Portfolio Modeling and Evaluation: Beating the Market

ABSTRACT During the period of 2005 to 2010, the market portfolio (P1) and one suggested portfolio (P3) post a positive absolute return of 0.80% and 0.82% respectively which underperformed the active fund portfolio (P2) 0.91%. This report follows various modeling methods in order to back test the performance of the active fund portfolio and compare its performance with that of two other portfolios. The findings indicate that, even though P2 achieves the highest return on the overall performance, the limitations such as the macro environment, the assumptions set, and the Shrinkage method used that accidentally downsizes some valuable stocks in out-­‐samples as they are closely correlated are being ignored. By contrast, P3 will probably offer a “middle-­‐choice” which will bring a promising and more stable return.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

TABLE CONTENT 1.INTRODUCTION 2.DATA 2.1.BASIC INFORMATION 2.2.DATA LIMITATIONS

3 3 3 4

3.METHODOLOGY 3.1. METHODS ON PORTFOLIO MODELING CONSTRUCTION 3.1.1. MARKET PORTFOLIO AND BLACK-­‐LITTERMAN MODEL 3.1.2. SHRINKAGE MIXED WITH BAYESIAN

4 4 4 8

3.2. PORTFOLIO CONSTRUCTION LIMITATIONS 3.3. METHODS ON EVALUATION

12 13

4. RESULTS 4.1. PERFORMANCE EVALUATION 4.2. SENSITIVITY TESTS

15 15 19

5.CONCLUSION REFERENCES

23 24

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

1.Introduction Platinum Fund (PF) is an actively managed investment portfolio composed by 10 large US stocks. This report is to estimate its performance by testing historical data on three portfolios namely, market portfolio (P1), optimal active portfolio incorporated with analyst opinions (P2) and suggested portfolio (P3). First, the data as well as its limitation are given which underlines the basic information of the fund. Second, this report elaborates methods commonly used of financial modeling, such as market portfolio, Black-­‐Litterman (BL) model, Shrinkage estimator and Bayesian method. Then, it empirically compares and evaluates the performance of three portfolios using six widely used ratios and sensitivity analyses. Last, it concludes with remarks on our findings and observations of the portfolios.

2.Data 2.1 Basic information

The original data source is the monthly share price of 10 US stocks from September 2005 to September 2013. Assuming risk free rate (RFR) and market return rate is constant over the whole period at 0.05% and 0.08% per month, respectively. Estimation refers to the period from September 2005 to September 2010 (which is known as In-­‐sample testing), and on the other hand, performance revaluation relates to period from October 2010 to September 2013 (Out-­‐of-­‐Sample testing). Additional information on these 10 stocks and analysts’ forecast is in Table 1. Table 1. Market Capitalisation and Analysts’ Forecast.
Market Capitalisation (USD Ticker WMT T PG JNJ WFC PFE KO ORCL INTC MRK Name Wal-Mart Stores AT&T Proctor and Gamble Johnson and Johnson Wells Fargo Pfizer Coca-Cola Oracle Intel Merk billions) 246.81 187.03 219.10 259.52 226.03 202.65 172.36 151.08 120.71 136.19 Analyst Forecast (per month) 0.00% 0.00% -0.10% 0.00% 0.10% 0.00% 0.00% -0.10% 0.00% 0.10%

Source: Author’s own processing

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

2.2. Data Limitation

The assumption on RFR at 0.05% and expected market return remain constant at 0.80% monthly for the whole period is not plausible in the real world, numerous factors would influence these rates to change on different extent, at different time. Another two assumptions worth-­‐mentioning, are the Shrinkage estimator and Bayesian degree of confidence indicator, which are also subjective. These assumptions come from empirical evidence, altered by professional judgment. In order to mitigate the disadvantage on limitation of assumptions, sensitivity tests are given in section 3. Inevitably, historical data are used in analysing and forecasting, and it should be noted that past performance can not guarantee the future result.

3.Methodology 3.1. Methods on portfolio modeling construction 3.1.1. Market Portfolio and Black-­‐Litterman model

The target in a portfolio management is to maximize return, ceteris paribus. In order to obtain the expected return the portfolio (E(rp)), proportion of each stock (P) and the expected return rate (E(R)) are calculated from individual stock as equation (1): E(rp)=PTE(R) (1) Firstly, the P is solved, as the variance (δ2) of the portfolio is determined by variance-­‐covariance VCV matrix (V) and P of the portfolio, excel Solver function can solve the minimum variance of portfolio or the using the formula in (3) and then P can be computed by applying (2).

