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Last lecture, we discussed the formation of risky portfolios based on the expected return and risk of the constituent assets. In doing this, we noted that:

Portfolio risk is a function of: The weight in which each asset is held; and,

The covariance of the returns on pairs of assets forming part of the portfolio.

The method we use to identify the efficient set of portfolios is known as the Markowitz Portfolio Selection Model. 3

However, Markowitz’s model suffers from several problems. Firstly, applying it is computationally intensive, even if:

Only a relatively small numbers of assets are involved and / or;

The process is automated. This is because implementation of the model involves estimating:

Expected returns; variances; and, (n2-n)/2 covariances. 4
To better understand the enormity of this task,
consider how many estimates are involved when
analyzing:
1. 10 stocks?
2. 50 stocks?
3. 100 stocks?
4. 1000 stocks?

## Estimating Expected Returns using Index Models

In order to estimate “m” which is the unexpected macroeconomic factor, the rate of return upon the wider index of securities are been utilized as a proxy of the factor. Based on the ASX values and the beta computed from the regression performed upon the historical data of 60 months of the 5 companies, the sensitivity of the assets with respect to the systematic risk present in the market of securities are obtained (Ahn et al.,  2018).

Based on the alpha values, beta values and the index risk premium, the values of the expected returns for the five companies can be estimated as follows:

 Index WES ASX AMP ANZ DMP 0.04624 0.090085 0.147962 0.084827 0.182444 0.046235 0.076346 0.10183 0.073792 0.151538 0.046234 0.071997 0.087058 0.070308 0.141671 0.046234 0.073266 0.091401 0.071323 0.144566 0.046234 0.065816 0.065857 0.065371 0.127548 0.046235 0.070595 0.082099 0.069198 0.138394 0.046235 0.071549 0.085305 0.069963 0.140541 0.046235 0.071108 0.083742 0.069614 0.139509 0.046236 0.069056 0.076556 0.067984 0.134748 0.046236 0.064644 0.061351 0.064463 0.124631 0.046237 0.068755 0.075365 0.067753 0.133983 0.046237 0.069167 0.07672 0.068086 0.134896 0.046237 0.083865 0.127282 0.079818 0.168553 0.046235 0.072285 0.087772 0.070556 0.142196 0.046235 0.063131 0.056284 0.063247 0.121234 0.046237 0.066232 0.066568 0.065747 0.128148 0.046238 0.061098 0.048835 0.061652 0.116356 0.046238 0.066135 0.066071 0.065679 0.127845 0.046237 0.069114 0.076537 0.068043 0.134774 0.046228 0.075782 0.101213 0.073261 0.150895 0.046226 0.068114 0.075305 0.067112 0.133566 0.046224 0.071905 0.088549 0.070126 0.142346 0.046222 0.073243 0.093677 0.071163 0.145668 0.046221 0.060151 0.04888 0.060697 0.115806 0.046221 0.073955 0.096168 0.071729 0.14732 0.046221 0.0633 0.059712 0.063212 0.123017 0.046222 0.070985 0.08584 0.069364 0.140463 0.046222 0.07907 0.113614 0.07582 0.158958 0.046223 0.066188 0.069152 0.065546 0.129388 0.046227 0.068721 0.077163 0.06761 0.134843 0.046225 0.063743 0.060349 0.063618 0.123596 0.046225 0.071554 0.087184 0.069855 0.141465 0.046226 0.075026 0.098899 0.07264 0.149304 0.046225 0.077786 0.108618 0.07483 0.155734 0.046218 0.072635 0.092363 0.070631 0.144657 0.046217 0.06674 0.072302 0.065912 0.131265 0.046216 0.081408 0.122817 0.077617 0.164881 0.04621 0.066418 0.072458 0.065578 0.131147 0.04621 0.073096 0.095472 0.070907 0.14646 0.04621 0.066601 0.073139 0.065721 0.131592 0.04621 0.064221 0.064943 0.063822 0.126137 0.046211 0.068603 0.079825 0.067331 0.136077 0.046211 0.071791 0.090734 0.06988 0.143349 0.046211 0.068386 0.079083 0.067158 0.135583 0.046211 0.068532 0.079529 0.067278 0.13589 0.04621 0.062211 0.05801 0.062219 0.121526 0.046209 0.062371 0.058777 0.062334 0.121998 0.046209 0.064848 0.067305 0.06431 0.127673 0.046209 0.07132 0.089559 0.069477 0.142489 0.046209 0.071338 0.089623 0.069492 0.142532 0.046209 0.070969 0.088248 0.069203 0.141634 0.046209 0.076001 0.105577 0.073219 0.153167 0.04621 0.076271 0.106391 0.073441 0.153728 0.046211 0.072561 0.093423 0.070492 0.145132 0.046211 0.070578 0.086528 0.068914 0.140556 0.04621 0.064377 0.06542 0.06395 0.126465 0.046215 0.065577 0.06865 0.064962 0.128773 0.046219 0.067169 0.073387 0.066278 0.132056 0.046216 0.067185 0.073989 0.066257 0.132361 0.046219 0.06395 0.062316 0.063707 0.124686 Summation of Expected Return 1.470246 0.408392 1.121318 0.662446 1.849002

