Treynor-Black ModelUsing 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 dayto-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] Page 1 Treynor-Black Model 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 2.1 Theory 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. Page 2 Use the estimates for the values of alpha. 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. Analysts may take several steps in formulating the active portfolio and evaluating its performance: a.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). Risk that is specific to an asset and can essentially only be eliminated by diversification is referred to as specific risk. 8. [1] 2. The quantitative performance measure for a single asset is alpha. Risk that cannot be eliminated by diversification because it is common to a large number of assets is referred to as systematic risk. determine the required rate of return of the asset. Estimate the best of each analyzed security and risk.3 Portfolio Construction The construct of the model is as follows [11]: 1. Page 3 . 5. [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. which the model treats as the passive portfolio.Treynor-Black Model 6. 2. For the purpose of efficient diversification. Determine the expected return and expected abnormal return (alpha) for each mispriced security. the market index portfolio is the baseline portfolio. 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). 9. The market index is used by the model as the passive portfolio 3. 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. From beta and macro forecast. which is determined in a subjective manner. and the variances to determine the optimal weight of each portfolio asset. beta. Determine how much benefit (alpha) of the underpriced asset remains as a result of nonsystematic risk. Individual Portfolio managers can estimate the expected risk and return parameters for a broad market (passively managed) portfolio. 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. 7. b. c. d. As a result. therefore all market participants must be paid to bear it. All returns are assumed to follow a normal distribution as with the original Markowitz portfolio selection model. 2. beta. The rate of return for each security. and variance of the active portfolio from the weights of the portfolio assets. with the assumption that the market portfolio M is the efficient portfolio. The rate of return on the ith security (ri). The intent of the model is to form an active portfolio of positions in the analyzed securities to be blended with the index portfolio.rf) + ei + αi (2) (1) Where αi represents the extra expected return (also referred to as the abnormal return) attributable to any perceived mispricing of the security.Treynor-Black Model 6. i.rf) + ei Where rf = Risk Free rate rM = expected return on market index ei = zero mean. is defined as [3]: ri = rf + βk (rM . Page 4 . firm specific disturbance Equation 1 represents the rate of return of all securities. assuming that all securities are fairly priced is given by: ri = rf + βi (rM . Figure 1: The optimization process with active and passive portfolios. Compute the alpha. that is analyzed. 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. A.[E(rM) . the following equation is used [2]: wA = {[E(rA) . M. rM)} We also know that E(rA) – rf = αA + βARM Cov(RA. This new efficient frontier identifies the optimal risky portfolio.rf] σM2 .Treynor-Black Model Figure 1 above shows graphically the optimization process with the active and passive portfolios. [11] To determine the location of the active portfolio A in Figure 1. constructed from mispriced securities. Portfolio A needs to be mixed with passive market index M to achieve optimal diversification. therefore their mutual correlation in the determination of the optimal allocation between the two portfolios must be taken into account.rf]Cov(rA. is on the efficient frontier and is tangent to the capital market line (CML). rM)} / { [E(rA). 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. The efficient frontier is shown as a red dashed line. They will view the marketindex portfolio as inefficient given their perceived superior analysis. shown as a black dashed line. must lie above the CML by design.rf + E(rM).rf] σA2 .