δ2= PTVP (2) P=!! !!! (! ! !!!) So, the pivotal part for portfolio optimization is to compute the VCV matrix, with which the proportion of optimal portfolio can be determined.

! !! (! ! !!!)

(3)

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

V=!!!(R-­‐R)T(R-­‐R)

!

(4)

In (4), R is the average of each stock in the portfolio and T is the number of the months of expected return. But estimation can not be totally precise, optimization will attach more weight to stock whose expected return is overestimated; on the other hand, it will attach much less for pair of stocks whose covariance is overestimated, and vice versa (Harris, 2013). Taking the 10-­‐stock portfolio for example, passive optimal portfolio seems unrealistic (Table 2).

Table 2. Weight of Optimal Market Portfolio.
Optimal market portfolio Ticker WMT T PG JNJ WFC PFE KO ORCL INTC MRK Name Wal-Mart Stores AT&T Proctor and Gamble Johnson and Johnson Wells Fargo Pfizer Coca-Cola Oracle Intel Merk Weights 51.61% -­‐14.59% 112.02% -­‐152.41% 11.76% -­‐103.70% 73.55% 135.19% -­‐88.13% 74.70% 100%
Source:
Author’s own processing

Fortunately, this predicament can be solved by BL model. This method begins with an assumption that investors choose their optimal portfolio, and this portfolio defines the benchmark and with no other information available, this benchmark is the best portfolio (Black; Litterman1992). The model composed by two steps: understanding the market and embedding analyst forecast.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

As BL assume benchmark portfolio is the best, it does not need to input the historical data (i.e. simple return rate) to derive an optimal portfolio, just using the benchmark (in our case, capitalization weight) as the portfolio’s optimal combination. Recall formula (1), what needed to solve here is the expected return rate of each stock. In BL model, one should try to derive what will be the implied return rate if the portfolio is weighted as the benchmark. E(R)=λVP + Rf (5) Derive λ=
! !" !!" !!

(6)

Rf: Risk free rate E(Rm)= Expected market return In (6), E(Rm) is 0.80% and historical average return rate is not considered because it is a noisy estimate, δ2 can be fixed by existing VCV matrix, and then λ can be computed. When λ is confirmed, implied return can be solved using (5). In table 3, it can be seen that portfolio appears to be more sensible by implementing the BL model step one, which is also know as the market capital portfolio. Table 3. Weight Comparison of P1 and Optimal Market Portfolio. Improved optimal market portfolio (P1) Ticker WMT T PG JNJ WFC PFE KO ORCL INTC MRK Name Wal-Mart Stores AT&T Proctor and Gamble Johnson and Johnson Wells Fargo Pfizer Coca-Cola Oracle Intel Merk Weights 12.84% 9.73% 11.40% 13.51% 11.76% 10.55% 8.97% 7.86% 6.28% 7.09% 100%
Source:
Author’s own processing

Optimal market portfolio Weights 51.61% -­‐14.59% 112.02% -­‐152.41% 11.76% -­‐103.70% 73.55% 135.19% -­‐88.13% 74.70% 100%

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

The second step of BL model is to incorporate analyst’s opinion. The logic is to manually increase/decrease the return rate for individual stocks based on analyst opinion. As several expected return (E(RAdj)) of these stocks changes, the whole VCV matrix will change correspondently as all stocks are correlated, and this will initiate changes in implied return , and eventually, the portfolio proportion of the investment.

E(RAdj) = E(R)+θδ (7) θ is a checking factor to show the sensitivity of one stock to other stocks. As δ can be solved by excel Solver function to generate a δ that will be the same as the analyst’s expected return of some specific stock. The E(R) is the implied market return rate computed above, and with θ and δ, the E(RAdj) can be computed. Recall formula (1), the new optimal portfolio proportion (P2) can be computed.Data of comparison of P1 and P2 is illustrated in Table 4 and Table 5: Table 4. Weight Comparison of P2 and P1 with Analyst forecast. Improved optimal BL model portfolio (P2) Ticker WMT T PG JNJ WFC PFE KO ORCL INTC MRK Name Wal-Mart Stores AT&T Proctor and Gamble Johnson and Johnson Wells Fargo Pfizer Coca-Cola Oracle Intel Merk Weights 14.60% 11.06% -8.46% 15.35% 20.23% 11.99% 10.20% -0.65% 7.14% 18.54% 100%
Source:
Author’s own processing

