Portfolio
Diversification
The Efficient Set for Two Assets
Risk

Return

0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50.00%
55%
60%
65%
70%
75%
80%
85%
90...
The Efficient Set for Two Assets
Risk

Return

0%
5%
10%
15%
20%
25%
30%
35%
40%
45%
50%
55%
60%
65%
70%
75%
80%
85%
90%
9...
return

Two-Security Portfolios with Various
Correlations
100%
stocks

ρ = -1.0

100%
bonds

ρ = 1.0
ρ = 0.2

σ





Re...
Perfect Positive Correlation
Corr(RA, RB) = 1
Perfect Negative Correlation
Corr(RA, RB) = −1
Zero Correlation
Corr(RA, RB) = 0
Portfolio Risk as a Function of the
Number of Stocks in the Portfolio
σ
Diversifiable Risk;
Nonsystematic Risk;
Firm Speci...
return

The Efficient Set for Many Securities

σP

Consider a world with many risky assets; we can still
identify the oppo...
return

The Efficient Set for Many Securities
(contd.)

minimum
variance
portfolio

σP

Given the opportunity set we can i...
return

The Efficient Set for Many Securities
(contd.)
c
effi

ie

t i er
ron
nt f

minimum
variance
portfolio

σP

The se...
return

Optimal Risky Portfolio with a
Risk-Free Asset

rf

σ

In addition to stocks and bonds, consider a world that
also...
return

Riskless Borrowing and Lending
CM

L

rf

σ

Now investors can allocate their money across
the T-bills and a balan...
return

Choice of Capital Market Line
CM

L

efficient frontier

rf

σP

With a risk-free asset available and the efficien...
return

Market Equilibrium
CM

L

efficient frontier

M
rf

σP
With the capital allocation line identified, all investors ...
return

Market Equilibrium (contd.)
CM

L

rf

σ
Just where the investor chooses along the capital allocation
line depends...
return

Market Equilibrium (contd.)
CM

L

Optimal
Risky
Portfolio

rf

σ

All investors have the same CML because they al...
return

The Separation Property
CM

L
efficient frontier

M
rf

σP
The Separation Property states that the market portfoli...
return

The Separation Property (contd.)
CM

L
efficient frontier

M
rf

σP
Investor risk aversion is revealed in their ch...
return

The Separation Property (contd.)
CM

L

Optimal
Risky
Portfolio

rf

σ
The separation property implies that portfo...
Dependence of Optimal Risky
return

Portfolio on Risk-Free Asset

1
f
0
f

r
r

L 0 CML 1
CM
First
Optimal
Risky
Portfolio...
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Лекц 14 Portfolio diversification

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Лекц 14 Portfolio diversification

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Лекц 14 Portfolio diversification

  1. 1. Portfolio Diversification
  2. 2. The Efficient Set for Two Assets Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50.00% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.08% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.00% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolo Risk and Return Combinations Portfolio Return % in stocks 12.0% 11.0% 10.0% 9.0% 8.0% 7.0% 6.0% 5.0% 0.0% 100% stocks 100% bonds 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) We can consider other portfolio weights besides 50% in stocks and 50% in bonds …
  3. 3. The Efficient Set for Two Assets Risk Return 0% 5% 10% 15% 20% 25% 30% 35% 40% 45% 50% 55% 60% 65% 70% 75% 80% 85% 90% 95% 100% 8.2% 7.0% 5.9% 4.8% 3.7% 2.6% 1.4% 0.4% 0.9% 2.0% 3.1% 4.2% 5.3% 6.4% 7.6% 8.7% 9.8% 10.9% 12.1% 13.2% 14.3% 7.0% 7.2% 7.4% 7.6% 7.8% 8.0% 8.2% 8.4% 8.6% 8.8% 9.0% 9.2% 9.4% 9.6% 9.8% 10.0% 10.2% 10.4% 10.6% 10.8% 11.0% Portfolo Risk and Return Combinations 12.0% Portfolio Return % in stocks 11.0% 10.0% 100% stocks 9.0% 8.0% 7.0% 6.0% 100% bonds 5.0% 0.0% 2.0% 4.0% 6.0% 8.0% 10.0% 12.0% 14.0% 16.0% Portfolio Risk (standard deviation) Note that some portfolios are “better” than others. They have higher returns for the same level of risk. These comprise the efficient frontier.
  4. 4. return Two-Security Portfolios with Various Correlations 100% stocks ρ = -1.0 100% bonds ρ = 1.0 ρ = 0.2 σ    Relationship depends on correlation coefficient -1.0 < ρ < +1.0 If ρ = +1.0, no risk reduction is possible If ρ = –1.0, complete risk reduction is possible
  5. 5. Perfect Positive Correlation Corr(RA, RB) = 1
  6. 6. Perfect Negative Correlation Corr(RA, RB) = −1
  7. 7. Zero Correlation Corr(RA, RB) = 0
  8. 8. Portfolio Risk as a Function of the Number of Stocks in the Portfolio σ Diversifiable Risk; Nonsystematic Risk; Firm Specific Risk; Unique Risk Portfolio risk Nondiversifiable risk; Systematic Risk; Market Risk n Thus diversification can eliminate some, but not all of the risk of individual securities.
  9. 9. return The Efficient Set for Many Securities σP Consider a world with many risky assets; we can still identify the opportunity set of risk-return combinations of various portfolios.
  10. 10. return The Efficient Set for Many Securities (contd.) minimum variance portfolio σP Given the opportunity set we can identify the minimum variance portfolio.
  11. 11. return The Efficient Set for Many Securities (contd.) c effi ie t i er ron nt f minimum variance portfolio σP The section of the opportunity set above the minimum variance portfolio is the efficient frontier.
  12. 12. return Optimal Risky Portfolio with a Risk-Free Asset rf σ In addition to stocks and bonds, consider a world that also has risk-free securities like T-bills
  13. 13. return Riskless Borrowing and Lending CM L rf σ Now investors can allocate their money across the T-bills and a balanced mutual fund
  14. 14. return Choice of Capital Market Line CM L efficient frontier rf σP With a risk-free asset available and the efficient frontier identified, we choose the capital market line with the steepest slope
  15. 15. return Market Equilibrium CM L efficient frontier M rf σP With the capital allocation line identified, all investors choose a point along the line—some combination of the risk-free asset and the market portfolio M. In a world with homogeneous expectations, M is the same for all investors.
  16. 16. return Market Equilibrium (contd.) CM L rf σ Just where the investor chooses along the capital allocation line depends on his risk tolerance. The big point though is that all investors have the same CML (Capital Market Line).
  17. 17. return Market Equilibrium (contd.) CM L Optimal Risky Portfolio rf σ All investors have the same CML because they all have the same optimal risky portfolio given the riskfree rate.
  18. 18. return The Separation Property CM L efficient frontier M rf σP The Separation Property states that the market portfolio, M, is the same for all investors—they can separate their risk aversion from their choice of the market portfolio.
  19. 19. return The Separation Property (contd.) CM L efficient frontier M rf σP Investor risk aversion is revealed in their choice of where to stay along the capital market line—not in their choice of the line.
  20. 20. return The Separation Property (contd.) CM L Optimal Risky Portfolio rf σ The separation property implies that portfolio choice can be separated into two tasks: (1) determine the optimal risky portfolio, and (2) selecting a point on the CML.
  21. 21. Dependence of Optimal Risky return Portfolio on Risk-Free Asset 1 f 0 f r r L 0 CML 1 CM First Optimal Risky Portfolio Second Optimal Risky Portfolio σ The optimal risky portfolio depends on the riskfree rate as well as the risky assets.
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