The expected return of stocks ≈ Mean

Risks of stocks ≈ Variance

Correlation of stocks ≈ Covariance

American Economist

A recipient of the 1990 Nobel Memorial Prize in Economic Science

The beginning of quantitative finance

Help investors get high profit with low risk

To obtain optimal portfolio

1995-2005

2005 - 2015

30 STOCKS IN AMRICAN SECURITIES MARKET

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

To obtain optimal portfolio

2005-2015

Each point: each stock

C: the stock of Citigroup Inc.

x-axis: risk

y-axis: profit

1995-2005

- 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

Markowitz Model works well in different decades.

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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.

THE MUTURAL RELATIONSHIP BETWEEN STOCKS

The relationship is linear.

The relationship is monotonic.

- 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.

Makowitz Model uses historical data

to predict the situation in the future.

to predict the situation in the future.

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.

- 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.

Based on our results,

we challenge the assumption of Markowitz Model.

we challenge the assumption of Markowitz Model.

Assumption:

Rational investors are risk adverse.

Rational investors are risk adverse.

- Research in behavioral economics
- Studies carried in behavioral finance

Assumptiion:

Nothing influences the value of the return.

Nothing influences the value of the return.

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

Assumption:

Companies never close down.

Companies never close down.

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

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