Analyst Forecast 0.00% 0.00% -0.10% 0.00% 0.10% 0.00% 0.00% -0.10% 0.00% 0.10%

market portfolio (P1) Weights 12.84% 9.73% 11.40% 13.51% 11.76% 10.55% 8.97% 7.86% 6.28% 7.09% 100%

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

From Table 4, analyst’s view is incorporated in P2, which leads to significant change of proportion of P1. Since P2 has embedded a forward-­‐looking perception into the portfolio, P2 should be more advanced than P1. Table 5. Expected Return and Standard Deviation Comparison of P1 and P2.
Improved optimal BL model portfolio (2) Expected return Standard deviation Sharpe ratio
Source:
Author’s own processing

market portfolio (P1) 0.80% 4.32% 17.36%

Fluctuation by % 13.19% 13.87% 0.18%

0.91% 4.92% 17.39%

With analyst’s opinion, expected return from P2 is larger than P1, but the standard deviation increased accordingly. Combined these two changes, Sharpe ratio rises slightly.

3.1.2. Shrinkage mixed with Bayesian The concept of Shrinkage method is to re-­‐arrange weight in the VCV matrix to mitigate the unrealistic outcome due to bad estimation, which is called shrinking the sample covariance matrix (Demigual et al., 2009). In practice, this method is empirically proved to be simply and effective. The formula is as below: Shrinkage VCV matrix= λ*Sample VCV+ (1-­‐λ)* Other matrix (8) In (8), other matrix is a diagonal matrix of only variances, and zero elsewhere. The shrinkage estimator is used to regularize ill-­‐posed inference problem. λ should be selected as long as it can make the Global Minimum Variance Portfolio entirely positive and in the example λ is set as 0.3. (Benninga, 2008). After implying the “shrank” VCV, a new implied return generated, meanwhile, capitalization weight remain unchanged; then a new adjusted return and a new weight portfolio is produced. Comparison of the adjusted and implied return is of these three portfolios is in Table 6.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Table 6. Comparison of adjusted/implied return of P3, P2 and P1.
P3 Final Adjusted Ticker WMT T PG JNJ WFC PFE KO ORCL INTC MRK Return 0.46% 0.58% 0.45% 0.54% 2.14% 0.80% 0.53% 0.75% 0.84% 0.84% New Implied Return 0.46% 0.60% 0.55% 0.55% 2.04% 0.80% 0.54% 0.85% 0.86% 0.74% Analyst Forecast 0.00% 0.00% -0.10% 0.00% 0.10% 0.00% 0.00% -0.10% 0.00% 0.10% P2 & P1 Implied Return 0.45% 0.63% 0.61% 0.63% 1.61% 0.86% 0.62% 0.94% 0.98% 0.77%

Source: Author’s own processing

The second step is to implement the Bayesian method, which is to improve our over-­‐dependent issue on analyst forecast. The idea of Bayesian adjustment was first initiated by Theil(1971). Black and Litterman explored the application of the ”mixed estimation” strategy method on 1991 (Black and Litterman, 1991). The concept of this method is to neutralize the market weight and analyst’s opinion weight. Formula is as below: Portfolio proportion: (1-­‐γ)*Market weights + γ*Analyst opinion weights (9) γ is how much it is believed in analyst’s opinion. In our portfolio, γ is set as 60% out of empirical evidence and professional judgment. After combining Shrinkage and Bayesian method on the foundation on the BL portfolio built previously, the portfolio of PF is finalized. The advantage for the PF portfolio is that it improves noisy VCV matrix, which is built on distorted estimation and solve the overconfidence problem on analyst’s opinion. Detail information on comparison of these portfolios provided in Table 7 and Table 8.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Table 7. Weight Comparison of P3, P2 and P1.
Shrinkage & Bayesian (P3) Ticker WMT T PG JNJ WFC PFE KO ORCL INTC MRK Name Wal-Mart Stores AT&T Proctor and Gamble Johnson and Johnson Wells Fargo Pfizer Coca-Cola Oracle Intel Merk Weights 13.20% 10.00% 8.12% 13.88% 12.84% 10.84% 9.22% 6.59% 6.46% 8.85% 100.00%
Source:
Author’s own processing

