Asset Alliance Principal Components Analysis FoFs
Content
Graph 1.
27.5%
15.0% 10.0%
INFORMING THE HEDGE FUND COMMUNITY
ISSUE 12 OCTOBER/NOVEMBER 2005
10.0% 17.5% 20.0%
US Equity Global Macro Fixed Income Arb. Source: Asset Alliance
CB Volatility Arb./ CB Arb. CTAs/Trend Follower, Systematic Event Driven and Risk Arb.
Principal Components Analysis of FoFs By Daniel Axmer, Analyst, Asset Alliance
Creating a Fund of Funds (FoFs) is predominantly being done on a bottom-up basis with top-down overlay conducted on the strategy level. The FoFs investment team usually formulate an outlook on the different strategies that they cover and from this make allocations to the hedge funds/strategies that are considered to best reflect their outlook. The portfolio construction is fine tuned by a discretionary call on the level of comfort of having different target weights for the different strategies, for example 35% in Equity Long/ Short, 15% in Event Driven, etc. In most cases FoFs have limits in terms of how much they can allocate to a specific strategy (as well as single manager). An important aspect that is often forgotten in the portfolio construction process is that a lot of different strategies may be driven by the same underlying factors that lead to some groups of managers performing in a similar manner. The main reason for this may be that they use comparable trading styles and/or trade in related markets/instruments. The fact that hedge funds are pursuing different strategies is not insurance that the returns they generate are not related in some way. There are numerous factors that influence the performance of FoFs and it can be hard to get a grasp of what these are by only analysing the aggregate return stream of the portfolio. A useful tool in gauging these factors is Principal Components Analysis (PCA), a quantitative method that can be used to determine how many common factors influence the overall portfolio performance. It also highlights which managers affect which factors and to what extent. In addition, PCA can be indicative of the underlying diversification of a portfolio. Implicitly, it would be desirable for a portfolio’s returns to be driven by several drivers that are uncorrelated to each other as opposed to only one or two that are responsible for the bulk of the performance. The FoF manager can also gain insight into whether there are any underlying bets/tilts in the portfolio that are sub-optimal. This article goes through how PCA can be utilised in FoFs management to increase the knowledge of how the underlying funds interact. Emphasis is placed on the interpretation of PCA and less so on the underlying calculations. PCA is a quantitative method used to simplify and find linear combinations in groups of data. The objective is to reduce the number of variables in the data and focus on the few factors that explain most of the variation of the underlying group. PCA uncovers the correlation structure of the group and decomposes the data into components, which are orthogonal, uncorrelated to each other, and explain all variance in the data set. The components are ranked in order of importance, i.e. how much they explain of the total variation. The first component accounts for the largest part of the explained variance, the second component for the second largest part of the explained variance, etc. Usually, the number of components is the same as the number of variables in the data set but, generally, the first few components explain the majority of the group’s variance. Table 1. Number of Managers per Strategy Strategy
Convertible Bond & Volatility Arb. CTA Event Driven & Risk Arb. Fixed Income Arb. Macro Long/Short Equity Total
Source: Asset Alliance
Graph 2.
Managers
6 4 8 7 4 11 40
The obstacles with PCA are that components are not directly observable and there may not even exist a real variable that reflects the component. One approach to identify the components is to run a correlation analysis against variables that are thought to influence the underlying data. Obviously it is important to have a representative and large database of market variables. Application and interpretation To get an understanding of how PCA can be used to analyse FoFs, a portfolio is created that consists of 40 managers representing six broad hedge fund strategies. The classifications and returns are acquired from a large, wellknown managed accounts platform. The fund classifications are as follows: Convertible Bond & Volatility Arbitrage, CTAs, Event Driven & Risk Arbitrage, Fixed Income Arbitrage, Global Macro and Long/Short Equity. Table 1 shows the number of managers represented in each strategy and Graph 1 on p.27 shows the weights according to strategies (equally for all managers at 2.5%).
