PORTFOLIO oPTIMIZATION

TO VALIDATE THE TIME VALIDITY OF MARKOWITZ MODEL

Markowitz Model

Mean-variance model

The expected return of stocks ≈ Mean
Risks of stocks ≈ Variance
Correlation of stocks ≈ Covariance

harry m. markowitz

American Economist
A recipient of the 1990 Nobel Memorial Prize in Economic Science

Value

The beginning of quantitative finance

Help investors get high profit with low risk

CONTRAST MODEL

To obtain optimal portfolio

DECADE one

1995-2005

DECADE two

2005 - 2015

data COLLECTION

30 STOCKS IN AMRICAN SECURITIES MARKET

NOTE

The closer time periods is, the better they are.
Choose stocks in different fields
Choose stocks as much as possible

data Analysis

use Matlab to deal with data

Efficient Frontier

To obtain optimal portfolio

2005-2015

Each point: each stock

C: the stock of Citigroup Inc.

x-axis: risk
y-axis: profit

1995-2005

RESULT ANALYSIS

TIME VALIDITY

Explain

X-axis: risk
Y-axis: profit
Each point: each stocks
Market: S&P 500 index
Cash: rate of deposit cash in bank
Equal: spending the same money on each stocks

Conclusion

Markowitz Model works well in different decades.

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1995-2005

2005-2015

NOTE

The scale of two graphs are different. The reason is as following:

In the graph from 2005 to 2015, the range of risk of most stocks is between 0 and 1 and the range of the expected return of most stocks is between -0.25 to 2.75.
These ranges are similar with those in the graph from 1995 to 2005.
The key reason is that there is an additional point in the graph from 2005 to 2015 --- C. C is the stock of Citigroup Inc. This point is on the right end of the portfolio line.

CORELATION

THE MUTURAL RELATIONSHIP BETWEEN STOCKS

Pearson's Correlation Coefficient

Assumption

The relationship is linear.

Spearman's Rank Correlation Coefficient

Assumption

The relationship is monotonic.

Analysis

Use Matlab to calculate both Pearson's correlation coefficients and Spearman's rank correlation coefficients between stocks for two separate decades.
We conclude that the correlation coefficients between stocks from 1995 to 2015 change.
The relationship between stocks is more complex rather than either linear or monotonic. We cannot further explore the relationship between stocks.

LIMITATION

Historical rECORD

HISTORICAL RECORD

Makowitz Model uses historical data
to predict the situation in the future.

Unpredictable Situations

Economic Crisis in 2008

Focus On The Porfolios of Stocks From 2005 to 2015

The risk of C is extremely high.
The risk of deposit cash is higher than investing stocks.
The correlation coefficient between Furniture and Real Estate sharply increases.

Contradiction

In the portfolio graph from 1995 to 2005, AAPL lies on the portfolio line. --- We can invest all the money on AAPL to get high profit with low risk.
In the portfolio graph from 2005 to 2015, AAPL does not lie on the line. --- It is not the optimal portfolio.

assumption

Based on our results,
we challenge the assumption of Markowitz Model.

Investors

Assumption:
Rational investors are risk adverse.

Criticism

Research in behavioral economics
Studies carried in behavioral finance

Value

Assumptiion:
Nothing influences the value of the return.

Criticism

Inverstors have to pay taxes or transaction cost.
Receive dividends influenses investors' decisions.
Investors have a credit limit.

Companies

Assumption:
Companies never close down.

Criticism

50 small companies closed down per day during Economic Crisis in 2008.
lLehman Brothers Holdings Inc, declared bankruptcy in 2008.

CONCLUSION

Markowitz Model is an relatively efficient model.
It offers people benefits on making decisions about portfolio