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Treynor-Black Model
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Using the Treynor-Black Model in Active Portfolio Management
Aruna Eluri, David S. Price, Kelly Walker
Course Project for IE590 Financial Engineering
Purdue University, West Lafayette, IN 47907-2023
August 1, 2011
Abstract
In 1973, Jack Treynor and Fischer Black published a mathematical model for security selection called the
Treynor-Black model. The model finds the optimum portfolio to hold in the situation where an investor
considers that most securities are priced effectively, but believes he has information that can be used to
predict an abnormal performance of a few of them. The theory behind the model is presented, along
with numerical examples to highlight specific realistic investment scenarios and how the model
performs for each, showing the advantages and disadvantages of the model.
1 Introduction
In developing investment strategy, there are two primary portfolio management styles – active and
passive. Active management is a portfolio management strategy that has a goal of outperforming the
market or some other investment benchmark index by making specific investment selections geared
towards outperforming the market. Passive management is a strategy of being content to invest in an
index fund that will closely replicate the investment weighting and returns of a specific index as the
investor is not seeking to create returns in excess of that benchmark. The first style requires much more
attention, effort, and diligence in obtaining and analyzing data, while the second requires much less day-
to-day attention by its very nature. [9]
To outperform the market and exploit market inefficiencies, the investment manager of an active
portfolio purchases undervalued assets such as stocks or he short sells assets that are overvalued, or a
combination of the two. Instead of always trying to increase value of a portfolio, the investor might also
be trying to reduce the risk with respect to a benchmark index fund. [9]
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The skill of the investment manager and the availability of relevant data and its interpretation will largely determine the performance of an actively-managed investment portfolio.
Approximately 20% of all mutual funds are pure index funds with the remainder being actively managed to some degree. In fact, 45% of all mutual funds are "closet indexers" funds whose portfolios mimic key indexes and whose performance is very closely correlated to an index and refer to themselves as “actively managed” purportedly to justify higher management fees. Results show that a large percentage of actively managed mutual funds rarely outperform their index counterparts over an extended period of time because of this. Often language in a prospectus of a closet indexer will state things such as "80% of holdings will be large cap growth stocks within the S&P 500", therefore the majority of their performance, less the larger fees, will be directly dependent upon the performance of the growth stock index they are benchmarking. [9]
Because indexes themselves have no expenses whatsoever, it is possible that an active or passively managed mutual fund where the securities that comprise the mutual fund are outperforming the benchmark, could underperform compared to the benchmark index due to mutual fund fees and/or expenses. The demand for actively-managed continues to exist, however, because many investors are not satisfied with a benchmark return. In addition, active management looks like an attractive investment strategy to investors in volatile or declining markets or when investing in market segments that are less likely to be profitable when considered as whole. [9]
Gaining knowledge about the future performance of assets is an extremely large endeavor requiring significant staff to research each industry and each company which is impractical for most, if not all, investment firms. The Treynor-Black model, however, assumes that individual portfolio managers possess information of the future performance of certain securities that is not reflected in the current price or projected market return of the asset, thus presenting the radical step of the model which maintains the overall quantitative framework of the efficient market approach to portfolio selection while also introducing a critical violation of the efficient markets theory. By blending a portfolio of these assets with an index fund, the investment manager can produce a portfolio that can outperform the benchmark while also keeping risk at a relatively low level. [1]
2 Theory
2.1 Assumptions
In using the Treynor-Black model, the following assumptions are made. [11]
1. Analysts have a limited ability to find a select number of undervalued securities while the rest
are assumed to be fairly priced (i.e. the security markets are nearly efficient).
2. There is a high degree of co-movement among security prices.
3. The “independent” returns of different securities are almost, but not quite, statistically
independent.
4. The costs of buying and selling are ignored in order to treat the portfolio problem as a single
period problem.
5. Individual Portfolio managers can estimate the future performance of certain securities that is
not reflected in the current price or projected market return of the asset.
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6. Individual Portfolio managers can estimate the expected risk and return parameters for a broad
market (passively managed) portfolio.
7. The quantitative performance measure for a single asset is alpha, which is determined in a
subjective manner.
8. All returns are assumed to follow a normal distribution as with the original Markowitz portfolio
selection model.
9. For the purpose of efficient diversification, the market index portfolio is the baseline portfolio,
which the model treats as the passive portfolio.
2.2 Logic of the Model
The Treynor-Black model is a derivative of the Markowitz (1952) efficient frontier and in its basic form is
an application of the diagonal version of the Capital Asset Pricing Model (CAPM) by Sharpe (1964). [10]
Recognition that the risk of holding an asset can be decomposed into two types: systematic (or market)
and specific (or idiosyncratic or residual) is a fundamental aspect of the Treynor-Black model. Risk that
cannot be eliminated by diversification because it is common to a large number of assets is referred to
as systematic risk, therefore all market participants must be paid to bear it. Risk that is specific to an
asset and can essentially only be eliminated by diversification is referred to as specific risk. As a result, a
risk premium for bearing the specific risk of an asset may be competed away by those best able to
mitigate it through diversification with adequately functioning asset markets (which the Treynor-Black
model assumes). [1]
2.3 Portfolio Construction
The construct of the model is as follows [11]:
1. Security analysts in an active investment management organization can analyze in depth only a
limited number of stocks while the rest are assumed to be fairly priced.
