Chap 5 Portfolio Theory

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Chap 5 Portfolio Theory
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5.1 Trading off Expected return and risk

• How to invest our wealth?
• To maximize the expected return
• To minimize the riskVariance return.

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5.2 One risky asset and one risk-free asset

• Suppose that there is one risky asset, e.g., a
  mutual fund with expected return 0.15 and sd
  (standard deviation) of the return 0.25 and one
  risk-free asset, a 30-day T-bill with expected
  return 0.06 and sd 0. If a fraction w of our
  wealth is invested in the risky asset, then what
  is the expected return and risk?

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• Question1 Suppose you want an expected return of
  0.10, what should w be?
• Question 2 Suppose you want sd0.05, what should
  w be?
• What is the conclusion can be drawn from this
  simple example? Finding an optimal portfolio can
  be obtained by
• 1. Finding the optimal portfolio of risky
  assets
• 2. Finding the appropriate mix of the
  risk-free asset and the optimal portfolio from
  step one.

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• Example In Feb 2001 issue of Paine Webbers
  Investment Intelligence, he said that the chart
  shows that a 20 municipal /80 SP 500 mix
  sacrificed only 0.42 annual after-tax return
  relative to a 100 SP 500 portfolio, while
  reducing risk by 13.6 from 14.91 to 12.88.
  Webbers point is correct, but for a investor,
  what is over-emphasize?

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• Question usually, the risk-free rate is known.
  (Treasury bill rates are published in most
  newspapers.) But, how to estimate E(R) and
  Var(R)?

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5.3 Two risky assets

• Suppose the two risky assets have returns R1 and
  R2 and we mix them in proportion w and 1-w.
• Example

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• Questions How to estimate the means, variances
  and covariance of R1 and R2?
• (Under stationary assumption.)

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5.4 Risk-efficient portfolio with N risky assets
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mu(0.08, 0.03, 0.05), Sigma 0.3, 0.02,
0.01,
0.02, 0.15, 0.03
0.01, 0.03, 0.18
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5.5 Combining two risky asset with a free asset

• From Fig 5.3, we see that the dotted line lies
  above the dashed line. This means that the dotted
  lines gives a higher expected return than the
  dashed line under given risk. The bigger the
  slope of the line (Sharpe ratio) the better,
  why? The point T on the parabola represents the
  portfolio with the highest Sharpe ratio. It is
  the optimal portfolio for the purpose of mixing
  with the risk-free asset. This portfolio is
  called tangency portfolio since its line is
  tangent to the parabola.

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• Key result The optimal or efficient portfolios
  mix the tangency portfolio of two risky assets
  with the risk-free asset. Each efficient
  portfolio has two properties
• it has a higher expected return than any other
  portfolio with the same (or smaller) risk.
• It has a smaller risk than any other portfolio
  with the same (or smaller) return.

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5.5.1 Tangency portfolio with two risky assets

• How to find the tangency portfolio?

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