Combining Individual Securities Into Portfolios Chapter 4

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Combining Individual SecuritiesInto
Portfolios(Chapter 4)

• Individual Security Return and Risk
• Portfolio Expected Rate of Return
• Portfolio Variance
• Combination Lines
• Combination Line Between a
• Risky Asset and a Risk-Free Asset

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Individual Security Return and Risk

• Expected Rate of Return
• where
• E(rA) Expected rate of return on security (A)
• rA,i i(th) possible return on security (A)
• hi probability of getting the i(th) return
• Variance and Standard Deviation

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Portfolio Expected Rate of Return

• A weighted average of the expected returns on the
  portfolios component securities.
• where
• E(rp) Expected rate of return on portfolio (p)
• m Number of securities in portfolio (p)
• xj Weight of security (j)
• Note The contribution of each security to
  portfolio expected return depends on
• 1. the securitys expected return
• 2. the securitys weight

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Portfolio Variance

• To compute the variance of a portfolio, you need
• (1) the covariances of every pair of securities
  in
• the portfolio, and
• (2) the weight of each security.
• Example (Three Security Portfolio)

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• Take each of the covariances in the matrix and
  multiply it by the weight of the security
  identified on the row (security j) and then again
  by the weight of the security identified on the
  column (security k). Then, add up all of the
  products.

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The Covariance Between a Security and Itself is
Simply Its Own Variance
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Each Element Above the Diagonal is Paired With an
Identical Element Below the Diagonal
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Using the Correlation Coefficient Instead of
Covariance

• Recall
• As a Result
• Therefore

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COMBINATION LINES

• A curve that shows what happens to the risk and
  expected return of a portfolio of two stocks as
  the portfolio weights are varied.
• Example 1 (Perfect Negative Correlation)

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• Now, for various weights, portfolio risk and
  return can be calculated

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• Students are encouraged to prove that the
  following portfolio standard deviations and
  expected rates of return are indeed correct for
  the weights given.
• Note With perfect negative correlation, we can
  create a riskless portfolio by taking positive
  positions in both stocks.

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COMBINATION LINE(Perfect Negative Correlation)
Expected Rate of Return ()
xA .5, xB .5
xA -.3, xB 1.3
All (B)
xA .75, xB .25
xA 1.5, xB -.5
All (A)
Standard Deviation of Returns ()
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• Example 2 (Perfect Positive Correlation)
• E(rA) 5 E(rB) 11
• ?(rA) 2.236 ?(rB) 6.708
• Cov(rA,rB) 15 ?A,B 1.00

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• Note When the stocks standard deviations are
  not equal and the stocks are perfectly positively
  correlated, we can always create a riskless
  portfolio by selling one of the two stocks short.

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COMBINATION LINE(Perfect Positive Correlation)
Expected Rate of Return ()
xA -.3, xB 1.3
xA .5, xB .5
All (B)
All (A)
xA 1.5, xB -.5
xA 1.75, xB -.75
Standard Deviation of Returns ()
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• Example 3 (Zero Correlation)

E(rA) 5 E(rB) 11 ?(rA) 2.236
?(rB) 6.708 Cov(rA,rB) 0 ?A,B 0
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COMBINATION LINE(Zero Correlation)
Expected Rate of Return ()
xA -.3, xB 1.3
xA .5, xB .5
All (B)
xA .9, xB .1
All (A)
xA 1.5, xB -.5
Standard Deviation of Returns ()
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PATHS OF COMBINATION LINES

• E(rp) is influenced by E(rj) and xj
• ?(rp) is influenced by ?(rj), ?j,k, and xj
• ?j,k determines the path between two securities
• Moving along the path occurs by varying the
  weights.

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COMBINATION LINES
Expected Rate of Return ()
? -1.00
Stock (B)
? 1.00
Stock (A)
? 0
Standard Deviation of Returns ()
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Combination Line Between a Risky Stock (or
Portfolio) and a Risk-Free Bond

• Example
• Risky Stock (A) E(rA) 10, ?(rA) 20
• Risk-Free Bond (B) E(rB) 6, ?(rB) 0
• Note on Risk

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Combination Line Between a Risky Stock (or
Portfolio) and a Risk-Free Bond (Continued)
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Combination Line When one of the Assets is
Risk-Free
Expected Rate of Return ()
Borrowing
Lending
xA 1.5, xB -.5
xA 1.0, xB 0
xA .5, xB .5
xA 0, xB 1.0
Standard Deviation of Returns ()
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Combination Line Between a Risky Stock (or
Portfolio) and a Risk-Free Bond (Continued)

• Note When one of the two investments is
  risk-free, the combination line is always a
  straight line.
• Lending When you buy a bond, you are lending
  money to the issuer.
• Borrowing Here, we assume that investors can
  borrow money at the risk-free rate, and add to
  their investment in the risky asset.