2. The market index is used by the model as the passive portfolio
3. The macro forecasting unit of the investment management firm provides forecasts of the
expected rate of return and variance of the market index portfolio
4. The intent of security analysis is to form an active portfolio of the limited number of securities
based on perceived mispricing of the analyzed securities.
5. Analysts may take several steps in formulating the active portfolio and evaluating its
performance:
a. Estimate the best of each analyzed security and risk. From beta and macro forecast,
determine the required rate of return of the asset.
b. Determine the expected return and expected abnormal return (alpha) for each
mispriced security.
c. Determine how much benefit (alpha) of the underpriced asset remains as a result of
nonsystematic risk.
d. Use the estimates for the values of alpha, beta, and the variances to determine the
optimal weight of each portfolio asset.
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6. Compute the alpha, beta, and variance of the active portfolio from the weights of the portfolio
assets.
The rate of return on the ith security (ri), assuming that all securities are fairly priced is given by:
ri = rf + βi (rM - rf) + ei (1)
Where
rf = Risk Free rate
rM = expected return on market index
ei = zero mean, firm specific disturbance
Equation 1 represents the rate of return of all securities, with the assumption that the market portfolio
M is the efficient portfolio. This model also assumes that the forecast for the passive portfolio has
already been made, so that the expected return on market index and its variance σ2M have been
assessed.
The intent of the model is to form an active portfolio of positions in the analyzed securities to be
blended with the index portfolio. The rate of return for each security, i, that is analyzed, is defined as
[3]:
ri = rf + βk (rM - rf) + ei + αi (2)
Where αi represents the extra expected return (also referred to as the abnormal return) attributable to
any perceived mispricing of the security.
Figure 1: The optimization process with active and passive portfolios.
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Figure 1 above shows graphically the optimization process with the active and passive portfolios. The
efficient frontier is shown as a red dashed line, representing the universe of all securities assuming that
they are all fairly priced (when all alphas are zero). The market index, M, is on the efficient frontier and
is tangent to the capital market line (CML), shown as a black dashed line. Analysts do not need to know
this frontier in practice as they need only to be aware of the market-index portfolio to construct a
portfolio that will produce a capital allocation line that lies above the CML. They will view the market-
index portfolio as inefficient given their perceived superior analysis. Therefore the new active portfolio,
A, constructed from mispriced securities, must lie above the CML by design. [11]
To determine the location of the active portfolio A in Figure 1, we need its expected return and standard
deviation, which is defined as:
σA = [ βA2
σM2
+ σ2
(eA)] ½ (3)
The alpha value that is forecast for A should be positive, therefore it is expected to plot above the black dashed CML line with the expected return defined as:
E(rA) = αA + rf + βA[ E(rM) – rF] (4)
The optimal combination of the active portfolio, A, with the passive portfolio, M, can be produced by applying simple optimal risky portfolio construction techniques from two component assets. The active portfolio is not perfectly correlated with the market-index portfolio, therefore their mutual correlation in the determination of the optimal allocation between the two portfolios must be taken into account, as evident from the efficient frontier that passes through M and A as a solid line in Figure 1. This new efficient frontier identifies the optimal risky portfolio, P, which combines portfolios A and M. As shown, it is the tangency point of the CAL to the efficient frontier. Portfolio A needs to be mixed with passive market index M to achieve optimal diversification, therefore active portfolio A is not the ultimate efficient portfolio in this example. The portfolio return, when investing proportion w in the active portfolio and 1-w in the market index, will be:
rp = w*rA + (1–w)*rM (5)
To find the weight, w, which will provide the best or steepest CAL, the following equation is used [2]:
wA = {[E(rA) - rf] σM2 - [E(rM) - rf]Cov(rA, rM)} /
{ [E(rA)- rf] σM 2 + [E(rM) - rf] σA
2 - [E(rA)- rf + E(rM)- rf]Cov(rA, rM)} (6)
We also know that
E(rA) – rf = αA + βARM Where RM = E(rM) – rf (7)
Cov(RA, RM) = βA σM2 Where RA = E (rA) – rf (8)
σA2 = βA
2 σM2 + σ2
(eA) (9)
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[E(rA) - rf ] + [E(rM) - rf ] = αA + βARM) + RM = αA + RM(1+ βA) (10)
The expression for the optimal weight in portfolio A may be determined by substituting these
expressions into Equation 6, dividing both numerator and denominator by σM2 and collecting terms,
giving the equation below:
w* = αA / , αA (1- βA) + RM * *σ2 (eA) / σM
2] } (11)
Assume βA = 1 which implies that the systematic risk of the active portfolio is average, then the optimal
weight wo is:
wo = , αA /σ2 (eA)} / { RM / σM
2} (12)
Or in other words the optimal weight is the relative “advantage” of portfolio A as measured by the ratio: of the alpha to market excess return, divided by the ”disadvantage” of A, that is, the ratio of the nonsystematic risk of A to the market risk. The beta of the active portfolio is expected to be in the proximity of 1.0 and the optimal weight, w*, to be close to w0. However, as βA increases, the term w* increases because as systematic risk of active portfolio A, βA, increases, the benefit from diversifying it with the index, M, decreases, thus making it more beneficial to take advantage of the mispriced securities. 3 Estimating the alpha and beta in the Treynor-Black Model
3.1 Forecasting to obtain alpha (αi)
Alpha (αi) is described as the expected abnormal return of an analyzed security i. An expected abnormal
return of 20% would be great to have, but the question remains about the analyst’s accuracy in his or
her expectations of a security. A regression of the forecasts can be used on the realized alphas by using
the following equation:
αf = a0 + a1α + Ɛ (13)
where α = R - βRM and a0 and a1 are the potential bias in the forecasts. In practical terms, this becomes
the following for the ith asset: αi = ri - (rm-rf)βi -rf .