The returns are collected net of fees and covers the five-year period starting in January 2000 and ending in December 2004. Funds with managed account track records shorter than five years are backfilled using the performance of the manager’s flagship fund, which the managed account is replicating. When selecting the managers, emphasis has been placed on selecting as diverse a portfolio as possible with managers within each strategy pursuing different sub-strategies. For example, the Long/Short Equity segment consists of managers with long bias, short bias, variable bias, market neutral/pair trading, and statistical arbitrage. Analysis The output from the PCA can be seen in Graph 2. The bars indicate how much of the total variation is explained by each of the first 10 components. The first component explain 25% of the variation of the FoFs, the second component explains 18%, etc. The line shows the cumulative proportion of variance explained. On aggregate, the first 10 components explain 82% of the portfolio variance. The PCA has effectively reduced the dimensions in the data set and it is now easier to analyse and interpret the underlying structure. Instead of trying to find patterns straight from the 40 different return streams we can now focus on a few key factors and try to get an understanding of what drives the portfolio. In this case, it is clearly important to try and relate real world variables to at least the first two components (which make 43% of total variation) and if
30%
90% 80% 70%
25%
25%
20% 18% 15%
Proportion of Variance (rhs) Cumulative Proportion (lhs)
60% 50% 40%
10%
9%
30% 8% 6% 20% 4% 4% 3% 3% 2% 10% 0%
5% Variances
0% Comp. 1 Comp. 2 Comp. 3 Comp. 4 Comp. 5 Comp. 6 Comp. 7 Comp. 8 Comp. 9 Comp. 10 Source: Asset Alliance
possible extend it to the first five in which case we would get a picture of what drives 66% of the portfolio. So how do we find out what’s behind the components? The first step would be to see who the managers are that contribute the most (positive) and the least (negative) to each component. Graph 3 on p.28 shows the loadings for the top five positive managers and top five negatively related managers to the first component.
The most negative managers consist of two L/S equity, two CTAs and one macro manager. Of the two equity managers, one has a short bias and the other one employs a statistical arbitrage strategy that tends to perform best in down markets. The CTAs and the macro managers employ systematic strategies and the commonality between them is that they tend to be long event risk. The three L/S equity managers on the positive side all utilize long biased and/or small cap
Table 2.
Comp 1 Small - Large Cap High - Low Book Momentum Equity Market Turnover Implied Volatility Treasuries High Yield Emerging Markets Debt Interest Rates US Dollar Credit Spread
Source: Asset Alliance
Comp 2
Comp 3
Comp 4 -46% 58%
Comp 5
-47% 80% -71% -46% 75% 60% 42% 54% -48% -45%
Graph 3.
First Principal Component L/S Equity 9 Macro 4 L/S Equity 7 -16% -15% FI Arb 3 FI Arb 1 L/S Equity 2 L/S Equity 5 L/S Equity 3 CTA 1 CTA 4 19% 21% 27% 29% 43% 10% 20% 30% 40% 50%
Graph 4.
Second Principal Component -57% -40% -32% -31% CTA 2 CTA 3 Macro 4 CTA 1 -24% L/S Equity 11 L/S Equity 8 L/S Equity 10 L/S Equity 7 L/S Equity 9 CTA 4 2% 4% 5% 8% 12% 0% 10% 20%
-31% -30% -29%
exposure and 14 have negative exposure to the component. The second principal component accounts for 18% of the total variation and the top negative/positive managers is depicted in Graph 4 on p.29. All CTAs and a macro manager are the most negatively exposed to this component. The CTAs and the macro manager all employ systematic strategies. On the opposite side we have five L/S equity managers, however none of these managers appeared in the top five positive for the first component. The managers consist of two short bias managers and three statistical arbitrage managers. It is also worth noting that they are only slightly correlated. It is less evident to draw conclusions about this component, except that trend following managers get hurt. What it may imply is that the component is positively related to markets trading sideways with low volatility, which is when trend followers are expected to lose money. The slight positive exposure of the L/S equity managers is hard to draw any conclusion from, except that it is not expected to be related to the direction of equity markets. The correlation results show positive correlation to a rise in interest rates (+42%) and negative correlation to global treasuries (-46%). Equity markets have different correlations varying from slightly negative to slightly positive which confirms the lack of equity direction of the component. The
positive exposure to a rise in interest rates is not significant enough to draw the conclusion that the component is a proxy for that. The manager exposure does not support that either, so for this component we’ll have to find more factors to add to our database and re-run the correlations. In terms of the rest of the components only the third, fourth, and maybe the fifth, is worthwhile to attempt to define. The third component is positively related to L/S equity managers, especially statistical arbitrage and managers with variable long/short net exposure. The negative managers are a mix, but predominately fixed income arbitrage, CTAs and macro managers. The most highly correlated factor to the third component is a momentum factor (-47%). The negative correlation suggests that the component could be a reversal factor, which makes sense according to the make up of the positive and negative managers. The fourth component is positively exposed to the spread between growth stocks and value stocks (58%) and negatively exposed to the spread between small cap and large cap stocks (-46%). So the component indicates a scenario when the largest companies outperform the smallest companies. The fifth component is positively exposed to the US dollar (+54%) and negatively exposed to global treasuries (-45%). Hence it could reflect a situation where investors liquidate global securities and move in to a safe haven of money market instruments or treasuries denoted in USD.
