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Spction A

1 Download 10-years of monthly stock price history for two large market capitalisation stocks from Yahoo Finance (click the Investing tab followed by the Historical Prices tab). Create a time series of monthly returns from this time series of monthly stock prices. Calculate the annualised mean return, standard deviation and correlation of the stocks. Calculate the weights on the minimum variance portfolio consisting of the two stocks, which we denote by stock X and stock Y, using the following formulae
2. Calculate the expected return and standard deviation of this minimum variance portfolio. Plot the two assets and the minimum variance portfolio on a diagram and plot the efficient frontier consisting of portfolios made up of these two assets. (You only have three points on this curve, but you can draw a smooth curve that joins these three points).

3. Report on section A of the assignment. Include in your report an introduction, a method section.
Section B
1. Download 10-years of monthly price history for a US mutual fund from Yahoo Finance (for example any Vanguard or Fidelity mutual fund would be fine but you can choose another US mutual fund if you would prefer). From Kenneth French's data library (httplimbaluck.dartmouth.edu/pages/faculty/ken.french/data library.html) download the corresponding ten years of data on the Fama-French three factors plus the risk-free rate.
2. Using LINEST in Excel run a CAPM regression to determine the alpha and the appraisal ratio of the mutual fund.
3. Using LINEST in Excel run a Fama-Fama type regression to detemiine the alpha and the appraisal ratio of the mutual fund when the additional two factors of Small Minus Big and High Minus Low are taken into account.
4. Calculate other performance measures for the mutual fund such as the Sharpe ratio.
5.  Include in your report an introduction, a method section a results section and conclusions.

## Application of Methods

Return on any investment can be referred as the reward, provided to the investors, for bearing the risk of the investment. Amongst all the investment alternatives, stock prices are the most volatile in nature and therefore, the risk on investment in stocks is higher in comparison to other investment alternatives. On the other hand, due to the higher risk, the investors use to get higher returns on the stock investments. Hence, it is most of the investors preferred to invest in the stocks irrespective of the higher risk level, incorporated with such investments. However, in the last few decades, the stock markets all around the world have experienced many drastic falls, caused by economic depression and other related factors. Considering those incidents, the investors have become very cautious about stock investment (Bodie et al. 2014).

To help the investors, many techniques and methods have been developed within the last few decades. With the help of such methods, investors can reduce the risk level of their investments and lower down the losses, due to any unexpected fall in stock prices. Amongst all the methods, portfolio distribution is considered as the most effective and appreciated method of optimizing the investment returns. Under this method, the total investment amount is distributed amongst two or more investment alternatives. The alternatives may be selected from different classes of investment or it may be from same group. However, most of the portfolios consist with a set of investments with higher return and risk level and some investment options with lower risk and return level.

In this report, the stocks of Apple Inc. and Microsoft, which have the largest market caps in the US stock market, are selected for portfolio construction. The two stocks are having different risk and return levels. The report demonstrates the methods of portfolio construction by using the historical data of these two stocks. It also compares the individual risk and returns of the two stocks with the risk and return of the investment portfolio, made with these two stocks. Thus, it helps to explain the effectiveness of the portfolio for optimizing the returns.

Different methods can be applied for constructing the portfolio and comparing the outcomes of the portfolio with the individual purpose. The methods, applied in this report for the said purpose, are discussed below:

Return of any stock or stock portfolio can be calculated under two aspects:

• Monthly Return:(Stock Price of 2nd Month – Stock Price of 1st Month)/Stock Price of 1st Month
• Annualized Return: Annualized return is the annualized form of geometric mean of the monthly returns of any stock for a series of months. The formula of annualized return is given below:

Annualized Return: (r1 x r2 x…..rn)1/n – 1

Where, r = Monthly return for the month 1,2,….n

n= Numbers of months

Risk Level of Stocks:

The risk level or the volatility of the stock prices can be measured by the following two statistical metrics:

• Standard Deviation: Standard deviation is used to measure the historical volatility of any stock. The formula of standard deviation is as follows:

Standard Deviation: [ Σ(ri - μ)2/ n]1/2

Where, ri = Monthly returns on for the month 1,2,…n

μ = Average of the monthly returns for the period of n months

n = Number of periods

• Variance:Variance is another statistical metrics, which use to measure the variability from the average risk level of any stock. The variance can be calculated as per the following formula:

Variance: Square Root of Standard Deviation or, [ Σ(ri - μ)2/ n]

According to the modern portfolio theory, the assets classes should be distributed within a portfolio on the basis of the different risk levels of the asset classes. Therefore, the weightage computation of the individual stocks includes only the risk measurement metrics, which are variance and standard deviation. Apart from these metrics, the process also requires the correlation between the two stocks, for which, the covariance is needed to be computed also. The formulae of these metrics are given below:

• Covariance: [Σ(rx – μx) x (ry – μy)] / (n – 1)

## Computation of Portfolio Weightage for each Stock

Where, rx = Monthly Returns of 1st Stock for the month of 1,2,….n

ry= Monthly Returns of 2nd Stock for the month of 1,2,….n

μx =  Average monthly returns of 1st Stock for the period of n months

μy =  Average monthly returns of 2nd  Stock for the period of n months

n = Number of Periods

• Correlation: COV (x,y) / sx sy

Where, COV (x,y) = Covariance of the two stocks

sx = Standard Deviation of the 1st Stock

sy = Standard Deviation of the 2nd  Stock

• Weights of each Stock:

Weight of 1st Stock (Wx) = (σ2y – ρxy σx σy) / (σ2x+ σ2y – 2 ρxy σx σy)

Weight of 2nd Stock = 1 - Wx

Where, σ2x = Variance of 1st Stock

σ2y = Variance of 2nd Stock

σx = Standard Deviation of 1st Stock

σy = Standard Deviation of 2nd Stock

ρxy = Correlation of both stocks

For the purpose of this report, it is necessary to compute the risk and returns of the portfolio along with the individual returns and risks. It would help to compare the performances of the portfolio and the individual stocks. The portfolio performance can be measured through the following statistical metrics:

• Expected Portfolio Returns: (w1x r1) + (w2 x r2)

Where, w1 = Weightage of 1st stock

w2 = Weightage of 2nd stock

r 1= Average Annualized Return of the 1st Stock

r2 = Average Annualized Return of the 2nd Stock

• Portfolio Variance:  (w12x σ2x) + (w22x σ2y) + [2 x w1 x w2 x COV (x,y)]

