Modern Portfolio Theory and the
Markowitz Model
Alex Carr
Nonlinear Programming
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
John Burr Williams
 Theory of Investment Value, 1938
 Present Value Model
 Discounted Cash Flow and Dividend based
 Assets have an intrinsic value
 Present value of it’s future net cash flows
 Dividend distributions and selling price
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
 Portfolio Selection: Efficient Diversification of Investments,
 Nobel Prize
 Markowitz Efficient Frontier and Portfolio
 Expected return of an asset is the mean
 Risk of an asset is the variability of an asset’s historical
 Reduce the risk of an individual asset by diversifying the
 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
Model Assumptions
1. Risk of a portfolio is based on the
variability of returns from the said
2. An investor is risk averse.
3. An investor prefers to increase
4. The investor's utility function is concave
and increasing.
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.
 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
Risk and Diversification
 Optimal: 25-30 stocks
 Smooth out unsystematic risk
 Less risk than any individual asset
 Assets that are not perfectly positively
 Foreign and Domestic Investments
 Mutual Funds
 None
 Small
 Medium
 Strong
−0.09 to 0.0
−0.3 to −0.1
−0.5 to −0.3
−1.0 to −0.5
0.0 to 0.09
0.1 to 0.3
0.3 to 0.5
0.5 to 1.0
Expected Return
 Individual Asset
 Weighted average of historical returns of that asset
 Portfolio
 Proportion-weighted sum of the comprising asset’s returns
Mathematical Model
The Process
 First:
 Determine a set of Efficient Portfolios
 Second:
 Select best portfolio from the Efficient Frontier
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
Plotting the Graph
 All possible combinations
of the assets form a region
on the graph
 Left Boundary forms a
 This region is called the
Markowitz Bullet
Determining the Efficient Frontier
 The left boundary makes
up the set of most efficient
 The half of the hyperbola
with positive slope makes
up the efficient frontier
 The bottom half is
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
 Each curve to the left
represents higher
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
Capital Market Line
 E(RP)= IRF + (RM - IRF)σP/σM
 Slope = (RM – IRF)/σM
 Tangent line from intercept point on efficient
frontier to point where expected return equals riskfree rate of return
 Risk-return trade off in the Capital Market
 Shows combinations of different proportions of riskfree assets and efficient portfolios
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
Efficient Frontier with CML
 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
 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

similar documents