The variance of the forecast is reflected as shown below:
σ2α
f = σ2α + σ2
Ɛ (14)
The ratio of explained variance to total variance, or the squared correlation coefficient is defined as:
ρ2 = σ2α / (σ2
α + σ2 Ɛ) (15)
This measures the quality of the forecast. Any new forecast (αf) will be adjusted by the squared
correlation coefficient (ρ2αf), which will minimize forecast error.
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3.2 Estimating Beta
Beta can be described as the unpredictability of an asset compared to the market. It relates an asset’s
return and the fluctuation of the market. Beta can be estimated using regression analysis against a
market index. If the Beta is equal to zero, the asset does not change according to the market. A positive
Beta means the return of an asset follows the market’s return and a negative Beta means the return of
an asset performs in the opposite direction of the market’s return.
The regression analysis for an asset can be performed in most common spreadsheets with a y range of
the return for asset i and the x range the market index returns (e.g. the S&P 500 return). This produces
an R-square (R2), a standard error, and a regression coefficient (beta) for the asset. The formula for Beta
is defined below to estimate the Beta value of security i.
βi= Cov(Ri,Rm)/Var(Rm) (16)
4 Numerical Example
The simplicity and benefits of the Treynor-Black model can be best highlighted by an example. In the example below, the numbers have been chosen to illustrate some important features of the Treynor-Black model but do not necessarily reflect realism.
Consider a portfolio that can be constructed from the following four assets:
Table 1 - Asset Parameters
The expected annual return, ri, is a measure of total return for the ith asset, and the annual standard deviation of total return, s I, is the measure of total risk. A single systematic risk factor, referred to as market risk, is used to capture all systematic risk for computational ease. The term beta, b I, is used to specify the amount of systematic risk contained in the ith asset.
For these four assets, additional information is needed in order to apply the Treynor-Black model. In particular, the risk-free rate of return (rf), the market rate of return (rm), and the market risk (σm), need to be established. For this example, refer to the values in the table below.
Asset
(i) 1 2 3 4
Annual Return
ri 25% 30% 18% 15%
Annual Risk
σi 35% 45% 18% 15%
Beta
βi 0.0 2.0 0.5 0.5
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Table 2 - Financial Environment Parameters
Before applying the Treynor-Black model, it is important to note the key differences of the four assets, which are as follows: [1]
1. The first asset has both high risk and high return, but its beta value of zero shows it is uncorrelated with the market. An asset such as this is often viewed by investment fund managers as an ideal addition to a portfolio it will help diversify risk because its lack of correlation with other assets. While it may seem intuitive that this high return asset will play prominently in percentage of portfolio allocation, we will see that this isn’t always the case.
2. The second asset has the characteristic of a highly leveraged asset as can be seen from its beta value of two. It has the same ratio of risk and return as Asset 1, but with higher risk and return values.
3. The third and fourth assets have more modest risk and return levels then the first two assets, with the fourth having slightly lower values for each than the third. Both also have a modest beta of 0.5. Another key difference is that the last two assets do exhibit higher return-to-risk ratios than the first two assets.