BL model portfolio (P2) Weights 14.60% 11.06% -8.46% 15.35% 20.23% 11.99% 10.20% -0.65% 7.14% 18.54% 100.00%

Improved optimal market portfolio (P1) Weights 12.84% 9.73% 11.40% 13.51% 11.76% 10.55% 8.97% 7.86% 6.28% 7.09% 100.00%

P3 is the naturalization of P2 and P1, especially for those stocks with analyst’s opinion. Table 8. Expected Return and Standard Deviation Comparison of P3, P2 and P1.
Improved Shrinkage & Bayesian (P3) Expected return Standard deviation Sharpe ratio 0.82% 3.10% 24.78% BL model portfolio (P2) 0.91% 4.92% 17.39% optimal market portfolio (P1) 0.80% 4.32% 17.36% Fluctuation P3 vs P2 -9.74% -37.05% 42.49% Fluctuation P3 vs P1 2.17% -28.33% 42.75%

Source: Author’s own processing

From Table 8, P3 is more superior than P1, because it has bigger return and less risky. Comparison between P3 and P2, is depend on investor’s risk preference, as even P3 is less profitable than P2 by 9.74%, its standard deviation is significantly lower than P2 by 37.05%, with a highest Sharpe ratio of 24.78%.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

An alternative profitable portfolio can be constructed on the foundation of P3, if short sale in risk free asset is permitted. “Investors satisfy their risk preferences by holding different amounts of their risky tangency portfolio, P*, and the risk free asset” (Harris, 2013). Assume the P3 is P*, then the efficient set of is the ray emanating from RFR passing through P*. Figure 1 as below is for illustration. Figure1. Efficient Set with Risk Free Asset

Efficient Set with Risk Free Asset
1.800% 1.600% 1.400% Expected Return 1.200% 1.000% 0.800% 0.600% 0.400% 0.200% RFR 0.000% 0.000% 1.000% 2.000% 3.000% 4.000% 5.000% 6.000% 7.000% 8.000% Standard Deviation
Source:
Author’s own processing

Add RFA No RFA P*

As can be seen from Figure 1 that investing beyond P* can harvest higher return at the expense of bigger risk and “Investing beyond point P* represents borrowing and investing more than 100% of your wealth in risky assets” (Wang, 2013). Assuming that this super-­‐portfolio standard deviation is no larger than that of P2, of 4.92%, and ask excel Solver function to compute the portfolio weight by maximizing the expected return. Result of P3 & Risk Free Asset (RFA) and comparison for P3 and P2 is in Table 9.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Table 9. Result Comparison of P3 & RFA, P3 and P2. Shrinkage & Super-portfolio of RFA & P3 Weight of P3/P2 Weight of RFR Total Weight Expected return Standard deviation Sharpe ratio 4.92% 24.78% 3.10% 24.78% 4.92% 17.39% 58.87% 0.00% 0.00% 42.49% 158.87% -58.87% 100.00% 1.27% Bayesian (P3) 100% 0 100% 0.82% BL model portfolio (2) 100% 0 100% 0.91% 55.26% 40.15% Fluctuation P3 & RFA vs P3 Fluctuation P3 & RFA vs P2

Source: Author’s own processing

If with same risk as P2, P3 & RFA will be more profitable by 40.15%. If P3 & RFA maintains the same Sharpe ratio with P3, its expected return will increase by 55.26% compared to P3, but the risk also increase significantly by 58.87%.

3.2. Portfolio Construction Limitations

There are two significant limitations for market portfolio. One is that the weight will be attached unreasonably to the assets in the VCV matrix due to biased estimation. The other disadvantage is that it may lack of the perception of forward looking. Fortunately, these deficiencies can be improved by BL model. But for BL model, the concern is, too much trust being reposed on analyst’s forecast, which could lead to big loss if the opinion is wrong. And indeed, many literatures discover analyst’s forecasts are either inefficient (Easterwood and Nutt, 1999), optimistically biased, or being part of the game with management (Bartov., et al 2002).So Shrinkage and Bayesian method are introduced to solve these two problems by implementing them in P2, and constructing the portfolio of PF (P3).

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Unfortunately, there is still disadvantage for Shrinkage and Bayesian method as there are two parameters λ and γ, embedded in the modeling construction, which are relied on investor’s judgment. It will inevitably increase risk as long as it is subjective. Hence P3 calls for even more comprehensive considerations.

