation, in terms of risk, would be to combine assets that move in opposite directions to one another (. have perfectly negative correlation).
However, diversification benefits are still available for assets with less than perfect positive correlation, though these benefits will decrease as the coefficient of correlation between the assets increases.
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Portfolio Variance and Standard Deviation
We will now go on to illustrate this by considering the (exhaustive) list of possible values of r1,2 discussed earlier and the resulting effects upon diversification:
Case 1: Perfect positive correlation (12=+1);
Case 2: Perfect negative correlation (12=-1); and,
Case 3: Non-perfect correlation (-1<12 <+1).
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Portfolio Variance and Standard Deviation
We will continue with the example of the 2 asset portfolio comprising 50% of Asset A and 50% of Asset B, where the standard deviations of these assets are both 12%.
As the weighting of the assets remains constant, so too will the portfolio’s expected return. However, we will show that the risk of the portfolio decreases as the correlation between the assets approaches –1.
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Portfolio Variance and Standard Deviation
Case 1: Perfect Positive Correlation
The variance and standard deviation of the portfolio if the assets are perfectly positively correlated is calculated as:
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Portfolio Variance and Standard Deviation
Case 2: Perfect Negative Correlation
The variance and standard deviation of the portfolio if the assets are perfectly negatively correlated is calculated as:
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Portfolio Variance and Standard Deviation
Case 3: Non-Perfect Correlation
The variance and standard deviation of the portfolio if the assets are non-perfectly correlated with AB= is calculated as:
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Portfolio Variance and Standard Deviation
The weights of each asset in a 2-asset portfolio resulting in the LEAST possible risk can be calculated using the minimum variance portfolio equation:
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Minimum Variance Portfolio Example
Share Current
Weights
Expected
return E[ri] Standard
deviation i
1
2
1,2 =
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Constructing a Diversified Portfolio
A number of commonsense procedures can be useful in constructing a diversified portfolio. These procedures are detailed below and involve selecting assets that are relatively unrelated.
1. Diversify across industries: Investing in a number of different stocks within the same industry does not generate a diversified portfolio since the returns of firms within an industry tend to be highly correlated. Diversification benefits can be increased by selecting stocks from different industries.
2. Diversify across industry groups: Some industries themselves can be highly correlated with other industries and hence diversification benefits can be maximized by selecting stocks from those industries that tend to move in opposite directions or have very little correlation with each other.
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Constructing a Diversified Portfolio
3. Diversify across geographical regions: Companies whose operations are in the same geographical region are subject to the same risks in terms of natural disasters and state or local tax changes. These risks can be diversified by investing in companies whose operations are not in the same geographical region.
4. Diversify across economies: Stocks in the same country tend to be more highly correlated than stocks across different countries. This is because many taxation and regulatory issues apply to all stocks in a particular country. International diversification provides a means for diversifying these risks.
5. Diversify across asset classes: Investing across asset classes such as stocks, bonds, and real property also produces diversification benefits. The returns of two stocks tend to be more highly correlated, on average, than the returns of a stock and a bond or a stock and an investment in real estate.
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Diversification with Real Data
Portfolio features:
10 stocks
equal weights of each stock
Portfolio experiences:
average returns BUT
< average risk
2003 to 2007
Stock Returns Std Dev
BHP % %
RIO % %
CBA % %
WBC % %
QBE % %
WDC % %
LEI % %
QAN % %
TLS % %
WOW % %
Average % %
Portfolio % %
ASX200 % %
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Next Week….
An important application of the material discussed in this lecture is in the derivation of the Capital Asset Pricing Model (CAPM). We will discuss the CAPM, which provides the basis for determining an appropriate discount rate to use in valuing investment proposals, in next week’s lecture.
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Chart1
Return on Asset 1
Return on Asset 2
Perfect Positive Correlation Between the Returns on Assets 1 and 2
Sheet1
Return on Asset 1 Return on Asset 2
Sheet1
Return on Asset 1
Return on Asset 2
Perfect Positive Correlation Between the Returns on Assets 1 and 2
Sheet2
Return on Asset 1 Return on Asset 2
Sheet3
Return on Asset 1 Return on Asset 2
Chart4
Return on Asset 1
Return on Asset 2
Perfect Negative Correlation Between Returns on Assets 1 and 2
Sheet1
Return on Asset 1 Return on Asset 2
Sheet1
Return on Asset 1
Return on Asset 2
Perfect Positive Correlation Between the Returns on Assets 1 and 2
Sheet2
Return on Asset 1 Return on Asset 2
Sheet2
Return on Asset 1
Return on Asset 2
Perfect Negative Correlation Between Returns on Assets 1 and 2
Sheet3
Return on Asset 1 Return on Asset 2
Chart5
Return on Asset 1
Return on Asset 2
Correlation of Between Returns on Assets 1 and 2
Sheet1
Return on Asset 1 Return on Asset 2
Sheet1
Return on Asset 1
Return on Asset 2
Perfect Positive Correlation Between the Returns on Assets 1 and 2
Sheet2
Return on Asset 1 Return on Asset 2
Sheet2
Return on Asset 1
Return on Asset 2
Perfect Negative Correlation Between Returns on Assets 1 and 2
Sheet3
Return on Asset 1 Return on Asset 2
Sheet3
Return on Asset 1
Return on Asset 2
Correlation of Between Returns on Assets 1 and 2
