CAPITAL ASSET PRICING MODEL (CAPM) AND THE FAMA FRENCH THREE FACTOR MODEL

Question One

Assumptions of Capital Asset Pricing Model (CAPM)

Sharp (1964) and Lintner (1965), made a number of assumptions with regard to the formulation of the Capital Asset Pricing Model. These assumptions and their implications in the real world are outlined as follows;

In the first place, each individual investor can invest any portion of his capital resources in specific risk free assets such as treasury bonds that pay a given exogenously determined interest rates.

Implication

The implication of the assumption is that the risk averse investors would have a range of portfolio of assets to choose from that will ensure that the overall portfolio risk is minimized. In addition it is possible for investors to evaluate the market risk premium of their portfolio of assets. Secondly, the investor can invest any portion of his capital resources in a number of risky securities that are traded in the market.

Implications

The implication of the assumption is that investors who are risk takers would be able to choose a portfolio of risky assets that would ensure that they maximize their portfolio returns with a given level of risk due to the fact that market securities are more risky compared to the risk free securities.

In addition, investors participate in a purely competitive market in which there are no transaction costs as well as personal and corporate taxes. This means that investors trade their securities in a perfectly competitive market.

Implications

The implication for this assumption is that securities markets are perfectly efficient and all investors have all the available stock information which implies that there are no insider dealings. In addition, the assumption implies that investors do not incur transaction brokerage costs.

Additionally, the paper by Sharpe (1964) and Lintner (1965) makes an assumption that investors may borrow any amount of money so as to invest in any risky assets that earn a higher return compared to the risk free assets.

Implications

The implication of the assumption is that investors may borrow money from the capital market or reduce their proportions of funds allocated for risk free assets so as to invest in risky assets and increase their overall portfolio returns. Moreover, investors can make purchases and sales transactions of their securities at a certain discrete transaction points.

Implications

The implication of this assumption is that the investors use a standardized single period transaction horizon which implies that the returns of the market securities can only be compared to the returns on the risk free securities after a certain period e.g. six months or after one year. The papers also make an assumption that investors’ capital consist of the cash and other resources that are specifically set aside for profitable investments in market securities.

Implications

The implication is that there is no competition for investors’ funds with respect to the need for liquidity transaction requirements as well as investments requirements. In addition, the papers make an assumption that each investor trading in market securities and risk free securities will have a given unique utility indifference curve that specifies their risk return preference for a given portfolio.

Implication

The implication of the assumption is that investors can choose to select portfolios with a higher market returns for the same risk or portfolios with a lower risk for the same market return. Finally, the papers by Sharpe and Lintner, assume that investors will always be rational and would always seek to maximize their wealth or portfolio value

Implication

The implication of this assumption is that if any two compositions of assets have the same expected returns investors would prefer the one with a smaller variance while if any two mixture of assets have the same variance, then investors would prefer the one with a greater market returns.

Question Two

Validity of the Capital Asset Pricing Model (CAPM) Assumptions

Investments in Risk-Free Assets

The assumption that each investor can invest a portion of his capital in certain risk free assets is valid in the real world. This is because there are numerous risk free assets that are available today such as treasury stocks and bonds. However, the breach of this assumption would imply that investors will only be able to select portfolios that consist of risky assets only and moreover, there would be no risk premium returns as well as the capital market line for investors.

Investments in Risky Assets

The assumption that investors can invest a portion of their capital in a number of risky assets is valid both in the ideal and the real world. This is because various capital markets have a number of securities that earn a higher market return although they are much risky compared to the risk free assets.

The implication for the CAPM if this assumption is breached is that portfolio return would be equal to the risk free return only and that investors would not be able to earn the risk premium on any given portfolio of assets.

Perfectly Competitive Securities Markets

The assumption that investors trade securities in perfectly competitive markets is not valid in the real world. This is because there is no capital market that is perfectly efficient since in most securities market, there are is much transaction costs in the form of brokerage costs and other personal and corporate taxes.

However, the implication if this assumption is breached is that the capital market line would not be linear. This is because, investors would be making their portfolio choices while considering the effect of the transaction costs and taxes rather than based on only the risk- return tradeoff.

