Performance Measures
(A) Stock Funds
(B) Market Timers
Performance Measures
• Here are some performance measures that have been used
(Refer Chapter 24 of text):
• 1. Sharpe’s Measure: (Rp - Rf)/Sigma_p
• 2. M-Square (an economic interpretation of the Sharpe
ratio)
• 3. Jensen’s alpha: Alpha_p = Rp - [Rf + Beta_p(Rm-Rf)]
• 4. Treynor’s Square : Alpha_p/Beta_p
• Treynor’s Measure: (Rp-Rf)/Beta_p
• 5. Appraisal Ratio: (Rp-Rf)/(volatility of non-market risk
in portfolio)
Jensen’s Alpha (1/6)
• Jensen’s alpha measures the extra return that the portfolio
earns after adjusting for its “beta” risk.
• The beta here does not have to refer to only the market
beta, but to all factors that are important to understanding
the allocation of the fund.
• An example: Suppose an actively managed fund has the
following allocation. It allocates 20% to small cap, and
80% to large cap. How do we evaluate the manager?
Jensen’s Alpha (2/6)
• 1. Get the historical return series for the fund, and calculate
the excess return (Rp-Rf).
• 2. Get the passive portfolio’s that can serve as the
benchmark. You could use one passive portfolio (say, S&P
500). Or you may even want to use multiple passive
portfolio’s (for example, add a small-cap index like the
Rusell 2000). Calculate the excess return for each of these
benchmarks.
• 3. Run a regression of (Rp-Rf) on the excess returns on the
benchmarks, eg. for our example, when we know the
portfolio manager is investing in both large cap and small
cap stocks, we want to two passive indices, (S&P500-Rf)
and (Rusell2000-Rf).
• 4. Examine the intercept. If the intercept is positive and
statistically significant, the manager has outperformed his
benchmark. If its negative, he has underperformed his
benchmark.
Problem with Jensen’s Alpha (3/6)
• 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.
• Consider, in the next slide, an example of the alpha of PEP.
We will conclude that PEP would have to beat the market
by 1.5% a month for 5 years before we are certain it has a
positive alpha.
Jensen’s Alpha and PEP (4/6)
• Over the period, 1997-2001, the cumulative return on PEP
over this period was 21%, in comparison with an S&P 500
return of –10%.
• A regression of the monthly excess return against that of
S&P 500 gives an alpha of 0.56% (or about 7%
annualized) with a t statistic of 0.60. As the t-statistic is
less than 2, we cannot say with confidence that PEP has a
positive alpha.
• Despite the fact that has beaten the market by 7%
annualized over 5 years, we still cannot say statistically
that PEP has outperformed the market.
• There are two questions we need to ask:
– Why is it statistically so difficult to conclude that PEP
has beaten the market?
– By how much would PEP have to outperform the S&P
for us to be certain of the result?
Cumulative Returns (5/6)
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
9 7 l-97 -9 8 l-98 -9 9 l-99 -0 0 l-00 -0 1 l-01
n
n
n
n
n
Ju Ja
Ju Ja
Ju Ja
Ju Ja
Ju
Ja
PEP
S&P 500
Jensen’s Alpha (6/6)
• The cumulative return graph on the previous page
illustrates graphically why it is so difficult to statistically
conclude that PEP outperformed the market. In particular,
PEP is much more volatile than the S&P 500.
• The annualized volatility of PEP is 28%, in comparison
with S&P’s volatility of 19%.
• The higher the volatility, the greater the alpha would have
to be before we can conclude that PEP has beaten the
market.
• Given the volatility, we can observe (by experimentation)
that PEP would have to beat the market by about 1.5% a
month for 5 years for us to be certain that PEP has
outperformed.
• However, given the last five year return history, we cannot
say with any certainty that PEP has truly outperformed the
market.
Treynor’s Square
• We define this as the ratio of the alpha of the portfolio to
the beta of the portfolio:
• Treynor’s Square = Alpha_p/Beta_p.
• The logic behind this ratio is that we should require a
higher alpha from a portfolio of a higher beta. This
measure is useful as it allows us to rank managers by their
risk-adjusted performance, after adjusting for the beta risk
they take.
Treynor’s Measure
• An alternative way of expressing this measure is: (RpRf)/Beta_p). This measure is called Treynor’s Measure.
• Treynor’s Measure = (Rp-Rf)/Beta_p.
• This is equivalent to calculating: Alpha_p/Beta_p + (RmRf).
• Thus, if you use either the Treynor’s square, or Treynor’s
measure to rank portfolios, you will get the same results.
• As the measure uses the Jensen’s Alpha, it has the same
limitations that we already discussed regarding Jensen’s
alpha.
An Example for Treynor Measures
• Suppose you have a choice between investing in two
managers. Which would you prefer?
• Manager A: Alpha= 2%, Beta =0.9.
• Manager B: Alpha=3%, Beta=1.6.
• The Treynor Square measure for A is 2/0.9=2.22. The
Treynor Square measure for B is 3/1.6=1.875. Because A
has a higher measure, you’ll prefer A.