[E(rA). when investing proportion w in the active portfolio and 1-w in the market index.rf]Cov(rA. which is defined as: σA = [ βA2 σM2 + σ2 (eA)] ½ (3) The alpha value that is forecast for A should be positive. The market index. therefore active portfolio A is not the ultimate efficient portfolio in this example. M. which will provide the best or steepest CAL. can be produced by applying simple optimal risky portfolio construction techniques from two component assets. A. will be: rp = w*rA + (1–w)*rM (5) To find the weight. representing the universe of all securities assuming that they are all fairly priced (when all alphas are zero). we need its expected return and standard deviation. As shown. as evident from the efficient frontier that passes through M and A as a solid line in Figure 1. with the passive portfolio.rf] σM 2 + [E(rM) . w. 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. it is the tangency point of the CAL to the efficient frontier. P. RM) = βA σM2 σA2 = βA2 σM2 + σ2 (eA) Page 5 Where RM = E(rM) – rf Where RA = E (rA) – rf (7) (8) (9) (6) . which combines portfolios A and M. The active portfolio is not perfectly correlated with the market-index portfolio. Therefore the new active portfolio. The portfolio return. as βA increases. In practical terms.βA) + RM * *σ2 (eA) / σM2] } (11) Assume βA = 1 which implies that the systematic risk of the active portfolio is average. the term w* increases because as systematic risk of active portfolio A. the benefit from diversifying it with the index. decreases. 3 3. αA (1. However. divided by the ”disadvantage” of A. thus making it more beneficial to take advantage of the mispriced securities. that is.1 Estimating the alpha and beta in the Treynor-Black Model Forecasting to obtain alpha (αi) Alpha (αi) is described as the expected abnormal return of an analyzed security i. The beta of the active portfolio is expected to be in the proximity of 1. giving the equation below: w* = αA / . Any new forecast (αf) will be adjusted by the squared correlation coefficient (ρ2αf). the ratio of the nonsystematic risk of A to the market risk.0 and the optimal weight. then the optimal weight wo is: wo = . which will minimize forecast error.βRM and a0 and a1 are the potential bias in the forecasts. or the squared correlation coefficient is defined as: ρ2 = σ2α / (σ2α + σ2 Ɛ) (15) This measures the quality of the forecast. An expected abnormal return of 20% would be great to have. 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 . this becomes the following for the ith asset: αi = ri .Treynor-Black Model [E(rA) . βA. dividing both numerator and denominator by σM2 and collecting terms.(rm-rf)βi -rf . increases. Page 6 . αA /σ2 (eA)} / { RM / σM2} (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. The variance of the forecast is reflected as shown below: σ2αf = σ2α + σ2Ɛ (14) The ratio of explained variance to total variance. but the question remains about the analyst’s accuracy in his or her expectations of a security.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.rf ] + [E(rM) . M. w*. to be close to w0. s I. the market rate of return (rm). If the Beta is equal to zero. For these four assets. The term beta. Consider a portfolio that can be constructed from the following four assets: Table 1 . It relates an asset’s return and the fluctuation of the market.5 The expected annual return. 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. In particular. the risk-free rate of return (rf). and the annual standard deviation of total return.0 2 30% 45% 2.g. b I. A single systematic risk factor. In the example below. is the measure of total risk. Beta can be estimated using regression analysis against a market index. and the market risk (σm). The formula for Beta is defined below to estimate the Beta value of security i. This produces an R-square (R2).5 4 15% 15% 0.0 3 18% 18% 0. is a measure of total return for the ith asset. is used to capture all systematic risk for computational ease. is used to specify the amount of systematic risk contained in the ith asset. ri. the S&P 500 return).2 Estimating Beta Beta can be described as the unpredictability of an asset compared to the market. referred to as market risk. the numbers have been chosen to illustrate some important features of the TreynorBlack model but do not necessarily reflect realism.Asset Parameters Asset (i) Annual Return ri Annual Risk σi Beta βi 1 25% 35% 0. need to be established.Rm)/Var(Rm) 4 Numerical Example (16) The simplicity and benefits of the Treynor-Black model can be best highlighted by an example. and a regression coefficient (beta) for the asset. Page 7 . βi= Cov(Ri. 