Calculation of portfolio risk and return using Markowitz model Calculation of optimal risk portfolio using a single index model; analysis using regression output and ANOVA table The excel file can be viewed at: https://drive.google.com/file/d/0B6nSS-YiZgneZTQ5blpWNkY3N2M/view?usp=sharing

1. Portfolio Mean and Variance
Analysis
By Prashant Keswani

2. Stock Portfolio Return
HUL RIL
Bharti
Airtel
Mindtree
Maruti
Suzuki
India Ltd
HDFC
Bank
Sensex
Month
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
HUL RIL
Bharti
Airtel
Mindtree
Maruti
Suzuki
India Ltd
HDFC
Bank
Sensex
Month
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
Monthly
Returns
Mar-08
Apr-08 9.1% 15.5% 8.8% 39.9% (10.6%) 14.8% 9.6%
May-08 (4.9%) (8.2%) (2.5%) 0.2% 3.0% (10.4%) -6.5%
Jun-08 (13.1%) (12.8%) (17.7%) (16.6%) (19.2%) (26.2%) -18.9%
Jul-08 16.3% 5.4% 10.7% (5.5%) (6.9%) 9.3% 6.5%
Aug-08 2.4% (3.1%) 4.8% (5.9%) 13.1% 16.6% 3.6%
Sep-08 2.5% (8.9%) (6.2%) (11.1%) 5.7% (3.8%) -10.8%
Oct-08 (11.8%) (29.6%) (17.3%) (8.4%) (17.9%) (16.7%) -24.7%
Nov-08 6.4% (17.4%) 3.4% (17.5%) (5.1%) (10.1%) -10.9%
Dec-08 5.9% 8.7% 6.6% 1.6% (2.9%) 8.4% 5.3%
Jan-09 4.4% 7.7% (11.4%) (13.5%) 9.8% (7.3%) -3.0%
Feb-09 (2.8%) (4.5%) 0.4% (0.9%) 18.7% (4.3%) -5.0%
Mar-09 (6.1%) 20.4% (1.7%) 4.2% 14.4% 9.4% 10.8%
Apr-09 (1.5%) 18.3% 19.7% 44.2% 5.2% 13.7% 17.0%
May-09 (1.5%) 26.3% 9.4% 21.4% 25.2% 31.0% 25.7%
Jun-09 15.6% (11.2%) (2.1%) 26.4% 4.3% 3.4% -1.7%
Jul-09 9.0% (3.3%) (48.8%) (1.0%) 32.6% 0.5% 8.0%
Aug-09 (10.8%) 2.4% 3.4% 13.1% 1.7% (2.0%) -0.2%
Sep-09 1.2% 9.8% (1.4%) 17.6% 18.3% 11.8% 9.1%
Oct-09 7.6% (12.3%) (30.2%) (2.7%) (17.4%) (1.3%) -7.5%
Nov-09 0.8% (45.0%) 2.6% 7.3% 11.3% 9.3% 6.9%
Dec-09 (7.2%) 2.5% 9.7% 7.8% (0.1%) (4.1%) 3.1%
Jan-10 (7.8%) (3.9%) (6.8%) (16.1%) (10.9%) (4.1%) -6.4%
Feb-10 (3.4%) (6.6%) (8.9%) (8.7%) 5.3% 4.5% 0.6%
