Dependencies of equity factor portfolios on
macroeconomic conditions: an analysis for the US
stock market
Rutger van der Werf
s3053725
11 December 2020
1
Introduction
Many investors hold equity factor portfolios as part of their total wealth. Fac-tor portfolios are frequently build around standard equity facFac-tors such as Value and Momentum. Since equity factors are not perfectly correlated, a portfo-lio consisting of multiple factors has a lower risk and a smoother return over time. Equity factors depend on macroeconomic environments and are cyclical. Multiple factors may be influenced in a similar way by (unexpected) changes in macroeconomic state variables. Therefore, it is important to determine the relationships between equity factors and these macroeconomic state variables in order to be able to build equity factor portfolios that satisfy certain preferences of investors. In this thesis we analyse the dependencies of equity factors and factor mimicking portfolios on macroeconomic state variables and macroeco-nomic regimes specified. We construct factor portfolios with the lowest possible dependencies on a chosen state variable and evaluate both the conditional and unconditional performance in chosen time periods.
In general, investors focus mainly on risk and return when constructing their investment portfolios. However, a disadvantage of this practice of focusing only on unconditional returns, volatilities and correlations of equities in portfolio con-struction is that possible relevant dependencies of equity returns on the state of the macro economy are not taken into account. Dependencies of equity returns on macroeconomic state variables or on the macro economy as a whole can be very important for investors, especially when the total wealth of an investor is considered. Since other types of investments such as bonds, options or annuities also have certain dependencies on macroeconomic variables, investors investing in multiple type of investments should consider the total effect on their wealth of changes in macroeconomic variables. Therefore, taking returns and volatilities of equity factors conditional on macroeconomic conditions into account when choosing the optimal investment policy could lead to portfolios more tailored to the specific preferences of investors.
Investors who aim to diversify their equity factor portfolios often try to bal-ance factor weights or risk contributions. However, this procedure does not take into account that multiple factors could depend on the same macroeco-nomic environments. Amenc et al. (2019) states that a portfolio consisting of not perfectly correlated equity factors does not necessarily result in a balanced performance across macro regimes. Instead of looking at average correlations, correlations of factor returns conditional on macroeconomic environments have to be taken into account to improve the diversification of equity factor invest-ments.
con-sidered. Following a standard approach, we determine the shocks of the macroe-conomic state variables considered in this thesis by estimating a VAR(1)-model on all state variables and extracting the (orthogonal) residuals. Besides exam-ining shocks in individual macroeconomic state variables, in this thesis we also analyse the effect of simultaneous shocks in multiple state variables. Based on the seven state variables considered we create macroeconomic regimes to get an accurate reflection of the total macroeconomic environment. We use multiple state variables in the construction of new state variables that represent par-ticular macroeconomic regime specifications. Different combination methods of state variables are used in the construction of these specifications.
Risk-averse investors may prefer portfolios with the least possible dependency on unexpected shocks in a particular state variable. For example, an investor holding a substantial amount of bonds in his total investment portfolio possibly would like to hold an equity portfolio with a low dependence on unexpected shocks in the interest rate term spread to minimize the risk of very low (neg-ative) total investment returns in case of unfavourable interest rate changes. In order to construct portfolios with the least possible dependencies on unex-pected changes in particular macroeconomic conditions, following Amenc et al (2019), we derive minimum-regime dependent allocations, both based on individual macroeconomic state variables and on the created macroeconomic regimes. The portfolio weights of such an allocation are obtained by solving a specified minimization problem. The portfolios resulting from this minimization problem have by construction the lowest possible dependency on the specified macroeconomic variable or macroeconomic regime. We consider both situations where short-selling is allowed and situations where short-selling is not allowed. The performance of the portfolios obtained is evaluated using appropriate per-formance measures.
