Factor risk decomposition

By Bram Baneke

© June 2013

University of Amsterdam

Actuarial science

Thesis supervisor: J. Cui

Student number: 6131298

Content Content ...... 1 1. Introduction ...... 1 2. Related literature ...... 3 3.Data and Method ...... 5 3.1 Data ...... 5

3.2 Method ...... 7

4. Results ...... 9 4.1 Return contribution: basic model ...... 9

4.2 Return contribution: extended model ...... 11

4.3 Risk contribution: asset based ...... 13

4.4 Risk contribution: factor based ...... 14

5. Conclusion ...... 15 References ...... 17 Appendix ...... 18

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1. Introduction A pension fund wants to give their participants a good pension. In order to give the participants a high pension the pension fund needs to raise a high return. A good pension also has a very stable return or low risks. Since a diversified portfolio bears less risk, pension funds will diversify their portfolio. Prior to the financial crisis of 2007/2008 most pension funds diversified their portfolio based on asset classes, like stocks, public debt, corporate bonds, real estate, commodities, liquidity. An example of a typical 60/40 portfolio is the portfolio in graph 1, where 60% are stocks (globally diversified) and 40% are bonds.

Portfolio S&P500 FTSE AEX MSCI EM Barclays US investment grade Barclays US High yield Barclays US treasury

Graph 1: Example of a portfolio

The question is whether the diversification can provide the low risk desired by the pension fund. The diversification did not serve their needs in the crisis of 2007/'08. Though it is not possible to avoid all damage, an attentive risk-control can limit the damage. In particular, since most of the assets took losses at the same period, one's mind should be open to other dimensions by questioning what common factors are driving the asset returns. An information transformation mapping a portfolio of positions in individual securities categorized by asset classes, presents the portfolio as a mix of variables or factors of another nature. The new dimension may offer macro- economic or general financial variables, but other time-series of any nature as well. This thesis tries to shed lights on understanding factor exposures of pension funds’ portfolio.

In order to analyze the factor exposure, this study will make a factor-return-analysis and a factor- risk- decomposition. The research question of this thesis consists of two parts. The first thing to

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be studied will be what risk factors are driving the portfolio returns. This first part will be called the return contribution. The second part of this question is how much risk (as measured by return volatility) can be attributed to each factor. This part is called the risk contribution.

One can link a set of measurable factors via the market conditions, which were mentioned above, to the return on a portfolio in a reduced form. This could be done with the portfolio return being the dependent variable, the standard deviation could also serve as the dependent variable. This study shall not do so; here the portfolio risk will be analyzed by using the covariance matrix as is current in the modern portfolio-theory.

The section 2 will discuss the literature on the subject. In section 3 the method and the data will be presented and discussed. In section 4 will include the results and the interpretation of the results. In section 5 is the conclusion.

Main finding of this thesis is the large exposure to the world market factor. Also, in order to make a factor based exposure analysis, the relation between the factor and the portfolio should be very solid. The factor exposure analysis gives an alternative insight in the risks a portfolio is exposed to.

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2. Related literature In order to make a factor exposure analysis, the return decomposition and the risk decomposition are important. These will be discussed in his section in subsequent order.

Many asset pricing theories have developed over the years. One of the first theories was the capital asset pricing model (CAPM). The CAPM is a theory based on one factor, the beta of an asset. This model was found to be inefficient. Therefore a later model, arbitrage pricing theory (APT), was developed by Ross (1976). This theory is based on multiple factors and explains the returns of assets by a linear relation with multiple factors. If the price of the asset doesn’t match with the expected price by the APT, an arbitrage opportunity will present itself. The arbitrageur will then take this opportunity. If multiple arbitrageurs will do this, then the opportunity will disappear.

Ang, Goetzmann and Schaefer (2009) evaluated active management of the Norwegian government pension fund. A benchmark is used by the fund to invest. Active management tries to take advantage of opportunities in the market by deviating from this benchmark. First they researched the theoretical possibility of active management by examining the Efficient Market Hypothesis, since active management is much more useful when markets are not efficient. Later they researched the excess returns produced by active management and the relation between these returns and some factors. Finally they take a closer look at the characteristics of the fund and how the fund can profit from these characteristics.

