Risk-Adjusted Performance
Measurement
1.
There are many types of performance measures but
most measures now adjust for portfolio or asset risk or
compare the performance between similar portfolios.
2.
The two general types of risk-adjusted measures are:
a.
Ratios of mean return to risk
b.
Regressions measuring mean return after adjustment for
risk.
3. Ratio measures
a.
Sharpe’s Ratio = [Rp - Rf ] / p
This is a reward to portfolio standard deviation p. Reward to
risk is measured as the premium above the risk-free rate.
a1. M2 = Rp* - RM – where Rp* = levered (if portfolio has low
standard deviation) or de-levered (for high standard
deviation) portfolio such that it has the same standard
deviation as the market portfolio.
The measure ranks performance the same as Sharpe’s ratio
but it is expressed as a return as opposed to a ratio.
Some feel it is easier to interpret, especially for
practitioners
Illustrated below.
Here, portfolio P* has a negative M2 measure and a Sharpe
ratio below that of the market M (its CAL is below the
CML).
b.
Treynor’s Ratio = [Rp - Rf ] / p
Similar to Sharpe but only uses systematic risk, . That is, put
 on the x-axis in the graph above and the slope of the
line would be the Treynor measure. The comparison line
would be the SML instead of the CML.
c. Appraisal ratio =  / e
Where  is a portfolio return adjusted for systematic risk (see
next section) and e is the standard deviation of unsystematic
risk (residual).
This measure gives a return per unit of extra risk one absorbs
by not holding the market portfolio.
If the extra risk will be diversified away because the portfolio
or asset will be added to a diversified portfolio, then this
measure can be misleading.
Alpha Measures - use regression
1. Jensen’s model based on CAPM -> Rp = Rf + p[Rm -Rf]
- Run the regression [Rp - Rf ] =  + p[Rm -Rf] + e
where the funds’ portfolio return is Rp, the risk-free
return is Rf, the expected market return is Rm, alpha
is , beta is B, and e is the regression error term.
2. Treynor/Mazuy version of this model includes a squared
term to account for nonlinear effects.
-Run [Rp - Rf ] =  + 1[Rm -Rf] + 2[Rm -Rf]2 + e
This approach handles the non-linearities that are caused by
market timing. Other alpha-type performance models add
other indexes to control for other systematic return factors.
Just as M2 can be seen as the distance between a portfolio’s
return and the CML, alpha is the distance between the
portfolio’s return and the SML.
The graph assumes that the portfolio can be leveraged up or
down so that any return along the lines containing P or Q can
be reached. It suggests that alpha can be misleading when
we compare alphas for portfolios with very different betas
because the gap between the portfolio’s line and the SML
increases as beta increases so two points on the same
portfolio line will have different alphas even though the
difference could be eliminated by leveraging one up or down.
Nevertheless, alpha is probably the most common
performance measure used because it can handle as many
factors as one would like to include. A ratio measure works
best for a single (market) factor.
3. Fama-French Model - adds small stock (SMB) and book-tomarket factors (HML) to Jensen’s model. SMB is the
difference in return between a portfolio of the smallest stocks
and a portfolio of the largest stocks (net investment is zero).
HML is the difference in return between a portfolio of high
book value-to-market value stocks and a portfolio of low book
value-to-market value stocks (net investment zero).
[Rp - Rf ] =  + 1[Rm -Rf] + 2[SMB] + 3[HML] + e
4. For the alpha models, one regresses the fund’s return
minus the risk free rate on the factors. Significant positive
(negative) performance is indicated if the intercept (the alpha
coefficient) is positive (negative) and statistically significant (tstatistic greater than 1.65).
Example: For Fidelity Value Fund
[Rp1 - Rf ] = -0.11 + 0.88[Rm -Rf]
(0.70) (21.7)
basic CAPM
[Rp1 - Rf ] = -0.15 +0.96[Rm -Rf] + 0.23[SMB] + 0.33[HML]
(1.0) (23.5)
(4.4)
(5.1)
Fama-French Model: Alpha = -0.15 with a t-statistic of 1.0
which means that performance is not statistically significant.