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. the asset does not change according to the market. refer to the values in the table below.Treynor-Black Model 3. For this example. a standard error. additional information is needed in order to apply the Treynor-Black model. but with higher risk and return values. it is important to note the key differences of the four assets. The first asset has both high risk and high return. but its beta value of zero shows it is uncorrelated with the market. of the overall portfolio can be computed as: [1] wi = ai/ σ 2(ei) and sharei = (ai/ σ 2(ei))/ ∑(aj/ σ 2(ej)) for j = 1 to 4 (20) (19) Page 8 . we will see that this isn’t always the case. along with the square of specific risk (αi /σ2(ei)). the weight.5. The second asset has the characteristic of a highly leveraged asset as can be seen from its beta value of two. sharei. with the fourth having slightly lower values for each than the third. which are as follows: [1] 1.(rm-rf)βi -rf and σ 2(ei) = σi2 . wi. are computed for each asset using the following formulas below: [1] αi = ri . From the information above. for each asset. While it may seem intuitive that this high return asset will play prominently in percentage of portfolio allocation. and its percentage contribution. 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.σm2β i2 (18) (17) From this. 3. alpha (αi).Treynor-Black Model Table 2 . the projected return of the security over-and-above its market rate risk-adjusted return.Financial Environment Parameters Assumptions risk free rate of return r f market rate of return r m market risk σ m Value 5% 10% 20% Before applying the Treynor-Black model. It has the same ratio of risk and return as Asset 1. Another key difference is that the last two assets do exhibit higher return-to-risk ratios than the first two assets. 2. The third and fourth assets have more modest risk and return levels then the first two assets. Both also have a modest beta of 0. 4. 3. Asset 2’s high rate of return was offset by its high risk. Other basic properties of the portfolio asset selection using the Treynor-Black model include the following: [1] 1. thus limiting its share to only 22.30%.00% 4.2 Performance Evaluation Page 9 .25% 1. the Treynor-Black weights and associated shares can be computed as shown in the table below: Table 3 .The shares are expressed in terms of percentage of shares. in spite of its lowest alpha.30% 2 15.25% 3. While Asset 1’s zero beta keeps its alpha high. While it might be more intuitive for a high risk/high return asset to be weighted more heavily in a portfolio. Stability – The model is not overly sensitive to small changes to the input parameters. Decentralized Decision Making – The relative allocation among the existing or remaining assets does not change as assets are added or removed.00% From the table above. 3.00% 12. extending the model to decentralized applications. The exception is for those rare instances of high return coupled with low risk.85 100.86% Totals 15.24% 4. it is left with a great amount of specific risk and thus a correspondingly small share of only 10. not monetary amounts. thus making the relative allocation among assets independent of the amount of money to be invested. the Treynor-Black model actually tends to favor assets with low risk and low return.69 29.Treynor-Black Computed Values Asset (i) alpha αi specific risk σ2(e i) Weight α i/σ2(e i) Share wi 1 20. Without any knowledge of the assets in any other group.57% 4 7. we can note the following: 1. 2. the allocation decision within a group of partitioned assets can be made.Treynor-Black Model Using these equations. 2.00 37.50% 1.53 22. the smallest of the portfolio.27% 3 10. Monetary Independence .63 10.25% 6. The reason it is a favored holding is because of its very low level of specific risk. The greatest beneficiary of the Treynor-Black model is Asset 4. thus drawing a distinction between the Treynor-Black model and more complex portfolio optimization models which are not as well-behaved.27% of the portfolio.50% 2. leading to a very stable portfolio selection. which are the Sharpe ratio. The quality of the computation.2. If not. Assets with higher Sharpe ratios provide more return for the same risk. Sharpe measure. The Treynor ratio compares excess return over the risk-free rate to the additional risk taken as shown below. however.4.2. Pyramid schemes. It is defined as: [4] S = (rp – rf)/σp 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. Jensen’s Alpha. 