Investors Borrow to Invest in Risky Assets

This assumption that investors are free to borrow any amount of money so as to invest in risky assets is valid in the real world. This is because; the capital market allows investors to borrow so as to invest in more risky assets. Alternatively, investors may reduce their proportions of funds that are invested in the risk free assets so as to increase their portfolio of risky assets that earn higher income. However, if this assumption is breached, the implication is that investors may not be able to move along their indifference curve so as to obtain an optimal portfolio of assets.

Single Transaction Period Horizon

This assumption is valid in the real world due to the fact that comparisons of the assets returns and risk is usually done after a specific period of time such as annually. The implication if this assumption is breached is that investors and analysts might find it difficult to compare and analyze the returns and risks of both the risky assets as well as the risk free assets.

Investors’ Funds for Capital Investment is Distinct

This assumption is much valid in the real world due to the fact that almost all investors have cash that is dedicated for liquid transactions as well as cash for investment purposes.

The implication if this assumption is breached is that investors may decide to use the funds available for the purchase of securities for their own personal transactions and thus reduce the proportion of securities in their portfolio.

Investors make Rational Investment Decisions

This assumption is valid in the real world due to the fact that each investor would always want to maximize his portfolio returns subject to a given risk preference. If this assumption is breached is that the investors’ indifference curves, the implication would however not be concave. This implies that there would be instances when investors would choose assets that have a higher variance for any given mix of assets that have the same return.

Question Three

CAPM Regression Model for “Fama French” 30 Year Data Series: Telecommunications Industry Security

Using the excess return model; R_pt – R_ft = α + b*(R_mt – R_ft) + ε_t an empirical test of the CAPM using the data file “Fama French”, the following CAPM regression model was derived from the results shown in table 1 (Appendix 1). R_pt – R_ft = 0.19 + 0.43 (R_mt – R_ft) + ε_t

(0.16)  (0.02)

With an adjusted R square of 57%, the model seems to be statistically significant due to the fact that 57% of the variations in the excess returns of the market portfolio are explained by the movement in the Telecommunication security market premium.

Hypothesis Testing

Given that H_ (0): α = 0

H_ (1): α = 0

In addition, the t-statistic from the regression results indicate that α has a t-statistic of 1.21. Therefore, since t = 1.21 < than t 0.95 = 1.96 then the CAPM regression model agrees with the null hypothesis which implies that both the model and its intercept α are statistically significant.

CAPM Regression Model for “Fama French” 30 Year Data Series: Health Industry Security

The following CAPM regression model was derived from the results shown in table 2 (Appendix 1) using the excess return model.

R_pt – R_ft = 0.08 + 0.42 (R_mt – R_ft) + ε_t

(0.17)  (0.02)

With an adjusted R square of 50%, the model seems to be relatively statistically significant due to the fact that 50% of the variations in the excess returns of the market portfolio are explained by the movements in the Health security market premium.

Hypothesis testing

Given that H_ (0): α = 0

H_ (1): α = 0

The t-statistic from the regression results indicate that α has a t-statistic of 0.47. Therefore, since t = 0.47 < t 0.95 = 1.96 then the null hypothesis is accepted and that both the model as well as its intercept α are statistically significant at 95% confidence level.

 

 

CAPM Regression Model for “Fama French” First 10 Year Data Series: Telecommunications Industry Security

The following CAPM Regression model was derived from the results shown in table 3 (Appendix 1) using the excess return model.

R_pt – R_ft = -0.07 + 0.67 (R_mt – R_ft) + ε_t

(0.25)  (0.04)

Given that the adjusted R square for this model is 69%, then the model seems to be statistically significant due to the fact that 69% of the variations in the excess market portfolio returns are explained by the variations in the Telecommunications industry security market return premium.

Hypothesis Testing

Given that H_ (0): α = 0

H_ (1): α = 0

The t-statistic from the regression results indicate that α has a t-statistic of -0.26. Therefore, since t = (-0.26) < t 0.95 = 1.96, then the null hypothesis is accepted and that both the model as well as it’s α will be significant at 95% confidence level.

CAPM Regression Model for “Fama French” First 10 Year Data Series: Health Industry Security

The model below was derived from the results depicted in table 4 (Appendix 1) using the excess market portfolio return model.

