Chapter 6: Portfolio Selection

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Chapter 6 Portfolio Selection

• ObjectiveTo present the basics of modern
  portfolio selection process
• Top down investment approach
• Capital allocation decision
• Risk-free vs. risky asset
• Asset allocation
• Decision on broad asset classes such as real
  estate, stocks, bond, foreign stocks, etc.
• Security selection

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Chapter 6 Portfolio Selection

• Section 6.7 is technical. If you understand the
  lecture and lecture note, that will be
  sufficient.
• Skip pp. 224-231, and pp. 233-237.

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1. Allocating capital between risky risk free
assets

• Consider a risky portfolio P and the risk-free
  asset F.
• Examine risk/return tradeoff
• Risk aversion determines optimal choice
• We fix the composition of portfolio P. We only
  focus on the allocation between the risky
  portfolio P and risk-free asset F.

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2. The Risk-Free Asset

• The only risk free asset in real terms is the
  perfectly price-indexed bond.
• T-bills are commonly viewed as the risk-free
  asset.
• Money market funds are the most accessible
  risk-free asset for most investors.

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3. Portfolios of one risky asset and one
risk-free asset

• Qu What are the risk-return combinations of P
  and F available to investors?
• E(rp) 15 and ?p 22
• rf 7 and ?rf 0
• Portfolio weight y in P and (1-y) in F Return
  on complete portfolio, rc
• rc yrp (1 - y)rf

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Expected returns and variance for combinations

• rc yrp (1 - y)rf
• E(rc) yE(rp) (1 - y)rf (1)
• rf yE(rp) rf
• 0.07 y(0.15 - 0.07), and
• ?cy?p
• 0.22y (2)
• y?c/?p from (2). Substitute this into (1)
• E(rc) rf ?cE(rp) rf/?p. (YabX)
• Slope reward-to-variability ratio

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CAL (Capital Allocation Line)
The straight line shows all the risk-return
combinations (investment opportunity set)
available to investors.
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Leveraged position

• Borrow at the risk-free rate and invest in stock
• Suppose you have 100,000, and you borrow
  40,000. You invest 140,000 in P.
• rc (-.4) (.07) (1.4) (.15) .182
• ?c (1.4) (.22) .308
• S (.182-.07)/.308 .36

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CAL with Higher Borrowing Rate
With borrowing, the slope is (.15-.09)/.22 .27.
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Risk tolerance and asset allocation

• Certainty equivalent of portfolio Ps expected
  return for two different investors

E(rp)15
P
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CAL with risk preferences

• The more risk averse investors choose to hold
  less of
• the risky assets and more of the risk-free
  assets.
• The asset allocation process (1) determine the
  CAL
• (2) find the optimal point along that line.

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• Assume UE(r) - ½A?2. Then the optimal
  portfolio weight on risky portfolio P is given by
• y E(rp)-rf/(A?2p)

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5. Passive strategy

• A passive strategy describes a portfolio decision
  that avoids any direct or indirect security
  analysis.
• A passive strategy involves in two passive
  portfolios risk-free T-bills and a market
  portfolio.
• The capital allocation line representing such a
  strategy is called the capital market line. (CML)

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6. Diversification and portfolio risk

• Market risk, systematic risk, nondiversifiable
  risk
• Unique risk, non-systematic risk, diversifiable
  risk,
• idiosyncratic risk, firm-specific risk

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7. Portfolio of two risky assets

• Previously, we considered naive
  diversification. Now we consider efficient
  diversification to form portfolios with the
  lowest risk for a given level of expected return.
  
• Consider two risky assets D and E. Let w
  denote portfolio weight on asset D. Then
• rp wrD (1-w)rE
• E(rp) wE(rD) (1-w)E(rE)
• ?p2w2?D2 (1-w)2?E2 2w(1-w)cov(rD,rE)
• cov(rD, rE) ?1 ?2 ?DE, by definition

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• Suppose E(rD).10, ?D .15,
• E(rE).14, ?E.20, and ?DE.2
• Find out E(rp) and ?p if w0.1.
• Find out E(rp) and ?p if w0.2.

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Portfolio opportunity set of risky assets

• Minimum variance portfolio (MVP).
• The lower the correlation, the greater the
  diversification benefits.

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Minimum-variance portfolio

• Solving the minimization problem we get

• Numerically

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Mean and variance of MVP

• Using the weights w and (1-w) we determine
  minimum-variance portfolios characteristics

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8. Asset allocation with stocks, bonds, and bills

• We approach the problem in three steps.
• Determine the risk-return opportunities
  available to the investor minimum variance
  portfolio
• Identify the optimal portfolio of risky assets
  by finding the portfolio with the steepest slope
  CAL
• Risk aversion determines the optimal mix of
  risky and riskless assets.

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Step 1 The minimum-variance frontier of risky
assets
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• Minimum variance frontier The optimal
  combinations result in lowest level of risk for a
  given return
• The part of the frontier that lies above the
  global minimum variance portfolio and upward is
  called the efficient frontier. Efficient
  frontier portfolios are dominant.

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Step 2 Optimal portfolio P and CAL (with
riskless asset)
Tangency portfolio P has the largest slope,
E(rp-rf)/?p
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• Separation property The property that portfolio
  choice problem is separated into two independent
  parts (1) determination of the optimal risky
  portfolio, and (2) the personal choice of the
  best mix of the risky portfolio and risk-free
  asset.
• Implication of the separation property the
  optimal risky portfolio P is the same for all
  clients of a fund manager.

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Step 3 Efficient frontier with lending
borrowing
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Portfolio selection risk aversion(without
riskless asset)