1. Alex Carr Nonlinear Programming Modern Portfolio Theory and the Markowitz Model

2. Louis Bachelier Father of Financial Mathematics The Theory of Speculation, 1900 The first to model the stochastic process, Brownian Motion Stock options act as elementary particles

3. John Burr Williams Theory of Investment Value, 1938 Present Value Model Discounted Cash Flow and Dividend based Valuation Assets have an intrinsic value Present value of it’s future net cash flows Dividend distributions and selling price

4. Harry Markowitz Mathematics and Economics at University of Chicago Earlier Models Lacked Analysis of Risk Portfolio Selection in the Journal of Finance, 1952 Primary theory of portfolio allocation under uncertainty Portfolio Selection: Efficient Diversification of Investments, 1959 Nobel Prize Markowitz Efficient Frontier and Portfolio

5. Foundations Expected return of an asset is the mean Risk of an asset is the variability of an asset’s historical returns Reduce the risk of an individual asset by diversifying the portfolio Select a portfolio of various investments Maximize expected return at fixed level of risk Minimize risk at a fixed amount of expected return Choosing the right combination of stocks

6. Model Assumptions 1. Risk of a portfolio is based on the variability of returns from the said portfolio. 2. An investor is risk averse. 3. An investor prefers to increase consumption. 4. The investor's utility function is concave and increasing.

7. Model Assumptions 5. Analysis is based on single period model of investment. 6. An investor either maximizes his portfolio return for a given level of risk or maximum return for minimum risk. 7. An investor is rational in nature.

8. Risk Standard deviation of the mean (or return) Systematic Risk: market risks that cannot be diversified away Interest rates, recessions and wars Unsystematic Risk: specific to individual stocks and can be diversified away Not correlated with general market moves

9. Risk and Diversification

10. Diversification Optimal: 25-30 stocks Smooth out unsystematic risk Less risk than any individual asset Assets that are not perfectly positively correlated Foreign and Domestic Investments Mutual Funds

11. Correlation Negative Positive None−0.09 to 0.0 0.0 to 0.09 Small−0.3 to −0.10.1 to 0.3 Medium−0.5 to −0.30.3 to 0.5 Strong−1.0 to −0.50.5 to 1.0

12. Expected Return Individual Asset Weighted average of historical returns of that asset Portfolio Proportion-weighted sum of the comprising asset’s returns

13. Mathematical Model

14. The Process First: Determine a set of Efficient Portfolios Second: Select best portfolio from the Efficient Frontier

15. Risk and Return Either expected return or risk will be the fixed variables From this the other variable can be determined Risk, standard deviation, is on the Horizontal axis Expected return, mean, is on the Vertical axis Both are percentages

16. Plotting the Graph All possible combinations of the assets form a region on the graph Left Boundary forms a hyperbola This region is called the Markowitz Bullet

17. Determining the Efficient Frontier The left boundary makes up the set of most efficient portfolios The half of the hyperbola with positive slope makes up the efficient frontier The bottom half is inefficient

19. Indifference Curve Each curve represents a certain level of satisfaction Points on curve are all combinations of risk and return that correspond to that level of satisfaction Investors are indifferent about points on the same curve Each curve to the left represents higher satisfaction

20. Optimal Portfolio The optimal portfolio is found at the point of tangency of the efficient frontier with the indifference curve This point marks the highest level of satisfaction the investor can obtain The point will be different for every investor because indifference curves are different for every investor

21. Capital Market Line E(R P )= I RF + (R M - I RF ) σ P / σ M Slope = (R M – I RF )/ σ M Tangent line from intercept point on efficient frontier to point where expected return equals risk- free rate of return Risk-return trade off in the Capital Market Shows combinations of different proportions of risk- free assets and efficient portfolios

22. Additional Use of Risk-Free Assets Invest in Market Portfolio But CML provides greatest utility Two more choices: Borrow Funds at risk-free rate to invest more in Market Portfolio Combinations to the right of the Market Portfolio on the CML Lend at the risk-free rate of interest Combinations to the left of the Market Portfolio on the CML

23. Efficient Frontier with CML

24. Criticisms There are a very large number of possible portfolio combinations that can be made Lots of data needs to be included Covariances Variance Standard Deviations Expected Returns Asset returns are, in reality, not normally distributed Large swings occur much more often 3 to 6 standard deviations from the mean

25. Criticisms Investors are not “rational” Herd Behavior Gamblers Fractional shares of assets cannot usually be bought Investors have a credit limit Cannot usually buy an unlimited amount of risk-free assets