Alex Carr Nonlinear Programming Modern Portfolio Theory and the
Markowitz Model
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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
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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 its future net cash
flows Dividend distributions and selling price
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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
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Foundations Expected return of an asset is the mean Risk of an
asset is the variability of an assets 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
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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.
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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.
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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
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Risk and Diversification
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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
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Correlation Negative Positive None0.09 to 0.0 0.0 to 0.09
Small0.3 to 0.10.1 to 0.3 Medium0.5 to 0.30.3 to 0.5 Strong1.0 to
0.50.5 to 1.0
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Expected Return Individual Asset Weighted average of historical
returns of that asset Portfolio Proportion-weighted sum of the
comprising assets returns
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Mathematical Model
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The Process First: Determine a set of Efficient Portfolios
Second: Select best portfolio from the Efficient Frontier
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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
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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
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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
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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
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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
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Capital Market Line E(R P )= I RF + (R M - I RF ) P / M Slope =
(R M I RF )/ 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
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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
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Efficient Frontier with CML
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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
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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