4. the better as the Sharpe ratio is used to characterize how well the return of an asset compensates the investor for the risk taken. In this section.2 The Treynor Ratio (21) 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). [4] T = (rp – rf)/βp Where: (22) Page 10 . for example. the Treynor Ratio.Treynor-Black Model There are several tools available for an investor to evaluate the performance of a selected portfolio. and the M2 method. therefore investors are often times advised to select investments with high Sharpe ratios. This led to the Modigliani-Modigliani (M2) Risk Adjusted Performance measure as discussed in Section 4.1 The Sharpe Ratio The Sharpe ratio (also known as the Sharpe Index. 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 disadvantage of the Sharpe ratio is that many people find it difficult to interpret because it is a dimensionless figure. [5] 4. is dependent upon the quality of the inputs. we’ll cover the primary tools used.2. 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. and reward-to-variability ration) is a measure of the excess return per unit of risk in an investment asset. providing the returns are normally distributed. the standard deviation value does not have as much meaning and can skew the data. 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. 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. It stems from the Sharpe ratio. 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.2.Treynor-Black Model 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. 4.8% for the second. thus indicating abnormal returns which is something investors seek.3 The Jensen Alpha An alternative method of ranking portfolios is Jensen’s alpha. the better the performance of the portfolio under analysis.2% versus 6. If one has an M2 of 6. it can be seen that the higher the Treynor ratio. it’s not readily apparent to most investors how much worse the 2nd portfolio is compared to the first.4 The M2 method (23) The Modigliani-Modigliani measure (also known as M2. If an asset is expected to have returns even higher than that of the risk adjusted return. For example.5 and another has one of -0. the difference is much easier to interpret. if one portfolio has a Sharpe ratio of 0. but is in terms of percentages. that asset will have a positive alpha. often in conjunction with the Sharpe ratio and the Treynor ratio. [8] Page 11 . In the capital asset pricing model (CAPM). [6] 4.2. [7] Jensen's alpha is computed as follows: [4] ΑJ = rp – [rf + βp*(rm – rf)] 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.50. Jensen’s alpha determines the abnormal return of an asset over the theoretical expected return. making it easier and more intuitive to interpret than the Sharpe ratio which is a dimensionless ratio. M2. 1 σm is the standard dev. 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 efficiency of the Treynor Black model depends critically on the ability to predict abnormal returns. extending the model to decentralized applications.2. because total volatility matters. The Treynor-Black model is conceptually easy to implement. then the Sharpe measure is appropriate. then the Treynor or the Jensen Alpha measure is appropriate because systematic risk matters. Page 12 . 2. Dimensionless ratios are difficult for most investors to interpret.Treynor-Black Model The M2 figure for a portfolio is given as follows: [8] M2 = S*σm + rf Where: S is the Sharpe Ratio from Equation (5) in Section 4. the allocation decision within a group of partitioned assets can be made. of the excess returns for a benchmark portfolio rf is the average risk free rate 4.5 Comparison of Measures The appropriate measure to use depends on the investment assumptions. thus drawing a distinction between the Treynor-Black model and more complex portfolio optimization models which are not as well-behaved. Moreover. The disadvantages of the Treynor-Black model include the following [1]: 1. thus making the relative allocation among assets independent of the amount of money to be invested. Monetary Independence . thus making the M2 measure which is in terms of percentages more intuitive for investors to understand. however. Stability – The model is not overly sensitive to small changes to the input parameters. which can be compared to that of the market.The shares are expressed in terms of percentage of shares. 