Optimal Risky Portfolios: Efficient Diversification

[Pages:22]Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

Prof. Alex Shapiro

Lecture Notes 7

Optimal Risky Portfolios: Efficient Diversification

I. Readings and Suggested Practice Problems II. Correlation Revisited: A Few Graphical Examples

III. Standard Deviation of Portfolio Return: Two Risky Assets

IV. Graphical Depiction: Two Risky Assets V. Impact of Correlation: Two Risky Assets VI. Portfolio Choice: Two Risky Assets VII. Portfolio Choice: Combining the Two Risky Asset

Portfolio with the Riskless Asset VIII. Applications

IX. Standard Deviation of Portfolio Return: n Risky Assets X. Effect of Diversification with n Risky Assets XI. Opportunity Set: n Risky Assets XII. Portfolio Choice: n Risky Assets and a Riskless Asset

XIII. Additional Readings

Buzz Words:

Minimum Variance Portfolio, Mean Variance Efficient Frontier, Diversifiable (Nonsystematic) Risk, Nondiversifiable (Systematic) Risk, Mutual Funds.

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Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

I. Readings and Suggested Practice Problems

BKM, Chapter 8.1-8.6. Suggested Problems, Chapter 8: 8-14

E-mail: Open the Portfolio Optimizer Programs (2 and 5 risky assets) and experiment with those.

II. Correlation Revisited: A Few Graphical Examples

A. Reminder: Don't get confused by different notation used for the same quantity:

Notation for Covariance: Cov[r1,r2] or [r1,r2] or 12 or 1,2 Notation for Correlation: Corr[r1,r2] or [r1,r2] or 12 or 1,2

B. Recall that covariance and correlation between the random return on asset 1 and random return on asset 2 measure how the two random returns behave together.

C. Examples

In the following 5 figures, we Consider 5 different data samples for two stocks: - For each sample, we plot the realized return on stock 1

against the realized return on stock 2. - We treat each realization as equally likely, and calculate

the correlation, , between the returns on stock 1 and stock 2, as well as the regression of the return on stock 2 (denoted y) on the return on stock 1 (x). [Note: the regression R2 equals 2]

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Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

1. A sample of data with = 0.630:

y = 0.9482x + 0.0506 R2 = 0.3972

35% 30%

25%

Return on Stock 2

20%

15%

10%

5%

-15% -10%

0% -5% 0%

-5%

5% 10%

-10% Return on Stock 1

15%

20%

25%

2. A sample of data with = -0.714:

Return on Stock 2

20%

15%

10%

5%

0%

-5%

0%

-5%

5%

10%

15%

-10% -15% -20%

y = -0.8613x + 0.0726 R2 = 0.51

Return on Stock 1

20%

25%

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Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

3. Sample with = +1:

Return on Stock 2

6% 6% 5% 5% 5% 5% 5% 5% 5% -10% -5% 0%

y = 0.02x + 0.05 R2 = 1

5% 10% 15% 20% 25% 30% Return on Stock 1

4. Sample with = -1:

Return on Stock 2

15%

10%

5%

-10%

0%

0% -5%

-10%

-15%

-20%

-25%

10%

20%

30%

y = -0.8x + 0.05 R2 = 1

Return on Stock 1

40%

5. Sample with 0:

15%

10% 5%

y = 0.009x + 0.0468 R2 = 0.0001

Return on Stock 2

0%

-5%

0%

5%

10%

15%

20%

25%

30%

-5%

-10%

Return on Stock 1

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Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

D. Real-Data Example

Us Stocks vs. Bonds 1946-1995, A sample of data with = 0.228:

STB

Stocks and Bonds

(Annual returns on S&P 500 and long term US govt bonds.) Raw Data

S&P500 LT Gov't

T-bill Inflation

1946 -8.07% -0.10% 0.35% 18.16%

1947 5.71% -2.62% 0.50% 9.01%

1948 5.50% 3.40% 0.81% 2.71%

1949 18.79% 6.45% 1.10% -1.80%

1950 31.71% 0.06% 1.20% 5.79%

1951 24.02% -3.93% 1.49% 5.87%

1952 18.37% 1.16% 1.66% 0.88%

1953 -0.99% 3.64% 1.82% 0.62%

1954 52.62% 7.19% 0.86% -0.50%

1955 31.56% -1.29% 1.57% 0.37%

1956 6.56% -5.59% 2.46% 2.86%

1957 -10.78% 7.46% 3.14% 3.02%

1958 43.36% -6.09% 1.54% 1.76%

1959 11.96% -2.26% 2.95% 1.50%

1960 0.47% 13.78% 2.66% 1.48%

1961 26.89% 0.97% 2.13% 0.67%

1962 -8.73% 6.89% 2.73% 1.22%

1963 22.80% 1.21% 3.12% 1.65%

1964 16.48% 3.51% 3.54% 1.19%

1965 12.45% 0.71% 3.93% 1.92%

1966 -10.06% 3.65% 4.76% 3.35%

1967 23.98% -9.18% 4.21% 3.04%

1968 11.06% -0.26% 5.21% 4.72%

1969 -8.50% -5.07% 6.58% 6.11%

1970 4.01% 12.11% 6.52% 5.49%

1971 14.31% 13.23% 4.39% 3.36%

1972 18.98% 5.69% 3.84% 3.41%

1973 -14.66% -1.11% 6.93% 8.80%

1974 -26.47% 4.35% 8.00% 12.20%

1975 37.20% 9.20% 5.80% 7.01%

1976 23.84% 16.75% 5.08% 4.81%

1977 -7.18% -0.69% 5.12% 6.77%

1978 6.56% -1.18% 7.18% 9.03%

1979 18.44% -1.23% 10.38% 13.31%

1980 32.42% -3.95% 11.24% 12.40%

1981 -4.91% 1.86% 14.71% 8.94%

1982 21.41% 40.36% 10.54% 3.87%

1983 22.51% 0.65% 8.80% 3.80%

1984 6.27% 15.48% 9.85% 3.95%

1985 32.16% 30.97% 7.72% 3.77%

1986 18.47% 24.53% 6.16% 1.13%

1987 5.23% -2.71% 5.47% 4.41%

1988 16.81% 9.67% 6.35% 4.42%

1989 31.49% 18.11% 8.37% 4.65%

1990 -3.17% 6.18% 7.81% 6.11%

1991 30.55% 19.30% 5.60% 3.06%

1992 7.67% 8.05% 3.51% 2.90%

1993 9.99% 18.24% 2.90% 2.75%

1994 1.31% -7.77% 3.90% 2.67%

1995 37.43% 31.67% 5.60% 2.74%

Excess over T-bill

S&P500 LT Gov't -8.42% -0.45% 5.21% -3.12% 4.69% 2.59% 17.69% 5.35% 30.51% -1.14% 22.53% -5.42% 16.71% -0.50% -2.81% 1.82% 51.76% 6.33% 29.99% -2.86% 4.10% -8.05% -13.92% 4.32% 41.82% -7.63% 9.01% -5.21% -2.19% 11.12% 24.76% -1.16% -11.46% 4.16% 19.68% -1.91% 12.94% -0.03% 8.52% -3.22% -14.82% -1.11% 19.77% -13.39% 5.85% -5.47% -15.08% -11.65% -2.51% 5.59% 9.92% 8.84% 15.14% 1.85% -21.59% -8.04% -34.47% -3.65% 31.40% 3.40% 18.76% 11.67% -12.30% -5.81% -0.62% -8.36% 8.06% -11.61% 21.18% -15.19% -19.62% -12.85% 10.87% 29.82% 13.71% -8.15% -3.58% 5.63% 24.44% 23.25% 12.31% 18.37% -0.24% -8.18% 10.46% 3.32% 23.12% 9.74% -10.98% -1.63% 24.95% 13.70% 4.16% 4.54% 7.09% 15.34% -2.59% -11.67% 31.83% 26.07%