From the information above, alpha (αi), the projected return of the security over-and-above its market rate risk-adjusted return, along with the square of specific risk (αi /σ2(ei)), are computed for each asset using the following formulas below: [1]
αi = ri - (rm-rf)βi -rf (17)
and
σ 2(ei) = σi2 - σm
2β i2 (18)
From this, the weight, wi, for each asset, and its percentage contribution, sharei, of the overall portfolio can be computed as: [1]
wi = ai/ σ 2(ei) (19)
and
sharei = (ai/ σ 2(ei))/ ∑(aj/ σ 2(ej)) for j = 1 to 4 (20)
Assumptions Value
risk free rate of return r f 5%
market rate of return r m 10%
market risk σm 20%
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Using these equations, the Treynor-Black weights and associated shares can be computed as shown in the table below:
Table 3 - Treynor-Black Computed Values
From the table above, we can note the following:
1. While Asset 1’s zero beta keeps its alpha high, it is left with a great amount of specific risk and thus a correspondingly small share of only 10.30%, the smallest of the portfolio.
2. Asset 2’s high rate of return was offset by its high risk, thus limiting its share to only 22.27% of the portfolio.
3. The greatest beneficiary of the Treynor-Black model is Asset 4, in spite of its lowest alpha. The reason it is a favored holding is because of its very low level of specific risk.
While it might be more intuitive for a high risk/high return asset to be weighted more heavily in a portfolio, the Treynor-Black model actually tends to favor assets with low risk and low return. The exception is for those rare instances of high return coupled with low risk.
Other basic properties of the portfolio asset selection using the Treynor-Black model include the following: [1]
1. Monetary Independence - The shares are expressed in terms of percentage of shares, not monetary amounts, thus making the relative allocation among assets independent of the amount of money to be invested.
2. Decentralized Decision Making – The relative allocation among the existing or remaining assets does not change as assets are added or removed. Without any knowledge of the assets in any other group, the allocation decision within a group of partitioned assets can be made, extending the model to decentralized applications.
3. Stability – The model is not overly sensitive to small changes to the input parameters, leading to a very stable portfolio selection, thus drawing a distinction between the Treynor-Black model and more complex portfolio optimization models which are not as well-behaved.
4.2 Performance Evaluation
Asset
(i) 1 2 3 4 Totals
alpha
αi 20.00% 15.00% 10.50% 7.50%
specific risk
σ2(ei) 12.25% 4.25% 2.24% 1.25%
Weight
αi/σ2(ei) 1.63 3.53 4.69 6.00 15.85
Share
wi 10.30% 22.27% 29.57% 37.86% 100.00%
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There are several tools available for an investor to evaluate the performance of a selected portfolio. In this section, we’ll cover the primary tools used, which are the Sharpe ratio, the Treynor Ratio, Jensen’s Alpha, and the M2 method.
4.2.1 The Sharpe Ratio
The Sharpe ratio (also known as the Sharpe Index, Sharpe measure, and reward-to-variability ration) is a
measure of the excess return per unit of risk in an investment asset. It is defined as: [4]
S = (rp – rf)/σp (21)
Where:
rp is the average return of the portfolio
rf is the average risk free rate
σp is the standard deviation of the portfolio return
The higher the Sharpe ratio, the better as the Sharpe ratio is used to characterize how well the return of an asset compensates the investor for the risk taken. Assets with higher Sharpe ratios provide more return for the same risk, therefore investors are often times advised to select investments with high Sharpe ratios. The quality of the computation, however, is dependent upon the quality of the inputs. Pyramid schemes, for example, may have a high Sharpe ratio if using the fund returns as a basis instead of using the actual asset returns as a basis.
A principal advantage of the Sharpe ratio is that it can be computed directly from any observed series of returns without needing additional information about profitability, providing the returns are normally distributed. If not, the standard deviation value does not have as much meaning and can skew the data.
A disadvantage of the Sharpe ratio is that many people find it difficult to interpret because it is a dimensionless figure. This led to the Modigliani-Modigliani (M2) Risk Adjusted Performance measure as discussed in Section 4.2.4. [5]
4.2.2 The Treynor Ratio
The Treynor ratio (also known as the reward-to-volatility ratio or Treynor measure) is a measurement of
the returns earned in excess of that which could have been earned on an investment that has no
diversifiable risk (such as Treasury Bills).
The Treynor ratio compares excess return over the risk-free rate to the additional risk taken as shown below. [4]
T = (rp – rf)/βp (22)
Where:
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rp is the average return of the portfolio
rf is the average risk free rate
βp is the weighted average β of the portfolio
From this, it can be seen that the higher the Treynor ratio, the better the performance of the portfolio under analysis. The value by itself is of not much value unless compared with other portfolios as it is a ranking criterion only like the Sharpe ratio. [6]
4.2.3 The Jensen Alpha
An alternative method of ranking portfolios is Jensen’s alpha. Jensen’s alpha determines the abnormal
return of an asset over the theoretical expected return.