To gain insights in the effect of (unexpected) shocks in the macroeconomic state variables considered on the return of factors, we regress the returns of the equity factors considered in this thesis on unexpected shocks in the macroeco-nomic state variables considered. Based on the results of these (time series) regressions, we obtain insights on the dependencies of equity factor returns on macroeconomic state variable shocks. More importantly, we also obtain in-sights on the dependencies of equity factor portfolios on unexpected changes in macroeconomic state variables. These dependencies possibly can explain the differences in conditional and unconditional returns and volatilities of different equity factor portfolios.
monthly observations of these variables for the United States between April 1990 and March 2020 are constructed and used for the analysis. The macroeconomic regime specifications constructed in this thesis are Risk Tolerance, Macro Out-look, Macro Stability and Risk-On Conditions. To be able to evaluate the out of sample performance of the minimum dependency portfolios constructed, we construct factor portfolios based on monthly data from April 1990 to November 2016. Subsequently, we evaluate the performance of the resulting portfolios us-ing appropriate performance measures for the time period from December 2016 to March 2020. As indicated before, to obtain more inference on the performance of the portfolios constructed in this way we construct multiple macroeconomic regimes and we use multiple definitions of macroeconomic states.
Instead of investing in standard equity factors, investors can opt to construct and invest in factor mimicking portfolios representing particular equity factors. In order to analyse the dependencies of the returns of these factor mimicking port-folios on the macroeconomic state variables specified in this thesis, we construct minimum dependency portfolios using factor mimicking portfolios representing particular equity factors. We construct these portfolios based on monthly data from January 1997 to August 2016 and evaluate the performance of the portfo-lios obtained for the time period from September 2016 to December 2019. To compare the dependencies of both factor mimicking portfolio returns and standard equity factor returns on the macroeconomic state variables specified more directly, we also construct minimum dependency portfolios using the cor-responding standard equity factors using the same estimation period and out of sample period. We evaluate and compare the performance of the portfolios obtained. The standard equity factors considered in this comparison are size, value, high profitability and momentum. Concerning factor mimicking port-folios, mimicking portfolios of the factors size, value, quality and momentum are considered, both within industries and regions (industry and region neu-tral) and across industries. The quality factor mimicking portfolio serves in the analysis as the analogous portfolio of the high profitability factor. Factor mimicking portfolios are by construction neutral to some chosen factors, which should lead to more significant dependencies of portfolio returns on shocks in particular macroeconomic state variables. To obtain inferences on the effect of these shocks on the return of the factor mimicking portfolios considered, we perform time series regressions of monthly factor mimicking portfolio returns on unexpected shocks in the macroeconomic state variables described earlier for the time period from January 1997 to December 2019.
Fama- French equity factors, which could be a consequence of these more signifi-cant dependencies found. Another noticeable result is that the way of specifying macroeconomic regimes to represent the macro economy has a significant influ-ence on the factor weights in the minimum dependency portfolios constructed for these regimes and on the conditional and unconditional performance of the resulting portfolios.
The remainder of this thesis is organized as follows. In the next section we give a review of literature investigating relationships between equity factors and macroeconomic conditions. In section three the methodologies used in this the-sis are specified. Section four describes the data analysed in this thethe-sis. The macroeconomic regimes constructed in this thesis are also introduced in this sec-tion. In section five we present the results of the minimum dependency portfolios constructed for both single macroeconomic state variables and macroeconomic regime specifications. The results of the comparison of minimum dependency portfolios constructed using standard equity factors and the minimum depen-dency portfolios constructed using factor mimicking portfolios are also presented in this section. In the last section of this thesis we give the main conclusions from all findings.
2
Literature review
A large strand of literature has investigated the relationships between equity fac-tors and certain macroeconomic variables in the past. In particular, relations between the Fama-French equity factors and macroeconomic conditions have been investigated intensively. Both dependencies between equity factors and single macroeconomic state variables and dependencies between equity factors and macroeconomic regimes have been topic of research frequently. Regarding the former, Johnson, Reilly and Wright (2007) analyses the effect of interest rate shocks on common stocks of multiple sectors and industries for the US stock market using the concept of empirical duration. They find that interest rate sensitivities of stocks vary considerably for sectors and industries and also vary over time. Franco, Monnier and Rulik (2017) investigates the interest rate risk and possible characteristics of this risk of portfolios with different levels of volatility using a five-factor model based on the Fama-French factors. The au-thors find significant and robust positive interest sensitivities for low volatility portfolios, whereas mixed results are found for medium volatility portfolios and high volatility portfolios.