Where, w1 = Weightage of 1st stock

w2 = Weightage of 2nd stock

σ2x = Variance of 1st Stock

σ2y = Variance of 2nd Stock

COV (x,y) = Covariance of the two stocks

• Standard Deviation of Portfolio: √σ2xy

Where, σ2xy = Portfolio Variance

The formulae, stated above, is applied on the 10 years’ monthly stock prices of Apple Inc. and Microsoft to derive the individual return and risk levels of the stocks and construction of portfolio in the following table:

 Apple Inc. Microsoft Date Stock Price Monthly Returns Monthly Return + 1 Stock Price Monthly Returns Monthly Return + 1 1/3/2006 9.783042 21.68675 2/1/2006 8.873533 -9.30% 90.70% 20.77078 -4.22% 95.78% 3/1/2006 8.125975 -8.42% 91.58% 21.03361 1.27% 101.27% 4/3/2006 9.119697 12.23% 112.23% 18.6682 -11.25% 88.75% 5/1/2006 7.743774 -15.09% 84.91% 17.57696 -5.85% 94.15% 6/1/2006 7.419876 -4.18% 95.82% 18.08137 2.87% 102.87% 7/3/2006 8.804867 18.67% 118.67% 18.67115 3.26% 103.26% 8/1/2006 8.790615 -0.16% 99.84% 20.01728 7.21% 107.21% 9/1/2006 9.973495 13.46% 113.46% 21.30243 6.42% 106.42% 10/2/2006 10.504688 5.33% 105.33% 22.36171 4.97% 104.97% 11/1/2006 11.875428 13.05% 113.05% 22.94617 2.61% 102.61% 12/1/2006 10.991832 -7.44% 92.56% 23.33694 1.70% 101.70% 1/3/2007 11.107141 1.05% 101.05% 24.11848 3.35% 103.35% 2/1/2007 10.962033 -1.31% 98.69% 22.09246 -8.40% 91.60% 3/1/2007 12.037377 9.81% 109.81% 21.85719 -1.06% 98.94% 4/2/2007 12.930043 7.42% 107.42% 23.4806 7.43% 107.43% 5/1/2007 15.701322 21.43% 121.43% 24.14675 2.84% 102.84% 6/1/2007 15.811448 0.70% 100.70% 23.18686 -3.98% 96.02% 7/2/2007 17.070766 7.96% 107.96% 22.8092 -1.63% 98.37% 8/1/2007 17.941408 5.10% 105.10% 22.68387 -0.55% 99.45% 9/4/2007 19.883505 10.82% 110.82% 23.26024 2.54% 102.54% 10/1/2007 24.609837 23.77% 123.77% 29.06346 24.95% 124.95% 11/1/2007 23.608341 -4.07% 95.93% 26.6167 -8.42% 91.58% 12/3/2007 25.663155 8.70% 108.70% 28.20103 5.95% 105.95% 1/2/2008 17.53718 -31.66% 68.34% 25.82454 -8.43% 91.57% 2/1/2008 16.197536 -7.64% 92.36% 21.63058 -16.24% 83.76% 3/3/2008 18.591795 14.78% 114.78% 22.56896 4.34% 104.34% 4/1/2008 22.536884 21.22% 121.22% 22.6803 0.49% 100.49% 5/1/2008 24.454365 8.51% 108.51% 22.60416 -0.34% 99.66% 6/2/2008 21.693451 -11.29% 88.71% 21.95764 -2.86% 97.14% 7/1/2008 20.593491 -5.07% 94.93% 20.52892 -6.51% 93.49% 8/1/2008 21.96423 6.66% 106.66% 21.86892 6.53% 106.53% 9/2/2008 14.725739 -32.96% 67.04% 21.38811 -2.20% 97.80% 10/1/2008 13.939312 -5.34% 94.66% 17.89421 -16.34% 83.66% 11/3/2008 12.006284 -13.87% 86.13% 16.31312 -8.84% 91.16% 12/1/2008 11.057907 -7.90% 92.10% 15.68383 -3.86% 96.14% 1/2/2009 11.677202 5.60% 105.60% 13.79596 -12.04% 87.96% 2/2/2009 11.570964 -0.91% 99.09% 13.11886 -4.91% 95.09% 3/2/2009 13.6193 17.70% 117.70% 14.92219 13.75% 113.75% 4/1/2009 16.302477 19.70% 119.70% 16.45747 10.29% 110.29% 5/1/2009 17.595482 7.93% 107.93% 17.07699 3.76% 103.76% 6/1/2009 18.453167 4.87% 104.87% 19.43131 13.79% 113.79% 7/1/2009 21.168736 14.72% 114.72% 19.22694 -1.05% 98.95% 8/3/2009 21.793213 2.95% 102.95% 20.26399 5.39% 105.39% 9/1/2009 24.013863 10.19% 110.19% 21.1436 4.34% 104.34% 10/1/2009 24.421976 1.70% 101.70% 22.79596 7.81% 107.81% 11/2/2009 25.900249 6.05% 106.05% 24.2839 6.53% 106.53% 12/1/2009 27.302084 5.41% 105.41% 25.1674 3.64% 103.64% 1/4/2010 24.883207 -8.86% 91.14% 23.26829 -7.55% 92.45% 2/1/2010 26.510475 6.54% 106.54% 23.78359 2.21% 102.21% 3/1/2010 30.446495 14.85% 114.85% 24.29792 2.16% 102.16% 4/1/2010 33.826706 11.10% 111.10% 25.33487 4.27% 104.27% 5/3/2010 33.281258 -1.61% 98.39% 21.49931 -15.14% 84.86% 6/1/2010 32.588116 -2.08% 97.92% 19.17439 -10.81% 89.19% 7/1/2010 33.329193 2.27% 102.27% 21.50765 12.17% 112.17% 8/2/2010 31.495928 -5.50% 94.50% 19.66204 -8.58% 91.42% 9/1/2010 36.76252 16.72% 116.72% 20.51654 4.35% 104.35% 10/1/2010 38.994835 6.07% 106.07% 22.34284 8.90% 108.90% 11/1/2010 40.312454 3.38% 103.38% 21.29164 -4.70% 95.30% 12/1/2010 41.79073 3.67% 103.67% 23.52532 10.49% 110.49% 1/3/2011 43.962147 5.20% 105.20% 23.3736 -0.64% 99.36% 2/1/2011 45.76173 4.09% 104.09% 22.53669 -3.58% 96.42% 3/1/2011 45.152802 -1.33% 98.67% 21.52771 -4.48% 95.52% 4/1/2011 45.362682 0.46% 100.46% 21.97709 2.09% 102.09% 5/2/2011 45.064697 -0.66% 99.34% 21.34451 -2.88% 97.12% 6/1/2011 43.489254 -3.50% 96.50% 22.18942 3.96% 103.96% 7/1/2011 50.590412 16.33% 116.33% 23.38423 5.38% 105.38% 