Example: Given the following data assess the performance of
the UCONN fund.
UCONN
S&P500
Sharpe’s Measure
-1.23
-0.92
M2
-7.56
0.00
Alpha
-3.67
0.00
Beta
1.11
1.00
-34.88
-13.32
Appraisal ratio
-0.13
0.00
e
19
0.00
p
32
21
Treynor
Measure
Performance Attribution
1.
Rather than focus on risk-adjustment, some prefer to
simply ascertain where a particular portfolio obtains its
performance in comparison to a “bogey” portfolio.
2.
Although this is often associated with practitioners, there
are theoretical reasons to focus solely on total returns
and their composition.
For example, it turns out that when there is asymmetry in
information between the investor (observer of returns)
and the investment manager, total returns may often be
a better measure of performance.
Recall from the paper on unconditional mean variance
portfolio optimization in the presence of conditioning
information that when the manager knows he will be
measured unconditionally, he will be “conservative” in his
portfolio selection so that he makes sure that his
performance looks good to the unconditional observer.
Here, “conservative” means that he does not take full
advantage of his information because he is constrained
by the observer who defines his performance.
3.
The act of risk adjustment will usually be
counterproductive in these cases because risk will be
defined improperly. Risk would need to be defined
conditional on the manager’s information but the
information asymmetry makes this impossible.
4.
As a result, total return is a reasonable performance
measure.
5. A common performance attribution system decomposes
performance into three components:
a. Asset allocation – equity, bonds, money market
b. industry selection – within each asset class
c. Security selection – within each industry
6. Performance is defined as the return difference between
the managed portfolio P, and the benchmark “bogey” B. Each
asset class is defined by I, for example, airline bonds
Return on P = Rp = i wpirpi
Return on B = RB = i wBirBi
The difference in returns is
Rp - RB = i [wpirpi - wBirBi ]
This can be further decomposed into an asset allocation part
and an asset selection part as follows
Rp - RB = i [wpirpi - wBirBi ] = i [(wpi – wBi)rBi + wpi(rpi - rBi )]
The first term measures how the deviation from the
benchmark’s asset weights affects the performance.
The second term measures how security selection within each
asset class i, provides better or worse returns than the
benchmark’s return on asset class i.
Example: Suppose we have the following data for UCONN:
P Weight
B Weight
P Return B Return
Bonds:
Corporate
.10
.20
3.0
4.0
Treasury
.10
.20
2.0
3.0
Small
.30
.20
-12.0
-15.0
Large
.20
.30
-8.0
-10.0
Money M
.30
.10
1.0
1.0
Stocks:
How did the UCONN manager perform? What do you attribute
his or her performance to, i.e., decompose the performance.
Christopherson, Ferson,
and Glassman (1998)
In footnote 6, they say that manager's alpha is a conditional function of
the covariances of a managers portfolio asset weights (X) and market
return Rm and residual return (u). To see this:
All variables are conditional on the information set Z.
I am using vectors so there are no i subscripts.
r is the return vector on the assets in the portfolio.
b is the asset beta.
As in the footnote, the asset returns follow the return model
r = bRm + u
So the portfolio return is
rX = bRmX + uX
Take expectations wrt. Z
E[rX] = E[bRmX] + E[uX]
If the manager has more information than Z, then X is random so you
cannot take the expectation through a product of two random
variables so use
E(XY) = Cov(X,Y) + E(X)E(Y)
then
E[rX] = Cov(r,X) + E(r)E(X)
Also note that the manager’s expected return is random because he
has more information than Z so we substitute for r using r = bRm + u,
take iterated expectations, and use the fact that E(u) = 0.
E[rX] = Cov((bRm + u),X) + E(bRm + u)E(X)
EE[rX]= E[Cov((bRm + u),X)] + E(bRm)E(X)
EE[rX]= E[Cov((bRm, X) + Cov(u,X)] + bE(Rm)E(X)
First note that bE(Rm)E(X) is the unconditional portfolio return, i.e., the
return one could expect if you just held the manager’s average portfolio
weights, E(X) and never changed them. You should just get the market
related returns from those weighted holdings, given each holding’s beta
b.