N Mean Std.Dev. Std.Err.Mean

50 13.16% 16.57%

2.34%

50 5.83% 10.54% 1.49%

50 4.84% 3.18% 0.45%

50 4.43% 3.82% 0.54%

50 8.31% 17.20% 2.43%

50 0.99% 10.13% 1.43%

Corr(Stocks, Bonds)=

0.228

0.265

Return on S&P 500

60%

y = 0.3592x + 0.1106

50%

R2 = 0.0522

40%

30%

20%

10%

0% -10% -5% 0% 5% 10% 15% 20% 25% 30% 35% 40% 45%

-10%

-20%

-30%

Return on US Gov't Bonds

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Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

III. Standard Deviation of Portfolio Return: Two Risky Assets

A. Formula

2

[r

p

(t)]=

w 2 1, p

[r1

(t )

]2

+

w 2 2,p

[r

2

(t

)

]2

+

2

w1, p

w2 , p

[r1

(t),

r

2

(t)]

[r p (t)] = 2[r p (t)]

where [r1(t), r2(t)] is the covariance of asset 1 s return and asset 2 s return in period t, wi,p is the weight of asset i in the portfolio p, 2[rp(t)] is the variance of return on portfolio p in period t.

B. Example

Consider two risky assets. The first one is the stock of Microsoft. The second one itself is a portfolio of Small Firms. The following moments characterize the joint return distribution of these two assets.

E[rSmall] = 1.912, E[rMsft] = 3.126, [rSmall] = 3.711, [rMsft] = 8.203,

>rMsft, rSmall]= 12.030

A portfolio formed with 60% invested in the small firm asset and 40% in Microsoft has standard deviation and expected return given by:

2[rp] = wSmall,p2 2[rSmall] + wMsft,p2 2[rMsft] + 2 wSmall,p wMsft,p [rSmall,rMsft] = 0.62 ? 3.7112 + 0.42 ? 8.2032 + 2 ? 0.6 ? 0.4 ? 12.030 = 4.958 +10.766 + 5.774 = 21.498

[rp] = 2[r p] = 21.498 = 4.637

E[rp] = wSmall,p E[rSmall] + wMsft,p E[rMsft] = 0.6 ? 1.912 + 0.4 ? 3.126 = 2.398

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Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

IV. Graphical Depiction: Two Risky Assets

A. Representation in the "Mean-Variance Space"

The standard deviation, p, of a return on a portfolio consisting of asset 1 and asset 2, and the portfolio's expected return, Ep, can be expressed in terms of w1, the weight of asset 1.

When plotting in the Mean-Variance plane p and Ep for all possible values of w1, we get a curve.

The curve is known as the portfolio possibility curve -, or as the portfolio frontier -, or as the set of feasible portfolios-, or as the opportunity set - with two risky assets.

An Algorithm to Plot the Portfolio Frontier: 1. Pick a value for w1 (and then w2 = 1- w1) 2. Compute expected return and standard deviation:

E[rp ] = w1E[r1 ] + w2E[r2 ] = w1E[r1 ] + (1 - w1 )E[r2 ]

p =

w12

2 1

+

w22

2 2

+

2w1w2 1,2

=

w12

2 1

+

(1 -

w1

)

2

2 2

+

2w1 (1 -

w1 ) 1,2

3. Plot a single point {p, E[rp]} 4. Repeat 1-3 for various values of w1

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Foundations of Finance: Optimal Risky Portfolios: Efficient Diversification

B. Example (cont.)

To get the portfolio possibility curve using the small-firm portfolio and Microsoft equity (i.e., to "get all possible p's"), the standard deviation of return on a portfolio consisting of the small firm portfolio (asset 1) and Microsoft equity (asset 2) and its expected return can be indexed by the weight of the small firm portfolio within portfolio p: w1= wSmall,p.

wSmall,p -0.2 0.0 0.2 0.4 0.6 0.8 1.0 1.2

wMsft,p 1.2 1.0 0.8 0.6 0.4 0.2 0.0 -0.2

[rp(t)] 9.574% 8.203% 6.889% 5.675% 4.637% 3.919% 3.711% 4.093%

E[rp(t)] 3.369% 3.126% 2.883% 2.641% 2.398% 2.155% 1.912% 1.670%

Figure here

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