4.2. 5. Treynor Black Model uses much less quantitative information than a fuller optimization method that requires the matrix of all pair-wise asset correlations. If the portfolio is just one portion of a larger portfolio. [4] 5 Advantages and Disadvantages (24) The advantages of the Treynor-Black model include the following [1]: 1. leading to a very stable portfolio selection. not monetary amounts. it is useful even when some of its simplifying assumptions are relaxed. If the portfolio represents the entire investment of an individual or entity. 3. rigorous tests of their individual performance. however. [1] 1) Risk-based product definition – Necessary because the model assumes that all correlation between products is captured by the systematic risk factors. The risk of the portfolio also quickly decreases when the number of securities analyzed increases. The Treynor-Black model lends itself as a valuable strategic planning tool for a multi-product or multidivisional financial investment firm because of its inherent modularity and modest data requirements. The Treynor-Black model demonstrates the following [11]: Page 13 . 3. but by blending with a passive index fund. leaving specific risk to be distributed independently from asset to asset. Using the model can be viewed as a three-step process that may need to pass through several iterations before it is complete. the specific risk can be obtained directly from a regression analysis. Using the least squares betas and diagonal market model. The outcome of this model heavily relies on security analysts’ ability to correctly forecast abnormal returns. the specific risk of the analyzed securities can be dramatically reduced. 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. 6 Results and Conclusions In this paper. Security analysts must submit quantifiable forecasts. 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. This leaves analysts under heavy scrutiny as to how reliable and accurate their ability to predict is. 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. while the M2 value quickly increases. thus a more experienced analysis will tend to fare better than a beginner. Using analysis techniques on each security can produce a portfolio that can out-perform the market. 3) Estimation of product-level specific risk – With sufficient and relevant time series of data. 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. 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 entire portfolio is also continuously subjected to performance evaluation that may engender greater exposure of managers to outside pressures. so they will be exposed to continuous.Treynor-Black Model 2. This model is also highly dependent on the analysts experience with predicting abnormal returns. Decentralized organizations can incorporate this model relatively easily. 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. which is essential to efficiency in complex organizations. 3. Page 14 .Treynor-Black Model 1. pages 923-933 Page 15 . J. 5th Edition.wikipedia.ohio-state.cba.edu/~kho_1/Ch20_performance.ppt http://en.cob.ppt http://www.wikipedia. 1973. How to Use Security Analysis to Improve Portfolio Selection.edu/~pbolster/3927/treynor. Investments.edu/~hwhite/pub_files/hwcv-112.org/wiki/Treynor_ratio http://en.wikipedia.wikipedia. and Marcus.net/~millerrisk/Papers/TreynorBlackRevisited. ISBN 0-390-32002-1.ucsd.wikipedia. pages 66-88 [11] Bodie.org/wiki/Sharpe_ratio http://en. McGraw-Hill Companies. Kane. and F.earthlink.org/wiki/Active_management Treynor. 2001.pdf http://web. January. Black.neu. L.Treynor-Black Model 7 [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] References http://home.htm http://weber.org/wiki/Jensen%27s_alpha http://en.org/wiki/Modigliani_Risk-Adjusted_Performance http://en. Journal of Business. 0137 0.0013 0.(rm-rf)βi – rf Specific Risk σ2(e i) Weight Numerator σ2 e i = σi 2 .