Ang, Goetzmann and Schaefer (2009) try to find evidence of violations of the EMH. They find that several investment strategies outperform the market and therefore violate the EMH. The factors they use in their evaluation are:

• Term risk • Credit risk • FX carry factor • Liquidity factor • Value factor • Size factor

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• Momentum factor • Volatility risk factor

Ang, Goetzmann and Schaefer (2009) test if active management makes an excess return. The NBIM uses a benchmark for their investment. Active management means deviating from this benchmark. Therefore, the active management returns will be the deviation of the returns of the portfolio caused by the active management.

This thesis has some differences with their study. The major difference is the focus. Whereas they focus on the active returns, this thesis focuses on the whole portfolio. Therefore the regression will be on the portfolio returns minus the risk free rate instead of the portfolio returns minus the benchmark as done in Ang, Goetzmann and Schaefer (2009) paper. Given our focus, two additional factors must be added to the analysis: i.e., the global equity risk factor and the inflation risk factor. As the CAPM and Fama and French (1993) suggests, the world market performance could be included to represent the global equity risk factor. An explanation for the absence of this factor in the study of active management study is the benchmark. This already includes the world market performance. The inflation risk factor should have an effect on the returns of the bonds (Chen Roll and Ross, 1986).

The modern portfolio theory (MPT) by Markowitz (1952) showed the effect of an introduction or increase of an asset in the portfolio mix on the aggregate portfolio return and risk, rather than looking at the risk and return of the asset on its own. The MPT considers hedging effects between components in a portfolio, which a diversified portfolio offers. It enables risk control by mix- management. The MPT uses the individual assets in the portfolio as the “factors”, and this study shall use the MPT-method replacing the individual assets by the factors.

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3. Data and Method This section is about the data and the method of this study. First the data will be discussed. The data set includes the portfolio data and the factor data, which will be discussed in that order. Later the method will be explained.

3.1 Data

In this study a dummy pension portfolio will be used as an illustration for the factor risk decomposition. This portfolio tries to represent a typical Dutch pension fund. All of the stock and bond data come from Datastream and are on monthly base in the period 1988-2012. The stock based assets included are:

• S&P500 index, that will represent the USA stocks of the portfolio

• FTSE100 index, that will represent the English stocks of the portfolio

• AEX index, that will represent the Euro stocks of the portfolio

• MSCI Emerging Markets Unadjusted United States Dollar, that will represent the emerging markets

The portfolio also includes bonds. The bond indexes used to illustrate this are:

• Barclays United States Corporate Investment Grade

• Barclays United States Treasury

• Barclays United States Corporate High Yield

60% of the portfolio will be stocks. The stocks will consist of two parts; the 90% developed equity (EQD) and the 10% emerging equity (EQE). The developed equity will be 55% S&P500 index, 30% FTSE100 index and 15%AEX index. The emerging equity will consist of the MCSI emerging markets index.

The 40% bonds will consist of 60% government bonds (TREASURY) and 40% non-government bonds (BNONGOV). The government bonds are represented by the Barclays United States Treasury. The non-government bonds consist of 90% cooperate bonds investment grade (Barclays

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United States Corporate Investment Grade) and 10 % high yield bonds (Barclays United States Corporate High Yield).

The characteristics of the time series are presented in table 1. BONDS TREASURY EQD EQE PORTF corporate Mean -0.226 -0.282 0.226 0.493 -0.145 Median -0.093 -0.211 0.833 0.907 0.219 Maximum 6.376 4.690 9.360 15.970 5.795 Minimum -8.321 -5.042 -16.829 -35.027 -12.523 Std. Dev. 1.604 1.347 4.159 7.078 2.668

Table 1 Return relative to risk-free 1988-2012 (monthly data)

Corporate bonds performed 0.22% less favorable than the risk-free reference-investment (three- months US-treasury bills known as T-bills). Treasury preformed 0.28% worse than the risk-free reference-investment. Since treasury includes all maturities, also the reference-investment, the long-term US- bonds didn’t perform well. Both the developed equity and the emerging equity outperformed the risk-free reference-investment.