In the capital asset pricing model (CAPM), returns are expected to be “risk adjusted” meaning risk has
already been taken into account as riskier assets should have higher expected returns than less riskier
ones. If an asset is expected to have returns even higher than that of the risk adjusted return, that asset
will have a positive alpha, thus indicating abnormal returns which is something investors seek. [7]
Jensen's alpha is computed as follows: [4]
ΑJ = rp – [rf + βp*(rm – rf)] (23)
Where:
rp is the average return of the portfolio
rf is the average risk free rate
rm is the average return on the market index
βp is the weighted average β of the portfolio
Jensen’s Alpha is widely used to evaluate mutual fund and portfolio manager performance, often in conjunction with the Sharpe ratio and the Treynor ratio.
4.2.4 The M2 method
The Modigliani-Modigliani measure (also known as M2, M2, and the Modigliani Risk Adjusted
Performance measure) is a tool for measuring the risk–adjusted returns of a portfolio relative to some
benchmark such as the market or an index fund. It stems from the Sharpe ratio, but is in terms of
percentages, making it easier and more intuitive to interpret than the Sharpe ratio which is a
dimensionless ratio. For example, if one portfolio has a Sharpe ratio of 0.5 and another has one of -0.50,
it’s not readily apparent to most investors how much worse the 2nd portfolio is compared to the first. If
one has an M2 of 6.2% versus 6.8% for the second, the difference is much easier to interpret. [8]
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The M2 figure for a portfolio is given as follows: [8]
M2 = S*σm + rf (24)
Where:
S is the Sharpe Ratio from Equation (5) in Section 4.2.1
σm is the standard dev. of the excess returns for a benchmark portfolio
rf is the average risk free rate
4.2.5 Comparison of Measures
The appropriate measure to use depends on the investment assumptions. If the portfolio represents
the entire investment of an individual or entity, then the Sharpe measure is appropriate, which can be
compared to that of the market, because total volatility matters.
If the portfolio is just one portion of a larger portfolio, then the Treynor or the Jensen Alpha measure is
appropriate because systematic risk matters.
Dimensionless ratios are difficult for most investors to interpret, however, thus making the M2 measure
which is in terms of percentages more intuitive for investors to understand. [4]
5 Advantages and Disadvantages
The advantages of the Treynor-Black model include the following [1]:
1. The Treynor-Black model is conceptually easy to implement. Moreover, it is useful even when some of its simplifying assumptions are relaxed.
2. Treynor Black Model uses much less quantitative information than a fuller optimization method that requires the matrix of all pair-wise asset correlations.
3. Monetary Independence - The shares are expressed in terms of percentage of shares, not monetary amounts, thus making the relative allocation among assets independent of the amount of money to be invested.
4. Decentralized Decision Making – The relative allocation among the existing or remaining assets does not change as assets are added or removed. Without any knowledge of the assets in any other group, the allocation decision within a group of partitioned assets can be made, extending the model to decentralized applications.
5. Stability – The model is not overly sensitive to small changes to the input parameters, leading to a very stable portfolio selection, thus drawing a distinction between the Treynor-Black model and more complex portfolio optimization models which are not as well-behaved.
The disadvantages of the Treynor-Black model include the following [1]:
1. The efficiency of the Treynor Black model depends critically on the ability to predict abnormal returns.
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2. Implementation of this model requires that security analyst forecasts be subjected to statistical analysis and that the properties of the forecasts be explicitly used when new forecasts are input to the optimization process.
3. Security analysts must submit quantifiable forecasts, so they will be exposed to continuous, rigorous tests of their individual performance. The entire portfolio is also continuously subjected to performance evaluation that may engender greater exposure of managers to outside pressures.
6 Results and Conclusions
In this paper, it has been shown how an investor can benefit from active portfolio management
techniques to outperform the market even when the investment firm only has special knowledge of a
subset of all securities available. Using analysis techniques on each security can produce a portfolio that
can out-perform the market, but by blending with a passive index fund, the specific risk of the analyzed
securities can be dramatically reduced.
The Treynor-Black model lends itself as a valuable strategic planning tool for a multi-product or multi-
divisional financial investment firm because of its inherent modularity and modest data requirements.
Using the model can be viewed as a three-step process that may need to pass through several iterations
before it is complete. [1]
1) Risk-based product definition – Necessary because the model assumes that all correlation
between products is captured by the systematic risk factors, leaving specific risk to be
distributed independently from asset to asset.
2) Determination of product-level risk-adjusted excess returns (alphas) – The opinion of the analyst
must be turned into an estimate of the excess return.
3) Estimation of product-level specific risk – With sufficient and relevant time series of data, the
specific risk can be obtained directly from a regression analysis.
The simplicity of the Treynor-Black model lends itself to producing this new optimal portfolio that will
outperform the market while mitigating risk of the securities forecasted to produce abnormally high
returns compared to market price.
