6- 1 © admn 3116, anton miglo admn 3116: financial management 1 lecture 7: optimal portfolios anton...

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© ADMN 3116, Anton Miglo

ADMN 3116: Financial Management 1

Lecture 7: Optimal Portfolios

Anton Miglo

Fall 2014

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Topics

Portfolios Diversification Sharpe ratio Optimal Portfolio Excel: CORR, Solver Additional readings: M ch. 5, 6

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Investment Opportunities

Cash Equivalents (T-Bills, Money Market Funds) Fixed Income (Bonds, Government and Corporate) Equities (Common Stock, Preferred Shares) Mutual Funds (FE, DSC, LL, NL, F-class) Segregated Funds (Principal protection) Exchange Traded Funds (ETFs)

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Investment Performance

Asset Allocation

91.5%

Other2.1%

Market Timing1.8%

Security Selection

4.6%

Source: “Determinants of Portfolio Performance II, An Update” by Gary Brinston, Brian D. Singer and Gilbert L. Beebower, Financial Analysts Journal

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Investment Choices

A B

C

AverageRetur

n

Risk

15%

5%

20%

20%5%

AverageReturn

Risk

Risk-averse

Risk-neutral

Risk-loving

D

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Portfolios

Portfolios are groups of assets, such as stocks and bonds, that are held by an investor.

One convenient way to describe a portfolio is by listing the proportion of the total value of the portfolio that is invested into each asset.

These proportions are called portfolio weights. Portfolio weights are sometimes expressed in percentages. However, in calculations, make sure you use proportions

(i.e., decimals).

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Expected Returns

The expected return on a portfolio is a linear combination, or weighted average, of the expected returns on the assets in that portfolio.

The formula, for “n” assets, is:

In the formula: E(RP) = expected portfolio return wi = portfolio weight for portfolio asset iE(Ri) = expected return for portfolio asset i

n

1iiiP REwRE

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Risk

For a portfolio of two assets, A and B, the variance of the return on the portfolio is:

Where: σA = the standard deviation of asset A

σ B = the standard deviation of asset B

corr(RA , RB ) the correlation between A and B

(Important: Recall Correlation Definition!)

)RCORR(Rσσw2wσwσwσ BABABA2B

2B

2A

2A

2p

σ

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Diversification

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Investment mistakes

1. “Put all eggs in one basket”

2. Superfluous or Naive Diversification (Diversification for diversification’s sake)

a. Results in difficulty in managing such a large portfolio

b. Increased costs (Search and transaction)3. Many investors think that diversification is

always associated with lower risk but also with lower return

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Diversification

Correlation: The tendency of the returns on two assets to move together. Imperfect correlation is the key reason why diversification reduces portfolio risk as measured by the portfolio standard deviation.

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Portfolio of two positively correlated assets

Asset A

0

15

30

-15

Asset B

0

15

30

-15

Asset C=1/2A+1/2B

0

15

30

-15

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Portfolio of two negatively correlated assets

-10

15

15

40

4040

15

0

-10

Asset A

0

Asset B

-10

0

Asset C=1/2A+1/2B

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The Home BiasCountry Share in World Market

ValueProportion of Domestic Equities in Portfolio

France 2.6% 64.4%

Germany 3.2% 75.4%

Italy 1.9% 91.0%

Japan 43.7% 86.7%

Spain 1.1% 94.2%

Sweden 0.8% 100.0%

United Kingdom 10.3% 78.5%

United States 36.4% 98.0%

Canada 4.7% 90.5%Calculations in next few slides are intended for education purposes only. They are based on publicly available data and are not recommended for scientific research

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International Correlation Structure and Risk Diversification

Security returns are much less correlated across countries than within a country. This is so because economic, political,

institutional, and even psychological factors affecting security returns tend to vary across countries, resulting in low correlations among international securities.

Business cycles are often high asynchronous across countries.

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Summary Statistics for Monthly Returns 1980-2011

Stock Market Correlation Coefficient Average return (%)

Risk (%)

CN FR GM JP UK

Canada (CN) 1.03 5.55

France (FR) 0.38 1.32 7.01

Germany (GM) 0.33 0.66 1.18 6.74

Japan (JP) 0.24 0.42 0.36 1.01 6.31

United Kingdom (UK)

0.58 0.54 0.49 0.42 1.19 5.20

United States 0.70 0.45 0.37 0.24 0.57 1.11 4.56

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Example of International Portfolio for a Canadian Investor

Belgian market 0.37%

Hong Kong market 14.66%

Italian market 9.25%

Dutch market 14.15%

Swedish market 20.26%

Canadian market 41.31%

Total 100.00%

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Gains from International Diversification

International portfolio

Canadian shares

Average Return 1.23% 1.03%

Risk 4.27% 5.55%

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Excel functions used

CORR Solver

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Portfolio with Two Assets

Expected StandardInputs Return Deviation

Risky Asset 1 11.0% 18.0%Risky Asset 2 7.0% 13.0%Correlation 10.0%

0%

2%

4%

6%

8%

10%

12%

14%

0% 5% 10% 15% 20% 25% 30%

Exp

ecte

d R

etu

rnStandard Deviation

Efficient Set--Two Asset Portfolio

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Optimal Portfolio

For a 2-asset portfolio:

)R,CORR(Rσσw2wσwσw

r -)E(Rw)E(Rw

σ

r-)E(RRatio Sharpe

BABABA2B

2B

2A

2A

fBBAA

P

fp

Now we have to choose the weights of assets that maximizes the Sharpe Ratio.

We could use calculus or Excel.

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Optimal Portfolio

DataInputs:

ER(S): 0.11 X_S: 0.300STD(S): 0.18

ER(B): 0.07 ER(P): 0.082STD(B): 0.13 STD(P): 0.110

CORR(S,B): 0.10R_f: 0.04 Sharpe

Ratio: 0.381

Suppose we enter the data into a spreadsheet.

Using formulas for portfolio return and standard deviation, we compute Expected Return, Standard Deviation, and a Sharpe Ratio. Then use Solver to find optimal portfolio.

6- 1 © admn 3116, anton miglo admn 3116: financial management 1 lecture 7: optimal portfolios anton...

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