The standard deviation is the characteristic for risk. As was to be expected, corporate bonds show a greater standard deviation than public debt, equities are more volatile than bonds, and shares in the emerging markets have a greater standard deviation than the developed world.

The factors in this study are:

CS Credit factor, the difference in returns between high and low rated bonds. Low rated bonds in general will have higher risk and therefore have higher returns, this factor is expected to be positive. This factor is also expected to have an effect on bonds. TS Term Factor, which is the difference in the returns of short term treasury and the yield of long term treasury. Since long term bonds in general have higher yields, this factor is expected to be positive most of the time. This factor tends to have an effect on bonds. WORLD Morgan Stanly Capital International world market performance. The factor is

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the index minus the risk free rate. INFL U.S.-Inflation MOM Momentum, this is the strategy of buying past winners and selling past losers. It’s the difference in return between past winners and past losers. Size Size factor, this is the difference between the returns of small companies minus the return of large companies. Small companies tend to have higher returns. Also one of the Fama and French factors, and in that study it has an effect on stocks. Value Value factor, this factor captures the difference in returns between high book-to-value stocks and low book-to-value stocks. Also one of the Fama and French factors and in that study it has an effect on stocks. Also known as the book-to-market effect. FXCARRY Foreign Exchange Carry-Trade, this is the difference in returns between currency strategies with high returns and strategies with low returns.

Table 2 Factors

The characteristics of the factors are presented in table 3.

CF TF WRLD FXCARRY Value Size INFL MOM Mean 0.22 0.37 0.10 0.17 0.26 0.16 0.23 0.59 Median 0.27 0.52 0.66 0.26 0.21 -0.06 0.23 0.66 Maximum 6.19 11.34 10.34 2.61 13.84 22.00 1.21 18.39 Minimum -8.23 -6.30 -21.14 -2.88 -12.60 -16.39 -1.93 -34.74 Std. Dev. 1.11 2.31 4.48 0.79 3.12 3.30 0.33 4.94 Table 3 Factor-characteristics 1988-2012 (monthly data)

As shown in table 3 all the factors have a positive mean. The momentum and the world factor have the highest standard deviation. Most of the data come from Kennedy French website. Others come from Datastream. A correlation matrix of the factors is shown in table A.3. The biggest positive correlation is between the credit factor and the world market factor (0.433).

3.2 Method

In order to measure the return contribution of the assets, this study will use Ordinary Least Squares regressions. In the return contribution this study tries to search for factors affecting the

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return of assets. This study will have to factor models: the basic model and the extended model. The regression equation of the basic model is:

= + + + +

𝑅𝑅𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 − 𝑅𝑅𝑓𝑓 𝛼𝛼 𝛽𝛽1 ∗ 𝐶𝐶𝐶𝐶 𝛽𝛽2 ∗ 𝑇𝑇𝑇𝑇 𝛽𝛽3 ∗ 𝑊𝑊𝑅𝑅𝐿𝐿𝐿𝐿 𝜀𝜀 Where Rport is the portfolio return, Rf is the risk free rate, CF is the credit factor, TF is the term factor and WRLD is the world market factor. The extended model has the equation:

= + +

𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑓𝑓 𝑖𝑖 𝑖𝑖 th 𝑅𝑅 − 𝑅𝑅 𝛼𝛼 𝛽𝛽 ∗ 𝐹𝐹 𝜀𝜀 Where Fi is the i factor of the extended model.

Later the modern portfolio theory will be used to determine the risk contribution. The risk contribution calculation will be based on the coefficients from the return contribution and the covariance matrix of the factors.

The risk contribution calculation will follow the formula based on the Modern Portfolio Theory of Markowitz:

= 𝜕𝜕𝜎𝜎𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝𝑝 𝑪𝑪′Σ 𝐶𝐶𝑖𝑖 𝐶𝐶𝑖𝑖 th 𝑖𝑖 Where Ci is the i factor coefficient from 𝜕𝜕the𝐶𝐶 return contribution√𝑪𝑪′Σ 𝑪𝑪regression, C is the vector of all the coefficients and Σ is the covariance matrix of the factors.

