F303 Intermediate Investments

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F303 Intermediate Investments

• Class 4
• Asset Allocation
• Andrey Ukhov
• Kelley School of Business
• Indiana University

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Outline of Todays Class

• Construction of the Minimum-Variance Set
• Efficient Set Theorem
• Implications of the Efficient Frontier to
  investors
• Selection of the Optimal Portfolio

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Some questions

• How can the Markowitz approach be used?
• How can we build the optimal portfolio given the
  infinite number of portfolios that can be
  composed from the existing securities?
• How can the investor combine a riskless asset
  with a set of risky assets? What will happen in
  this case?

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The Case of 2 Assets
E(R)
Asset 2
Asset 1
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Consider the problem

• Consider a situation where you have three stocks
  to choose from Stock A, Stock B and Stock C.
• You can invest your entire wealth in one of these
  three securities. Or you could purchase two
  securities, investing 10 in A and 90 in B, or
  20 in A and 80 in B, or 70 in A and 30 in B,
  or 50 in each, or 60 in A and 40 in B, or
• there is a huge number of possible combinations
  and this in a simple case when considering two
  securities.
• Imagine the different combinations you have to
  consider when you have thousands of stocks

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and the Solution

• The investor does not need to evaluate all the
• possible portfolios.
• The answer is provided by the efficient set
  theorem.
• Any investor will choose the optimal portfolio
  from the
• set of portfolios that
• Maximize expected return for a given level of
  risk and
• Minimize risks for a given level of expected
  returns.

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Minimum-Variance Frontier
E(R)
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Minimum-Variance Frontier

• The outcome of risk-return combinations generated
  by portfolios of risky assets that gives you the
  minimum variance for a given rate of return.
• Intuitively, any set of combinations formed by
  two risky assets with less than perfect
  correlation will lie inside the triangle XYZ and
  will be convex.

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The Efficient Frontier

• Investors will never want to hold a portfolio
  below the minimum variance point.
• They will always get higher returns along the
  positively sloped part of the minimum-variance
  frontier.
• The Efficient Frontier is the set of
  mean-variance combinations from the
  minimum-variance frontier where for a given risk
  no other portfolio offers a higher expected
  return.

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Efficient Frontier
E(R)
F
A
C
Efficient Frontier (portfolios lying between
points E and F)
E
D
B
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Efficient Frontier

• The concept of Efficient Frontier narrows down
  the different portfolios from which the investor
  may choose.
• Refer to the previous slide
• For example, portfolios at points A and B offer
  the same risk, but the one at point A offers a
  higher return (for the same risk).
• No rational investor will hold portfolio at point
  B and therefore we can ignore it. In this case, A
  dominates B. In the same way, C dominates D.

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Selection of the Optimal Portfolio

• How will the investor go about selecting the
  optimal portfolio?
• Investors will have to consider their
  indifference curves.
• Put the investors indifference curves and the
  efficient frontier and go for the portfolio on
  the farthest northwest indifference curve, where
  the indifference curve is tangent to the
  efficient frontier.

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Indifference Curves for a Risk-Averse Investor
E(R)
Tangent Portfolio
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Portfolio Selection for a Highly Risk-Averse
Investor
E(R)
Tangent Portfolio
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Indifference Curves for a Low Risk-Averse Investor
E(R)
Tangent Portfolio
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Introducing the Risk-free Asset

• We now consider expanding the Markowitz approach
  by considering investing not just in risky assets
  but also in a risk-free asset.
• The risk-free asset has a certain payoff. There
  is no uncertainty about the terminal value of
  this type of asset.
• Remember the of the risk-free asset is
  zero the covariance between the risk-free asset
  and any risky asset is zero.

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Combining Risky and Risk-free Assets

• Let us say that the investor has reached a
  decision regarding the composition of the optimal
  risky portfolio.
• His next decision deals with how much of his/her
  wealth will be channeled to the risky portfolio
  (let us say, y) and how much to invest in the
  risk-free asset (1-y).

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Combining Risky and Risk-free Assets

• With proportion y in the risky asset and (1-y)
  in the risk-free asset, the expected return of
  our portfolio, which we shall call the Complete
  Portfolio, will be
• The risk of the portfolio is given by
• We can plot the risk-return outcomes of various
  combinations, giving us the Capital Allocation
  Line.

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Capital Allocation Line
E(R)
Capital Allocation Line (CAL)
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Capital Allocation Line

• The CAL has a slope of
• and we can represent the entire line with
• The slope gives us the Reward-to-Variability
  Ratio.
• CAL shows one simple fact increasing the amount
  invested in the risky asset increases the
  expected return by a certain risk premium.

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Asset Allocation Process

• The Asset Allocation Process can be divided into
  different stages
• First, determine the Capital Allocation Line
• Second, find the highest Reward-to-Variability
  Ratio
• Third, the investor must find the point of
  highest utility on CAL.

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Achieving the Highest Reward-to-Variability Ratio
Capital Allocation Line (CAL)
Efficient Frontier with risky assets only
E(R)
A
Minimum Variance Portfolio
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Capital Allocation Line and the Efficient Frontier
Capital Allocation Line (CAL)
Portfolio A maximizes the reward-to-variability
ratio
E(R)
B
A
Efficient Frontier with risky assets only
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Optimal Risky Portfolio

• The tangent portfolio, Portfolio A (in the
  previous slide), is the optimal risky portfolio.
  This portfolio should be offered to investors
  regardless of their degree of risk aversion.
• The crucial point the optimal portfolio A is the
  same for all investors.
• The investors different degree of risk aversion
  will then decide the actual position on the CAL.

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Separation Property

• The portfolio choice problem is separated into
  two independent tasks
• First task Determining the optimal risky
  portfolio (the portfolio made up of risky
  assets)
• Second task The allocation between the risk-free
  asset (T-bills) versus the risky portfolio
  depends on the investors personal preferences
  for risk-taking (his utility function).

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Optimal Portfolio Involving Risk-free Lending
Capital Allocation Line (CAL)
E(R)
B
A
C
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Optimal Portfolio Involving Risk-free Borrowing
Capital Allocation Line (CAL)
E(R)
C
B
A
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Key Points to Remember

• Minimum Variance Frontier
• Efficient Frontier with risky assets
• Choosing the Optimal Portfolio of risky assets
• The Capital Allocation Line and the
  Reward-to-Variability Ratio
• How the Efficient Frontier changes with the
  introduction of a risk-free asset
• Separation Property.