Mar-10 1.3% 9.9% 11.7% 10.6% (3.2%) 13.4% 6.6%
Apr-10 0.1% (3.9%) (4.3%) (2.3%) (9.6%) 3.1% 0.0%
May-10 (0.9%) 1.2% (12.1%) (4.2%) (3.3%) (5.3%) -3.4%
Jun-10 12.7% 4.0% 0.4% 0.0% 15.1% 1.6% 4.5%
Jul-10 (5.9%) (7.1%) 16.6% (3.5%) (15.8%) 11.1% 1.1%
Aug-10 5.3% (9.0%) 6.6% (5.5%) 4.8% 0.2% 0.3%
Sep-10 16.5% 7.3% 11.8% 2.0% 14.7% 16.3% 11.3%
Oct-10 (4.5%) 11.1% (11.0%) 0.0% 7.7% (8.2%) -0.3%
Nov-10 1.6% (9.9%) 10.7% (2.0%) (8.2%) 0.5% -3.7%
Dec-10 4.5% 7.2% (0.6%) 11.4% (0.2%) 2.5% 5.0%
Jan-11 (13.2%) (13.1%) (11.1%) (12.5%) (11.8%) (12.9%) -11.1%
Feb-11 4.0% 5.0% 3.9% (24.6%) (3.7%) 0.3% -3.3%
Mar-11 0.9% 8.6% 8.0% 6.6% 4.7% 14.3% 8.1%
Apr-11 0.2% (6.3%) 6.0% (6.5%) 4.4% (2.2%) -1.7%
May-11 6.9% (3.1%) (1.2%) (4.4%) (7.0%) 4.2% -3.7%
Jun-11 12.4% (5.7%) 5.6% 11.7% (5.6%) 4.8% 1.7%
Jul-11 (5.5%) (7.8%) 10.6% 5.5% 4.3% (2.6%) -4.1%
Aug-11 (0.9%) (5.6%) (7.5%) (17.5%) (9.6%) (3.1%) -9.1%
Sep-11 6.0% 3.4% (6.5%) 2.2% (1.0%) (1.0%) -3.0%
Oct-11 10.3% 8.6% 3.5% 14.8% 4.1% 4.7% 8.9%
Nov-11 5.6% (11.3%) (1.6%) 1.3% (13.9%) (9.7%) -8.1%
Dec-11 3.0% (11.0%) (11.0%) (2.2%) (5.0%) (3.3%) -6.7%
Jan-12 (7.0%) 17.7% 6.4% 8.2% 29.1% 15.0% 10.7%
Feb-12 0.2% 0.4% (4.3%) 8.0% 5.8% 5.5% 3.3%
Mar-12 7.8% (8.6%) (3.6%) 4.9% 7.4% 0.4% -1.8%
Apr-12 1.6% (0.4%) (7.9%) 20.7% 1.5% 4.3% -0.6%
May-12 2.5% (5.3%) (2.6%) 7.1% (19.2%) (6.7%) -6.6%
Jun-12 6.5% 4.5% 1.0% 2.4% 5.7% 11.4% 7.5%
Jul-12 2.8% 0.7% (1.6%) (2.2%) (3.1%) 4.3% -1.2%
Aug-12 11.0% 3.9% (18.1%) 4.5% 0.5% 1.2% 1.1%
Sep-12 5.1% 8.4% 7.8% 0.2% 18.6% 5.7% 7.4%
Oct-12 0.5% (3.8%) 1.6% (0.3%) 6.4% 0.9% -1.5%
Nov-12 (1.7%) (1.4%) 25.3% 9.1% 2.6% 10.9% 4.6%
Dec-12 (2.5%) 5.7% (6.0%) (5.2%) 1.0% (3.5%) 0.4%
Jan-13 (9.9%) 5.7% 7.1% 15.4% 6.3% (5.2%) 2.0%
Feb-13 (6.1%) (8.1%) (4.7%) 9.0% (14.3%) (2.8%) -5.3%
Mar-13 5.0% (5.0%) (9.8%) 6.3% (5.7%) (0.2%) -1.0%