mar-ket. The authors find that correlations can differ considerably across macroeco-nomic regimes. Correlations seem to increase during bad macroecomacroeco-nomic times, diminishing the diversification effect when it is most needed. The findings of the authors suggest that in general equity factors have a substantial exposure to macroeconomic risks, with dependencies differing across equity factors. For example, the value factor has a positive sensitivity to the interest rate term spread, just as the low investment factor. However, the momentum factor and high profitability factor are found to have a negative sensitivity on this macroe-conomic variable. None of the equity factors investigated shows a significant positive dependency on the macroeconomic variables dividend yield and sys-tematic volatility. The authors also find that standard portfolio allocation ap-proaches are not able to exploit differences in dependencies across equity factors. Several studies have analysed the dependencies of equity factors on macroe-conomic environments. Ilmanen, Maloney and Ross (2014) investigates how investable return sources relate to macroeconomic conditions. The authors find that most investments are negatively affected by lower economic growth, es-pecially when low economic growth comes together with high inflation. They also find that stocks and bonds generally have opposite sensitivities to eco-nomic growth, suggesting that investors seeking effective diversification should combine these investments in their portfolios. Gupta et al. (2014) finds that the equity factors Value, Momentum and Size are pro-cyclical, while Quality and Low Volatility perform strong in weak macroeconomic environments. In a similar study, Ung and Luk (2016) finds that Momentum performs best in bull markets, Low Volatility and Quality have the best performance in bear markets, and Value and High Yield have the best performance in recovery periods. The authors also find that combining various factors in portfolios reduces risk and enhance return. Bender et al (2017) analyses the linkages between equity factor performance and different groupings of predictors, among which economic con-ditions and financial concon-ditions. The findings of the authors suggest that the underlying causal links are mostly time-varying, which makes the investment horizon chosen for dynamical factor allocation critical. Regarding dynamical factor allocation, Jain and Varsani (2018) investigates the static and dynamic allocation of equity factors in both developed and emerging markets using the MSCI World Index. The authors find that multi-factor allocations outperform their benchmarks in terms of annualized returns, Sharpe ratios and information ratios. Their results also suggest that dynamic allocation strategies outperform static allocation strategies. However, the authors establish that the conclusions could vary depending on the starting universe and selection of factors and there may not be any outperformance in alternative cases.
in the next section we specify the methodology used in this thesis, including the construction of equity portfolios with the lowest possible dependency on a particular macroeconomic state variable or macroeconomic regime.
3
Methodology
In this thesis we adapt the consensus view in the financial world that market prices reflect all current expectations about the future, which is also called the efficient market hypothesis. Consequently, in order to obtain inference on the relationships between macroeconomic state variables and equity factor returns, we consider unexpected shocks in macroeconomic variables. We determine un-expected shocks in macroeconomic variables by estimating a VAR(1)-model on a set of macroeconomic variables and subsequently extracting the residuals of the model estimated. More specifically, defining Xt= (X1,t, X2,t, ..., Xn,t)0 as a
vector of n macroeconomic state variables at time t, we estimate the following VAR(1)-model:
Xt= v + AXt−1+ t, (1)
t = 1, ..., m, where Xt = (X1,t, X2,t, ..., Xn,t)0, v is nx1, A is nxn and t ∼
N (0, σ2I
n). The model is estimated without restrictions being placed on the
coefficients in matrix A. We define the vector of obtained residuals of the model at time t by Zt= (SX1,t, SX2,t, ..., SXn,t)0. The S in front of the abbreviation
indicates that the shock in the respective state variable is considered.
3.1
Minimum dependency portfolios
As stated earlier, some investors may prefer investment portfolios with the low-est possible dependency on a particular macroeconomic state variable or on macroeconomic conditions as a whole. For that matter, we are interested in constructing minimum dependency portfolios on single macroeconomic state variables and on macroeconomic regimes. Our objective is to construct an eq-uity factor portfolio for which the expected return conditional on a specific macroeconomic state differs least from the unconditional expected return of the portfolio. We consider portfolios with fixed investment weights for equity fac-tors in each time period. Mathematically, defining w = (w1, w2, ..., wn) as a
vector of portfolio weights of equity factors, we can find minimum dependency portfolios by solving the following minimization problem:
w = argmin
N
X
i=1
(w0xi− w0x0)2,
where xiis the vector of mean returns conditional on state i (for example a
returns. The portfolios resulting from this minimization problem have by con-struction the lowest possible dependency on the specified macroeconomic state variable or macroeconomic regime.
