This document discusses portfolio management and various risk-adjusted performance measures. It provides information on how to construct an optimal portfolio based on an investor's risk preferences, time horizon, and tax status. It also discusses Markowitz portfolio theory and how diversification across assets with low correlations can reduce a portfolio's risk. Finally, it summarizes various risk-adjusted return ratios like the Sharpe ratio, Treynor ratio, and Jensen's alpha that are used to evaluate portfolio performance based on risk.

1. UZ Investments & Portfolio
Management
Dr Z Mazhambe

2. Investor’s Portfolio
•There is no “one” perfect portfolio for every
client. To create a portfolio that is right for an
investor, we need to know:
•The investor’s risk preferences
•The investor’s time horizon
•The investor’s tax status

3. Risk
• Securities are priced as if the market in general is “risk averse”. That
is, the typical investor appears to prefer a less risky alternative to a
more risky alternative.
• So in order to induce investors to hold risky investments, the
investment must be priced so as to reward the investor for the risk he
takes on.
• This reward is called the risk premium associated with the expected
return of risky securities, and projects.

4. Risk versus Return
• That is:
• E(Return of a risky venture)
= The reward for waiting plus compensation for taking on risk.
= Risk free return plus a risk premium.

5. Investment fundamentals
• A portfolio
• Diversified collection of stocks, bonds
and other assets.
• Individual investments are often
evaluated on how they change the
characteristics of the portfolio.
• Risk
• Chance of economic loss.
• Sometimes measured as a variation in
return.
• Expected Return
• Anticipated gain of a specific period of
time.
• Often evaluated as compensation for
taking certain types of risks.

6. Portfolio management
• Selected securities viewed as a single unit
• How efficient are financial markets in processing new information?
• How and when should it be revised?
• How should portfolio performance be measured?
The Investment Decision Process – Portfolio
Management

7. Markowitz Portfolio Theory
• Combining stocks into portfolios can reduce
standard deviation below the level obtained from a
simple weighted average calculation.
• Correlation coefficients make this possible.
• The various weighted combinations of stocks that
create this standard deviations constitute the set of
efficient portfolios.

8. Portfolio – 2 Asset
A 2-asset portfolio:
)
R
,
CORR(R
σ
σ
x
2x
σ
x
σ
x
r
-
)
E(R
x
)
E(R
x
σ
r
-
)
E(R
Ratio
Sharpe
)
R
,
CORR(R
σ
σ
x
2x
σ
x
σ
x
σ
:
Variance
Portfolio
)
E(R
x
)
E(R
x
)
E(R
:
Return
Portfolio
B
S
B
S
B
S
2
B
2
B
2
S
2
S
f
B
B
S
S
P
f
p
B
S
B
S
B
S
2
B
2
B
2
S
2
S
2
P
B
B
s
s
p











10. Calculating Portfolio Risk and Return
• The expected risk is calculated as
• Where:
• A = first asset
• B = second asset
• w = weights (respectively)
• σ = standard deviation of assets
• ρ = correlation coefficient of the two assets
AB
B
A
B
A
B
B
A
A
P ρ
σ
σ
w
w
σ
w
σ
w
σ 2
2
2
2
2




11. Markowitz Portfolio Theory
• Quantifies risk
• Derives the expected rate of return for a portfolio of
assets and an expected risk measure
• Shows that the variance of the rate of return is a
meaningful measure of portfolio risk
• Derives the formula for computing the variance of a
portfolio, showing how to effectively diversify a
portfolio

12. Assumptions of
Markowitz Portfolio Theory
1. Investors consider each investment alternative as being presented by
a probability distribution of expected returns over some holding
period.

13. Assumptions of
Markowitz Portfolio Theory
2. Investors minimize one-period expected utility, and their utility
curves demonstrate diminishing marginal utility of wealth.
• I.e., investors like higher returns, but they are risk-averse in seeking those
returns
• And, again, this is a one-period model (i.e., the portfolio will need to be
rebalanced at some point in the future in order to remain optimal)

14. Assumptions of
Markowitz Portfolio Theory
3. Investors estimate the risk of the portfolio on the basis of the
variability of expected returns.
• I.e., out of all the possible measures, variance is the key measure of risk

15. Assumptions of
Markowitz Portfolio Theory
4. Investors base decisions solely on expected return and risk, so their
utility curves are a function of only expected portfolio returns and the
expected variance (or standard deviation) of portfolio returns.
• Investors’ utility curves are functions of only expected return and the variance
(or standard deviation) of returns.
• Stocks’ returns are normally distributed or follow some other distribution that
is fully described by mean and variance.

16. Assumptions of
Markowitz Portfolio Theory
5. For a given risk level, investors prefer higher returns to lower returns.
Similarly, for a given level of expected returns, investors prefer less
risk to more risk.

17. Portfolio Standard Deviation Calculation
• Any asset of a portfolio may be described by two
characteristics:
• The expected rate of return
• The expected standard deviations of returns
• A third characteristic, the covariance between a pair of
stocks, also drives the portfolio standard deviation
• Unlike portfolio expected return, portfolio standard deviation is
not simply a weighted average of the standard deviations for the
individual stocks
• For a well-diversified portfolio, the main source of portfolio risk
is covariance risk; the lower the covariance risk, the lower the
total portfolio risk

18. Covariance of Returns
• Covariance is a measure of:
• the degree of “co-movement” between two stocks’ returns, or
• the extent to which the two variables “move together” relative to their
individual mean values over time

19. Covariance of Returns
For two assets, i and j, the covariance of rates of return is defined as:
 
   
 
 
   
 










S
s
s
j
js
i
is
j
i
j
j
i
i
P
R
E
R
R
E
R
or
dP
R
E
R
R
E
R
ij
,
ij


2

20. Assumptions of Markowitz Theory:
• The Portfolio Theory of Markowitz is based on the following
assumptions:
• (1) Investors are rational and behave in a manner as to maximise their
utility with a given level of income or money.
• (2) Investors have free access to fair and correct information on the
returns and risk.
• (3) The markets are efficient and absorb the information quickly and
perfectly.
• (4) Investors are risk averse and try to minimise the risk and maximise
return.