R_pt – R_ft = 0.28 + 0.59 (R_mt – R_ft) + ε_t

(0.24)  (0.04)

The adjusted R square for the model is 70% which implies that almost 60% of the variations in the excess market portfolio returns are explained by the movement in the Health industry security market return premium.

Hypothesis Testing

Given that H_ (0): α = 0

H_ (1): α = 0

The t-statistic from the regression results indicate that α has a t-statistic of 1.17. Therefore given that the value of t = 1.17 < t 0.95 = 1.96, then both the model as well as its intercept α are statistically significant.

CAPM Regression Model for “Fama French” Second 10 Year Data Series: Telecommunications Industry Security

The model below was derived from the results depicted in table 5 (Appendix 1) using the excess return market portfolio model.

R_pt – R_ft = 0.33 + 0.30 (R_mt – R_ft) + ε_t

(0.26)  (0.05)

The adjusted R square for the model is 53% which implies that almost 53% of the variations in the excess market portfolio returns are explained by the movement in the Telecommunications industry security market return premium.

Hypothesis Testing

Given that H_ (0): α = 0

H_ (1): α = 0

The t-statistic from the regression results indicate that α has a t-statistic of 1.17. Therefore given that the value of t = 1.29 < t 0.95 = 1.96, then both the model as well as its intercept α are statistically significant.

CAPM Regression Model for “Fama French” Second 10 Year Data Series: Health Industry Security

The model below was derived from the results depicted in table 6 (Appendix 1) using the excess return market portfolio model.

R_pt – R_ft = 0.60 + 0.36 (R_mt – R_ft) + ε_t

(0.29)  (0.04)

The adjusted R square for the model is 40% which implies that almost 40% of the variations in the excess market portfolio returns are explained by the movement in the Health industry security market return premium.

Hypothesis Testing

Given that H_ (0): α = 0

H_ (1): α = 0

The t-statistic from the regression results indicate that α has a t-statistic of 2.09. Therefore given that the value of t = 2.09 < t 0.95 = 1.96, then both the model as well as its intercept α are not statistically significant which implies that the null hypothesis is rejected.

CAPM Regression Model for “Fama French” Final 10 Year Data Series: Telecommunications Industry Security

The model below was derived from the results depicted in table 7 (Appendix 1) using the excess return market portfolio model.

R_pt – R_ft = 0.016 + 0.35 (R_mt – R_ft) + ε_t

(0.27)  (0.03)

The adjusted R square for the model is 60% which implies that almost 60% of the variations in the excess market portfolio returns are explained by the movement in the Telecommunications industry security market return premium.

Hypothesis Testing

Given that H_ (0): α = 0

H_ (1): α = 0

The t-statistic from the regression results indicate that α has a t-statistic of 0.06. Therefore given that the value of t = 0.06 < t 0.95 = 1.96, then both the model as well as its intercept α are statistically significant at 95% confidence level.

CAPM Regression Model for “Fama French” Second 10 Year Data Series: Health Industry Security

The following model was derived from the results depicted in table 8 (Appendix 1) using the excess return market portfolio model.

R_pt – R_ft = -0.61 + 0.36 (R_mt – R_ft) + ε_t

(0.32)  (0.04)

The adjusted R square for the model is 46% which implies that almost 46% of the variations in the excess market portfolio returns are explained by the movement in the Health industry security market return premium.

Hypothesis Testing

Given that H_ (0): α = 0

H_ (1): α = 0

The t-statistic from the regression results indicate that α has a t-statistic of -1.88. Therefore given that the value of (t = -1.88) < t 0.95 = 1.96, then both the model as well as its intercept α are statistically significant.

Validity of the Empirical Model to the CAPM Theory

This model agrees with the Capital Asset Pricing Model theory in various aspects. In the first place, the standard error of the model using the 30 year data series indicate that the value of 3.00 and 3.23 for the telecommunication and health industry respectively is a bit higher compared to the standard deviation of a portfolio that consist of both industries which is computed at 2.89. This therefore, agrees with the CAPM theory that if the correlation coefficient between the securities of the two industries is less than 1, then the portfolio that comprises both securities would have a much smaller variance compared to the variance of each individual security. This is the effect of diversification. In addition, the model is statistically significant, which implies that its validity to the CAPM theory is much higher. This is evidenced by the positive beta coefficient.

