2. Factors to be Considered When Evaluating
Portfolios
• Differential risk levels:
– If a unit trust provides its investors with the total return
of 15%, is their performance good or bad?
– Not possible to answer without knowing the level of
risk involved
– High returns v s. risk aversion
– Performance evaluation on the basis of the risk
adjusted returns
3. Factors to be Considered When Evaluating
• Benchmarks:
Portfolios, cont.
– How do we determine whether portfolio performance
is superior or inferior?
– Need for comparison: comparable alternative
– That alternative needs to be the legitimate portfolio:
benchmark portfolio
– Pay attention to choose the adequate benchmark:
FTSE 100 index cannot be a benchmark if you invest
in FTSE small companies (BELEX15 i akcije van
indeksa)
– Customised benchmarks can be constructed (npr.
Bankarski sektor)
4. Factors to be Considered When Evaluating
Portfolios, cont
• Constraints on portfolio manager:
– some unit or investment trusts have set constraints,
such as: not allowed to short-sell or to invest in small
stocks or emerging markets
– investment policy determines the risk and the return
of the portfolio
• Other factors to be considered
– evaluation of a manager v s. evaluation of a portfolio
performance
– diversification issues
– past is not a guarantee for the future performance
5. Risk-adjusted Measures of Performance
• Composite measures of portfolio performance
– incorporate return and risk in the evaluation
– two kinds of risk to be estimated: portfolio’s market
risk (beta) and total risk (standard deviation)
– the relevant measure of risk depends whether the
client has other assets apart from the portfolio
– Sharpe, Treynor, Jensen alpha and Information ratio
measures date from 1960s and are still in use
6. Example to be Used
We will show the application of all the measures on the example
below:
Return SD
Market 0.04798 0.7318
Portfolio A 0.0793 0.8315
Portfolio B 0.06388 0.7498
Risk-free 0.032
7. The Sharpe Ratio
• Reward-to-variability ratio (RVAR)
• The Sharpe measure evaluates portfolios that are adjusted
for their total risk, measured in terms of standard deviation
(total risk), not just the systematic risk
• The Sharpe ratio for portfolio p is given by:
• The Sharpe ratio for the market portfolio is defined as:
S
r -
r m
sˆ
m
f
m
=
• Measure is based on the ex post (historical) CML
8. The Sharpe performance measure, cont.
• The denoiminator (imenilac) measures the portfolios
excess return - risk premium
• The numerator (brojilac) is the total risk of the
portfolio
• The measure shows excess return per unit of total
risk
• We can use Sharpe ratio to rank portfolios. The
higher the value of the Sharpe ratio, the better the
portfolio performance is.
9. Sharpe Performance Measure, the Example
Return SD Sharpe Ratio
Market 0.04798 0.7318 0.0218
Portfolio A 0.0793 0.8315 0.0568
Portfolio B 0.06388 0.7498 0.0425
Risk-free 0.032
The result of the descending order ranking is: A, B, Market.
10. Graphical Presentation of the Sharpe Example
Rp
A
We can see from the graph that the
slope of the CML is the Sharpe
ratio of the market and the slope of
the other two lines above CML is
the Sharpe ratio for portfolios A
and B.
B CML
M the slope of CML is:
Rf
sp
11. The Treynor Ratio
• Similar to Sharpe’s measure
• Reward-to-volatility ratio (RVOL)
• Treynor distinguishes between the total risk and the
systematic risk
• Assumption: portfolios are well diversified, i.e.
diversifiable risk can be ignored
• The Treynor ratio for portfolio q is defined as:
• The Traynor ratio for the market is:
r r T
q f
b q
q
-
=
r -
r T = -
m r r
m f
m f
m
=
b
12. The Treynor Performance Measure, cont.
• The measure is showing the excess return per unit
of the systematic risk (beta)
• The higher the Treynor ratio, the better the
investment performance is. Portfolio q will
outperform the market if:
r -
r T > -
q r r
m f
q f
b
q
=
• As a benchmark Treynor uses ex post SML
13. Treynor Performance Measure, the Example
Return SD Beta Treynor
Market 0.04798 0.7318 1.00 0.01598
Portfolio A 0.0793 0.8315 0.8223 0.05746
Portfolio B 0.06388 0.7498 0.9322 0.034198
Risk-free 0.032
The result of the descending order ranking is: A, B, Market.
Note that it is the same as ranking obtained with
Sharpe ratio!
14. Rp
Graphical presentation of the Treynor
A
B SML
M the slope of SML is:
Rf
bp
example
We can see from the graph that
the slope of the SML is the
Treynor ratio of the market and
the slope of the other two lines
above SML is the Treynor ratio for
portfolios A and B.
15. Sharpe’s Measure vs. Treynor’s Measure
• Sharpe and Treynor measures are similar
• Definition of risk determines which measure to
use
• Ranking order of portfolios according to Sharpe
and Treynor ratios can be different
– if portfolio is perfectly diversified, the rankings will be
the same, as it appears to be the case in our example
– if portfolio is not well diversified, Treynor’s ranking will
be higher than Sharpe’s
16. Definition of Jensen’s Alpha
a is measuring the average rate of return
above the return corresponding to the
given level of risk (according to CAPM)
• Positive alpha indicates that portfolio is
performing better than the market on the
risk-adjusted basis and vice versa .
17. Definition of Jensen’s Alpha, cont.
• Therefore, Jensen’s portfolio performance measure can
be expressed as:
[ ( )] p p f p M f a = R - R +b R - R
• Where alpha is the difference between the actual excess
return of the portfolio p and the risk premium on that
portfolio according to the CAPM
• Alpha is estimated by regressing the excess returns of a
portfolio on the excess returns of the market
• It is important that it is statistically significant (t ≥ 2)
18. Information (Appraisal) Ratio
• The source of portfolio’s excess return is the deviation
from the market portfolio. This deviation is reflected in
the extra amount of unique risk that the investor is willing
to undertake.
• The information ratio is the ratio of the excess return
divided by the standard deviation of that excess return,
where alpha is used as a measure of excess return:
a p
a s
p IR =
• The higher the information ratio, the better the
performance of portfolio is
19. Limitations of the Portfolio Performance
Measures
• Derived from CAPM - depend on the
assumptions
• Market portfolio proxied by market index
– benchmark error
• Depending on which index you use as a proxy
for market portfolio, beta can change
• Global investing enhances the problem of
benchmark error
• Period for evaluation should be long – 10 years
or longer
20. Da se podsetimo: Beta i prinos
• Beta measures the sensitivity of stock’s return to movements in the
market return.
• Market beta is equal to 1.
• Defensive vs. aggressive stocks
• If the market return increases by 1%, then the following 5 stocks
will exhibit the changes in returns as in the table below:
Stock Beta Change in the
stock return
The stock is:
A -0.5 -0.5% Negatively correlated and LESS
volatile than the market
B 0 0% Not correlated with the market
C 0.5 0.5% Positively correlated and LESS volatile
than the market
D 1 1% Perfectly positively correlated with the
market
E 1.5 1.5% Positively correlated and MORE
volatile than the market
