Modern Portfolio Theory provides a framework for constructing investment portfolios to maximize expected return based on a given level of market risk. It assumes investors aim to reduce risk through diversification among assets with low correlations. Markowitz models show how to combine assets to obtain an efficient portfolio with the highest return for a given risk. Mean-variance optimization identifies the portfolio on the efficient frontier with the best risk-return tradeoff. However, the theory relies on historical data and assumptions that may not always hold in real markets.

1. Modern Portfolio Theory
Trading Software and Programming
Acedo  Fabia  Reyes  Sorbito  Vidamo

2. Modern Portfolio Theory
• Markowitz provides the tools
for identifying the portfolio
which give the highest return
for a particular level of risk
(mean-variance portfolio
theory)
• Total risk of the portfolio can be
reduced by diversification – this
can be achieved by investing in
assets that have low positive
correlation, or better still, a
Harry Markowitz negative correlation

3. Markowitz model assumptions
• Investors consider each investment alternative as
being presented by a probability distribution of
1 expected returns over some holding period.
• Investors maximize one-period expected
utility, and their utility curves demonstrate
2 diminishing marginal utility of wealth.
• Investors estimate the risk of the portfolio on the
basis of the variability of expected returns.
3

4. Markowitz model assumptions
• Investors base decisions solely on expected return and
risk, so their utility curves are a function of expected
return and the expected variance (or standard deviation)
4 of returns only.
• For a given level of risk, investors prefer higher returns
to lower returns. Similarly, for a given level of expected
5 returns, investors prefer less risk to more risk.
Under these assumptions, a single asset or portfolio of assets is
considered to be efficient if no other asset or portfolio of
assets offers higher expected return with the same (or lower)
risk, or lower risk with the same (or higher) expected return.

5. Mean variance optimization
Find the portfolio that that minimizes variance
for a given level of return, or maximizes return
for a given level of risk
Basic philosophy: don’t put all your eggs in one
basket! 
Key assumption: returns are normally
distributed

6. Mean variance optimization
Expected
Standard return of
deviation of each asset
returns of
each asset
Correlation
of returns
between
assets
The Efficient Frontier

7. General Formulas
Given: Wi is the percent of the portfolio in asset i
Ri is the possible rate of return for asset i
E(Ri) is the expected rate of return for asset i
Pi is the probability of the possible rate of return Ri
n
Expected Return E RPort Wi E Ri
i 1
n
Measure of Risk
2 2
Variance Pi Ri E Ri
i 1
n
2
Standard Deviation P Ri
i E Ri
i 1

8. General Formulas
Given: Ri is the possible rate of return for asset i
E(Ri) is the expected rate of return for asset i
σi is the standard deviation in rates of return for asset i
Covariance
(2 asset portfolio Covij E Ri E Ri Rj E Rj
i and j )
Covij
rij
Correlation
i j

9. General Formulas
Given: wi is the weights of the individual assets in the portfolio
σ2i is the variance of the rate of return for asset i
Covij is covariance bet the rates of return for
assets i and j, where Covij = rij σi σj
n n n
2 2 2
Portfolio Variance port w i i wi w j Covij
i 1 i 1 j 1
n n n
Portfolio
Standard Deviation port wi2 i
2
wi w j Covij
i 1 i 1 j 1

10. Optimization problem
cov11 cov12  cov1n
cov21 cov22  cov2 n
V
   
covn1 covn 2  covnn

11. Efficient frontier

12. Criticisms
Estimates relies on historical return data
and probability, assumption of normality
Assumes all investors are rational, risk
averse, maximize utility
Assumes markets are efficient: assets are
fairly valued or correctly priced
No consideration for transaction cost or
taxes

13. References
• Bodie, Kane and Marcus. Essentials of
Investments, Eight Edition
• Brown,K. and Reilly, F. Investment Analysis and
Portfolio Management
• Elton, E. and Gruber, M. Modern Portfolio
Theory, 1950 to Date. Working Paper Series 1997.
• Wilmott, P. Paul FAQs Quantitative Finance
• www.investingdaily.com
• Berkelaar, Arjan. Portfolio Optimization powerpoint
presentation