The index model that is being utilized to measure the expected return is reformulated with the following formula that can be represented as:

Expected Return =  + { +  ( )} +

According to the analysis it is found that based on the index models all the five companies are performing close to each other and their securities are financially viable (Arora and Gakhar, 2017). Moreover, it is seen that within the index model the highest level of expected return is expected over the 60 months period from DMP Company while WES has second highest level of expected return followed by AMP (1.121318), ANZ (0.662446) and ASX (0.408392) respectively.

The respective alpha and beta values of the five companies can be interpreted from the table where the values of them for all the five companies are represented as follows:

 Parametric Factors WES ASX AMP ANZ DMP Beta 0.002876 0.348312 0.644287 0.311414 0.526035 Alpha 0.015415 0.017943 0.00872 0.008056 0.035576

The beta coefficient is the measurement of volatility where it gives a clear idea regarding the movement of a stock or any other assets or portfolios with respect to an index which is considered as a benchmark (Ahn et al., 2018). The alpha value on the other hand is the asset’s return on the respective investment made upon it in comparison to the expected return adjusted to its corresponding risk associated with it. Hence, alpha is termed as the active return on investment. Thus alpha and beta together helps effectively to interpret, illustrate, compare as well as analyze the performance of any company. Based on alpha value of the five companies it can be seen that DMP has the value of alpha as 0.035576, which represents that the securities of the company has outperformed its index benchmark by 0.036 % while in case of ASX the alpha value is 0.017943 which incorporates the fact that the stocks of this company has outperformed its market index by 0.018 %. In the same way, for WES, AMP, and ANZ the outperformance percentages are 0.015 %, 0.0087 %, and 0.008 % respectively. Clearly among the five companies, all of them have alpha value close to 0 which reflects the fact that the funds have performed exactly up to the market expectations and there is addition of value. Notably, DMP have the highest alpha value which reveals the fact that for this company the return of its securities are the highest that didn’t took place due to mere movement within the greater market rather it had outperformed its benchmark index.

## Beta and Alpha Values

The future results of the performance of the securities are not guaranteed to be perfectly tracked based on the past performance as revealed by the backward estimated risk ratios like alpha and beta. The beta value of WES, ASX, AMP, ANZ or DMP are 0.002876, 0.348312, 0.644287, 0.311414 and 0.526035 respectively. This reflects the fact that the WES is 99.714 % more volatile than the market, while ASX 65.16 % less volatile than the market. In the same way, AMP is 35.57 % less volatile and ANZ is 68.86 % less volatile than the market and DMP is 47.39 % less volatile than the overall market. Thus beta offers a risk reward measurement where the extent of risk associated with an expected level of return is being found. The beta value differs within the 5 companies where as a proxy for risk revealing the possibility of the stocks of the five companies to lose their value. All are securities are found to be less volatile in the market since all of their values are less than 1. However, among all the companies WES is the least volatile one and DMP is the most volatile one and is closer to the overall market line which reflects the fact that the price of the security moves 47.39 % less exactly in the direction of the market movement (Arora and Gakhar, 2017).

Based on the risk free rate, market risk premium and respective factor coefficients for sensitivity which is beta values of small minus big (SML) and high minus low (HML), the results of final computation of the expected returns can be represented as follows:

 Fama-French Three Factor Model WES ASX AMP ANZ DMP 0.032228 0.169549 0.216096 0.139482 0.178865 0.026765 0.055732 0.071204 0.058174 0.073258 0.028991 0.018343 0.035155 0.024297 0.050377 0.031197 0.023468 0.046861 0.019004 0.053832 0.03192 -0.06417 -0.05819 -0.07083 -0.04747 0.027752 0.001799 0.010273 0.011098 0.026805 0.029926 -0.01617 -0.00957 -0.02988 -0.01694 0.028948 0.020367 0.038151 0.036907 0.065813 0.031213 -0.02239 -0.01057 -0.02153 0.00399 0.027461 -0.05631 -0.06157 -0.03635 -0.03024 0.03115 -0.01375 0.000905 -0.00328 0.026715 0.029541 -0.01945 -0.01301 -0.01261 0.003843 0.027069 0.07999 0.090677 0.042273 0.037769 0.028534 0.009796 0.020177 0.010868 0.026915 0.028879 -0.08225 -0.08955 -0.07154 -0.06469 0.031665 -0.04769 -0.04035 -0.04012 -0.01586 0.030717 -0.08712 -0.08904 -0.06324 -0.04405 0.029803 -0.02527 -0.01596 0.00291 0.028779 0.030287 -0.00566 0.008415 0.00952 0.037444 0.026447 0.07599 0.105135 0.086144 0.121704 0.029643 -0.01628 0.003658 -0.01746 0.020532 0.02564 0.015292 0.028888 0.011106 0.03075 0.024732 0.041696 0.062447 0.042052 0.069045 0.024309 -0.10038 -0.11063 -0.09352 -0.08844 0.02781 0.044548 0.075028 0.035216 0.077029 0.022667 -0.06173 -0.06808 -0.05068 -0.04663 0.02729 0.019773 0.043929 0.019071 0.056591 0.023709 0.087717 0.11199 0.075821 0.097076 0.028711 -0.01819 0.003604 -0.00765 0.035713 0.026605 0.012953 0.032197 0.03414 0.067888 0.026949 -0.07414 -0.0765 -0.07548 -0.06363 0.022086 0.007425 0.006751 0.008536 0.005915 0.021959 0.060622 0.072485 0.072253 0.081381 0.027301 0.060642 0.085615 0.03886 0.06575 0.024183 0.047348 0.073117 0.050878 0.085826 0.022125 -0.02501 -0.02173 -0.02169 -0.01005 0.020839 0.12602 0.156951 0.117105 0.142022 0.016031 -0.00644 -0.00901 0.012635 0.012433 0.021088 0.049343 0.072338 0.043644 0.071962 0.018252 -0.02443 -0.02683 -0.02341 -0.0221 0.017911 -0.04668 -0.05446 -0.04112 -0.04363 0.017381 0.013105 0.017678 0.026857 0.033995 0.02057 0.026132 0.039878 0.018127 0.035102 0.020719 -0.00843 -0.00103 -0.01361 0.000286 0.019532 0.017685 0.030628 0.031573 0.052262 0.017981 -0.05445 -0.06154 -0.03824 -0.03627 0.018562 -0.03745 -0.03561 -0.01392 0.00094 0.017903 -0.00192 0.006961 0.027116 0.047278 0.017303 0.026409 0.032838 0.024605 0.029743 0.020002 0.03196 0.048938 0.028798 0.050838 0.02273 0.035429 0.062468 0.033368 0.07228 0.023209 0.094289 0.136344 0.091102 0.143093 0.023009 0.080252 0.115416 0.066804 0.108356 0.020468 0.055005 0.078129 0.060205 0.089142 0.020759 0.012549 0.02373 0.004898 0.020432 0.020161 -0.02778 -0.02191 -0.01438 0.003863 0.019785 -0.03604 -0.04066 -0.02956 -0.02876 0.019881 -0.00964 -0.0102 0.007994 0.011755 0.019489 0.007681 0.014881 0.033042 0.047656 0.020466 -0.03131 -0.0326 -0.00396 0.004716

According to the analysis it is found that based on the Fama French Three Factor Model, in the long run the small companies are able to outperform the large firms as the value companies are able to beat the growth firms. Thee formula that is being incorporated to compute the expected return can be represented as follows:

Expected Return = Risk – free rate + market Risk Premium + SMB + HML

Reformulating it in the form of an equation –

Where,

R = Expected return rate

= The risk –free rate

= The sensitivity factor co-efficient

= The premium for market risk

SMB = the excess returns of small cap companies over that of the large firms termed as small minus the big.

HML = the excess returns of the value stocks having high book to price ratio over that of the growth stocks which possess lower book to price ratio and hence termed as high minus low.

ASX, AMP and ANX companies are found to perform moderately as it is being found from the index model analysis of expected return.  However, the monthly expected returns of WES remained close to 0.04  0.05 as per the index model analysis and same for DMP, while the Fama French Three Factor Model reveals that both of the companies remained adhered with positive returns throughout the long run period of about 60 months. But considering the risk return factors of alpha and beta as well as the expected return valuation it is been seen that DMP has performed better than WES over the period of 60 months. In fact, DMP is the better value firm among all the other companies. Since it is a value company compared to other growth companies hence it is able to perform better in the long run as it is found after analysis of the time-series data of over 60 months tenure.

Conclusion

Most of the diversified portfolio returns are being explained by the results obtained from the model where the simple index models are not able to compute them with accuracy. It can be concluded that it’s true that the risk of a security and its expected returns are related where the return of a security is equivalent to summation of the risk free return and the risk premium associated with it inclusive of the beta of the securities. Moreover, higher returns are associated with riskier investments and hence it is important to adjust the outperformance tendency as it is found that compared to other firms DMP is able to over perform than the growth firms since it is a value firm.

References

Ahn, H., Ichimura, H., Powell, J.L. and Ruud, P.A., 2018. Simple estimators for invertible index models. Journal of Business & Economic Statistics, 36(1), pp.1-10.

Arora, D. and Gakhar, D.V., 2017. Fama French Three Factor Model: A Study of Nifty Fifty Companies.

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