For comparison, the risk contribution will also be calculated based on assets. In this case the vector C is the weights of the assets in the portfolio. The matrix Σ will be the covariance matrix of the assets.

The risk contribution provides an inside look in the risk. If a factor has a high number in the risk contribution, the risk of the portfolio is heavy loaded on this factor. A low number means the factor has a low contribution in the portfolio’s risk. A negative number means the factor is reducing the risk, this is also known as a hedge.

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4. Results

4.1 Return contribution: basic model First the results of the basic model will be shown and discussed. This basic model will consist of three factors (credit factor, term factor and world market) and is part of the return contribution analysis. As mentioned in the method section an ordinary least squares regression has been done, where the asset returns and the portfolio returns are the dependent variable and the factors are the independent variables.

Equity Equity Bonds non- Bonds Portfolio developed emerging government government

Credit factor 0.28 0.61 0.49 -0.14 0.23 (2.40) (2.0) (6.0) (-2.09) (3.2) Term factor 0.07 -0.14 0.39 0.33 0.17 (1.3) (-1.1) (11.0) (10.8) (5.4) World 0.81 1.10 0.11 0.02 0.53 market (31.6) (16.4) (6.0) (1.1) (33.3) R2 0.82 0.57 0.39 0.38 0.83

Table 4 Basic model regression results: coefficients and (t-values). Significant factors are bold

Now the results of the basic model regression will be discussed. As shown in table 4, the term factor is not significant in the equity developed regression. The explanation behind this is the term factor is theoretically more related to bonds. The credit factor is positive and significant related to the equity developed. A high credit factor could indicate low returns in relative save bonds. This could stimulate participation in stocks, which could stimulate the stock indices in equity developed. As expected, the world market factor is very significant (t-value is 31.6). This can be explained by the fact that the equity developed is a part of the world market and therefore the relation is very significant in this regression. The R2 is 0.82, this is sufficient. The high R2 can be related to the very high significance of the world market factor.

In the third column of table 4 the results of the emerging equity regression are displayed. The credit factor in this regression is less significant than the developed equity. However table 4

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shows the coefficient is higher. The term factor is not significant in the emerging equity regression, since this factor is more related to bonds. The world market factor is significant and positive. However, in the developed equity it is more significant. An explanation for this could be the larger volatility in the emerging markets. Another explanation could be a bigger influence of the developed equity on the world market. The R2 of this regression is lower than the one in the developed equity. The smaller significance of the world market factor is probably the reason for this.

In the fourth column of table 4 the non-government bonds regression is shown. All three factors in this regression are significant. The credit factor could be explained by de variety in credit ratio’s of the non-government bonds. Since the credit factor is the difference in the returns between the 5-year constant maturity treasury bonds and BAA rated bonds, the non-government bonds could have higher returns when the credit factor is large. The term factor is very significant in this regression. The world market factor is also significant, the tendency of the world market is likely to have an effect on the corporate bonds, due to the relation between corporate bonds and corporate stocks. Noticeable is the lower coefficient and t-value of this factor relative to the equity regressions. The R2 of this is regression is 0.39, which is quite low.

The fifth column in table 4 displays the results of the government bonds regression. The credit factor coefficient is negative and significant. If the credit spread is high it could indicate low returns in the 5-year constant maturity treasury bonds, in that case the government bonds will have low returns. The term factor is very significant as was expected. The world market factor is not significant. The government bonds are not likely to be related with the world stock index. The R2 of this regression is 0.38.

The last column of table 4 shows the results of the portfolio regression. All the factors in this regression are positive and significant. Remarkable is the significance of the world market factor, in this regression the t-value of this factor is higher than any t-value in the other regressions. The R2 of 0.83 is also higher than any R2 of the other regression. Since all of the factors in the basic model are significant and the R2 is sufficient, a risk contribution analysis can be done based on this model. However the basic model will be extended with additional factors.

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4.2 Return contribution: extended model In the extended model more factors are included in the regression. The extra factors are: size, value, momentum, fx carry and inflation.

