Optimal Risky Portfolios

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• Optimal Risky Portfolios

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Review

• Mix one risky asset with the risk-free asset

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Example

• Suppose as an investor, you want to invest 10000
• There are 2 assets to pick from SP500 index and
  the risk-free T-bill
• For SP500, the annual return from 1980-2005 is
  around 10.5, standard deviation is 15 annually
• For the risk-free, one-year Treasury security is
  about 3

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Possible Combinations for SP500 and T-bill
E(r)
E(rp) 10.5
P
E(rc) 8
C
rf 3
F
?
0
?c
15
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New question

• Now introduce two risky assets, how can we find
  the optimal mix between the two to form a new
  portfolio p so that we can improve upon the
  reward-variability ratio (Sharpe ratio)

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A new asset real estate

• Average annual housing price increases from
  1995-2004 around NYC is 7.5, standard deviation
  is 8
• Correlation coefficient between SP500 and
  housing return is 0.3

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Two-Security Portfolios p withDifferent
Correlations
E(r)
10.5
? -1
? - .3
7.5
? -1
? 1
St. Dev
8
15
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Minimum-Variance Combination
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s 22 - Cov(r1r2)

W1
s2
s?2
- 2Cov(r1r2)

1
2
W2
(1 - W1)
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Minimum-Variance Combination ? -.03
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Minimum -Variance Return and Risk with ? -.3
rp .277(.105) .723(.075) .08
?
(.277)2(.15)2 (.723)2(.08)2
p
1/2
2(.277)(.723)(-.3)(.15)(.08)
s
.06
p
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Optimal Risk Portfolio

• To achieve the optimal portfolio, we need to find
  the weights for both risky assets in the
  portfolio, the way to find them is to maximize
  the following
• Where p stands for the optimal portfolio.
• The optimal portfolio weights are given

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Solution for the weights of the optimal risky
portfolio with two risky assets (asset 1 and 2)
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Example

• Risk-free rate as 3
• Risky asset one has expected return as 10.5,
  standard deviation as 15
• The return-risk tradeoff for the above 2 assets
• Reward-variability ratio is 0.5

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Calculation
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The new risky portfolio p

• By investing 33 in risky asset SP500 and 67 in
  housing, the new risky portfolio has expected
  return as 0.3310.50.677.5 8.5, the
  standard deviation as

?p w12?12 w22?22 2W1W2 Cov(r1r2)1/2 6.1

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• 2. the second step, combining the new risky
  portfolio and the risk-free asset, the return and
  risk tradeoff (CAL line) line becomes