Chapter 6 Mathematics of Diversification

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Chapter 6 - Mathematics of Diversification

• Business 4179

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Key Chapter Concepts

• The expected return of a linear combination is a
  weighted average of the component expected
  returns.
• The variance of a linear combination is not a
  simple weighted average of the component
  variances.usually, the variance of a combination
  of assets is lower than a simple weighted average

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Portfolio ReturnsSimply the Weighted Average of
Expected Returns
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K. Hartviksen
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Other Key Concepts

• The idea of a minimum variance portfolio
• The important role of correlation
• The concept of covariance
• Utility of the single index model rather than
  classical Markowitz Portfolio Optimization

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Covariance

• The formula for the covariance between the
  returns on the stock and the returns on the
  market is
• Covariance is an absolute measure of the degree
  of co-movement of returns. The correlation
  coefficient is also a measure of the degree of
  co-movement of returnsbut it is a relative
  measurethis is why it is on a scale from 1 to
  -1.

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Correlation Coefficient

• The formula for the correlation coefficient
  between the returns on the stock and the returns
  on the market is
• The correlation coefficient will always have a
  value in the range of 1 to -1.

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Perfect Negatively Correlated Returns over Time
Returns
A two-asset portfolio made up of equal parts of
Stock A and B would be riskless. There would be
no variability of the portfolios returns over
time.
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Returns on Stock A
Returns on Stock B
Returns on Portfolio
1994
1995
1996
Time
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Grouping Individual Assets into Portfolios

• The riskiness of a portfolio that is made of
  different risky assets is a function of three
  different factors
• the riskiness of the individual assets that make
  up the portfolio
• the relative weights of the assets in the
  portfolio
• the degree of comovement of returns of the assets
  making up the portfolio
• The standard deviation of a two-asset portfolio
  may be measured using the Markowitz model

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Diversification Potential

• The potential of an asset to diversify a
  portfolio is dependent upon the degree of
  comovement of returns of the asset with those
  other assets that make up the portfolio.
• In a simple, two-asset case, if the returns of
  the two assets are perfectly negatively
  correlated it is possible (depending on the
  relative weighting) to eliminate all portfolio
  risk.
• This is demonstrated through the following chart.

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Covariance Matrix

• Is a tabular presentation of the pairwise
  combinations of all portfolio components
• See page 133 of your text.

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Disadvantage of the Markowitz Model

• It is data and computationally intensive
• The number of covariances you need is

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The Sharpe Single-Index Model

• By calculating a coefficient that relates each
  portfolio component to a single benchmark (the
  market portfolio) we can reduce the information
  required to the number of securities in the
  portfolio
• We use the stocks beta coefficient

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Estimating the Beta using historical returns

• If we regress the realized returns on the stock
  with those realized on the market portfolio we
  will get an idea of the relative performance of
  the stock compared to the market.
• The beta coefficient is the slope of the
  regression (characteristic) line.

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Historical Beta Estimation
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Characteristic Line

• The characteristic line is a regression line that
  represents the relationship between the returns
  on the stock and the returns on the market over a
  period of time.
• The slope of the Characteristic Line is the Beta
  Coefficient
• The degree to which the characteristic line
  explains the variability in the dependent
  variable (returns on the stock) is measured by
  the coefficient of determination. (also known as
  the R2 (r-squared)).
• If the coefficient of determination equals 1.00,
  this would mean that all of the points of
  observation would lie on the line. This would
  mean that the characteristic line would explain
  100 of the variability of the dependent variable.

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Forecasting Returns

• Sometimes a company changes its operating
  activities such that the past is no longer a good
  predictor of future performance.
• In this case the analyst can used subjective
  forecasts for returns on the stock and the market
  to estimate the beta coefficient.
• An example of forecast data follows

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Predicting Stock Returns(ex ante returns)
Expected return is the weighted average of the
possible returns that have been predicted.
K. Hartviksen
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Measuring Risk of the Individual Security

• Risk is the possibility that the actual return
  that will be realized, will turn out to be
  different than what we expect (or have forecast).
• This can be measured using standard statistical
  measures of dispersion for probability
  distributions. They include
• variance
• standard deviation
• coefficient of variation

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Standard Deviation

• The formula for the standard deviation when
  analyzing population data (realized returns) is

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Standard Deviation

• The formula for the standard deviation when
  analyzing forecast data (ex ante returns) is
• it is the square root of the sum of the squared
  deviations away from the expected value.

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Forecasting Risk and Return for the Individual
Asset
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Using Forecasts to Estimate Beta

• The formula for the beta coefficient for a stock
  s is
• Obviously, the calculate a beta for a stock, you
  must first calculate the variance of the returns
  on the market portfolio as well as the covariance
  of the returns on the stock with the returns on
  the market.

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Beta of a Portfolio

• The Beta of a portfolio is simply the weighted
  average of the betas of the stocks that make up
  the portfolio.

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Variance of a Portfolio

• The variance of a portfolio is a function of the
  beta-squared of the portfolio and the market
  variance

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Variance of a Portfolio

• Of course the term in the brackets is equal to
  the portfolio beta

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Variance of a Portfolio

• And according to the work of Evans and Archerthe
  unsystematic risk of a portfolio (as measured in
  the second term) approaches zero and the number
  of securities in the portfolio increases

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Variance of a Portfolio

• Hence the variance of the portfolio depends on
  the portfolios beta squared times the variance
  of the market portfolio

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This Chapter in Conclusion

• Helps review Markowitz portfolio theory
• Shows the computational challenges of MPT
• Illustrates the Sharpe single factor model and
  its advantages in practice. (CAPM)
• Extends the Sharpe model to multi-index models
  (APT)

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Question 6 - 1

• Selling stock short brings cash in rather than
  requiring a cash outflow. In the absence of
  margin requirements (which is arguably true for
  large institutional investors), this means there
  is no initial investment, and any gain on no
  investment is an infinite return.

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Question 6 - 3

• The two-security portfolio is preferable, as it
  has higher expected return per unit of risk.

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Question 6 - 4

• Covariance is the expected value of the product
  of two numbers. Each of the two numbers is a
  value minus its mean. Some values lie below the
  mean, some above. Consequently, each number can
  be positive or negative, and the product can
  therefore be positive or negative. Depending on
  the nature of the dispersion around the means,
  the expected value of the product can be positive
  or negative.

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Question 6 - 5

• By the commutative law for multiplication, ab
  ba. This means the order of the two products
  inside the expected value operator can be
  reversed and

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Question 6 - 6

• Where

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Question 6 - 7

• 0.25 (4).5 (6).5 1.225

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Question 6 - 8

• The size of the error term approaches zero as the
  number of portfolio components increases.

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Question 6 - 9

• Standard deviations can only be positive, so a
  negative correlation means the covariance is also
  negative.

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Question 6 - 10

• In a prediction model, R squared can only
  increase if additional explanatory variables are
  added. You cannot lose predictive ability by
  including additional data.

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Problem 6 - 1
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Problem 6 - 2
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Problem 6 - 3
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Problem 6 - 4
The error term approaches zero, so
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Problem 6 - 6
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Problem 6 - 7
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Problem 6 - 8
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Problem 6 - 9