8/1/2011 49.858402 -1.45% 98.55% 22.84476 -2.31% 97.69% 9/1/2011 49.403648 -0.91% 99.09% 21.37617 -6.43% 93.57% 10/3/2011 52.443115 6.15% 106.15% 22.87053 6.99% 106.99% 11/1/2011 49.517658 -5.58% 94.42% 22.13419 -3.22% 96.78% 12/1/2011 52.471619 5.97% 105.97% 22.463 1.49% 101.49% 1/3/2012 59.141342 12.71% 112.71% 25.5521 13.75% 113.75% 2/1/2012 70.27829 18.83% 118.83% 27.6452 8.19% 108.19% 3/1/2012 77.677429 10.53% 110.53% 28.09811 1.64% 101.64% 4/2/2012 75.660187 -2.60% 97.40% 27.88907 -0.74% 99.26% 5/1/2012 74.850441 -1.07% 98.93% 25.591 -8.24% 91.76% 6/1/2012 75.662781 1.09% 101.09% 26.81839 4.80% 104.80% 7/2/2012 79.129791 4.58% 104.58% 25.83648 -3.66% 96.34% 8/1/2012 86.558243 9.39% 109.39% 27.19903 5.27% 105.27% 9/4/2012 86.800262 0.28% 100.28% 26.26356 -3.44% 96.56% 10/1/2012 77.460548 -10.76% 89.24% 25.1869 -4.10% 95.90% 11/1/2012 76.502007 -1.24% 98.76% 23.68552 -5.96% 94.04% 12/3/2012 69.55999 -9.07% 90.93% 23.7656 0.34% 100.34% 1/2/2013 59.537144 -14.41% 85.59% 24.42402 2.77% 102.77% 2/1/2013 58.031689 -2.53% 97.47% 24.94023 2.11% 102.11% 3/1/2013 58.197346 0.29% 100.29% 25.66691 2.91% 102.91% 4/1/2013 58.21312 0.03% 100.03% 29.69502 15.69% 115.69% 5/1/2013 59.518211 2.24% 102.24% 31.52941 6.18% 106.18% 6/3/2013 52.477612 -11.83% 88.17% 31.20418 -1.03% 98.97% 7/1/2013 59.888763 14.12% 114.12% 28.76494 -7.82% 92.18% 8/1/2013 64.905457 8.38% 108.38% 30.38691 5.64% 105.64% 9/3/2013 63.510681 -2.15% 97.85% 30.27773 -0.36% 99.64% 10/1/2013 69.63195 9.64% 109.64% 32.21558 6.40% 106.40% 11/1/2013 74.509857 7.01% 107.01% 34.95329 8.50% 108.50% 12/2/2013 75.173126 0.89% 100.89% 34.29327 -1.89% 98.11% 1/2/2014 67.077225 -10.77% 89.23% 34.68745 1.15% 101.15% 2/3/2014 70.934898 5.75% 105.75% 35.38163 2.00% 102.00% 3/3/2014 72.35025 2.00% 102.00% 37.85678 7.00% 107.00% 4/1/2014 79.541603 9.94% 109.94% 37.31188 -1.44% 98.56% 5/1/2014 85.802269 7.87% 107.87% 38.07734 2.05% 102.05% 6/2/2014 88.17572 2.77% 102.77% 38.7842 1.86% 101.86% 7/1/2014 90.709122 2.87% 102.87% 40.14211 3.50% 103.50% 8/1/2014 97.739883 7.75% 107.75% 42.51729 5.92% 105.92% 9/2/2014 96.071152 -1.71% 98.29% 43.38767 2.05% 102.05% 10/1/2014 102.984459 7.20% 107.20% 43.93984 1.27% 101.27% 11/3/2014 113.898628 10.60% 110.60% 45.02692 2.47% 102.47% 12/1/2014 105.710335 -7.19% 92.81% 43.74609 -2.84% 97.16% 1/2/2015 112.203506 6.14% 106.14% 38.04826 -13.02% 86.98% 2/2/2015 123.510986 10.08% 110.08% 41.59133 9.31% 109.31% 3/2/2015 119.636238 -3.14% 96.86% 38.56564 -7.27% 92.73% 4/1/2015 120.328499 0.58% 100.58% 46.1346 19.63% 119.63% 5/1/2015 125.784088 4.53% 104.53% 44.73514 -3.03% 96.97% 6/1/2015 121.101463 -3.72% 96.28% 42.14803 -5.78% 94.22% 7/1/2015 117.113991 -3.29% 96.71% 44.5824 5.78% 105.78% 8/3/2015 109.361488 -6.62% 93.38% 41.82056 -6.19% 93.81% 9/1/2015 106.975632 -2.18% 97.82% 42.53167 1.70% 101.70% 10/1/2015 115.898346 8.34% 108.34% 50.58444 18.93% 118.93% 11/2/2015 115.225647 -0.58% 99.42% 52.57969 3.94% 103.94% 12/1/2015 102.524529 -11.02% 88.98% 53.67288 2.08% 102.08% 1/4/2016 94.810341 -7.52% 92.48% 53.29559 -0.70% 99.30% 2/1/2016 94.688271 -0.13% 99.87% 49.57613 -6.98% 93.02% 3/1/2016 106.73362 12.72% 112.72% 53.81466 8.55% 108.55% 4/1/2016 91.799339 -13.99% 86.01% 48.59201 -9.70% 90.30% 5/2/2016 98.388046 7.18% 107.18% 52.00301 7.02% 107.02% 6/1/2016 94.190842 -4.27% 95.73% 50.20743 -3.45% 96.55% 7/1/2016 102.673927 9.01% 109.01% 55.61378 10.77% 110.77% 8/1/2016 105.102364 2.37% 102.37% 56.7305 2.01% 102.01% 9/1/2016 111.987015 6.55% 106.55% 56.86872 0.24% 100.24% 10/3/2016 112.472404 0.43% 100.43% 59.15927 4.03% 104.03% 11/1/2016 110.0429 -2.16% 97.84% 59.89687 1.25% 101.25% 12/1/2016 115.320023 4.80% 104.80% 61.76555 3.12% 103.12% 1/3/2017 120.826149 4.77% 104.77% 64.26042 4.04% 104.04% 2/1/2017 136.990005 13.38% 113.38% 63.98 -0.44% 99.56% 3/1/2017 140.880005 2.84% 102.84% 65.1 1.75% 101.75% Annualized Mean Return 2.25% 0.92% Standard Deviation 9.30% 7.01% Variance 0.86% 0.49% Covariance 0.002943838 Correlation 0.45165057 Weights 26% 74% Portfolio Return 1.262% Portfolio Variance 0.00440947 Standard Deviation 6.64%

From the table, it can be stated that Apple Inc. has provided 2.25% annualized return in the period of last 10 years with a risk level of 9.30%. On the other hand, the risk level of Microsoft stocks is 7.01%, but it has generated only 0.92% return over the period.

