2. Performance and the Market Line
E(Ri)
E(RM)
RF
Undervalued
ML
M
Overvalued
RiskM
Note: Risk is either β or σ
Riski
3. Performance and the Market Line
(cont.)
E(Ri)
B
ML
A
M
E(RM)
C
E
RFR
D
RiskM
Note: Risk is either β or σ
Riski
4. The Treynor Measure
The Treynor measure calculates the risk premium per
unit of risk (βi)
Note that this is simply the slope of the line between the
RFR and the risk-return plot for the security
Also, recall that a greater slope indicates a better riskreturn tradeoff
Therefore, higher Ti generally indicates better
performance
5. The Sharpe Measure
The Sharpe measure is exactly the same as the
Treynor measure, except that the risk measure is
the standard deviation:
6. Sharpe vs Treynor
The Sharpe and Treynor measures are similar,
but different:
S uses the standard deviation, T uses beta
S is more appropriate for well diversified portfolios,
T for individual assets
For perfectly diversified portfolios, S and T will give
the same ranking, but different numbers (the ranking,
not the number itself, is what is most important)
7. Sharpe & Treynor Examples
Portfolio
X
Y
Z
Market
Return
15%
8%
6%
10%
RFR
5%
5%
5%
5%
Beta
2.50
0.50
0.35
1.00
Std. Dev.
20%
14%
9%
11%
Trenor
0.0400
0.0600
0.0286
0.0500
Sharpe
0.5000
0.2143
0.1111
0.4545
Risk vs Return
Return
15%
10%
Y
5%
X
M
Z
0%
0.00
1.00
Beta
1.50
2.00
Risk vs Return
15%
Return
0.50
2.50
X
M
10%
5%
Z
Y
0%
0%
5%
10%
Std. Dev.
15%
20%
8. Jensen’s Alpha
α>0
Jensen’s alpha is a measure of
the excess return on a
portfolio over time
A portfolio with a consistently
positive excess return
(adjusted for risk) will have a
positive alpha
A portfolio with a consistently
negative excess return
(adjusted for risk) will have a
negative alpha
α=0
α<0
Risk Premium
0
Market Risk Premium
9. Modigliani & Modigliani (M2)
M2 is a new technique (Fall 1997) that is closely
related to the Sharpe Ratio.
The idea is to lever or de-lever a portfolio (i.e.,
shift it up or down the capital market line) so that
its standard deviation is identical to that of the
market portfolio.
The M2 of a portfolio is the return that this
adjusted portfolio earned. This return can then
be compared directly to the market return for the
period.
10. Calculating M2
The formula for M2 is:
M 2 = σ M ( R i − R f ) + R f
σi
As an example, the M2 for our example portfolios is calculated
below:
( 0.20)( 0.15 − 0.05) + 0.05 = 0.105
)( 0.08 − 0.05) + 0.05 = 0.074
= (0.11
0.14
)( 0.06 − 0.05) + 0.05 = 0.062
= (0.11
0.09
M 2 = 0.11
X
M2
Y
M2
Z
Recall that the market return was 0.10, so only X outperformed.
This is the same result as with the Sharpe Ratio.
11. Fama’s Decomposition
Fama decomposed excess return into two main
components:
Risk
Selectivity
Manager’s risk
Investor’s risk
Diversification
Net selectivity
Excess return is defined as that portion of the
return in excess of the risk-free rate
12. Fama’s Decomposition (cont.)
T o ta l R is k P re m iu m
R is k P re m iu m D u e to R is k
M a n a g e r 's R i s k
I n v e s t o r 's R i s k
R is k P r e m iu m D u e to S e le c tiv ity
D iv e rsific a tio n
N e t S e le c tiv ity
13. Fama’s Decomposition: Risk
This is the portion of the excess return that is
explained by the portfolio beta and the market
risk premium:
14. Fama’s Decomposition: Investor’s
Risk
If an investor specifies a particular target level of
risk (i.e., beta) then we can further decompose
the risk premium due to risk into investor’s risk
and manager’s risk.
Investors risk is the risk premium that would
have been earned if the portfolio beta was
exactly equal to the target beta:
RPInvestorRisk = βT ( RM − R f )
15. Fama’s Decomposition: Manager’s
Risk
If the manager actually takes a different level of
risk than the target level (i.e., the actual beta was
different than the target beta) then part of the risk
premium was due to the extra risk that the
manager’s took:
RPManagerRisk = ( β i − βT ) ( RM − R f )
16. Fama’s Decomposition: Selectivity
This is the portion of the excess return that is not
explained by the portfolio beta and the market
risk premium:
Since it cannot be explained by risk, it must be
due to superior security selection.
17. Fama’s Decomposition: Diversification
This is the difference between the return that
should have been earned according to the CML
and the return that should have been earned
according to the SML
If the portfolio is perfectly diversified, this will
be equal to 0
18. Fama’s Decomposition: Net Selectivity
Selectivity is made up of two components:
Net Selectivity
Diversification
Diversification is included because part of the manager’s
skill involves knowing how much to diversify
We can determine how much of the risk premium comes
from ability to select stocks (net selectivity) by
subtracting diversification from selectivity
19. Additive Attribution
Fama’s decomposition of the excess return was the first attempt at
an attribution model. However, it has never really caught on.
Other attribution systems have been proposed, but currently the most
widely used is the additive attribution model of Brinson, Hood, and
Beebower (FAJ, 1986)
Brinson, et al showed that the portfolio return in excess of the
benchmark return could be broken into three components:
Allocation describes the portion of the excess return that is due to
sector weighting different from the benchmark
Selection describes the portion of the excess return that is due to
choosing securities that outperform in the benchmark portfolio
Interaction is a combined effect of allocation and selection.
20. Additive Attribution (cont.)
The Brinson model is a single period model,
based on the idea that the total excess return is
equal to the sum of the allocation, selection, and
interaction effects.
Note that Rt is the portfolio return, Rt bar is the
benchmark return, and At, St, and It are the
allocation, selection, and interaction effects
respectively:
R t − R t = A t + St + I t
21. Additive Attribution (cont.)
The equations for each of the components of
excess return are:
A t = ∑ ( w i,t − w i,t )( R i,t − R t )
N
i =1
St = ∑ w i , t ( R i , t − R i , t )
N
i =1
I t = ∑ ( w i,t − w i,t )( R i,t − R i,t )
N
i =1
22. Additive Attribution (cont.)
So, looking at the formulas it should be obvious that:
Allocation measures the relative weightings of each sector in
the portfolio and how well the sectors performed in the
benchmark versus the overall benchmark return. A positive
allocation effect means that the manager, on balance, overweighted sectors that out-performed in the index and underweighted the under-performing sectors.
Selection measures the sector’s different returns versus their
weightings in the benchmark. A positive selection effect means
that the manager selected securities that outperformed, on
balance, within the sectors.
Interaction measures a combination of the different weightings
and different returns and is difficult to explain. For this reason,
many software programs allocate the interaction term into both
allocation and selection.
