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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