However, on the basis of the individual standard deviation and variances of the stocks of Apple and Microsoft, the weight of Apple and Microsoft would be 26% and 74% in the portfolio. As per the calculation methods, described above, the portfolio would generate 1.26% return for the same period with a risk level of 6.64%.

Efficient frontier line is a graphical representation, which denotes the returns of portfolio at different risk levels of any portfolio for different combinations of weightings. Thus, it helps to determine the optimum return from any portfolio (Zia 2013). The efficient frontier line of the portfolio, constructed above in shown below:

The graph also denotes that the weighting combination, calculated above, would be the ideal combination for the portfolio, constructed with the stocks of Apple Inc. and Microsoft.

Conclusion:

It is clear from the above discussions and calculations that portfolio investment in always a better alternative comparing to investment in any single stock. Apple Inc. might have generated higher returns than the portfolio returns, but, the returns of Microsoft were lower than the portfolio returns. Moreover, the individual risk levels of both the stocks, denoted through the standard deviation, are also higher than the standard deviation of the portfolio. It indicates that through the portfolio diversification the investors can effectively reduce the risk level of the investment and earn optimum returns without bearing higher risk (Guerard et al. 2015).

However, it should be noted that the calculations, done above, are based on the historical prices of the stocks. Historical data often incorporates various forms of errors. There are many factors, which have significant impact on the stock prices, but are not assessed from the historical data. For example, future planning of the companies, change in the preferences of the customers, future planning of government etc. can affect the future performances of the companies (Saunders and Cornett 2014). Historical data only includes only the factors, affected the stock prices in the past. Therefore, if Apple Inc. would plan for such financial strategies, which might be able to generate higher returns, than the present return rates, then in the portfolio, the weightage of the company should be higher than the weightage percentage, calculated above (DeFusco et al. 2015). Hence, though portfolio diversification is a very effective tool for optimizing returns, the diversification process should incorporate some factors, which might affect the future performances of the companies, apart from historical data.

Introduction:

In modern times, mutual funds have become very popular amongst the investors. This new form of investment has been developed on the basis of modern portfolio theory. Mutual fund is a combination of stocks, which provides higher returns at higher risks, and bonds, which pay low returns at minimum risk level. The total fund of the investment is divided into these two forms of investments. Thus, the mutual fund can provide returns in the period of economic depression and can generate higher returns during uprisings in the stock markets. The dividing ratios between the two investments, depends on the market scenario.

In this report, the performance of a mutual fund has been measured under different methods to explain the evaluation process of funds. Vanguard 500 Index Mutual funds has been selected for this purpose. The report includes various models to evaluate the prices of the fund over the last year.

The most popular tool for evaluating the performances of fund is appraisal ratio. It can be described as the measurement metrics, used to assess the investment picking ability of the fund (Jagric et al. 2015). The formula of appraisal ratio is as follows:

Appraisal Ratio = α / σ

Where, α = Alpha or Adjusted Return of any fund with zero market return

σ = Standard Deviation of the Fund

The standard deviation can be computed from the series of fund returns over a certain period. However, alpha is based on the relation between the fund return and benchmark index returns. Therefore, it requires regression models to compute the alpha of any fund in respect to the benchmark index and other relative factors (Kevin 2015). There are different models, used to compute the alpha of the stocks or portfolio funds.

Capital Asset Pricing Model:

Capital Asset Pricing Model (CAPM) is most well known and widely used model, which estimate the investment return on basis of the correlation between the market index and the individual stocks. It includes the systematic risks in form of Beta (β) to compute the future stock prices (Spronk et al. 2016). The equation of CAPM model is shown below:

Re = Rf + β(Rm – Rf) + α

or, Re – Rf = β(Rm – Rf) + α

Where, Re = Return on Stock or Fund Portfolio

Rf = Risk-Free Rate of Long-Term Bond Rate

Rm = Market Index Return

β = Measurement metrics of the volatility of the investment in respect to market index

α = Alpha or return from investment when market index return is zero

Fama-French three factor model is an upgraded version of CAPM, which incorporates two additional factors for estimating the investment return. According to the concept, return of any single stock uses to be affected from the movements of small caps and the stocks with higher book-to-market ratio also along with the market index in a whole. Hence, the model correlates the investment returns with the stocks with small capitalization and stocks with higher book-to-market ratio in addition to the market index (Cosio et al. 2015). The equation of the concept is stated below:

Re = Rf + β(Rm – Rf) + bs*SMB + bv*HML + α

or, Re – Rf = β(Rm – Rf) + bs*SMB + bv*HML + α

Where, Re = Return on Stock or Fund Portfolio

Rf = Risk-Free Rate of Long-Term Bond Rate

Rm = Market Index Return

β = Measurement metrics of the volatility of the investment in respect to market index

SMB = Small Market Cap minus Big Market Cap

bs = Coefficient of Fund Return in relation to SMB

HML = High Book-to-Ratio minus Low Book-to-Ratio

bv = Coefficient of Fund Return in relation to HML

α = Alpha or return from investment when market index return, SMB and HML are zero

Sharpe’s ratio is another performance metrics, used widely to evaluate the performance of any fund portfolio. It depicts whether any investment is providing relatively higher returns in comparison to the risk, associated with the fund. As per the method, any fund may provide higher return, but if the returns are relatively higher than its risk, then such fund cannot be considered as better investment alternative (Rutkauskas et al. 2015). The equation of sharpe ratio is as follows:

Sharpe Ratio: (Rp – Rf) / σp

Rp = Expected Return of any Fund

Rf = Risk-Free Rate

σp = Standard Deviation of the fund

The appraisal ratio of Vanguard Mutual Fund on the basis last 10 years performance is calculated under CAPM in the following table:

 Year Price Monthly Performance Market Premium re rm-rf 1/3/2006 94.730927 2.97% 2/1/2006 94.971848 0.25% -0.28% 3/1/2006 96.149284 1.24% 1.42% 4/3/2006 97.431389 1.33% 0.77% 5/1/2006 94.609161 -2.90% -3.59% 6/1/2006 94.729774 0.13% -0.33% 7/3/2006 95.30468 0.61% -0.72% 8/1/2006 97.555717 2.36% 2.03% 9/1/2006 100.055435 2.56% 1.86% 10/2/2006 103.308212 3.25% 3.16% 11/1/2006 105.25174 1.88% 1.74% 12/1/2006 106.726593 1.40% 0.85% 1/3/2007 108.320251 1.49% 1.44% 2/1/2007 106.187195 -1.97% -1.90% 3/1/2007 107.36805 1.11% 0.67% 4/2/2007 112.111511 4.42% 3.54% 5/1/2007 116.009682 3.48% 3.23% 6/1/2007 114.0672 -1.67% -1.96% 7/2/2007 110.540451 -3.09% -3.75% 8/1/2007 112.196709 1.50% 0.82% 9/4/2007 116.37719 3.73% 3.17% 10/1/2007 118.2146 1.58% 1.73% 11/1/2007 113.265198 -4.19% -4.92% 12/3/2007 112.48275 -0.69% -0.84% 1/2/2008 105.716309 -6.02% -6.33% 2/1/2008 102.278992 -3.25% -3.16% 3/3/2008 101.822807 -0.45% -0.91% 4/1/2008 106.773865 4.86% 4.64% 5/1/2008 108.145439 1.28% 1.88% 6/2/2008 99.006027 -8.45% -8.38% 7/1/2008 98.182587 -0.83% -0.72% 8/1/2008 99.6026 1.45% 1.53% 9/2/2008 90.742096 -8.90% -9.57% 10/1/2008 75.504311 -16.79% -17.15% 11/3/2008 70.087006 -7.17% -7.77% 12/1/2008 70.838203 1.07% 1.84% 1/2/2009 64.878899 -8.41% -7.93% 2/2/2009 57.964733 -10.66% -9.94% 3/2/2009 63.052597 8.78% 8.83% 4/1/2009 69.079681 9.56% 10.17% 5/1/2009 72.960373 5.62% 5.33% 6/1/2009 73.121719 0.22% 0.44% 7/1/2009 78.662811 7.58% 7.76% 8/3/2009 81.493774 3.60% 3.23% 9/1/2009 84.530174 3.73% 4.15% 10/1/2009 82.951462 -1.87% -2.49% 11/2/2009 87.913116 5.98% 5.64% 12/1/2009 89.624168 1.95% 2.80% 1/4/2010 86.39431 -3.60% -3.51% 2/1/2010 89.065491 3.09% 3.39% 3/1/2010 94.423714 6.02% 6.30% 4/1/2010 95.913734 1.58% 2.12% 5/3/2010 88.244492 -8.00% -7.86% 6/1/2010 83.611862 -5.25% -5.67% 7/1/2010 89.461433 7.00% 7.27% 8/2/2010 85.409019 -4.53% -4.81% 9/1/2010 93.022614 8.91% 9.56% 10/1/2010 96.5466 3.79% 4.02% 11/1/2010 96.5466 0.00% 0.63% 12/1/2010 102.986847 6.67% 6.76% 1/3/2011 105.41436 2.36% 2.05% 2/1/2011 109.01561 3.42% 3.49% 3/1/2011 109.047775 0.03% 0.57% 4/1/2011 112.262413 2.95% 2.95% 5/2/2011 110.976555 -1.15% -1.28% 6/1/2011 109.119339 -1.67% -1.68% 7/1/2011 106.885818 -2.05% -2.30% 8/1/2011 101.064323 -5.45% -5.95% 9/1/2011 93.930855 -7.06% -7.53% 10/3/2011 104.182289 10.91% 11.34% 11/1/2011 103.938843 -0.23% -0.29% 12/1/2011 105.001579 1.02% 0.86% 1/3/2012 109.689468 4.46% 5.03% 2/1/2012 114.413635 4.31% 4.46% 3/1/2012 118.169579 3.28% 2.96% 4/2/2012 117.413826 -0.64% -0.78% 5/1/2012 110.348061 -6.02% -6.27% 6/1/2012 114.886261 4.11% 3.85% 7/2/2012 116.460167 1.37% 0.86% 8/1/2012 119.0681 2.24% 2.58% 9/4/2012 122.135841 2.58% 2.69% 10/1/2012 119.8647 -1.86% -1.70% 11/1/2012 120.535919 0.56% 0.71% 12/3/2012 121.615723 0.90% 1.09% 1/2/2013 127.91082 5.18% 5.65% 2/1/2013 129.623459 1.34% 1.27% 3/1/2013 134.464661 3.73% 4.07% 4/1/2013 137.031021 1.91% 1.57% 5/1/2013 140.220398 2.33% 2.64% 6/3/2013 138.328369 -1.35% -1.12% 7/1/2013 145.344772 5.07% 5.63% 8/1/2013 141.112503 -2.91% -2.65% 9/3/2013 145.512527 3.12% 3.74% 10/1/2013 152.186478 4.59% 4.23% 11/1/2013 156.795334 3.03% 3.13% 12/2/2013 160.741287 2.52% 2.70% 1/2/2014 155.164963 -3.47% -3.20% 2/3/2014 162.241516 4.56% 4.57% 3/3/2014 163.576019 0.82% 0.39% 4/1/2014 164.760468 0.72% -0.13% 5/1/2014 168.607529 2.33% 2.17% 6/2/2014 172.068512 2.05% 2.69% 7/1/2014 169.68013 -1.39% 2.08% 8/1/2014 176.445618 3.99% 4.21% 9/2/2014 173.943329 -1.42% -2.00% 10/1/2014 178.158325 2.42% 2.35% 11/3/2014 182.937256 2.68% 2.41% 12/1/2014 182.473312 -0.25% -0.22% 1/2/2015 176.97673 -3.01% -2.95% 2/2/2015 187.124268 5.73% 6.11% 3/2/2015 184.15007 -1.59% -1.14% 4/1/2015 185.897812 0.95% 0.75% 5/1/2015 188.263535 1.27% 1.28% 6/1/2015 184.620239 -1.94% -1.62% 7/1/2015 188.460846 2.08% 1.41% 8/3/2015 177.074829 -6.04% -5.99% 9/1/2015 172.651535 -2.50% -3.27% 10/1/2015 187.193481 8.42% 7.82% 11/2/2015 187.729553 0.29% 0.46% 12/1/2015 184.721252 -1.60% -2.23% 1/4/2016 175.52832 -4.98% -5.90% 2/1/2016 175.273499 -0.15% -0.13% 3/1/2016 187.153244 6.78% 7.00% 4/1/2016 187.852646 0.37% 1.03% 5/2/2016 191.201889 1.78% 1.84% 6/1/2016 191.685089 0.25% -0.03% 7/1/2016 198.732101 3.68% 3.99% 8/1/2016 198.989456 0.13% 0.40% 9/1/2016 199.01828 0.01% 0.33% 10/3/2016 195.370102 -1.83% -2.08% 11/1/2016 202.596832 3.70% 4.76% 12/1/2016 206.570007 1.96% 1.84% Average 0.69% 0.67% Standard Deviation 0.0425 0.0436 Slope under CAPM 0.966065487 Alpha under CAPM 0.0004 Appraisal Ratio 0.95%