In this thesis we start by constructing portfolios with minimum dependency on a single macroeconomic state variable. To give an example of such a port-folio, we take a particular macroeconomic state variable. We define a state of this variable as a percentile of the unexpected shocks in this state variable de-rived by the VAR-(1) model earlier specified. As an example, we choose N = 5, which implies that xi, i = 1, ..., 5, is the vector of mean returns conditional on
unexpected shocks in the respective macroeconomic state variable being in the ith quintile.
Of course, the specification of states of macroeconomic state variables or macroe-conomic regimes and the number of states are subject to choice. In the next section of this thesis we will construct minimum dependency portfolios both on individual macroeconomic state variables and macroeconomic regimes. We will evaluate the performance of the portfolios obtained using various portfolio performance measures introduced later in this section.
As a starting point, we construct minimum dependency portfolios without im-posing restrictions on the portfolio weights. So, first we allow short-selling. As a next step, we restrict the portfolio weights to be non-negative for all equity factors when solving the relevant minimization problems in order to obtain port-folios with minimum dependency on macroeconomic state variables or regimes under the restriction of no short-selling.
Note that we derive minimum dependency portfolios with fixed investment weights in each time period. Therefore, in order to perform the investment strategy obtained by this method, the equity factor weights in the portfolio should be re-balanced after each period of time.
3.2
Equity factor returns regressions
We wish to obtain insights in the dependencies of equity factor returns on un-expected shocks in particular macroeconomic state variables. Understanding these dependencies enables us to explain the performance of particular equity factor portfolios. Moreover, we could construct new portfolios satisfying specific preferences of investors.
To gain insights on the effect of (unexpected) shocks in these macroeconomic state variables, we regress equity factor returns on macroeconomic state vari-ables shocks. We define Fi,t as the return of equity factor i at time t. After
Fi,t= αi+ βiSX1,t+ γiSX2,t+ ... + ξiSXn,t+ i,t, (2)
t = 1, ..., m, for factor i = 1, ..., k. Just as before, the S in front of the standard abbreviation of the macroeconomic state variable indicates that the shock in this state variable is considered.
Based on the results of these regressions, insights on the dependencies of eq-uity factor returns and eqeq-uity factor portfolio returns on macroeconomic state variables shocks are obtained as well as inference on differences in conditional and unconditional returns and volatilities of different equity factor portfolios. As indicated, new equity factor portfolios can be formed based on desired depen-dencies. Possibilities are constructing portfolios with the lowest coefficient on a particular state variable or portfolios with negative coefficients on particular state variables to possibly create hedge portfolios (due to positive correlation between the state of the macro economy and economic growth).
3.3
Performance evaluation of portfolios constructed
Based on the methodology specified above, we construct minimum dependency portfolios both based on single macroeconomic state variables and on macroe-conomic regimes. The macroemacroe-conomic regimes constructed in this thesis are introduced in the next section. To evaluate the performance of the obtained portfolios in both the estimation period and out of sample period chosen, we use the following performance measures (given as abbrevations):
portfolio optimization problem given in section 3.1.
3.4
Factor mimicking portfolios
Beside the Fama-French factors earlier mentioned, in this thesis we also anal-yse the dependencies of factor mimicking portfolios on unexpected shocks in macroeconomic state variables. Factor mimicking portfolio are stock portfolios constructed to represent a particular equity factor. These portfolios can be con-structed in multiple ways, one of them being the cross-sectional approach. In this approach, the following equation is estimated for a particular universe of stocks: ri= fw+ m X j=1 Xijfj+ i, (3)
i = 1, .., n, where ri is the excess return of stock i, fw is the intercept term,
fj is the return of (pure) factor j, i is the error term of stock i and n is the
total number of stocks in the universe chosen. The model is estimated using restricted least squares estimation. The pure factor returns can be written as follows: fj= n X i=1 Wijrj, (4)
where Wij is the weight of stock i in pure factor portfolio j.
