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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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© ADMN 3116, Anton Miglo
Topics
Portfolios Diversification Sharpe ratio Optimal Portfolio Excel: CORR, Solver Additional readings: M ch. 5, 6
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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
Diversification
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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
Gains from International Diversification
International portfolio
Canadian shares
Average Return 1.23% 1.03%
Risk 4.27% 5.55%
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© ADMN 3116, Anton Miglo
Excel functions used
CORR Solver
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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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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© ADMN 3116, Anton Miglo
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.