Portfolio Theory and Financial Engineering

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Portfolio Theory and Financial Engineering

• FIN 428
• Lecture Five Risk and Diversification
• Tuesday, January 23, 2007

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Background Assumptions

• Your portfolio includes all of your assets and
  liabilities
• The relationship between the returns for assets
  in the portfolio is important.
• A good portfolio is not simply a collection of
  individually good investments.
• As an investor you want to maximize the returns
  for a given level of risk.
• Given a choice between two assets with equal
  rates of return, most investors will select the
  asset with the lower level of risk.

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Evidence That Investors are Risk Averse

• Many investors purchase insurance for Life,
  Automobile, Health, and Disability Income. The
  purchaser trades known costs for unknown risk of
  loss
• Yield on bonds increases with risk
  classifications from AAA to AA to A.
• On the other hand
• Risk preference may have to do with amount of
  money involved - risking small amounts, but
  insuring large losses

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What do we mean by risk?

• Uncertainty of future outcomes
• Probability of a bad outcome

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Markowitz Portfolio Theory

• Quantifies risk
• Derives the expected rate of return for a
  portfolio of assets and an expected risk measure
• Shows that the variance of the rate of return is
  a meaningful measure of portfolio risk
• Derives the formula for computing the variance of
  a portfolio, showing how to effectively diversify
  a portfolio

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Assumptions

• Investors consider each investment alternative as
  being presented by a probability distribution of
  expected returns over some holding period.
• Investors maximize one-period expected utility,
  and their utility curves demonstrate diminishing
  marginal utility of wealth.
• Investors estimate the risk of the portfolio on
  the basis of the variability of expected returns.
• Investors base decisions solely on expected
  return and risk, so their utility curves are a
  function of expected return and the expected
  variance (or standard deviation) of returns only.
• For a given risk level, investors prefer higher
  returns to lower returns. Similarly, for a given
  level of expected returns, investors prefer less
  risk to more risk.

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Markowitz Portfolio Theory

• Using these five assumptions, a single asset or
  portfolio of assets is considered to be efficient
  if no other asset or portfolio of assets offers
  higher expected return with the same (or lower)
  risk, or lower risk with the same (or higher)
  expected return.

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Alternatives

• Variance or standard deviation of expected return
• Range of returns
• Returns below expectations
• Semivariance a measure that only considers
  deviations below the mean
• These measures of risk implicitly assume that
  investors want to minimize the damage from
  returns less than some target rate

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Expected Rates of Return

• For an individual asset - sum of the potential
  returns multiplied with the corresponding
  probability of the returns
• For a portfolio of investments - weighted average
  of the expected rates of return for the
  individual investments in the portfolio

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Computation of Expected Return for an Individual
Risky Investment
Exhibit 7.1
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Computation of the Expected Return for a
Portfolio of Risky Assets
Exhibit 7.2
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Variance (Standard Deviation) of Returns for an
Individual Investment

• Variance is a measure of the variation of
  possible rates of return Ri, from the expected
  rate of return E(Ri)
• Standard deviation is the square root of the
  variance

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Variance (Standard Deviation) of Returns for an
Individual Investment

• where Pi is the probability of the possible rate
  of return, Ri

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Variance (Standard Deviation) of Returns for an
Individual Investment

• Standard Deviation

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Variance (Standard Deviation) of Returns for an
Individual Investment
Exhibit 7.3
Variance ( 2) .000451 Standard Deviation (
) .021237
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Covariance of Returns

• A measure of the degree to which two variables
  move together relative to their individual mean
  values over time
• For two assets, i and j, the covariance of rates
  of return is defined as
• Covij ERi - E(Ri) Rj - E(Rj)

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Covariance and Correlation

• The correlation coefficient is obtained by
  standardizing (dividing) the covariance by the
  product of the individual standard deviations

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Covariance and Correlation

• Correlation coefficient varies from -1 to 1

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Portfolio Standard Deviation Formula
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Portfolio Standard Deviation Calculation

• Any asset of a portfolio may be described by two
  characteristics
• The expected rate of return
• The expected standard deviations of returns
• The correlation, measured by covariance, affects
  the portfolio standard deviation
• Low correlation reduces portfolio risk while not
  affecting the expected return

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Portfolios

• Portfolio One equally weighted between GE and
  DELL
• Recall

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Portfolios

• Portfolio Two (of GE and DELL)
• Compare with individual stocks return and
  volatility as well as those of portfolio one.
• Where do the portfolio weights come from? Is
  there a better set of weights?

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Risk-Return Plots
E(R)
2
With two perfectly correlated assets, it is only
possible to create a two asset portfolio with
risk-return along a line between either single
asset
rij 1.00
1
Standard Deviation of Return
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Risk-Return Plots
E(R)
f
2
g
With uncorrelated assets it is possible to create
a two asset portfolio with lower risk than either
single asset
h
i
j
rij 1.00
k
1
rij 0.00
Standard Deviation of Return
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Risk-Return Plots
E(R)
f
2
g
With correlated () assets it is possible to
create a two asset portfolio between the first
two curves
h
i
j
rij 1.00
rij 0.50
k
1
rij 0.00
Standard Deviation of Return
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Risk-Return Plots
E(R)
With negatively correlated assets it is
possible to create a two asset portfolio with
much lower risk than either single asset
rij -0.50
f
2
g
h
i
j
rij 1.00
rij 0.50
k
1
rij 0.00
Standard Deviation of Return
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Risk-Return Plots
Figure 7.13 (incorrect)
E(R)
rij -0.50
f
rij -1.00
2
g
h
i
j
rij 1.00
rij 0.50
k
1
rij 0.00
With perfectly negatively correlated assets it is
possible to create a two asset portfolio with
almost no risk
Standard Deviation of Return
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Risk-Return Plots
Figure 7.13 (corrected)
E(R)
rij -0.50
f
rij -1.00
2
g
h
i
j
rij 1.00
rij 0.50
k
1
rij 0.00
With perfectly negatively correlated assets it is
possible to create a two asset portfolio with
almost no risk
Standard Deviation of Return
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The Efficient Frontier

• The efficient frontier represents a set of
  portfolios with the maximum rate of return for a
  given level of risk, or the minimum risk for
  every level of return
• A single asset or portfolio of assets is
  considered to be efficient if no other asset or
  portfolio of assets offers higher expected return
  with the same (or lower) risk, or lower risk with
  the same (or higher) expected return.

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Efficient Frontier for Alternative Portfolios
Exhibit 7.15
Efficient Frontier
B
E(R)
A
C
Standard Deviation of Return
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The Efficient Frontier and Investor Utility

• An individual investors utility curve specifies
  the trade-offs he is willing to make between
  expected return and risk
• The slope of the efficient frontier curve
  decreases steadily as you move upward
• These two interactions will determine the
  particular portfolio selected by an individual
  investor
• The optimal portfolio has the highest utility for
  a given investor
• It lies at the point of tangency between the
  efficient frontier and the utility curve with the
  highest possible utility

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Selecting an Optimal Risky Portfolio
Exhibit 7.16
U3
U2
U1
Y
X
U3
U2
U1
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Before the Next Class

• Readings
• Chapter 8
• Topics to be discussed in the next class
• Capital Asset Pricing Model