• Intuition: Suppose you combine B with t-bills, with
weights (0.9/1.6= 0.5625) and 0.4375, respectively. In this
case, this combined portfolio will have a beta of 0.9, but an
alpha of 1.6876. Clearly, you will prefer A.
Appraisal Ratio (1/2)
•
•
•
•
Appraisal Ratio: Alpha_p/(Vol of non-market risk).
Suppose you run the following regression:
Rp-Rf = Alpha + Beta (Rm-Rf) + e.
Then, the volatility of “e” represents the non-market risk or
residual risk - or the extra risk you take over the
benchmark.
• Intuitively, the appraisal ratio trades off the extra return
you receive by investing in the active portfolio, versus the
extra risk you take.
• You can calculate the volatility of the idiosyncratic risk by:
• volatility of the non-market risk = sqrt[ (Vol of
Portfolio)^2 - (Beta*Vol of Market)^2 ].
Appraisal Ratio:An Example (2/2)
•
•
•
•
•
•
•
•
Portfolio P: Alpha=1.63, Beta=0.69, Vol = 6.17.
Portfolio Q: Alpha=5.28, Beta=1.40, Vol=14.89.
Benchmark: Vol =8.48.
The non-market risk taken by P is: SQRT(6.17*6.17 0.69*0.69*8.48*8.48)=1.95. The appraisal ratio for P is:
1.63/1.95 = 0.84.
The non-market risk taken by Q is:
SQRT(14.89*14.89 - 1.4*1.4*8.48*8.48)=8.98.
The appraisal ratio for Q is: 5.28/8.98=0.59.
Thus, P has a higher appraisal ratio and should be
preferred.
Final Comments (1/2)
• Note that the performance measures differ from each other,
and it is possible that they may also give different
rankings. The appropriate measure to use will depend on
your total portfolio.
• Here are some thumb rules to follow:
Thumb Rules (2/2)
• 1. Suppose you are only investing in 1 portfolio: P or Q. In
that case, choose the one with the highest Sharpe Ratio or
M Square.
• 2. Always choose a portfolio with positive alpha.
• 3. Comparison between two funds with the same alpha:
– Suppose you already have an index portfolio, and you
want to add one actively managed portfolio, P or Q. In
this case, choose the one with the higher appraisal ratio.
– If your portfolio consists of several actively managed
portfolios, then choose between P and Q by the Treynor
measure.
The TIAA-CREF Social Choice Fund
• As another example, consider the TIAA-CREF
Social Choice fund (http://www.tiaa-cref.org).
• The Social Choice fund invests in firms that do not
violate certain socially desirable objectives.
• The fund typically has a mix of equity and fixed
income instruments.
Sharpe Ratio and M-Square
• Comparing the TIAA-CREF fund to the index fund over
the last five years, we get.
• Riskfree rate = 2%.
• Social Choice Fund:
– Sharpe Ratio = 0.31, Vol = 12%,
– Cumulative 5-yr Fund Return = 44%.
• Index Fund:
– Sharpe Ratio = 0.09, Vol = 20.5%,
– Cumulative 5-yr Fund Return = 28%.
• Thus, the M Square for the Stock Fund is (0.31-0.09)(20.5)
= 4.53%/year.
• If we leveraged the fund by borrowing at a rate of 2% to
create a portfolio of the same volatility as the index fund,
then this portfolio would have outperformed this index by
4.53% /year.
Alpha, Treynor and Appraisal Ratio
• We estimate the alpha of the fund by regressing the
excess returns on the fund, (Rp – Rf), on the excess
returns of the benchmark, (Rm – Rf).
• From the regression results:
– Alpha of the Fund: 2.13% per year.
– Beta of the Fund = 0.58.
– Treynor measure: alpha/beta = 0.036.
– Appraisal Ratio = 1.08.
Market Timing
• What is market timing?
• Does timing work?
• How do we do performance evaluation when the
fund manager times the market?
• How well do the timing measures work in
practice?
Market Timing
• Market timing involves:
• 1. Shifting funds between a market-index and
cash.
• 2. Shifting funds between high beta stocks and
low beta stocks.
• Essentially, market timing attempts to anticipate
market up and down movements.
• Perhaps the most famous timing strategy is the
“Dow Theory”.
The Dow Theory
• The Dow Theory is named after the founding editor of the WSJ,
Charles Henry Dow. But we know of the Dow Theory, not from Dow
himself (who died in 1902), but from William Peter Hamilton.
• Hamilton was the editor of the journal for 27 years after Dow (taking
over in 1902, after the death of Charles Dow), and he wrote a series of
editorials forecasting major trends. The theory, that Hamilton attributes
to Dow, was further elaborated in Hamilton’s book, “The Stock Market
Barometer”, published in 1922.
• However, much of what we know comes from a book by Robert Rhea
(1932, The Dow Theory, Barron’s, New York.).
• For a brief history, see:
– http://www.e-analytics.com/cd.htm.
