Performance Measures (A) Stock Funds (B) Market Timers

1
Performance Measures (A) Stock Funds(B) Market
Timers
2
Performance Measures

• Here are some performance measures that have been
  used (Refer Chapter 24 of text)
• 1. Sharpes Measure (Rp - Rf)/Sigma_p
• 2. M-Square (an economic interpretation of the
  Sharpe ratio)
• 3. Jensens alpha Alpha_p Rp - Rf
  Beta_p(Rm-Rf)
• 4. Treynors Square Alpha_p/Beta_p
• Treynors Measure (Rp-Rf)/Beta_p
• 5. Appraisal Ratio (Rp-Rf)/(volatility of
  non-market risk in portfolio)

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Jensens Alpha (1/6)

• Jensens 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?

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Jensens Alpha (2/6)

• 1. Get the historical return series for the fund,
  and calculate the excess return (Rp-Rf).
• 2. Get the passive portfolios that can serve as
  the benchmark. You could use one passive
  portfolio (say, SP 500). Or you may even want to
  use multiple passive portfolios (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, (SP500-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.

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Problem with Jensens Alpha (3/6)

• Although Jensens 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 doesnt 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.

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Jensens Alpha and PEP (4/6)

• Over the period, 1997-2001, the cumulative return
  on PEP over this period was 21, in comparison
  with an SP 500 return of 10.
• A regression of the monthly excess return against
  that of SP 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 SP
  for us to be certain of the result?

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Cumulative Returns (5/6)
8
Jensens 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 SP 500.
• The annualized volatility of PEP is 28, in
  comparison with SPs 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.

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Treynors Square

• We define this as the ratio of the alpha of the
  portfolio to the beta of the portfolio
• Treynors 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.

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Treynors Measure

• An alternative way of expressing this measure is
  (Rp-Rf)/Beta_p). This measure is called Treynors
  Measure.
• Treynors Measure (Rp-Rf)/Beta_p.
• This is equivalent to calculating Alpha_p/Beta_p
  (Rm-Rf).
• Thus, if you use either the Treynors square, or
  Treynors measure to rank portfolios, you will
  get the same results.
• As the measure uses the Jensens Alpha, it has
  the same limitations that we already discussed
  regarding Jensens alpha.

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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 Alpha3, Beta1.6.
• The Treynor Square measure for A is 2/0.92.22.
  The Treynor Square measure for B is 3/1.61.875.
  Because A has a higher measure, youll 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.

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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 - (BetaVol of Market)2 .

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Appraisal RatioAn Example (2/2)

• Portfolio P Alpha1.63, Beta0.69, Vol 6.17.
• Portfolio Q Alpha5.28, Beta1.40, Vol14.89.
• Benchmark Vol 8.48.
• The non-market risk taken by P is SQRT(6.176.17
  - 0.690.698.488.48)1.95. The appraisal ratio
  for P is 1.63/1.95 0.84.
• The non-market risk taken by Q is
• SQRT(14.8914.89 - 1.41.48.488.48)8.98.
• The appraisal ratio for Q is 5.28/8.980.59.
• Thus, P has a higher appraisal ratio and should
  be preferred.

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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

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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.

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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.

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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.

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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.

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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?

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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.

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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 Hamiltons 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, Barrons,
  New York.).
• For a brief history, see
• http//www.e-analytics.com/cd.htm.

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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.

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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
  short-term 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.

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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.

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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).

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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.

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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 markets 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.

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Duplicating Hamiltons 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-Hold1.48, Ham11.10
• 1940-49 Buy-Hold3.21, Ham6.04
• 1950-59 Buy-Hold9.64, Ham9.91
• 1960-69 Buy-Hold7.71, Ham9.68
• 1970-79 Buy-Hold0.41, Ham6.74
• 1980-89 Buy-Hold12.63, Ham11.29
• 1990-97 Buy-Hold15.44, Ham16.24

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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 managers 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.

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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.

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Managers Timing Ability

• Because a managers 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 D1,
  if market gives a positive return, and D0 if the
  market gives a negative return.
• In a bull market, the managers beta will be
  (bc), 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.

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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.

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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.