3. Markowitz Model
Mean Variance
Analysis
•The process of weighing risk (variance) against expected
return.
•To make more efficient investment choices - seeking the
lowest variance for a given expected return, or seeking the
highest expected return for a given variance level.
Markowitz Model
•Markowitz model, also known as Mean-Variance Model is
based on the expected returns (mean) and the standard
deviation (variance) of different portfolios
•Helps to make the most efficient selection by analyzing
various portfolios of the given assets.
•It shows investors how to reduce their risk in case they
have chosen assets not “moving” together.

4. Markowitz Model
HUL RIL BhartiAirtel MindTree MarutiSuzuki HDFC Weights
HUL 0.0049560 0.0008444 0.0000416 0.0012679 0.0011206 0.0022712 0.0263960
RIL 0.0008444 0.0130694 0.0037334 0.0056200 0.0052610 0.0055585 0.1783812
BhartiAirtel 0.0000416 0.0056200 0.0133666 0.0048783 0.0031900 0.0053241 -0.0190796
MindTree 0.0012679 0.0056200 0.0048783 0.0155891 0.0031900 0.0059105 0.1255385
MarutiSuzuki 0.0011206 0.0052610 0.0007715 0.0031900 0.0131600 0.0050159 0.1780277
HDFC 0.0022712 0.0055585 0.0053241 0.0059105 0.0050159 0.0085790 0.5107362
Weights 0.0263960 0.1783812 -0.0190796 0.1255385 0.1780277 0.5107362 1
BETA 0.2584832 0.9685631 0.6115864 0.9305475 0.8951083 1.0000797
Rm (Expected) 13.50% (assumed)
RFR 8.00% (assumed)
MRP 5.50%
Expected Return 9.42% 0.1332710 0.1136373 0.1311801 0.1292310 0.1350044
Portfolio Return 13.25%
Portfolio Risk 0.2825272
Sharpe Ratio 0.1858876
Covariance Matrix

5. Single Index Model
• Single-index model assumes that there is only 1 macroeconomic factor that causes
the systematic risk affecting all stock returns
•This factor can be represented by the rate of return on a market index, such as the
S&P 500
•According to this model, the return of any stock can be decomposed into the
expected excess return of the individual stock due to firm-specific factors, commonly
denoted by its alpha coefficient (α), which is the return that exceeds the risk-free
rate, the return due to macroeconomic events that affect the market, and the
unexpected microeconomic events that affect only the firm
•The return of stock i is:
ri = αi + βirm + ei

6. Single Index Model
(A) Risk Parameters
SD Beta
SD of
Systematic Risk
SD of
Residual
Correlation
with SENSEX
Sensex 0.282579515 1 0.081573679 0 1
HUL 0.245925901 0.258483191 0.021085425 0.23684419 0.297008385
RIL 0.399363379 0.968563069 0.079009253 0.29332553 0.685330945
BhartiAirtel 0.403878862 0.611586371 0.049889351 0.3681687 0.427904989
MindTree 0.436164384 0.93054749 0.075908183 0.35097376 0.602877419
MarutiSuzuki 0.400745456 0.895108264 0.073017275 0.31350357 0.63117187
HDFC 0.323563407 1.000079686 0.08158018 0.15892596 0.873405417
(B) Covariance Matrix
Sensex HUL RIL BhartiAirtel MindTree MarutiSuzuki HDFC
BETA 1 0.258483191 0.96856307 0.611586371 0.93054749 0.895108264 1.000079686
Sensex 1 0.079851182 0.020640188 0.07734091 0.048835895 0.074305317 0.071475453 0.079857545
HUL 0.258483191 0.020640188 0.005335142 0.01999132 0.012623258 0.019206675 0.018475203 0.020641833
RIL 0.968563069 0.077340906 0.019991324 0.07490955 0.047300644 0.071969386 0.069228484 0.077347069
BhartiAirtel 0.611586371 0.048835895 0.012623258 0.04730064 0.029867368 0.045444119 0.043713413 0.048839786
MindTree 0.93054749 0.074305317 0.019206675 0.07196939 0.045444119 0.069144626 0.066511303 0.074311238
MarutiSuzuki 0.895108264 0.071475453 0.018475203 0.06922848 0.043713413 0.066511303 0.063978269 0.071481149
HDFC 1.000079686 0.079857545 0.020641833 0.07734707 0.048839786 0.074311238 0.071481149 0.079863909
( C ) Market Forecast and Alpha values
Expected Market Return 0.13
Rf 0.08
MRP 0.05
ALPHA 0.013580316 -0.013503 -0.011325084 0.021255731 0.010830176 0.015553917
Expected Return 0.026504476 0.03492515 0.019254235 0.067783106 0.055585589 0.065557901

7. Single Index Model
(D) Optimal Risk Portfolio
Optimal Risky Portfolio Sensex Active Port HUL RIL BhartiAirtel MindTree MarutiSuzuki HDFC
SD of Residual 0.236844195 0.29332553 0.368168704 0.350973765 0.313503574 0.158925964
σ2(e) 0.056095173 0.08603987 0.135548195 0.123182584 0.098284491 0.025257462
α/σ2(e) 0.242094208 -0.1569389 -0.083550237 0.172554679 0.110192116 0.61581471
W0(i) 1 0.26894378 -0.1743443 -0.092816415 0.191691936 0.122413025 0.68411193
[W0(i)]2 0.072330757 0.03039592 0.008614887 0.036745798 0.014984949 0.468009133
αA 0.023098593
σ2(eA) 0.025660353
W0A 5.087372816
W*(Riksy portf) -1.627716744 2.627716744
Beta 1 0.816006589 0.258483191 0.96856307 0.611586371 0.93054749 0.895108264 1.000079686
Risk Premium 0.05 0.063898922 0.026504476 0.03492515 0.019254235 0.067783106 0.055585589 0.065557901
SD 0.282579515 0.230586746
Sharpe Ratio 0.176941347 0.277114462