3.3. Methods on evaluation

Treynor (1965) measure, Sharpe (1966) ratio, M2 (1997) measure, Jensen’s Alpha (1968) measure, Information ratio (2013) as well as Omega ratio (2006) are used to measure the performance of each portfolio or compare it to its benchmark. S&P 500 is downloaded for embedment in the 6 measures ration for benchmark comparison of the market. Monthly returns, and standard deviation for each fund category during the period of October 2010 and September 2013 are calculated. The beta used in the analysis is the adjusted beta which is more applicable in real life, which is known as βAdjusted =0.33+0.67βRaw.

The Treynor ratio (also known as Reward-­‐to-­‐Volatility Ratio, 1965) is the measurement of returns earned in excess of which could have been earned on a riskless investment per unit of market risk. In other words, it is a risk-­‐adjusted measure of return based on systematic risk (beta). The measure is calculated as: ������������������������������������������ ������������������������������������������ =
!! !!! !!

(10)

������! = Average return of the portfolio βp= Adjusted portfolio beta Sharpe (1966) conceived of a composite measure to evaluate the performance of mutual funds. The measure closely followed his earlier work on the Capital Asset Pricing Model (CAPM) dealing specifically with the Capital Market Line (CML). The Sharpe Measure of portfolio performance (designated Sharpe Ratio) is stated as follows: Sharpe Ratio = !"#$%#&% !"#$%&$'( !" !"#$%"&'" !"#!$$ !"#$!% = σp: Standard deviation of portfolio !!"#$!%&! !"#$ !"#$%&$ !! !!! !!

(11)

13

Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Modigliani and Modigliani (M2, 1997) transformed Sharpe performance measure and has the significant advantage of being in units of percent return (as opposed to the Sharpe ratio, dimensionless ratio of limited utility to most investors), which makes it dramatically more intuitive to interpret. M2 measure = (������! − ������! ) !! − (������! − ������! )

!

!

(12)

σM: Standard deviation of benchmark ������! : Average return of benchmark Often in conjunction with the Sharpe ratio and the Treynor ratio, Jensen’s alpha (1968) measures excess returns above (or below) the fund risk-­‐adjusted return as expected in a CAPM world (Elfakhani and Hassan, 2005). A positive/negative alpha implies that the portfolio is outperforming /underperforming its market premium benchmark, while a statistically zero alpha means that the portfolio performance is normal as expected in a CAPM setting. It is estimated using the following regression model: αp = r! -­‐(Rf+βp (r! -­‐Rf))

(13) Keating and Shadwick (2006) integrate the work of Jensen’s alpha. They proposed an Omega ratio which is involves partitioning returns into loss and gain above and below a given threshold; the ratio is then the ratio of the probability of having a gain by the probability of having a loss. It can be used to rank and evaluate portfolios unequivocally. Omega ratio= ! !!! !! !!! |!! !!! ! !"#(! !! )|! !! ! ! ! ! !!!

(14)

rp : Return of portfolio rb : Return of benchmark

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

The Information ratio (IR, also known as Appraisal ratio) is a measure of the risk-­‐adjusted return of a financial security (or asset or portfolio); it is defined as the expected active return divided by tracking error, where active return is the difference between the return of the security and the return of a selected benchmark index, and tracking error is the portfolio’s diversifiable risk. It can help one identify: efficiency (when comparing investments, those with lower information ratios use risk less efficiently) and consistency (the higher the information ratio, the more consistent a manager tends to be) (JP Morgan, 2013). The information ratio is:

IR = !

!,!

!

(15)

������ : Jensen’s Alpha ������!,! : Portfolio’s diversifiable risk

4. Results

4.1. Performance evaluation

The portfolio construction has been improved before analyses taken place, by using Black-­‐Litterman model to P2 and Shrinkage and Bayesian methods to P3. The expected returns and standard deviation of P2 are higher than that of P3, which are 1.32% (P3 1.19%) and 2.93% (P3: 2.70%) respectively. In the risk diversification process, P3 (8.79%) is much prior to P2 (22.47%) for having less proportion unsystematic risk. Through Shrinkage and Bayesian methods, the risk of the fund has been diversified to some extent. Detail is provided in Table 10.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Table 10. Comparison of in Performance and Risk of In-sample and Out-of-Sample.
Shrinkage & Bayesian (P3) Expected return Standard deviation Sharpe ratio Expected return Standard deviation Sharpe ratio Total risk Non-diversifiable risk
Diversifiable risk (DR) Proportion of DR Source: Author’s own processing