Therefore, the covariances that are left over should have values of zero
unless the manager changes his holdings over time in response to his
private information (which includes more than Z).
If the manager has extra information about the market return, then he
increases (decreases) the weights on low (high) beta stocks when his
information tells him that the market return will be low. He does the
reverse if his information tells him that the market return will be high.
Therefore, Cov(bRm,X) > 0.
Similarly, he reweights his portfolio if he has extra firm-specific
information by increasing (decreasing) the weight on stocks that his
information tells him will have unusually high (low) firm-specific returns.
Therefore, Cov(u,X) > 0.
6. Newer methods of performance measurement try to
determine whether an asset manager tends to buy assets
whose prices go up more (down less) than the assets they
sell. This is a covariance performance measure
1. Graham-Harvey, Grinblatt-Titman Measure
Cov(wi, ri) = i [ E(wiri) - E(wi) E(ri) ]
This is the covariance between the portfolio weights of each
asset i, and the return of asset i. The first term in the square
brackets can be thought of as the manager’s weighted
expected return for an asset in his portfolio given his weight
selection (based upon his information). The second term can
be though of as the expected return for a bogey portfolio
where the weights are not actively changed (covariance
between weights and returns is zero which implies 0= [E(wr)
– E(w)E(r)] so E(wr) = E(w)E(r)).
This can be rewritten as an “event-study” measure as follows
Cov(wi, ri) = i E[ (wi(ri - E(ri)) ]
In an event study, we calculate the residual return as (ri - E(ri))
and find an average residual where all weights equal 1/n or
sometimes we weight by market value of each security. The
asset’s expected return is given by a model like the CAPM.
The “portfolio change” measure can be written as
Cov(wi, ri) = i E[ (wi - E(wi))ri ]
With T time series observations on a manager’s asset weights
and returns, this measure can be estimated as follows
Cov(wi, ri) = i t [(wi,t – wi,t-1))rit ]/T
This is positive if the manager increases (decreases) his
weights between times t-1 and t on assets that have large
(small) returns in period t on average over all assets and time
periods. Conversely, the covariance is negative if the manager
changes asset weights inappropriately. If the manager holds a
passive (index) portfolio, the weights don’t change actively so
performance is zero.
2. Admati and Ross’s measure – I use this
(# shares of stock i)t = b0 + b1(stock i price + dividend)t+1
+ b2(stock i price)t + e
This measure controls for stock price at time t (it is
conditioned on the information reflected in the market price).
Performance is positive if b1 is positive and significant.
Here again, the regression feature of this approach allows one
to make adjustments to the model. For example, I get
# shares t = 4.71 + 0.04(stock price + dividend)t+1
– 0.05(stock price + dividend)t+1 (Sale Dummy)
+ 0.06(stock i price)t - 0.04(stock i price)t(Sale Dummy)
All estimates have t-statistics greater than 2.3.
When to Use Each Measure
1.
When the managed portfolio represents an investor’s
entire wealth.
Use the Sharpe Measure because the investor bears the full
standard deviation risk.
2. When the managed portfolio is to be mixed with a market
portfolio.
Use the appraisal ratio because it represents the improvement
in the Sharpe ratio for the overall portfolio due to the
addition of the managed portfolio.
3. When the managed portfolio is combined with many other
portfolios into a large investment fund.
Use the Treynor measure because the risk of each portfolio in
the group (i.e., the marginal risk it adds to the large fund)
can be approximated by its beta. Each portfolio’s
unsystematic risk will be diversified away when it is
grouped into the large fund.
4. If a time series of data on the holdings in the portfolio is
available.
Use the Admati and Ross measure (or Grinblatt-Titman)
because it does not require the assumption that the CAPM (or
any other pricing model) holds.
Also, it typically offers more statistical power than the other
measures. Indeed, it may take 10-30 years of data in order to
obtain enough statistical power to determine whether a
particular portfolio’s alpha or Sharpe ratio is statistically
significant. This is because the standard errors of these
measures is relatively large.
The advantage of the Admati and Ross measure is that every
portfolio holding offers an observation so that the large
number of potential observations drives down the standard
errors of the performance measure.