Equity Equity Bonds Bonds Portfolio developed emerging corporate government

Credit factor 0.22 0.45 0.45 -0.13 0.19 (1.9) (1.5) (5.4) (-1.8) (2.6) Term factor 0.07 -0.03 0.39 0.30 0.16 (1.1) (-0.3) (10.7) (9.9) (5.2) World market 0.79 1.00 0.09 0.03 0,51 factor (28.3) (14.2) (4.8) (1.7) (29.8) Size factor -0.01 0.40 0.03 -0.03 0,01 (0.4) (4.8) (1.2) (-1.9) (0.6) Value factor -0.05 -0.10 0.03 0.00 -0,03 (-1.4) (-1.1) (1.0) (0.0) (-1.3) Momentum -0.05 -0.07 -0.03 0.02 -0,03 factor (-2.1) (-1.3) (-1.9) (1.2) (-2.3) FX Carry 0.19 1.02 0.17 -0.10 0.17 factor (1.3) (2.8) (1.7) (-1.2) (1.9) Inflation -0.37 0.50 -0.19 -0.37 -0,29 factor (-1.1) (0.6) (-0.8) (-1.9) (-1.6) R2 0.83 0.61 0.40 0.40 0.84

Table 5 Extended model regression results: coefficients and (t-value). Significant factors are bold

The size factor is shown in the fifth row of table 5. This factor has a significant t-value on emerging equity. An explanation for the low t-value in the developed equity could be the fact that most of the stocks included in the developed equity are big stocks. But in emerging markets there will possibly be more small upcoming stocks and therefore the returns are related with the small- big factor. The size factor is a stock factor in the literature. There the low relevance for the bond regressions could be expected.

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The sixth row in table 5 shows the results of the value factor. None of the regressions show a significant value factor. Earlier studies show the value factor should affect the stock portfolios, however this is not the case in the regressions in this study.

The momentum factor is significant negatively related to the developed equity and the portfolio. The t-values in the corporate bond regression are almost significant. If the momentum factor is high, it indicates large differences between recent winners and losers. These large differences could suggest panic in the stock market, therefore the developed equity returns and the portfolio returns could go down. The positive relation in the government regression could be explained by the fact that the possible panic causes investors to buy less risky government bonds. However the momentum factor in this regression is not significant. Although this factor is significant the portfolio regression, the coefficient is very low (-0.03) and therefore the effect is marginal.

The coefficient of the fx carry factor is significant in the emerging equity regression. An explanation for this could be the bigger role of currency when investing in emerging markets. The fx carry factor is almost significant and positive in the portfolio regression. When the currency ratios are favorable, the portfolio is likely to go up. The fx carry doesn’t have an effect on the bonds, since all bonds are U.S. bonds.

The inflation factor has a negative, almost significant coefficient in the government bonds regression and the portfolio regression. Higher inflation makes investors demand a higher interest rate. A higher interest rate is likely to lower the bond index.

The R2 of the regressions in the extended model don’t increase significant with the regressions in the basic model. The largest increase in R2 is the emerging equity regression. This can be explained by the significant factors size and fx carry. Remarkable is the drop in both coefficients as in significance of the basic model factors in the extended model. Since only 4 factors out of possible 25 extra factors are significant, the question rises whether extending the basic model was useful.

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4.3 Risk contribution: asset based In graph 2 the asset based risk contribution is shown, based on the asset weights of the portfolio and the covariance matrix in table A.1, in the appendix.

Risk contribution 5% 1% equity developed

12% equity emerging

bonds corporate 82% bonds government

Graph 2: asset based risk contribution.

Clearly the equity has a higher risk contribution than the bonds. Notable is the risk contribution of the equity emerging. Although this type of equity is expected to be more risky relative to developed equity, there is only a small amount of emerging equity in the portfolio (6% of the portfolio is emerging equity). As could be expected, the relative save government bonds have the lowest marginal risk contribution.

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4.4 Risk contribution: factor based Graph 3 shows the results of the factor based risk analysis. Only the significant factors are included into the factor based risk contribution (credit factor, term factor, world factor a momentum factor). In the asset based risk contribution the weights of the portfolio are known.

Risk contribution

2% -1% 4% CF TF WRLD MOM

93%

Graph 3: Factor based risk contribution

As shown in graph 3, the world market factor has a very large risk contribution in this case. Since 93% of the risk can be related to this factor. The term factor serves as a hedge, since its risk contribution is negative. The credit factor and momentum factor also contribute risk, although it is very small compared to the world performance factor.