4
Data
In order to analyse the dependencies of equity factor returns on the macroeco-nomic environment, we first need a clear definition of macroecomacroeco-nomic conditions. In the existing literature, a wide range of single macroeconomic state variables is defined and used for an analysis of the relationships between factor returns and macroeconomic conditions. In the spirit of Amenc et al (2019), we construct seven macroeconomic state variables that satisfy various criteria for relevance. Monthly observations of these variables for the United States between April 1990 and March 2020 are obtained. We construct the following macroeconomic state variables (the abbreviations used are given in parentheses):
Moody’s Baa corporate bonds and Moody’s Aaa corporate bonds. Higher val-ues of this measure are associated with a higher risk compensation; Systemic volatility (SV) : the standard deviation of the daily returns on the NASDAQ index. Monthly values of this state variable are constructed by calculating the standard deviation of the daily returns in the respective month. Higher systemic volatility implies higher risk levels in the stock market; Aggregate effective bid-ask spread (ABAS): this state variable is based on the daily high price and daily low price of the NASDAQ index and is constructed using the methodology specified in Corwin and Schultz (2012). Defining dht and dlt as the daily high
price and daily low price of the NASDAQ index at time t respectively, the daily value of the state variable at time t is constructed using the following formula: ABASt=2(e α−1) 1+eα , where α = √ 2β−√β 3−2√2 − q γ 3−2√2, with β = ln(dht dlt) 2+ ln(dht+1 dlt+1) 2and γ = ln(max(dht,dht+1) min(dlt,dlt+1)) 2.
After constructing daily values, monthly values of the state variable are ob-tained by taking the average of the daily values of the state variable in the respective month. A higher aggregate effective bid-ask spread indicates a lower liquidity of the stock market due to higher differences between bid and ask prices for stocks; Aggregate price impact (API): This state variable is a version of the price impact estimator specified in Amihud (2002). Defining tvolt as the
daily trading volume of stocks in the NASDAQ index at time t and srttas the
daily return of stocks in the NASDAQ index at time t, daily values of this state variable are constructed using the following formula:
AP It=|srttvoltt|.
Again, monthly values of the state variable are obtained by taking the average of the relevant daily values of the state variable. Just like aggregate effective bid-ask spread, the state variable aggregate price impact is a measure of stock market liquidity, in this case focusing on the relationship between stock returns and stock trading volume. Higher values for aggregate price impact imply rel-atively more effect of trading on stock price changes and therefore indicate less stock market liquidity.
Relevant descriptive statistics of monthly observations of the seven constructed macroeconomic state variables for the time period between April 1990 and March 2020 are given in table 1.
Table 1: Descriptive statistics macroeconomic state variables April 1990 - March 2020
State variable mean stdev auto corr
min med max Agg. dividend yield 1.947 0.511 0.98 1.00 1.89 3.78 Short interest rate 0.2209 1.74 0.98 0.00 0.19 0.69 Term spread 1.451 1.05 0.98 -0.49 1.499 3.446 Default spread 0.95 2.18 0.94 0.52 0.88 3.49 Systemic volatility 19.30 2.39 0.81 9.51 17.15 59.89 Agg. effective bid-ask spread -0.16 0.25 0.32 -1.58 -0.11 0.33 Agg. price impact 1.53 1.79 0.90 0.08 0.50 10.9
The table reports descriptive statistics of monthly observations of the seven constructed macroeconomic state variables for the time period between April 1990 and March 2020. The statistics given are mean (mean), standard deviation (stdev), first-order autocorrelation (auto corr), minimum (min), median (med) and maximum (max). Apart from the autocorrelation the statistics are given in hundreds. Sources: Eikon database, Factsheet, Bloomberg
In table 2 correlations between the macroeconomic state variables are given.
Table 2: Correlations macroeconomic state variables April 1990 -March 2020
State variable AD SI Term Def. SV AE b.a
API Agg. div. yield 1 0.00 0.34 0.41 0.05 -0.07 0.37 Short int. rate 1 -0.55 -0.32 -0.01 0.05 0.72
Term spread 1 0.24 0.08 0.09 0.00
Default spread 1 0.62 -0.29 -0.19
Systemic volatility 1 -0.50 0.04
Agg. eff. b-a spread 1 0.02
Agg. price impact 1
As can be seen in table 1, all state variables except aggregate effective bid-ask spread have a very strong positive autocorrelation, implying that values of these variables are very persistent through time. The state variable aggregate effective bid-ask spread has also a clearly positive autocorrelation, but the persistence of the values of this variable through time is much lower than those of the other state variables considered.