Question Four

Background and Features of the Fama French Three Factor Model

The Fama French three-factor model is an asset valuation model that specifically expands on the Capital Asset Pricing Model by including in the model the size and value factor of a certain security in addition to the risk factor that is normally considered under CAPM. This model was formulated by Fama and French who sought to find a formula that would better measure market returns than is provided under CAPM.

The features of the Fama French three-factor model include the fact that the model incorporates the size of the security as well as the value of the security in addition to the risk factor that is normally considered under CAPM. In addition, the model assumes that the small cap stocks with a higher value tend to outperform most securities in the capital market. In essence, the three factor model states that higher value stocks tend to outperform growth stocks while small cap stocks tend to outperform large cap stocks. Finally, the model seems to provide a better evaluation tool compared to the CAPM Model.

Question Five

Fama French Three Factor Model: Telecommunications Industry Security

Using the excess return portfolio model, an empirical test was conducted to test the validity of the Fama French Model. The following regression model was obtained:

R_pt – R_ft = 0.26 + 0.47 (R _mt – R_ft) -0.34(SMB) _t -0.15(HML) _t + ε_t

(0.15)  (0.02)                         (0.06)                (0.05)

With an adjusted R square of 61%, the model seems to be statistically significant due to the fact that 61% of the variations in the excess return portfolio model are explained by the telecommunications security market return premium as well as the size and value of the securities in the portfolio.

Hypothesis Testing

Given that H_0: α = 0, b = 0, s = 0, h = 0

H_1: α = 0, b = 0, s = 0, h = 0

The t-statistic from the regression results indicate that t α = 1.67, t b = 20.9, t s = -6.2, t h = -2.76. When the statistical t is compared to the t 0.95 = 1.96, then the null hypothesis for the model intercept α will be the only one accepted. Therefore, the rest of the coefficients would not pass the test and will be considered not statistically significant.

Fama French Three Factor Model: Health Industry

Using the excess return portfolio model, an empirical test was conducted to test the validity of the Fama French three factor model based on the health industry security. The following model was obtained.

R_pt – R_ft = 0.04 + 0.57 (R _mt – R_ft) -0.64(SMB) _t -0.11(HML) _t + ε_t

(0.15)  (0.03)                         (0.06)                (0.05)

The model also has an adjusted R square of 61% which implies that 61 % of the variations in the excess return portfolio model are explained by the movements in the health industry market risk premium as well as the size and value of the securities in the portfolio.

Hypothesis Testing

Given that H_0: α = 0, b = 0, s = 0, h = 0

H_1: α = 0, b = 0, s = 0, h = 0

The t-statistic from the regression results indicate that t α = 0.25, t b = 20.9, t s = -10, t h = -2.10. When the values of the statistical t are compared to the t 0.95 = 1.96, only the model’s intercept passes the test due to the fact that tα = 0.25 < t 0.95 . The other coefficients failed to pass the hypothesis test and would thus be considered not statistically significant at 95% confidence level.

Implication of the Empirical Results to the Fama French Theory

The results of the empirical results tend to prove the validity of the Fama French theory. This is because, in the first place, the stock size coefficient in the model is negative and less than one. This implies that a small cap stock that is indicated by a small stock size coefficient, will lead to an increase in the excess portfolio return in the model. Moreover, the stock value coefficient in the model similarly implies that it would cause an increase in the value of the excess portfolio return.

Moreover, the Fama French three-factor model tends to explain a higher proportion of the variations in the stock’s beta compared to the CAPM. This is due to the fact that the coefficient of determination in the CAPM model which is 57% and 50% for the telecommunication and health industry respectively is much a lower compared to the coefficient of multiple determinations of 61% each for both the telecommunications and health industry respectively.

Comparison of the Fama French Three Factor Model and CAPM Results

The main difference between the results from the Fama French three factor model and the CAPM model is that the Fama French three factor model has a greater explanatory power with an adjusted R square of 61% compared to the adjusted R square of 50% and 57% for the telecommunications and the health industry respectively. There the Fama French model is a better evaluation tool for managers.