BL model portfolio (2) 0.91% 4.92% 17.39% 1.32% 2.93% 43.44% 0.00086 0.00067 0.00019 22.47%

Improved optimal market portfolio (P1) 0.80% 4.32% 17.36% 1.16% 2.68% 41.54% 0.00072 0.00066 0.00005 7.29%

For Estimation (In-Sample) 0.82% 3.10% 24.78% 1.19% 2.70% 42.13% 0.00073 0.00066 0.00006 8.79%

For Evaluation (Out-of-Sample)

Risk Diversification

Figure 2 displays the fund categories’ Treynor ratios. The ratio shows the portfolio excess return per unit of systematic risk (beta), and the higher it is the better is the performance. In the calculation, Treynor ratio of P2 is the highest (1.76%), follows by P3 (1.57%) and P1 (1.54%). Noticeably, Treynor ratio does not quantify the value added, and is only useful if the portfolios are fully diversified. Therefore, it may include a higher unsystematic risk, which is not priced in the market. Figure 2. Results of the Treynor Ratios and Measure Comparison.

Treynor ratio
1.80% 1.75% 1.70% 1.65% 1.60% 1.55% 1.50% 1.45% 1.40% P1
Source:
Author’s own processing

1.76%

1.54%

1.57%

P2

P3

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Figure 3 displays the results of the Sharpe ratios per fund category. The ratio shows the portfolio excess return per unit of total risk; the higher the ratio, the better is the fund’s performance. The Sharpe ratio of the P2 outperformed P3 (0.421) and P1 (0.415) with a Sharpe ratio of 0.434. Similarly, the M2 and Omega ratios of P2 both achieved the highest performance by 0.535% and 2.97 whereas P3 is 0.489% and 2.96 the P1 is 0.467% and 2.95. Clearly, these performance measures draw a similar estimation as the Treynor measure; however, the Sharpe ratio seeks to measure the total risk of the portfolio by using the standard deviation of returns rather than solely considering the systematic risk summarized by beta. As the numerator is the portfolio’s risk premium, this measure indicates the risk premium return earned per unit of total risk (Reilly and Brown, 2012). Likewise, this portfolio performance measure uses the CML to compare portfolios whereas the Treynor measure examines portfolio performance relating to the Security Market Line (SML). Figure 3. Results of the Sharpe Ratios and Measure Comparison.

Sharpe ratio
0.440 0.435 0.430 0.425 0.420 0.415 0.410 0.405 P1
Source:
Author’s own processing

0.434

0.421 0.415

P2

P3

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Jensen’s alpha is calculated and their relative significances are checked. Figure 4 displays the results of Jensen measures amongst the three fund categories. As can be seen, every one of the portfolios (P1, P2 and P3) has a positive alpha, which implies that all the portfolios outperform its market premium benchmark. The optimal active portfolio again, unsurprisingly, is the most outstanding one with an alpha of 0.539% while P3 is 0.404% and the market portfolio is 0.379%. Figure 4. Result of the Jensen’s alpha Measures.

Jensen’s Alpha
0.600% 0.500% 0.400% 0.300% 0.200% 0.100% 0.000% P1
Source:
Author’s own processing

0.539% 0.379% 0.404%

P2

P3

Interestingly, the Information ratio of P1, by the first time, beats against the P2 but the ranking of P3 stills remain unchanged. The result mostly due to the diversified risk for P2 is the largest and for P1 is the smallest. Information ratio confirms what conclusion can be drawn in Table 9 about the diversified risk. Details can be referred to Figure 5. As P2’s diversified risk is the highest, means it makes the investors burdened with numerous unsystematic risk.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Figure 5. Result of Information ratio Measures

Information ratio
0.600 0.500 0.400 0.300 0.200 0.100 0.000 P1
Source:
Author’s own processing

0.524 0.388

0.505

P2

P3

4.2. Sensitivity Test Figure 6 and 7 presents P2 and P3 versus P1while RFR changes (from 0.01% to 0.1%) considering its returns and standard deviation. It can be seen that, the main outliers stand out is P2 while P3 moves closely to P1.The tiny fluctuation indicated that from the above-­‐mentioned range, RFR make little impact on return and risk. Figure 6. The Impact of Change of Risk Free Rate on Returns.
1.350% 1.300% 1.250%