-40%
-30%
-20%
-10%
0% Loadings
-70%
-60%
-50%
-40%
-30% Loadings
-20%
-10%
Source: Asset Alliance
Source: Asset Alliance
strategies. The two Fixed Income managers are both running directional, long-biased strategies in emerging markets. With this information it is quite easy to draw the conclusion that the first component might be a factor that performs in line with equity markets and is short event risk. To check if that is the case, we measure the correlation of the components against our database, which contains numerous factors representing equities, treasuries, high yield bonds,
commodities, credit spreads, volatility, options strategies, etc. We find that the average correlation to equities is roughly 80% and to high yield and emerging market bonds, 75% and 60%, respectively. On the flipside, the correlation to implied volatility and credit spreads are –71% and -48% respectively. Thus, the main positive influence of our portfolio would be a positive performance in equity/high yield markets. On a manager-bymanager level, 26 managers have positive
In summary, this FoF will do well when equities and high yield bonds outperform. It may also be the case that a strengthening in the US dollar and US interest rates will have a positive effect. On the other hand, an increase in volatility, a widening in spreads, and lack of market momentum, will have a negative effect on the fund. The main correlations between the components and our factors are summarized in Table 2 above.
PCA can be an effective tool when investigating the performance drivers of FoFs. It is also very insightful to use when trying different manager/strategy combinations and monitoring how the portfolio drivers change. Like all investment tools, it’s most beneficial as part of a comprehensive risk management and investment process.
27.5%
15.0% 10.0%
INFORMING THE HEDGE FUND COMMUNITY
ISSUE 12 OCTOBER/NOVEMBER 2005
10.0% 17.5% 20.0%
US Equity Global Macro Fixed Income Arb. Source: Asset Alliance
CB Volatility Arb./ CB Arb. CTAs/Trend Follower, Systematic Event Driven and Risk Arb.
Principal Components Analysis of FoFs By Daniel Axmer, Analyst, Asset Alliance
Creating a Fund of Funds (FoFs) is predominantly being done on a bottom-up basis with top-down overlay conducted on the strategy level. The FoFs investment team usually formulate an outlook on the different strategies that they cover and from this make allocations to the hedge funds/strategies that are considered to best reflect their outlook. The portfolio construction is fine tuned by a discretionary call on the level of comfort of having different target weights for the different strategies, for example 35% in Equity Long/ Short, 15% in Event Driven, etc. In most cases FoFs have limits in terms of how much they can allocate to a specific strategy (as well as single manager). An important aspect that is often forgotten in the portfolio construction process is that a lot of different strategies may be driven by the same underlying factors that lead to some groups of managers performing in a similar manner. The main reason for this may be that they use comparable trading styles and/or trade in related markets/instruments. The fact that hedge funds are pursuing different strategies is not insurance that the returns they generate are not related in some way. There are numerous factors that influence the performance of FoFs and it can be hard to get a grasp of what these are by only analysing the aggregate return stream of the portfolio. A useful tool in gauging these factors is Principal Components Analysis (PCA), a quantitative method that can be used to determine how many common factors influence the overall portfolio performance. It also highlights which managers affect which factors and to what extent. In addition, PCA can be indicative of the underlying diversification of a portfolio. Implicitly, it would be desirable for a portfolio’s returns to be driven by several drivers that are uncorrelated to each other as opposed to only one or two that are responsible for the bulk of the performance. The FoF manager can also gain insight into whether there are any underlying bets/tilts in the portfolio that are sub-optimal. This article goes through how PCA can be utilised in FoFs management to increase the knowledge of how the underlying funds interact. Emphasis is placed on the interpretation of PCA and less so on the underlying calculations. PCA is a quantitative method used to simplify and find linear combinations in groups of data. The objective is to reduce the number of variables in the data and focus on the few factors that explain most of the variation of the underlying group. PCA uncovers the correlation structure of the group and decomposes the data into components, which are orthogonal, uncorrelated to each other, and explain all variance in the data set. The components are ranked in order of importance, i.e. how much they explain of the total variation. The first component accounts for the largest part of the explained variance, the second component for the second largest part of the explained variance, etc. Usually, the number of components is the same as the number of variables in the data set but, generally, the first few components explain the majority of the group’s variance. Table 1. Number of Managers per Strategy Strategy
Convertible Bond & Volatility Arb. CTA Event Driven & Risk Arb. Fixed Income Arb. Macro Long/Short Equity Total
Source: Asset Alliance
Graph 2.