References

Craig, MA 2006, ‘Multifactor models do not explain deviations from the CAPM’, Journal of Financial Economics, vol. 38, no. 1. pp. 3-28.

Harvey 2013, ‘The risk exposure of emerging equity markets’, World Economic Review, vol. 9, no. 1, pp. 19-50.

John, P 2014, The Fama French three factor model [online] Available from: <https://www.portfoliosolutions.com&gt; [Accessed on 11th December 2014]

Lintner, J 2013, ‘The valuation of risk assets and the selection of risky investments in stock portfolios and capital budget’, The Review of Economics and Statistics, vol. 47, no. 1, pp. 13-37

Lintner, J 2014, ‘The valuation of risk assets and the selection of risky investments in stock portfolios and capital budget’, The Review of Economic and Statistics, vol. 47, no. 1. pp. 13-37.

Perold, FA 2004, ‘The Capital Asset Pricing Model’, Journal of Economic Perspective, vol. 18, no. pp. 3–24.

Reinganum, MR 2012, ‘The anomalous stock market behaviour of small firms in January’, Journal of Financial Economics, vol. 38, pp. 89-104.

Sharpe, FW 2014, ‘A theory of market equilibrium under conditions of risk’, American Finance Association, vol. 19, no. 3, pp. 425-442.

Wilcox, JW 2001, ‘Taming frontier markets’, Journal of Portfolio Management, vol. 19, no. 1, pp. 51-55.

Wilcox, JW 2002, ‘Taming frontier markets’, Journal of Portfolio management, vol. 20, no. 1, pp. 50-57.

 

Appendix 1: Summary Output for the CAPM Regression

Table 1: Summary Output for “Fama French” CAPM Regression using the 30 Year Data Series: Movement of the Excess Return Market Portfolio with the Telecommunications Industry Security

Regression Statistics
Multiple R 0.75767723
R Square 0.57407479
Adjusted R Square 0.57288506
Standard Error 3.00194143
Observations 360
ANOVA
  df SS MS F
Regression 1 4348.330949 4348.331 482.5232
Residual 358 3226.171551 9.011652
Total 359 7574.5025
  Coefficients Standard Error t Stat P-value
Intercept 0.19310535 0.159011321 1.214413 0.225391
X Variable 1 0.42538976 0.019365466 21.96641 2.48E-68

Table 2: Summary Output for “Fama French” CAPM Regression using the 30 Year Data Series: Movement of the Excess Return Market Portfolio with the Health Industry Security

Regression Statistics
Multiple R 0.71137184
R Square 0.5060499
Adjusted R Square 0.50467015
Standard Error 3.23278693
Observations 360
ANOVA
  df SS MS F
Regression 1 3833.076232 3833.07623 366.7696
Residual 358 3741.426268 10.4509114
Total 359 7574.5025
  Coefficients Standard Error t Stat P-value
Intercept 0.08026852 0.172079981 0.46646053 0.64117
X Variable 1 0.41993212 0.021927163 19.1512289 8.72E-57

Table 3: Summary Output for “Fama French” CAPM Regression using the First 10 Year Data Series: Movement of the Excess Return Market Portfolio with the Telecommunication Industry Security.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.832764479
R Square 0.693496678
Adjusted R Square 0.690899192
Standard Error 2.712641994
Observations 120
ANOVA
  df SS MS F
Regression 1 1964.609175 1964.609 266.9877
Residual 118 868.2943372 7.358427
Total 119 2832.903513
  Coefficients Standard Error t Stat P-value
Intercept -0.06746535 0.252144638 -0.26757 0.7895
X Variable 1 0.665755428 0.040744511 16.33976 4.39E-32

Table 4: Summary Output for “Fama French” CAPM Regression using the First 10 Year Data Series: Movement of the Excess Return Market Portfolio with the Health Industry Security.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.839878081
R Square 0.705395191
Adjusted R Square 0.70289854
Standard Error 2.659468214
Observations 120
ANOVA
  df SS MS F
Regression 1 1998.316513 1998.317 282.5366
Residual 118 834.5869996 7.072771
Total 119 2832.903513
  Coefficients Standard Error t Stat P-value
Intercept 0.284483163 0.244083703 1.165515 0.246161
X Variable 1 0.592069084 0.035223706 16.80882 4.21E-33