σ m 2 β i 2 Weight wi αA βA σA 2 rp IBM 0.5422 1.0541 -0. Assumptions risk free rate of return r f market rate of return r m market risk σ m Value 0.0092 0.2359 0.0884 0.4182 0.0118 -0. Treynor-Black Parameters Beta βi alpha α i= ri .2 Using the Matlab code as listed in Appendix C.1069 0.0006 BAC 0. including that of the S&P 500 Index.0093 0.0016 0.0000 0.5370 383.8675 0.0144 0.0470 Page 16 .0000 0.0126 0.0117 0.0070 0.0012 DGX 0.0003 MSFT 0.0120 0.0558 293.0337 0.2144 0.1048 0. Inc Bank of America Stock Ticker IBM MSFT AAPL DGX BAC From the expected returns of each as listed in Appendix B.0001 0.1188 0.4948 143. the Treynor-Black parameters were computed as shown below.0011 0.0003 0.1807 0.3868 0.05 0.1 0.3377 0.1863 0.0028 0. To do so.1397 0.0007 AAPL 0.6347 0.2466 0.0000 0.4689 Totals 1030.0000 0.0428 -0. the following assumptions were made.0001 0.4530 0.0017 1623.3221 -226. the Treynor-Black parameters were computed.Treynor-Black Model Appendix A Examples of Computations Supposed that an investment firm has special knowledge about the expected returns of 5 select assets: Company IBM Microsoft Apple Computers Quest Diagnostics. 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.rf)] M = S*σm + rf 2 Value 25.Treynor-Black Model From this.rf]/βp α = rp .βA)w0 ] 1. Performance Measures Sharpe Ratio Treynor Ratio Jensen's Alpha M2 Equation S = [rp .4182 5.rf+/σp T = [rp .9028 33.2326 0.rf)/σm2] 1275.017982 The performance measures were then computed as shown below. [3] Computation of W0 and WA Value w 0 = [ αA/σA2] / [(rm .768 w A = w 0 / [ 1 + (1.6773% Page 17 .[rf+ βp*(rp . 00133926 0.01117222 0.06311428 -0.03677258 0.02058342 -0.00244300 -0.03086921 -0.00091463 0.00069225 0.05181554 0.03978159 0.02244339 -0.00121760 -0.02705392 -0.00797220 0.02324393 0.03458878 0.01462687 -0.02870960 -0.06056990 -0.01730592 -0.00277298 -0.03516682 0.01245412 0.03458878 0.00282973 Page 18 .00336381 -0.00015598 DGX -0.06244261 0.06542056 -0.02509751 0.01884701 -0.01394494 0.00300619 -0.05228398 -0.01276918 -0.02048281 AAPL 0.04026350 0.00945342 0.00164880 -0.00239874 -0.00308285 0.02058342 -0.01288692 0.01563235 0.00697248 -0.05736960 0.01722251 -0.01163201 0.00538274 0.00949914 -0.00653302 -0.04985755 0.00239874 -0.01378092 0.01111111 0.01045105 0.00383118 0.01102064 0.05418251 0.05614727 -0.01841360 -0.00383118 -0.00224888 0.01556420 -0.02143951 0.00764746 0.00957521 0.02600235 -0.00613793 0.03666667 0.01546213 0.04203869 -0.01762645 0.01132990 -0.01553021 0.01433692 0.05107151 0.03507271 -0.06542056 -0.03094317 0.06359993 0.01358885 0.01042772 0.00162758 -0.01416502 -0.02268250 -0.00035292 -0.00096221 -0.04034896 -0.07078189 0.00772153 -0.04792286 0.03247004 -0.00639225 0.04561404 -0.02704013 -0.01898943 0.01787143 0.00563380 0.00591250 -0.02806834 0.00740616 -0.01567398 0.00638041 0.00072072 -0.00684619 -0.01117222 0.01341234 -0.07078189 0. not percentages Date 13-Sep-12 6-Sep-12 30-Aug-12 23-Aug-12 16-Aug-12 9-Aug-12 2-Aug-12 26-Jul-12 19-Jul-12 12-Jul-12 5-Jul-12 28-Jun-12 21-Jun-12 14-Jun-12 7-Jun-12 31-May-12 24-May-12 17-May-12 10-May-12 3-May-12 26-Apr-12 19-Apr-12 12-Apr-12 5-Apr-12 29-Mar-12 22-Mar-12 15-Mar-12 8-Mar-12 1-Mar-12 23-Feb-12 16-Feb-12 9-Feb-12 2-Feb-12 26-Jan-12 19-Jan-12 IBM -0.02987013 0.02385852 0.02225000 0.01796875 0.00318220 0.01716767 -0.02757140 -0.00609864 -0.07097050 -0.02259887 -0.00991060 -0.05736960 0.00963597 -0.Treynor-Black Model Appendix B – Example Expected Return Data for select Assets and Index Values in decimal.01550661 -0.00590188 0.00192901 0.00135410 0.00401003 0.00370759 -0.00162532 0.02315542 -0.03426791 -0.01898943 -0.03728814 -0.01058476 -0.01961297 -0.00836470 -0.03358992 0.00124292 MSFT -0.00321384 0.00596340 0.00546227 -0.00932597 0.01901580 0.01316672 -0.06557377 -0.00308285 0.05181554 0.00401003 0.02685079 0.05614727 -0.03101439 0.00520059 0.03306452 -0.01366770 0.01498127 -0.02319696 -0.06147220 BAC S&P 500 Index -0.01040462 -0.00604211 -0.01498127 -0.06821589 0.01048951 0.00164880 0.03503675 0.01234786 0.01448389 0.00638524 -0.01776957 -0.01953852 -0.02364395 -0.04792286 0.03400735 -0.01102491 -0.00243704 0.00538274 0.04836649 0.03516682 0.02600235 -0.03008605 0.01872524 0.01709792 0.02472232 -0.03432956 0.01033837 -0.01901580 -0.07017544 0.00463599 0.00822737 -0.01985549 0.05073529 0.04896142 -0.04255319 -0.00237921 0.00520059 0.05418251 0.00546448 0.03898170 0.00181316 0.02933780 -0.01399310 0.00989399 -0.00740616 -0.00990737 0.02157411 -0.01922133 0.01448389 0.00141243 0.00040913 -0.02842085 -0. 