21. Assumptions of Markowitz Theory
• (5) Investors base decisions on expected returns and variance or
standard deviation of these returns from the mean.
• (6) Investors choose higher returns to lower returns for a given level
of risk.

22. Assumptions of Markowitz Theory
• A portfolio of assets under the above assumptions is considered
efficient if no other asset or portfolio of assets offers a higher
expected return with the same or lower risk or lower risk with the
same or higher expected return.
• Diversification of securities is one method by which the above
objectives can be secured.
• The unsystematic and company related risk can be reduced by
diversification into various securities and assets whose variability is
different and offsetting or put in different words which are negatively
correlated or not correlated at all.

23. Portfolio Performance
• A portfolio’s performance is the result of the performance of its
components
• The return realized on a portfolio is a linear combination of the returns on the
individual investments
• The variance of the portfolio is not a linear combination of component
variances

24. Portfolio Performance
• Portfolio variance is the essence of understanding the mathematics of
diversification
• The variance of a linear combination of random variables is not a weighted
average of the component variances

25. Portfolio Variance
• For an n-security portfolio, the portfolio variance is:
2
1 1
where proportion of total investment in Security
correlation coefficient between
Security and Security
n n
p i j ij i j
i j
i
ij
x x
x i
i j
   

 





26. Two-Security Portfolio
• For a two-security portfolio containing Stock A and Stock B, the
variance is:
2 2 2 2 2
2
p A A B B A B AB A B
x x x x
     
  

27. ASSUMPTION OF THE CAPM
• Assumptions of Capital Asset Pricing Model (CAPM)
• The capital asset pricing model (CAPM) is valid within a special set of assumption.
• These assumptions are
• • All investors have homogenous expectations about the assets.
• • Investor may borrow and lend unlimited amount of risk free asset.
• • The risk free borrowing and lending rates are equal.
• • The quantity of assets is fixed.
• • Perfectly efficient capital markets.
• • No market imperfections such like taxes and regulation and no change in the level of interest
rate exists.
• • There are no arbitrage opportunities.
• • There is a separation of production and financial stocks.
• • Returns (assets) are distributed by normal distribution.

28. The Sharpe Ratio
• The Sharpe ratio is a reward-to-risk ratio that focuses on total risk.
p
f
p
σ
R
R
ratio
Sharpe



29. The Sharpe measure relates return to total
risk. It can be used effectively with a portfolio
where unsystematic risk has been diversified
away.
Traditional Performance Measures
Sharpe measure

Ri  Rf
i
where = arithmetic mean return of security i
= risk free rate
= standard deviation of returns on security i
Ri
Rf
i

30. The Treynor Ratio
• The Treynor ratio is a reward-to-risk ratio that looks at systematic risk
only.
p
f
p
β
R
R
ratio
Treynor



31. Treynor Portfolio
Performance Measure
• Treynor recognized two components of risk
• Risk from general market fluctuations
• Risk from unique fluctuations in the securities in the portfolio
• His measure of risk-adjusted performance focuses on
the portfolio’s undiversifiable risk: market or systematic
risk

32. Treynor Portfolio
Performance Measure
• The numerator is the risk premium
• The denominator is a measure of risk
• The expression is the risk premium return per unit of
risk
• Risk averse investors prefer to maximize this value
• This assumes a completely diversified portfolio
leaving systematic risk as the relevant risk
 
i
i RFR
R
T




33. Treynor Portfolio
Performance Measure
• Comparing a portfolio’s T value to a similar measure for
the market portfolio indicates whether the portfolio
would plot above the SML
• Calculate the T value for the aggregate market as follows:
 
m
m
m
RFR
R
T




34. Jensen’s Alpha
• Jensen’s alpha is the excess return above or below the security market line.
It can be interpreted as a measure of how much the portfolio “beat the
market.”
• It is computed as the raw portfolio return less the expected portfolio return
as predicted by the CAPM.
Actual
return
CAPM Risk-Adjusted ‘Predicted’ Return
“Extra”
Return
 
 
 
R
R
E
β
R
R
α f
M
p
f
p
p 





35. Jensen’s Alpha Application challenges
• Although Jensen’s alpha is theoretically a very appealing
performance evaluation method, and also adjusts for risk -
in practice, it is difficult to use in practice.
• The reason why it doesn’t work is because, even if the
manager is skillful, the “alpha” is likely to be small, and
therefore it is difficult to statistically prove that the alpha is
positive. When the “alpha” is small, we require either large
amounts of data, or we require the manager to have a very
low volatility in his excess returns.
• For typical fund managers, we will thus not be able to
conclude that the manager has an alpha different from
zero.

36. Risk Adjusted Performance Measures
• Assumptions: (1) The SML are applicable to the
pricing of securities. (2) Borrowing and lending
takes place at the risk-free rate. (3)
Construction of the SML is a function of
publicly available information.
• Given the above assumptions, investors may
attempt to employ private information to
identify undervalued and overvalued securities.
One source of legal private information is the
output of unique techniques of analysis of
publicly available data.

37. Treynor versus Sharpe Measure
• Sharpe uses standard deviation of returns as the
measure of risk
• Treynor measure uses beta (systematic risk)
• Sharpe therefore evaluates the portfolio manager on the
basis of both rate of return performance and
diversification
• The methods agree on rankings of completely diversified
portfolios
• Produce relative, not absolute, rankings of performance