8. Regression Output
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.297008385
R Square 0.088213981
Adjusted R Square 0.072493532
Standard Error 0.06837103
Observations 60
ANOVA
df SS MS F Significance F
Regression 1 0.026231114 0.026231114 5.61141625 0.021195108
Residual 58 0.271126667 0.004674598
Total 59 0.297357781
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 0.013580316 0.008833331 1.537394626 0.12963466 -0.004101518 0.03126215 -0.004101518 0.03126215
X Variable 1 0.258483191 0.109117916 2.368842808 0.02119511 0.040059952 0.47690643 0.040059952 0.47690643
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.685330945
R Square 0.469678504
Adjusted R Square 0.46053503
Standard Error 0.084675787
Observations 60
ANOVA
df SS MS F Significance F
Regression 1 0.368305264 0.368305264 51.3676204 1.53554E-09
Residual 58 0.415859352 0.007169989
Total 59 0.784164616
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept -0.013503 0.010939857 -1.234293996 0.22207112 -0.035401503 0.008395503 -0.035401503 0.008395503
X Variable 1 0.968563069 0.135139772 7.167120787 1.5355E-09 0.698051427 1.239074712 0.698051427 1.239074712

9. Regression Output
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.427904989
R Square 0.183102679
Adjusted R Square 0.169018243
Standard Error 0.10628115
Observations 60
ANOVA
df SS MS F Significance F
Regression 1 0.146847891 0.146847891 13.0003553 0.000648599
Residual 58 0.655149607 0.011295683
Total 59 0.801997497
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept -0.011325084 0.013731205 -0.824769836 0.4128815 -0.038811077 0.016160909 -0.038811077 0.016160909
X Variable 1 0.611586371 0.169621222 3.605600544 0.0006486 0.272052608 0.951120134 0.272052608 0.951120134
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.602877419
R Square 0.363461183
Adjusted R Square 0.352486376
Standard Error 0.101317399
Observations 60
ANOVA
df SS MS F Significance F
Regression 1 0.339961079 0.339961079 33.1177739 3.44476E-07
Residual 58 0.595382488 0.010265215
Total 59 0.935343567
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 0.021255731 0.013089903 1.623826441 0.1098359 -0.004946557 0.047458019 -0.004946557 0.047458019
X Variable 1 0.93054749 0.161699238 5.754804417 3.4448E-07 0.606871301 1.25422368 0.606871301 1.25422368

10. Regression Output
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.63117187
R Square 0.398377929
Adjusted R Square 0.388005135
Standard Error 0.090500686
Observations 60
ANOVA
df SS MS F Significance F
Regression 1 0.314559821 0.314559821 38.4060376 6.43072E-08
Residual 58 0.475041705 0.008190374
Total 59 0.789601526
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 0.010830176 0.011692417 0.926256429 0.35815099 -0.012574739 0.034235091 -0.012574739 0.034235091
X Variable 1 0.895108264 0.144436121 6.197260491 6.4307E-08 0.605987958 1.18422857 0.605987958 1.18422857
SUMMARY OUTPUT
Regression Statistics
Multiple R 0.873405417
R Square 0.762837022
Adjusted R Square 0.758748006
Standard Error 0.045877974
Observations 60
ANOVA
df SS MS F Significance F
Regression 1 0.392664217 0.392664217 186.557564 8.93872E-20
Residual 58 0.122077734 0.002104789
Total 59 0.514741951
Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0%
Intercept 0.015553917 0.005927296 2.624116605 0.01108514 0.003689144 0.027418689 0.003689144 0.027418689
X Variable 1 1.000079686 0.073219739 13.65860768 8.9387E-20 0.853514456 1.146644916 0.853514456 1.146644916