The appraisal ratio of the sated fund under Fama-French Model is derived in the table below:

 Year Price Monthly Performance Market Premium SMB HML re rm-rf 1/3/2006 94.730927 2.97% 5.58% -0.89% 2/1/2006 94.971848 0.25% -0.28% -0.51% 0.21% 3/1/2006 96.149284 1.24% 1.42% 3.06% -0.38% 4/3/2006 97.431389 1.33% 0.77% -1.13% 2.01% 5/1/2006 94.609161 -2.90% -3.59% -2.00% 2.79% 6/1/2006 94.729774 0.13% -0.33% -0.81% 0.92% 7/3/2006 95.30468 0.61% -0.72% -3.54% 3.33% 8/1/2006 97.555717 2.36% 2.03% 0.71% -0.51% 9/1/2006 100.055435 2.56% 1.86% -1.50% -0.22% 10/2/2006 103.308212 3.25% 3.16% 2.05% 0.14% 11/1/2006 105.25174 1.88% 1.74% 0.56% 0.97% 12/1/2006 106.726593 1.40% 0.85% -0.94% 1.65% 1/3/2007 108.320251 1.49% 1.44% -0.36% -0.43% 2/1/2007 106.187195 -1.97% -1.90% 1.10% 0.69% 3/1/2007 107.36805 1.11% 0.67% -0.07% -0.64% 4/2/2007 112.111511 4.42% 3.54% -2.07% -0.71% 5/1/2007 116.009682 3.48% 3.23% 0.11% -0.04% 6/1/2007 114.0672 -1.67% -1.96% 0.76% -1.53% 7/2/2007 110.540451 -3.09% -3.75% -2.62% -4.39% 8/1/2007 112.196709 1.50% 0.82% -0.16% 1.59% 9/4/2007 116.37719 3.73% 3.17% -2.24% -3.29% 10/1/2007 118.2146 1.58% 1.73% 0.41% 4.71% 11/1/2007 113.265198 -4.19% -4.92% -2.47% -1.77% 12/3/2007 112.48275 -0.69% -0.84% 0.61% -3.13% 1/2/2008 105.716309 -6.02% -6.33% -1.29% 8.18% 2/1/2008 102.278992 -3.25% -3.16% 0.07% -4.27% 3/3/2008 101.822807 -0.45% -0.91% 0.75% -1.12% 4/1/2008 106.773865 4.86% 4.64% -2.45% -0.36% 5/1/2008 108.145439 1.28% 1.88% 3.23% -4.25% 6/2/2008 99.006027 -8.45% -8.38% -0.09% -8.75% 7/1/2008 98.182587 -0.83% -0.72% 2.49% 6.19% 8/1/2008 99.6026 1.45% 1.53% 3.29% 2.28% 9/2/2008 90.742096 -8.90% -9.57% 0.20% 3.26% 10/1/2008 75.504311 -16.79% -17.15% -2.87% -7.53% 11/3/2008 70.087006 -7.17% -7.77% -4.27% -5.73% 12/1/2008 70.838203 1.07% 1.84% 2.88% -0.06% 1/2/2009 64.878899 -8.41% -7.93% 0.91% -5.43% 2/2/2009 57.964733 -10.66% -9.94% -0.83% -8.27% 3/2/2009 63.052597 8.78% 8.83% 0.92% 5.07% 4/1/2009 69.079681 9.56% 10.17% 10.64% 19.72% 5/1/2009 72.960373 5.62% 5.33% 1.53% 7.27% 6/1/2009 73.121719 0.22% 0.44% 2.04% -4.75% 7/1/2009 78.662811 7.58% 7.76% 2.43% 3.44% 8/3/2009 81.493774 3.60% 3.23% -1.10% 7.34% 9/1/2009 84.530174 3.73% 4.15% 2.86% 0.58% 10/1/2009 82.951462 -1.87% -2.49% -4.14% -1.70% 11/2/2009 87.913116 5.98% 5.64% -3.12% -0.10% 12/1/2009 89.624168 1.95% 2.80% 5.70% -0.16% 1/4/2010 86.39431 -3.60% -3.51% 0.43% 3.51% 2/1/2010 89.065491 3.09% 3.39% 1.15% 0.56% 3/1/2010 94.423714 6.02% 6.30% 1.84% 1.56% 4/1/2010 95.913734 1.58% 2.12% 4.36% 2.58% 5/3/2010 88.244492 -8.00% -7.86% 0.32% -1.79% 6/1/2010 83.611862 -5.25% -5.67% -1.90% -1.69% 7/1/2010 89.461433 7.00% 7.27% -0.45% 1.04% 8/2/2010 85.409019 -4.53% -4.81% -2.75% -1.65% 9/1/2010 93.022614 8.91% 9.56% 3.57% -2.65% 10/1/2010 96.5466 3.79% 4.02% 0.41% -1.46% 11/1/2010 96.5466 0.00% 0.63% 3.45% -0.54% 12/1/2010 102.986847 6.67% 6.76% 0.82% 4.70% 1/3/2011 105.41436 2.36% 2.05% -2.05% 0.62% 2/1/2011 109.01561 3.42% 3.49% 1.64% 0.00% 3/1/2011 109.047775 0.03% 0.57% 2.05% -1.61% 4/1/2011 112.262413 2.95% 2.95% -0.79% -1.70% 5/2/2011 110.976555 -1.15% -1.28% -0.59% -0.69% 6/1/2011 109.119339 -1.67% -1.68% -0.56% -1.01% 7/1/2011 106.885818 -2.05% -2.30% -1.17% -1.16% 8/1/2011 101.064323 -5.45% -5.95% -2.76% -2.27% 9/1/2011 93.930855 -7.06% -7.53% -2.86% -2.04% 10/3/2011 104.182289 10.91% 11.34% 3.55% 3.23% 11/1/2011 103.938843 -0.23% -0.29% 0.17% -1.75% 12/1/2011 105.001579 1.02% 0.86% -0.80% 0.90% 1/3/2012 109.689468 4.46% 5.03% 2.80% 1.45% 2/1/2012 114.413635 4.31% 4.46% -1.89% 1.12% 3/1/2012 118.169579 3.28% 2.96% -0.50% 0.20% 4/2/2012 117.413826 -0.64% -0.78% -0.31% -2.18% 5/1/2012 110.348061 -6.02% -6.27% -0.25% -1.91% 6/1/2012 114.886261 4.11% 3.85% 0.72% 0.80% 7/2/2012 116.460167 1.37% 0.86% -2.52% -0.17% 8/1/2012 119.0681 2.24% 2.58% 0.50% 0.79% 9/4/2012 122.135841 2.58% 2.69% 0.72% 1.36% 10/1/2012 119.8647 -1.86% -1.70% -1.31% 3.43% 11/1/2012 120.535919 0.56% 0.71% 0.55% -1.40% 12/3/2012 121.615723 0.90% 1.09% 1.53% 3.56% 1/2/2013 127.91082 5.18% 5.65% 0.57% 0.68% 2/1/2013 129.623459 1.34% 1.27% 0.07% -0.04% 3/1/2013 134.464661 3.73% 4.07% 0.54% 0.08% 4/1/2013 137.031021 1.91% 1.57% -2.20% -0.05% 5/1/2013 140.220398 2.33% 2.64% 2.30% 0.91% 6/3/2013 138.328369 -1.35% -1.12% 1.44% -0.48% 7/1/2013 145.344772 5.07% 5.63% 1.26% 0.88% 8/1/2013 141.112503 -2.91% -2.65% 0.06% -2.17% 9/3/2013 145.512527 3.12% 3.74% 2.39% -1.70% 10/1/2013 152.186478 4.59% 4.23% -1.40% 1.01% 11/1/2013 156.795334 3.03% 3.13% 1.00% 0.24% 12/2/2013 160.741287 2.52% 2.70% -0.57% -0.39% 1/2/2014 155.164963 -3.47% -3.20% 0.83% -1.99% 2/3/2014 162.241516 4.56% 4.57% 0.27% -0.85% 3/3/2014 163.576019 0.82% 0.39% -1.04% 4.83% 4/1/2014 164.760468 0.72% -0.13% -3.91% 2.69% 5/1/2014 168.607529 2.33% 2.17% -1.79% -0.37% 6/2/2014 172.068512 2.05% 2.69% 2.83% 0.67% 7/1/2014 169.68013 -1.39% 2.08% -3.93% -0.16% 8/1/2014 176.445618 3.99% 4.21% 0.69% -1.11% 9/2/2014 173.943329 -1.42% -2.00% -3.67% -1.72% 10/1/2014 178.158325 2.42% 2.35% 3.08% -0.45% 11/3/2014 182.937256 2.68% 2.41% -2.30% -3.73% 12/1/2014 182.473312 -0.25% -0.22% 2.24% 1.15% 1/2/2015 176.97673 -3.01% -2.95% -0.73% -4.73% 2/2/2015 187.124268 5.73% 6.11% 1.00% -0.29% 3/2/2015 184.15007 -1.59% -1.14% 2.38% -0.88% 4/1/2015 185.897812 0.95% 0.75% -2.00% 4.17% 5/1/2015 188.263535 1.27% 1.28% 0.29% -3.53% 6/1/2015 184.620239 -1.94% -1.62% 2.68% -1.42% 7/1/2015 188.460846 2.08% 1.41% -4.16% -4.77% 8/3/2015 177.074829 -6.04% -5.99% 0.40% 2.62% 9/1/2015 172.651535 -2.50% -3.27% -2.97% 0.66% 10/1/2015 187.193481 8.42% 7.82% -1.73% 0.32% 11/2/2015 187.729553 0.29% 0.46% 2.86% -0.78% 12/1/2015 184.721252 -1.60% -2.23% -3.34% -3.28% 1/4/2016 175.52832 -4.98% -5.90% -4.64% 1.66% 2/1/2016 175.273499 -0.15% -0.13% 0.52% -0.73% 3/1/2016 187.153244 6.78% 7.00% 1.87% 3.84% 4/1/2016 187.852646 0.37% 1.03% 2.45% 5.44% 5/2/2016 191.201889 1.78% 1.84% -0.29% -1.83% 6/1/2016 191.685089 0.25% -0.03% -0.17% -1.51% 7/1/2016 198.732101 3.68% 3.99% 2.34% -1.47% 8/1/2016 198.989456 0.13% 0.40% 1.82% 3.17% 9/1/2016 199.01828 0.01% 0.33% 1.43% -0.27% 10/3/2016 195.370102 -1.83% -2.08% -3.98% 3.72% 11/1/2016 202.596832 3.70% 4.76% 4.78% 9.19% 12/1/2016 206.570007 1.96% 1.84% 0.23% 3.96% Average 0.69% 0.67% 0.12% 0.21% Standard Deviation 0.0425 0.0436 0.0233 0.0346 Slope under Fama-French 0.9661 0.6632 0.5281 Alpha under CAPM -0.00151 Appraisal Ratio -3.55%