Managers
6 4 8 7 4 11 40
The obstacles with PCA are that components are not directly observable and there may not even exist a real variable that reflects the component. One approach to identify the components is to run a correlation analysis against variables that are thought to influence the underlying data. Obviously it is important to have a representative and large database of market variables. Application and interpretation To get an understanding of how PCA can be used to analyse FoFs, a portfolio is created that consists of 40 managers representing six broad hedge fund strategies. The classifications and returns are acquired from a large, wellknown managed accounts platform. The fund classifications are as follows: Convertible Bond & Volatility Arbitrage, CTAs, Event Driven & Risk Arbitrage, Fixed Income Arbitrage, Global Macro and Long/Short Equity. Table 1 shows the number of managers represented in each strategy and Graph 1 on p.27 shows the weights according to strategies (equally for all managers at 2.5%).
The returns are collected net of fees and covers the five-year period starting in January 2000 and ending in December 2004. Funds with managed account track records shorter than five years are backfilled using the performance of the manager’s flagship fund, which the managed account is replicating. When selecting the managers, emphasis has been placed on selecting as diverse a portfolio as possible with managers within each strategy pursuing different sub-strategies. For example, the Long/Short Equity segment consists of managers with long bias, short bias, variable bias, market neutral/pair trading, and statistical arbitrage. Analysis The output from the PCA can be seen in Graph 2. The bars indicate how much of the total variation is explained by each of the first 10 components. The first component explain 25% of the variation of the FoFs, the second component explains 18%, etc. The line shows the cumulative proportion of variance explained. On aggregate, the first 10 components explain 82% of the portfolio variance. The PCA has effectively reduced the dimensions in the data set and it is now easier to analyse and interpret the underlying structure. Instead of trying to find patterns straight from the 40 different return streams we can now focus on a few key factors and try to get an understanding of what drives the portfolio. In this case, it is clearly important to try and relate real world variables to at least the first two components (which make 43% of total variation) and if
30%
90% 80% 70%
25%
25%
20% 18% 15%
Proportion of Variance (rhs) Cumulative Proportion (lhs)
60% 50% 40%
10%
9%
30% 8% 6% 20% 4% 4% 3% 3% 2% 10% 0%
5% Variances
0% Comp. 1 Comp. 2 Comp. 3 Comp. 4 Comp. 5 Comp. 6 Comp. 7 Comp. 8 Comp. 9 Comp. 10 Source: Asset Alliance
possible extend it to the first five in which case we would get a picture of what drives 66% of the portfolio. So how do we find out what’s behind the components? The first step would be to see who the managers are that contribute the most (positive) and the least (negative) to each component. Graph 3 on p.28 shows the loadings for the top five positive managers and top five negatively related managers to the first component.
The most negative managers consist of two L/S equity, two CTAs and one macro manager. Of the two equity managers, one has a short bias and the other one employs a statistical arbitrage strategy that tends to perform best in down markets. The CTAs and the macro managers employ systematic strategies and the commonality between them is that they tend to be long event risk. The three L/S equity managers on the positive side all utilize long biased and/or small cap
Table 2.
Comp 1 Small - Large Cap High - Low Book Momentum Equity Market Turnover Implied Volatility Treasuries High Yield Emerging Markets Debt Interest Rates US Dollar Credit Spread
Source: Asset Alliance
Comp 2
Comp 3
Comp 4 -46% 58%
Comp 5
-47% 80% -71% -46% 75% 60% 42% 54% -48% -45%
Graph 3.