Although the Treynor-Black Model is a very simple model to use, it has not been heavily used by analysts
in evaluating optimal portfolios. The outcome of this model heavily relies on security analysts’ ability to
correctly forecast abnormal returns. This leaves analysts under heavy scrutiny as to how reliable and
accurate their ability to predict is. Using the least squares betas and diagonal market model, however,
the mean return of the portfolio increases despite the predictive power of the security analyst as the
number of securities used in the Treynor-Black model increases. The risk of the portfolio also quickly
decreases when the number of securities analyzed increases, while the M2 value quickly increases. This
model is also highly dependent on the analysts experience with predicting abnormal returns, thus a
more experienced analysis will tend to fare better than a beginner.
The Treynor-Black model demonstrates the following [11]:
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1. Proper active management can add value even when using imperfect security analysis.
2. The Treynor-Black model is conceptually very easy to implement and is useful even when some
of its simplifying assumptions are relaxed.
3. Decentralized organizations can incorporate this model relatively easily, which is essential to
efficiency in complex organizations.
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7 References
[1] http://home.earthlink.net/~millerrisk/Papers/TreynorBlackRevisited.htm
[2] http://weber.ucsd.edu/~hwhite/pub_files/hwcv-112.pdf
[3] http://web.cba.neu.edu/~pbolster/3927/treynor.ppt
[4] http://www.cob.ohio-state.edu/~kho_1/Ch20_performance.ppt
[5] http://en.wikipedia.org/wiki/Sharpe_ratio
[6] http://en.wikipedia.org/wiki/Treynor_ratio
[7] http://en.wikipedia.org/wiki/Jensen%27s_alpha
[8] http://en.wikipedia.org/wiki/Modigliani_Risk-Adjusted_Performance
[9] http://en.wikipedia.org/wiki/Active_management
[10] Treynor, J. L. and F. Black, 1973, How to Use Security Analysis to Improve Portfolio Selection,
Journal of Business, January, pages 66-88
[11] Bodie, Kane, and Marcus, Investments, 5th Edition, 2001, McGraw-Hill Companies,
ISBN 0-390-32002-1, pages 923-933
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Appendix A
Examples of Computations
Supposed that an investment firm has special knowledge about the expected returns of 5 select assets:
Company Stock Ticker
IBM IBM
Microsoft MSFT
Apple Computers AAPL
Quest Diagnostics, Inc DGX
Bank of America BAC
From the expected returns of each as listed in Appendix B, including that of the S&P 500 Index, the
Treynor-Black parameters were computed. To do so, the following assumptions were made.
Using the Matlab code as listed in Appendix C, the Treynor-Black parameters were computed as shown
below.
Assumptions Value
risk free rate of return r f 0.05
market rate of return r m 0.1
market risk σm 0.2
Treynor-Black Parameters IBM MSFT AAPL DGX BAC Totals
Beta
βi 0.0144 0.0120 0.0117 0.0070 0.0118
alpha
αi= ri - (rm-rf)βi – rf 0.3377 0.1048 0.4530 0.1863 -0.3868
Specific Risk
σ2(ei) 0.0003 0.0007 0.0012 0.0006 0.0017
Weight Numerator
σ2 e i = σi2 - σm
2 βi2 1030.4948 143.5370 383.0558 293.3221 -226.8675 1623.5422
Weight
wi 0.6347 0.0884 0.2359 0.1807 -0.1397 1.0000
αA 0.2144 0.0093 0.1069 0.0337 0.0541 0.4182
βA 0.0092 0.0011 0.0028 0.0013 -0.0016 0.0126
σA2 0.0001 0.0000 0.0001 0.0000 0.0000 0.0003
rp 0.2466 0.0137 0.1188 0.0428 0.0470 0.4689
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From this, the proportion or weight of the new portfolio A of the new enhanced market portfolio is
computed (w0) as well as the proportion of the index fund (wa) as shown below. [3]
The performance measures were then computed as shown below.