04088206 -0.02069257 -0.01110180 0.02366559 0.04500978 -0.05860806 0.01591646 0.05365805 -0.00902256 0.07097922 -0.01558927 0.00911368 -0.03795380 0.00569801 0.01355140 0.00792830 0.02866741 -0.02164443 0.06654676 0.00674916 0.00982863 0.02173186 0.02284935 -0.02050480 -0.07028939 0.02014652 0.01787102 -0.03010984 0.01941748 -0.00779327 -0.03352490 0.01718884 0.02696182 -0.02228357 -0.07947598 0.01103984 -0.05099530 BAC S&P 500 Index 0.00015579 MSFT 0.01802362 0.09104704 0.01007361 0.00422037 -0.01034854 DGX 0.00224352 -0.04631218 -0.03940779 0.01811445 0.00700486 -0.00456311 0.01492537 0.02602304 -0.01744275 0.03779287 -0.00696721 -0.01504388 -0.01956088 0.02869486 0.00578496 0.02172549 -0.05229226 -0.00626468 0.00096131 Page 19 .02311978 0.01238135 -0.01614205 -0.02721088 -0.01903114 0.02205882 -0.01712729 -0.02309843 0.01028492 -0.04225460 0.00043362 -0.04540450 0.06803797 0.02813775 -0.02110775 -0.00662505 -0.01048810 -0.07925801 0.01819172 0.00385197 0.07009901 -0.Treynor-Black Model Date 12-Jan-12 5-Jan-12 29-Dec-11 22-Dec-11 15-Dec-11 8-Dec-11 1-Dec-11 24-Nov-11 17-Nov-11 10-Nov-11 3-Nov-11 27-Oct-11 20-Oct-11 13-Oct-11 6-Oct-11 29-Sep-11 22-Sep-11 15-Sep-11 8-Sep-11 1-Sep-11 IBM -0.02207842 -0.03730056 -0.00861027 -0.02968724 -0.05082742 -0.00158667 0.05744235 -0.00985396 0.03599378 0.00627082 -0.00000000 AAPL 0.02183406 -0.02160149 0.03212626 0.03947904 0.00370370 0.00321367 0.02383486 0.00211549 0.06155355 0.00289356 -0.01184308 0.00774603 0.04507513 0.00947517 -0.01744647 0.01281120 0.00585790 -0.00015215 -0.03749800 -0.00087413 0.01649742 -0.00216780 -0.01140855 0.01445631 0.01107011 0.01793098 0. Date = full_data(:.dat'). % Initialize data (in terms of %) r_f = 0. not % full_data = load('project_data_1. MSFT.05.M2 measure Security Selection .July 19.1). and S&P 500 % date in mm/dd/yyyy format % expected return in decimal.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 Ratio . DGX. 2011 clear all % Structure of data = date and expected returns of IBM. Financial Engineering. AAPL.Jensen's Alpha .The Treynor-Black Model % Course Project: % Course: % Author: IE590. Price .Sharpe Ratio .Treynor-Black Model Appendix C . Summer 2011 David S.10.Treynor-Black portfolio parameters and weights .Matlab Code Course_Project. % BAC. Page 20 . r_m = 0. 0. for i = 1:6 for j = 1:n % read data data(j.0. normalized to % number of data points.0.0.0].0].0.Treynor-Black Model sigma_m = 0. Page 21 .i) = full_data(j. % ******************************************************************* % Compute total expected return and other parameters.0]). % compute covariance matrix (6x6) V = cov(data).0.0.i+1).0.0.0. weight_num_total = 0. end.0.0.20.0]. for each asset and S&P 500 index % ******************************************************************* exp_return = [0. % Create matrix strictly of expected return data n = length(full_data). end.0. sigma = double([0.0. asset_cov = [0.0. beta = [0.0. end. % compute weight numerator weight_num(i) = alpha(i)/sigma2e(i). % add up weight but only for assets.Treynor-Black Model for i = 1:6 % Asset data starts at column 2 exp_return(i) = 0.i))).r_f)*beta(i)) .i). % compute beta beta(i) = V(i.((r_m .6)/sigma(i). Page 22 . not index fund if i < 6 weight_num_total = weight_num_total + weight_num(i). % compute alpha alpha(i) = exp_return(i) . % compute the standard deviation (sigma) for each asset sigma(i) = double(std(data(:. % compute specific weight sigma2e(i) = (sigma(i)^2) .r_f.((sigma_m^2)*(beta(i)^2)). for j = 1:n exp_return(i) = exp_return(i) + data(j. r_p_total = 0. for i = 1:5 % compute weight of each asset weight(i) = weight_num(i)/weight_num_total. alpha_a_total = alpha_a_total + alpha_a(i). Page 23 . sigma_a_sq_total = 0. beta_a_total = 0. % ******************************************************************* % Compute Treynor-Black parameters for each asset. end. %compute beta_a beta_a(i) = beta(i)*weight(i). %compute alpha_a alpha_a(i) = alpha(i)*weight(i). beta_a_total = beta_a_total + beta_a(i).Treynor-Black Model end. but not S&P 500 index % ******************************************************************* % initialize portfolio sigma alpha_a_total = 0. r_f)/(sigma_a_sq_total^0.Treynor-Black Model %compute sigmaa_2 sigma_a_sq(i) = (weight(i)^2) * (sigma(i)^2).r_f)/beta_a_total.(r_f+ (beta_a_total*(r_m . end. r_p_total = r_p_total + r_p(i).5). % compute Jensen's Alpha Jensens_Alpha = r_p_total . % compute Sharpe Ratio Sharpe_Ratio = (r_p_total . %compute r_p r_p(i) = weight(i) * exp_return(i). % compute Treynor Ratio Treynor_Ratio = (r_p_total . wa = w0/(1+(1-beta_a_total)*w0). the weights of the new portfolio and index w0 = (alpha_a_total/sigma_a_sq_total)/((r_m-r_f)/(sigma_m^2)). sigma_a_sq_total = sigma_a_sq_total + sigma_a_sq(i).r_f))). % compute w0 and wa. % compute M2 measure M2 = Sharpe_Ratio*sigma_a_sq_total + r_f. Page 24 .