The sharpe ratio of Vanguard Mutual Fund is calculated below:

 Year Price Monthly Performance re 1/3/2006 94.730927 2/1/2006 94.971848 0.25% 3/1/2006 96.149284 1.24% 4/3/2006 97.431389 1.33% 5/1/2006 94.609161 -2.90% 6/1/2006 94.729774 0.13% 7/3/2006 95.30468 0.61% 8/1/2006 97.555717 2.36% 9/1/2006 100.055435 2.56% 10/2/2006 103.308212 3.25% 11/1/2006 105.25174 1.88% 12/1/2006 106.726593 1.40% 1/3/2007 108.320251 1.49% 2/1/2007 106.187195 -1.97% 3/1/2007 107.36805 1.11% 4/2/2007 112.111511 4.42% 5/1/2007 116.009682 3.48% 6/1/2007 114.0672 -1.67% 7/2/2007 110.540451 -3.09% 8/1/2007 112.196709 1.50% 9/4/2007 116.37719 3.73% 10/1/2007 118.2146 1.58% 11/1/2007 113.265198 -4.19% 12/3/2007 112.48275 -0.69% 1/2/2008 105.716309 -6.02% 2/1/2008 102.278992 -3.25% 3/3/2008 101.822807 -0.45% 4/1/2008 106.773865 4.86% 5/1/2008 108.145439 1.28% 6/2/2008 99.006027 -8.45% 7/1/2008 98.182587 -0.83% 8/1/2008 99.6026 1.45% 9/2/2008 90.742096 -8.90% 10/1/2008 75.504311 -16.79% 11/3/2008 70.087006 -7.17% 12/1/2008 70.838203 1.07% 1/2/2009 64.878899 -8.41% 2/2/2009 57.964733 -10.66% 3/2/2009 63.052597 8.78% 4/1/2009 69.079681 9.56% 5/1/2009 72.960373 5.62% 6/1/2009 73.121719 0.22% 7/1/2009 78.662811 7.58% 8/3/2009 81.493774 3.60% 9/1/2009 84.530174 3.73% 10/1/2009 82.951462 -1.87% 11/2/2009 87.913116 5.98% 12/1/2009 89.624168 1.95% 1/4/2010 86.39431 -3.60% 2/1/2010 89.065491 3.09% 3/1/2010 94.423714 6.02% 4/1/2010 95.913734 1.58% 5/3/2010 88.244492 -8.00% 6/1/2010 83.611862 -5.25% 7/1/2010 89.461433 7.00% 8/2/2010 85.409019 -4.53% 9/1/2010 93.022614 8.91% 10/1/2010 96.5466 3.79% 11/1/2010 96.5466 0.00% 12/1/2010 102.986847 6.67% 1/3/2011 105.41436 2.36% 2/1/2011 109.01561 3.42% 3/1/2011 109.047775 0.03% 4/1/2011 112.262413 2.95% 5/2/2011 110.976555 -1.15% 6/1/2011 109.119339 -1.67% 7/1/2011 106.885818 -2.05% 8/1/2011 101.064323 -5.45% 9/1/2011 93.930855 -7.06% 10/3/2011 104.182289 10.91% 11/1/2011 103.938843 -0.23% 12/1/2011 105.001579 1.02% 1/3/2012 109.689468 4.46% 2/1/2012 114.413635 4.31% 3/1/2012 118.169579 3.28% 4/2/2012 117.413826 -0.64% 5/1/2012 110.348061 -6.02% 6/1/2012 114.886261 4.11% 7/2/2012 116.460167 1.37% 8/1/2012 119.0681 2.24% 9/4/2012 122.135841 2.58% 10/1/2012 119.8647 -1.86% 11/1/2012 120.535919 0.56% 12/3/2012 121.615723 0.90% 1/2/2013 127.91082 5.18% 2/1/2013 129.623459 1.34% 3/1/2013 134.464661 3.73% 4/1/2013 137.031021 1.91% 5/1/2013 140.220398 2.33% 6/3/2013 138.328369 -1.35% 7/1/2013 145.344772 5.07% 8/1/2013 141.112503 -2.91% 9/3/2013 145.512527 3.12% 10/1/2013 152.186478 4.59% 11/1/2013 156.795334 3.03% 12/2/2013 160.741287 2.52% 1/2/2014 155.164963 -3.47% 2/3/2014 162.241516 4.56% 3/3/2014 163.576019 0.82% 4/1/2014 164.760468 0.72% 5/1/2014 168.607529 2.33% 6/2/2014 172.068512 2.05% 7/1/2014 169.68013 -1.39% 8/1/2014 176.445618 3.99% 9/2/2014 173.943329 -1.42% 10/1/2014 178.158325 2.42% 11/3/2014 182.937256 2.68% 12/1/2014 182.473312 -0.25% 1/2/2015 176.97673 -3.01% 2/2/2015 187.124268 5.73% 3/2/2015 184.15007 -1.59% 4/1/2015 185.897812 0.95% 5/1/2015 188.263535 1.27% 6/1/2015 184.620239 -1.94% 7/1/2015 188.460846 2.08% 8/3/2015 177.074829 -6.04% 9/1/2015 172.651535 -2.50% 10/1/2015 187.193481 8.42% 11/2/2015 187.729553 0.29% 12/1/2015 184.721252 -1.60% 1/4/2016 175.52832 -4.98% 2/1/2016 175.273499 -0.15% 3/1/2016 187.153244 6.78% 4/1/2016 187.852646 0.37% 5/2/2016 191.201889 1.78% 6/1/2016 191.685089 0.25% 7/1/2016 198.732101 3.68% 8/1/2016 198.989456 0.13% 9/1/2016 199.01828 0.01% 10/3/2016 195.370102 -1.83% 11/1/2016 202.596832 3.70% 12/1/2016 206.570007 1.96% Average 0.69% Standard Deviation 0.0425 Sharpe Ratio 16%

From the above calculation, it can be stated that Vanguard Mutual Fund has been able to provide 0.95% excess return on the risks, associated with the market index. However, the Fama-French model implies that if SMB and HML factors are taken into consideration along with the market index, then the fund would fail to provide any additional returns over the associated risks. The sharpe ratio indicates that the fund would provide 16% additional return in relation to the risks, associated with the fund only.

Conclusion:

It is clear from the above discussions that performance of any mutual fund, depends on the variety of risks, associated with the fund. Sharpe ratio only incorporates the risk factor, related to the fund itself. Therefore, the outcomes of the ratio imply higher returns from the fund. However, after incorporating the market risk under CAPM, the return rate falls down. Under Fama-French three factor model, the performance ratio further reduces, as the model includes two additional risk factor to evaluate the fund.

Hence, it can be stated that though mutual funds are developed to reduce the investment risk, it is not excluded from the impacts of various risk factors, associated with any form of investments. Therefore, it is necessary to evaluate the mutual funds with the help of various evaluation techniques before taking any investment decision.

References:

Bodie, Z., Kane, A. and Marcus, A.J., 2014. Investments, 10e. McGraw-Hill Education

Cosio, R.M., Estrada, J. and Kritzman, M., 2015. New Frontiers in Portfolio Management

DeFusco, R.A., McLeavey, D.W., Pinto, J.E., Runkle, D.E. and Anson, M.J., 2015. Quantitative investment analysis. John Wiley & Sons.

Guerard, J.B., Markowitz, H. and Xu, G., 2015. Earnings forecasting in a global stock selection model and efficient portfolio construction and management. International Journal of Forecasting, 31(2), pp.550-560.

Jagric, T., Podobnik, B., Strasek, S. and Jagric, V., 2015. Risk-adjusted performance of mutual funds: some tests. South-eastern Europe journal of Economics, 5(2).

Kevin, S., 2015. Security analysis and portfolio management. PHI Learning Pvt. Ltd.

Rutkauskas, A.V., Stasytyte, V. and Borisova, J., 2015.Adequate Portfolio As a Conceptual Model Of Investment Profitability, Risk And Reliability Adjustment To Investor ‘s Interests. Economics and Management, (14), pp.1170-1174

Saunders, A. and Cornett, M.M., 2014. Financial institutions management. McGraw-Hill Education

Spronk, J., Steuer, R.E. and Zopounidis, C., 2016. Multicriteria decision aid/analysis in finance. In Multiple Criteria Decision Analysis (pp. 1011-1065). Springer New York

Zia, S.A., 2013. Investment Analysis & Portfolio Management

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