Table 5: Summary Output for “Fama French” CAPM Regression using the Second 10 Year Data Series: Movement of the Excess Return Market Portfolio with the Telecommunications Industry Security.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.7358754
R Square 0.5415126
Adjusted R Square 0.5336752
Standard Error 2.7256613
Observations 120
ANOVA
  df SS MS F
Regression 2 1026.622521 513.3113 69.09347
Residual 117 869.2198661 7.42923
Total 119 1895.842387
  Coefficients Standard Error t Stat P-value
Intercept 0.3320095 0.257039962 1.291665 0.199019
X Variable 1 0.304937 0.052032705 5.860487 4.35E-08

Table 6: Summary Output for “Fama French” CAPM Regression using the Second 10 Year Data Series: Movement of the Excess Return Market Portfolio with the Health Industry Security.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.637905779
R Square 0.406923783
Adjusted R Square 0.401897714
Standard Error 3.086848854
Observations 120
ANOVA
  df SS MS F
Regression 1 771.4633568 771.4634 80.96262
Residual 118 1124.37903 9.528636
Total 119 1895.842387
  Coefficients Standard Error t Stat P-value
Intercept 0.599681714 0.286468722 2.093358 0.038459
X Variable 1 0.35531713 0.039488793 8.997923 4.66E-15

Table 7: Summary Output for “Fama French” CAPM Regression using the Final 10 Year Data Series: Movement of the Excess Return Market Portfolio with the Telecommunications Industry Security.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.779549619
R Square 0.607697609
Adjusted R Square 0.604373013
Standard Error 3.0252627
Observations 120
ANOVA
  df SS MS F
Regression 1 1672.91843 1672.918 182.7884
Residual 118 1079.9613 9.152214
Total 119 2752.87973
  Coefficients Standard Error t Stat P-value
Intercept 0.016023084 0.276428938 0.057965 0.953875
X Variable 1 0.348886763 0.025805377 13.51993 9.88E-26

Table 8: Summary Output for “Fama French” CAPM Regression using the Final 10 Year Data Series: Movement of the Excess Return Market Portfolio with the Health Industry Security.

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.687874887
R Square 0.47317186
Adjusted R Square 0.468707215
Standard Error 3.505799683
Observations 120
ANOVA
  df SS MS F
Regression 1 1302.585223 1302.585 105.982
Residual 118 1450.294507 12.29063
Total 119 2752.87973
  Coefficients Standard Error t Stat P-value
Intercept -0.609372121 0.323177203 -1.88557 0.061812
X Variable 1 0.362752446 0.035236631 10.29475 4E-18

Appendix 2: Fama French Three Factor Model

Table 1: Summary Results for Fama French Three Factor Model: Telecommunications Industry Security

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.786856078
R Square 0.619142488
Adjusted R Square 0.615933015
Standard Error 2.846645067
Observations 360
ANOVA
  df SS MS F
Regression 3 4689.696323 1563.232 192.9109
Residual 356 2884.806177 8.103388
Total 359 7574.5025
  Coefficients Standard Error t Stat P-value
Intercept 0.2557664 0.15303518 1.671292 0.095543
X Variable 1 0.46890494 0.022427955 20.90716 6.78E-64
X Variable 2 -0.34431206 0.05528231 -6.22825 1.33E-09
X Variable 3 -0.14558302 0.052716972 -2.7616 0.00605

Table 2: Summary Results for Fama French Three Factor Model: Health Industry Security

SUMMARY OUTPUT
Regression Statistics
Multiple R 0.787283913
R Square 0.61981596
Adjusted R Square 0.616612161
Standard Error 2.844127087
Observations 360
ANOVA
  df SS MS F
Regression 3 4694.797535 1564.933 193.4629
Residual 356 2879.704965 8.089059
Total 359 7574.5025
  Coefficients Standard Error t Stat P-value
Intercept 0.039098143 0.154680088 0.252768 0.800594
X Variable 1 0.570019499 0.027220603 20.94074 4.94E-64
X Variable 2 -0.636800296 0.062449528 -10.197 1.37E-21
X Variable 3 -0.109502327 0.053236389 -2.05691 0.040423

 

 

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