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5. Conclusion First of all, with the financial factors of the examples given in this study, it will be clear that one can manage the most efficient (return, risk)-combination given the possibilities; it certainly is bounded, and it can be done worse. The very same technique can be used to investigate the sensitivity for external volatile phenomena like weather in a year, employment, war etc. The findings may be helpful in designing smart contingency plans, and even trace investment opportunities.

As could be expected, a large part of the risk can be attributed to the world market performance factor. In order to diversify more, one should study how the portfolio can be changed to reduce the world market factor and increase the risk contribution of other factors. Therefore it could be very useful to search for other investible factors and build a portfolio based on factors. How to build such a portfolio and what factors it should be built on can be the subject of further studies.

In this study volatility in the portfolio return is treated as undesirable. However, the return of a portfolio can also be a very large peak upwards. When this happens the volatility of a portfolio increases. But the pension fund will find these peaks upwards desirable.

One should be aware of the fact that a factor risk decomposition uses an estimation of the relation between the portfolio and the factors. In the modern portfolio theory the weights of assets in the portfolio is used. Instead of these known weights, the factor risk decomposition first estimates the relation between the portfolio and the factors. In this study this is called the return contribution. Since this relation is estimated, uncertainty will be added to the risk analysis. Therefore the relation between the portfolio returns and the factors should be subject for further research.

The results of the extended model are not as expected. As mentioned earlier, only 4 factors out of possible 25 extra factors are significant. Therefore one should research if this can be improved. Other factors could be included. Also a different period can be used. In this study monthly data from 1988-2012 was used. This period was chosen due to availability of the data. However new studies might have access to larger data and could extend the data.

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In this study the risk analysis of one fictive portfolio is done. Much more interesting would be to analyze a real pension fund portfolio. This portfolio used in this study isn’t much diversified using asset based risk analysis. A portfolio that is diversified based on the asset based risk analysis, could be analyzed by the factor based analysis and the other way around in further research.

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References Ang, A., Goetzman, W. N., and Schaefer, S.M. (2009) , “Evaluation of Active Management of the Norwegian Government Pension Fund - Global”. Available at http://www.regjeringen.no

Chen, N. F., Roll, R., and Ross, S. A. (1986), “Economic Forces and the Stock Market”, Journal of Business 59, 383‐403.

Fama, E. F., and French, K. R. (1993). "Common Risk Factors in the Returns on Stocks and Bonds". Journal of Financial Economics, 33 (1), 3–56.

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Appendix

Table A.1: Covariance matrix of the assets EQD EQE BNONGOV BTBILL EQD 17.2 20.7 2.3 -0.6 EQE 20.7 49.9 3.3 -1.7 BNONGOV 2.3 3.3 2.6 1.4 BTBILL -0.6 -1.7 1.4 1.8

Table A.2: Covariance matrix of the significant factors CF TF WRLD MOM CF 1.23 -1.18 2.15 -1.52 TF -1.18 5.32 -1.61 1.74 MSWRLD 2.15 -1.61 20.02 -5.24 MOM -1.52 1.74 -5.24 24.35

Table A.3: Correlation matrix of the factors

CF FXCARRY VALUE INFL MOM SIZE TF MSWRLD

CF 1.000 0.208 -0.082 0.023 -0.277 0.159 -0.462 0.433 FXCARRY 0.208 1.000 0.110 0.121 -0.127 0.010 -0.072 0.369 VALUE -0.082 0.110 1.000 0.043 -0.143 -0.324 0.069 -0.156 INFL 0.023 0.121 0.043 1.000 0.022 -0.011 -0.194 -0.022 MOM -0.277 -0.127 -0.143 0.022 1.000 0.040 0.153 -0.237 SIZE 0.159 0.010 -0.324 -0.011 0.040 1.000 -0.183 0.132 TF -0.462 -0.072 0.069 -0.194 0.153 -0.183 1.000 -0.156 MSWRLD 0.433 0.369 -0.156 -0.022 -0.237 0.132 -0.156 1.000