Computation of W0 and WA Value
w 0 = [ αA/σA2] / [(rm - rf)/σm
2] 1275.768
w A = w 0 / [ 1 + (1- βA)w0 ] 1.017982
Performance Measures Equation Value
Sharpe Ratio S = [rp - rf+/σp 25.9028
Treynor Ratio T = [rp - rf]/βp 33.2326
Jensen's Alpha α = rp - [rf+ βp*(rp - rf)] 0.4182
M2 M2 = S*σm + rf 5.6773%
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Appendix B – Example Expected Return Data for select Assets and Index
Values in decimal, not percentages
Date IBM MSFT AAPL DGX BAC S&P 500 Index
13-Sep-12 -0.00538274 -0.00520059 0.01448389 -0.02600235 -0.06542056 -0.02058342
6-Sep-12 0.01117222 0.03458878 0.04792286 -0.00401003 -0.03516682 0.00308285
30-Aug-12 0.05736960 0.07078189 0.05181554 0.00740616 0.05418251 0.05614727
23-Aug-12 0.00383118 0.00164880 0.01901580 -0.01898943 -0.01498127 -0.00239874
16-Aug-12 -0.00538274 -0.00520059 0.01448389 -0.02600235 -0.06542056 -0.02058342
9-Aug-12 0.01117222 0.03458878 0.04792286 -0.00401003 -0.03516682 0.00308285
2-Aug-12 0.05736960 0.07078189 0.05181554 0.00740616 0.05418251 0.05614727
26-Jul-12 0.00383118 0.00164880 0.01901580 -0.01898943 -0.01498127 -0.00239874
19-Jul-12 0.00772153 0.02319696 -0.01730592 0.03503675 -0.01111111 0.00040913
12-Jul-12 -0.01132990 -0.00836470 -0.05107151 0.00463599 -0.04255319 -0.02244339
5-Jul-12 -0.01462687 -0.03432956 0.01787143 0.01058476 -0.03507271 -0.02324393
28-Jun-12 -0.01563235 0.01102491 0.00653302 -0.00945342 0.00949914 -0.00162758
21-Jun-12 0.00141243 -0.02157411 -0.01550661 0.01358885 -0.02933780 -0.00336381
14-Jun-12 0.00609864 -0.03247004 -0.01776957 0.01163201 -0.03086921 -0.00181316
7-Jun-12 -0.00990737 -0.00192901 -0.00991060 0.00638524 0.00244300 -0.01716767
31-May-12 0.01366770 0.01567398 -0.00162532 -0.00300619 -0.00243704 0.01961297
24-May-12 0.01245412 0.00591250 0.07097050 -0.03101439 -0.03978159 0.01341234
17-May-12 0.01316672 -0.02685079 -0.02268250 0.00638041 -0.04896142 -0.00639225
10-May-12 -0.00133926 0.02315542 -0.02757140 0.00957521 0.00822737 -0.00318220
3-May-12 0.01288692 -0.00546448 -0.01985549 0.04836649 0.00224888 0.01416502
26-Apr-12 0.04034896 0.03306452 0.06311428 0.01953852 -0.04985755 0.02704013
19-Apr-12 -0.04026350 -0.03426791 -0.06056990 -0.06359993 -0.02364395 -0.01922133
12-Apr-12 0.00370759 -0.01040462 -0.02225000 0.01234786 0.01841360 -0.01276918
5-Apr-12 -0.00277298 -0.02259887 0.03400735 0.00035292 -0.00563380 0.00096221
29-Mar-12 -0.01553021 -0.01884701 -0.00684619 -0.01546213 -0.03728814 -0.01722251
22-Mar-12 0.00604211 -0.00697248 -0.01762645 0.00121760 -0.00135410 0.01042772
15-Mar-12 -0.00091463 -0.01872524 0.02987013 -0.02509751 0.03358992 0.01394494
8-Mar-12 0.03008605 0.00072072 0.03094317 0.02806834 0.05073529 0.02705392
1-Mar-12 0.02385852 -0.00963597 0.02870960 0.05228398 -0.04561404 -0.00546227
23-Feb-12 0.03666667 -0.00989399 -0.06244261 -0.00237921 -0.06557377 -0.00764746
16-Feb-12 0.01399310 -0.01048951 0.03677258 0.02842085 0.07017544 0.01709792
9-Feb-12 0.00797220 0.02472232 0.04203869 -0.01556420 0.06821589 0.01102064
2-Feb-12 0.00596340 -0.01378092 -0.00321384 -0.01045105 0.02143951 0.00069225
26-Jan-12 0.00613793 0.01433692 0.00932597 0.00590188 0.03898170 0.01033837
19-Jan-12 0.00124292 0.02048281 0.00015598 0.06147220 -0.01796875 0.00282973
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Date IBM MSFT AAPL DGX BAC S&P 500 Index
12-Jan-12 -0.00385197 0.01184308 0.00982863 0.01956088 0.07925801 0.01281120
5-Jan-12 0.01028492 0.07009901 0.00774603 0.01048810 0.06654676 0.02968724
29-Dec-11 -0.00792830 -0.01712729 0.02696182 -0.02228357 -0.04631218 -0.00861027
22-Dec-11 0.00911368 -0.02207842 -0.00422037 -0.00627082 -0.03795380 0.00043362
15-Dec-11 -0.02164443 -0.02160149 -0.02869486 -0.01903114 -0.01941748 -0.02173186
8-Dec-11 0.02311978 0.00674916 0.05365805 0.05860806 0.07947598 0.03599378
1-Dec-11 0.02813775 0.05082742 -0.02110775 0.00224352 0.00087413 0.00015215
24-Nov-11 -0.00985396 -0.00626468 -0.02309843 -0.02602304 -0.04507513 0.00585790
17-Nov-11 0.01591646 0.03947904 0.07028939 -0.00158667 -0.09104704 0.00947517
10-Nov-11 0.02366559 0.00779327 0.04088206 0.00578496 -0.00902256 0.01649742
3-Nov-11 0.01140855 -0.01614205 -0.03352490 0.02014652 -0.02205882 -0.00211549