The Dow Theory and Hamilton
• Hamilton believed both in informationally efficient
markets (“The market movement reflects al the real
knowledge available..”) as well as in the “irrational
exuberance” of individual investors,
• “..the pragmatic basis for the theory, a working hypothesis,
if nothing more, lies in human nature itself. Prosperity will
drive men to excess, and repentance for the consequences
of those excesses will produce a corresponding
depression.”
The Dow Theory: Basics
• Market movements may be decomposed into primary,
secondary and tertiary trends.
• The primary trend is the long-term movement of prices,
lasting from several months to several years.
• Secondary or intermediary trends are caused by shortterm deviations of prices from the underlying trend line.
These deviations are eliminated via corrections, and prices
revert back to trend values.
• Tertiary are daily fluctuations of little importance.
• Primary trends are further classified into bull and bear
markets.
The Bull Market
• Bull Markets have three stages: “first, is the
revival of confidence in the future of business,
second is the response of stock prices to the
known improvement in corporate earnings, and
the third is the period when speculation is rampant
and inflation apparent.”
The Bear Market
• Bear Markets also have three stages, “the first
represents the abandonment of the hopes on which
the stocks were purchased at inflated prices; the
second reflects selling due to decreased business
and earnings, and the third is caused by distressed
selling of sound securities, regardless of value”,
(Rhea, The Dow Theory, 1932).
On Identifying the Primary Trend
• The objective of market timing is to identify the
primary trend (bull or bear market). Here are some
rules:
• 1. The trend must be confirmed by movement in
two different market sectors. Movement in one
sector alone is not reliable.
• 2. A big move followed by a period of quiescence
usually identifies the beginning of a primary trend
in that direction.
How well does the theory do?
• Comparing the strategy to a buy and hold strategy
over 27 years, Hamilton beats the market until
1926, when the strong bull market took over.
However, on a risk adjusted basis, his portfolio
has a higher Sharpe ratio (0.559 vs 0.456) over the
entire period. His average arithmetic return is
10.73% vs. the market’s 10.75%.
• Hamilton died in 1929.
• How well would the theory do today? This was
investigated by researchers at NYU and Yale [see
http://www.stern.nyu.edu/~sbrown].
Duplicating Hamilton’s Strategy
• Neural network: train a program to “learn” the strategy.
This strategy is then applied to 1930-1997. Over the whole
period, 5.48 vs 9.87
•
•
•
•
•
•
•
1930-39: Buy-Hold=1.48, Ham=11.10
1940-49: Buy-Hold=3.21, Ham=6.04
1950-59: Buy-Hold=9.64, Ham=9.91
1960-69: Buy-Hold=7.71, Ham=9.68
1970-79: Buy-Hold=0.41, Ham=6.74
1980-89: Buy-Hold=12.63, Ham=11.29
1990-97: Buy-Hold=15.44, Ham=16.24
Measuring Market Timing
• Analyzing the market timing ability of Hamilton was easy,
because we knew his calls from his editorials. However,
the typical fund manager does not advertise how he is
timing - so how do we infer his timing ability?
• The first question: how do we model the value added by
the manager via market timing?
• We can view the manager’s ability to market time as an
embedded put option. In other words, by investing in a
manager with timing ability, we are buying an index fund
with an embedded put.
– In fact, it may be argued that the real test of a market
timer is in a volatile (non-trending) market.
Two Tests
• 1.Treynor and Mazuy Test: Add a square term to
the usual regression:
• Rp - Rf = a + b(Rm-Rf) + c(Rm-Rf)^2 + e.
• 2. Henriksson and Merton:
• Rp - Rf = a + b(Rm-Rf) + c(Rm-Rf)D + e.
• We will mainly consider the second test.
Manager’s Timing Ability
• Because a manager’s timing ability allows you to avoid a
downturn in a bear market, we consider the following
regression:
• Rp-Rf = a + b (Rm-Rf) + c(Rm-Rf)D + e, where D=1, if
market gives a positive return, and D=0 if the market gives
a negative return.
• In a bull market, the manager’s beta will be (b+c), but in a
bear market, the beta will be (b).
• In other words, we test whether the manager increases his
‘beta’ or the weight on the market in a bull market, and
decreases it in a bear market.
• If we run a regression and estimate a positive “c”, then it
shows that the manager has timing abilities.
• See the attached spreadsheet for an example of the test on
a sample strategy.
Implementation Problems
• The typical problem with implementation is that the data
we use may not match with the timing horizon of the
portfolio manager.
• If the manager times on a daily basis, but we only have
monthly data it will be difficult to capture the timing
ability. Suppose, for example, over the month the market
moved up. Even if the timer was successful, it is difficult
to distinguish the timing ability from the buy and hold
strategy.
Some Recent Results
• Goetzmann, Ingersoll and Ivkovic at Yale recently
applied the test to asset allocation funds.
[“Monthly Measurement of Daily Timers”, 1998].
• Of the 23 Asset Allocation Funds, they found 2
funds that showed significant market timing skills.