First Principal Component L/S Equity 9 Macro 4 L/S Equity 7 -16% -15% FI Arb 3 FI Arb 1 L/S Equity 2 L/S Equity 5 L/S Equity 3 CTA 1 CTA 4 19% 21% 27% 29% 43% 10% 20% 30% 40% 50%
Graph 4.
Second Principal Component -57% -40% -32% -31% CTA 2 CTA 3 Macro 4 CTA 1 -24% L/S Equity 11 L/S Equity 8 L/S Equity 10 L/S Equity 7 L/S Equity 9 CTA 4 2% 4% 5% 8% 12% 0% 10% 20%
-31% -30% -29%
exposure and 14 have negative exposure to the component. The second principal component accounts for 18% of the total variation and the top negative/positive managers is depicted in Graph 4 on p.29. All CTAs and a macro manager are the most negatively exposed to this component. The CTAs and the macro manager all employ systematic strategies. On the opposite side we have five L/S equity managers, however none of these managers appeared in the top five positive for the first component. The managers consist of two short bias managers and three statistical arbitrage managers. It is also worth noting that they are only slightly correlated. It is less evident to draw conclusions about this component, except that trend following managers get hurt. What it may imply is that the component is positively related to markets trading sideways with low volatility, which is when trend followers are expected to lose money. The slight positive exposure of the L/S equity managers is hard to draw any conclusion from, except that it is not expected to be related to the direction of equity markets. The correlation results show positive correlation to a rise in interest rates (+42%) and negative correlation to global treasuries (-46%). Equity markets have different correlations varying from slightly negative to slightly positive which confirms the lack of equity direction of the component. The
positive exposure to a rise in interest rates is not significant enough to draw the conclusion that the component is a proxy for that. The manager exposure does not support that either, so for this component we’ll have to find more factors to add to our database and re-run the correlations. In terms of the rest of the components only the third, fourth, and maybe the fifth, is worthwhile to attempt to define. The third component is positively related to L/S equity managers, especially statistical arbitrage and managers with variable long/short net exposure. The negative managers are a mix, but predominately fixed income arbitrage, CTAs and macro managers. The most highly correlated factor to the third component is a momentum factor (-47%). The negative correlation suggests that the component could be a reversal factor, which makes sense according to the make up of the positive and negative managers. The fourth component is positively exposed to the spread between growth stocks and value stocks (58%) and negatively exposed to the spread between small cap and large cap stocks (-46%). So the component indicates a scenario when the largest companies outperform the smallest companies. The fifth component is positively exposed to the US dollar (+54%) and negatively exposed to global treasuries (-45%). Hence it could reflect a situation where investors liquidate global securities and move in to a safe haven of money market instruments or treasuries denoted in USD.
-40%
-30%
-20%
-10%
0% Loadings
-70%
-60%
-50%
-40%
-30% Loadings
-20%
-10%
Source: Asset Alliance
Source: Asset Alliance
strategies. The two Fixed Income managers are both running directional, long-biased strategies in emerging markets. With this information it is quite easy to draw the conclusion that the first component might be a factor that performs in line with equity markets and is short event risk. To check if that is the case, we measure the correlation of the components against our database, which contains numerous factors representing equities, treasuries, high yield bonds,
commodities, credit spreads, volatility, options strategies, etc. We find that the average correlation to equities is roughly 80% and to high yield and emerging market bonds, 75% and 60%, respectively. On the flipside, the correlation to implied volatility and credit spreads are –71% and -48% respectively. Thus, the main positive influence of our portfolio would be a positive performance in equity/high yield markets. On a manager-bymanager level, 26 managers have positive
In summary, this FoF will do well when equities and high yield bonds outperform. It may also be the case that a strengthening in the US dollar and US interest rates will have a positive effect. On the other hand, an increase in volatility, a widening in spreads, and lack of market momentum, will have a negative effect on the fund. The main correlations between the components and our factors are summarized in Table 2 above.
PCA can be an effective tool when investigating the performance drivers of FoFs. It is also very insightful to use when trying different manager/strategy combinations and monitoring how the portfolio drivers change. Like all investment tools, it’s most beneficial as part of a comprehensive risk management and investment process.