27-Oct-11 0.03010984 -0.01744647 0.06155355 0.01802362 0.01492537 0.02050480
20-Oct-11 0.01718884 0.05744235 0.04540450 0.01110180 -0.01107011 0.01445631
13-Oct-11 0.00321367 -0.01811445 0.01793098 0.03940779 0.00370370 0.00456311
6-Oct-11 0.02284935 0.01504388 0.07097922 0.02866741 0.06803797 0.03749800
29-Sep-11 -0.02172549 -0.01238135 -0.03212626 -0.03730056 -0.01787102 -0.00662505
22-Sep-11 -0.00289356 -0.00696721 0.00216780 -0.02069257 -0.02721088 -0.00700486
15-Sep-11 -0.01744275 -0.04500978 -0.04225460 -0.01558927 -0.05229226 -0.03779287
8-Sep-11 0.01355140 -0.01007361 0.01103984 0.02383486 -0.00569801 0.01819172
1-Sep-11 0.00015579 0.00000000 -0.01034854 0.05099530 0.02183406 -0.00096131
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Appendix C - Matlab Code Course_Project.m
% Overview: Matlab script file for computing the following for a
% sample data set of expected returns for 5 assets and
% the S&P 500 Index:
% - Treynor-Black portfolio parameters and weights
% - Sharpe Ratio
% - Treynor Ratio
% - Jensen's Alpha
% - M2 measure
% Course Project: Security Selection - The Treynor-Black Model
% Course: IE590, Financial Engineering, Summer 2011
% Author: David S. Price - July 19, 2011
clear all
% Structure of data = date and expected returns of IBM, MSFT, AAPL, DGX,
% BAC, and S&P 500
% date in mm/dd/yyyy format
% expected return in decimal, not %
full_data = load('project_data_1.dat');
Date = full_data(:,1);
% Initialize data (in terms of %)
r_f = 0.05;
r_m = 0.10;
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sigma_m = 0.20;
% Create matrix strictly of expected return data
n = length(full_data);
for i = 1:6
for j = 1:n
% read data
data(j,i) = full_data(j,i+1);
end;
end;
% *******************************************************************
% Compute total expected return and other parameters, normalized to
% number of data points, for each asset and S&P 500 index
% *******************************************************************
exp_return = [0;0;0;0;0;0];
sigma = double([0;0;0;0;0;0]);
beta = [0;0;0;0;0;0];
asset_cov = [0;0;0;0;0;0];
weight_num_total = 0;
% compute covariance matrix (6x6)
V = cov(data);
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for i = 1:6
% Asset data starts at column 2
exp_return(i) = 0;
for j = 1:n
exp_return(i) = exp_return(i) + data(j,i);
end;
% compute the standard deviation (sigma) for each asset
sigma(i) = double(std(data(:,i)));
% compute beta
beta(i) = V(i,6)/sigma(i);
% compute alpha
alpha(i) = exp_return(i) - ((r_m - r_f)*beta(i)) - r_f;
% compute specific weight
sigma2e(i) = (sigma(i)^2) - ((sigma_m^2)*(beta(i)^2));
% compute weight numerator
weight_num(i) = alpha(i)/sigma2e(i);
% add up weight but only for assets, not index fund
if i < 6
weight_num_total = weight_num_total + weight_num(i);
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end;
end;
% *******************************************************************
% Compute Treynor-Black parameters for each asset, but not S&P 500 index
% *******************************************************************
% initialize portfolio sigma
alpha_a_total = 0;
beta_a_total = 0;
sigma_a_sq_total = 0;
r_p_total = 0;
for i = 1:5
% compute weight of each asset
weight(i) = weight_num(i)/weight_num_total;
%compute alpha_a
alpha_a(i) = alpha(i)*weight(i);
alpha_a_total = alpha_a_total + alpha_a(i);
%compute beta_a
beta_a(i) = beta(i)*weight(i);
beta_a_total = beta_a_total + beta_a(i);
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%compute sigmaa_2
sigma_a_sq(i) = (weight(i)^2) * (sigma(i)^2);
sigma_a_sq_total = sigma_a_sq_total + sigma_a_sq(i);
%compute r_p
r_p(i) = weight(i) * exp_return(i);
r_p_total = r_p_total + r_p(i);
end;
% compute w0 and wa, the weights of the new portfolio and index
w0 = (alpha_a_total/sigma_a_sq_total)/((r_m-r_f)/(sigma_m^2));
wa = w0/(1+(1-beta_a_total)*w0);
% compute Sharpe Ratio
Sharpe_Ratio = (r_p_total - r_f)/(sigma_a_sq_total^0.5);
% compute Treynor Ratio
Treynor_Ratio = (r_p_total - r_f)/beta_a_total;
% compute Jensen's Alpha
Jensens_Alpha = r_p_total - (r_f+ (beta_a_total*(r_m - r_f)));
% compute M2 measure
M2 = Sharpe_Ratio*sigma_a_sq_total + r_f;