Return

1.200% 1.150% 1.100% 1.050%

P1 P2 P3

Risk Free Rate

Source: Author’s own processing

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Figure 7. The Impact of Change of Risk Free Rate on Standard deviation. 3.000% 2.950% 2.900% 2.850% 2.800% 2.750% 2.700% 2.650% 2.600% 2.550% 2.500%

Standard Deviation

P1 P2 P3

Risk Free Rate

Source: Author’s own processing

Figure 8 and 9, by contrast, represents the P2 and P3 versus P1 while market return changes (from 0.1% to 1.6%) considering its returns and standard deviation. P2’s standard deviation sees a sharp decline for the market return in a range of 0.1% to 0.5% and its return also changes significantly by increasing 3.32% (from -­‐1.5% to 1.83%). It suggests that, the return of P2 has an exponential growth as the market return climbing within the range of 0.1% to 0.3% roughly. This may be explained by the power of macro environment, in a bear market for instance, the portfolio may end up with a disappointing result no matter how the portfolio is constructed; and when it turns out to be a bull market, the expected return of the portfolio will raise correspondingly, In this case particular, it inevitably involved a recession factor when calculating the year 2008 and 2009. As a result, since P2 embedded 100% of the analyst opinion, its return and standard deviation are more vulnerable compared to P3 and P1. The data of both returns and standard deviation become stable after 0.3%.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

Figure 8. Impact of Change of Market Return Rate on Returns
3.500% 3.000% 2.500% 2.000% 1.500% Return 1.000% 0.500% 0.000% -0.500% -1.000% -1.500% -2.000% P1 P2 P3

Market Return Rate
Source:
Author’s own processing

Figure 9. Impact of Change of Market Return Rate on Standard Deviation
18.000% 15.000% Standard Deviation 12.000% 9.000% 6.000% 3.000% 0.000% P1 P2 P3

0.10% 0.20% 0.30% 0.40% 0.50% 0.60% 0.70% 0.80% 0.90% 1.00% 1.10% 1.20% 1.30% 1.40% 1.50% 1.60% Market Return Rate

Source: Author’s own processing

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

The improved estimate is made closer to the value supplied by other information rather than the raw estimate using Shrinkage and Bayesian method. Table 10 lists varies of returns and standard deviation referring to the λ (Shrinkage) and γ (Bayesian) of P3. As mentioned in the early section, the λ of 0.3 and γ of 0.6 are chosen in our method. The results indicate that, when using different λ and γ, the outcomes of returns and standard deviation can change, and the sensitivity result for λ is gentler than Υ.

Table 10. Sensitivity test on λ and γ λ Expected return 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 2.362% 2.354% 2.361% 2.369% 2.377% 2.385% 2.392% 2.400% 2.409% 2.417% 2.425% Standard deviation 5.384% 5.377% 5.384% 5.391% 5.399% 5.407% 5.416% 5.426% 5.436% 5.446% 5.457% Expected return 1.162% 1.166% 1.170% 1.174% 1.178% 1.182% 1.186% 1.191% 1.195% 1.199% 1.203% γ Standard deviation 2.677% 2.679% 2.683% 2.686% 2.690% 2.693% 2.697% 2.702% 2.706% 2.711% 2.716%

Source: Author’s own processing

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

5. Conclusion

This report examines the performance of Platinum Fund with regard to P2 and P3 verses P1. Overwhelming results (5 ratios out of 6) indicate that P2 tends to be the best performer given its highest return on the overall performance, followed by the P3. On one hand, the best out-­‐coming of P2 heavily relies on the expertise of analyst and the limitations such as macro environment and the assumptions are being ignored which actually “helped” it outperformed the market. On the other hand, Shrinkage method may accidentally downsize some valuable stocks in out-­‐samples as they are closely correlated. In contrast, P3 can potentially perform well by somehow diversifies risks due to its ability to diversify unsystematic risks. Furthermore, if RFA can be involved; it can maximize the return by building a super-­‐portfolio with P3, as long as short position is allowed. To sum up, P3 will bring a promising return higher than the market portfolio, and a more stable income than totally relying on analysts’ forecast.

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Portfolio Modeling and Evaluation: Beating the Market Platinum Fund

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