The high positive autocorrelations of the state variables do not come as a sur-prise. For example, the term spread of interest rates usually changes only by small values on a monthly basis, just as the credit spread. Dividend yields are expected to be relatively stable over time. In our data this is reflected by the low standard deviation of the state variable aggregate dividend yield (0.00511). Since this state variable is a yearly aggregated measure, the calculations of the values of successive monthly observations of these state variable only differ by one month, which obviously contributes to a higher positive autocorrelation. Systemic volatility shows the highest volatility of all seven constructed state variables (0.0239). However, given the fact that the mean of this state vari-able (0.1930) is of a much larger magnitude than the mean of the other state variables, in relative terms to the mean this does not indicate that this state variable is more volatile than the other state variables.
Since we are interested in unexpected changes in the constructed state variables, we estimate the VAR(1)-model specified in the previous section for monthly val-ues of the seven constructed state variables between April 1990 and March 2020 and extract the unexpected shocks in these variables. More formally, we esti-mate equation (1) with
Xt= (DIVt, SIRt, T ERMt, DEFt, SVt, ABASt, AP It)0,
April 1990 defined as t = 1 and m = 361. The estimated coefficients of the model can be found in the appendix of this paper. The obtained shocks
Zt= (SDIVt, SSIRt, ST ERMt, SDEFt, SSVt, SABASt, SAP It)0
will play an important roll in determining relationships between the constructed macroeconomic state variables and the equity factors. They will be also be used in the construction of multiple macroeconomic regimes.
Table 3: Descriptive statistics standard equity factor returns US mar-ket April 1990 - March 2020
equity factor mean stdev auto corr min med max Value 0.075 3.1 0.17 -14.11 -0.13 12.87
Size 0.093 3.1 -0.04 -14.91 0.065 18.32 Momentum 0.54 4.8 0.05 -34.39 0.56 18.36 High profitability 0.33 2.2 0.17 -18.34 0.365 13.33 Low investment 0.18 2.4 0.08 -6.86 -0.02 9.56
The table reports descriptive statistics for the monthly returns of the five standard equity factors considered in this thesis for the US market for the time period between April 1990 and March 2020. The statistics given are mean (mean), standard deviation (stdev), first-order autocorrelation (auto corr), minimum (min), median (med) and maximum (max). Apart from the autocorrelation the statistics are given in hundreds. Source: Fama-French database.
Correlations between the returns of the equity factors are given in table 4.
Table 4: Correlations standard equity factor returns US market April 1990 - March 2020
equity factor value size Momentum High prof. Low inv.
Value 1 -0.05 -0.22 0.36 0.64
Size 1 -0.02 -0.45 -0.05
Momentum 1 0.07 0.03
High profitability 1 0.23
Low investment 1
The table reports the correlations between the monthly returns of the five standard equity factors considered in this thesis for the US market for the time period between April 1990 and March 2020. Monthly observations of the state variables are used.
Looking at table 3, we can see that in the period from April 1990 to March 2020 on average the momentum factor clearly experienced the highest return with an average monthly return of 0.54 percent. The other equity factors also had an positive return on average, but the magnitudes of these returns were much lower on average.
The autocorrelations of the five equity factor returns are all relatively close to zero, indicating that past values of these equity factor returns on itself are not good predictors of successive returns of the same equity factor.
high profitability factor to 0.048 for the momentum factor. So, the high prof-itability factor exhibits both the highest average return and highest volatility of all five factors, indicating that in this case the usual risk-return trade-off pre-vails. In our historical data set the high profitability factor has also experienced the highest maximum return and lowest minimum return of all factors, which is a further indication of higher risk for this factor.
The equity factor low investment has experienced the lowest maximum return and the highest (least negative) minimum return of all factors, indicating that the performance of this factor is relatively stable historically. This is confirmed by the low standard deviation (0.024) of the return of this factor. We expect that, as a result of this relatively stable performance, the factor low investment will have substantial weights in multiple minimum dependency portfolios con-structed in this paper.
The factor mimicking portfolios considered in this thesis are value, size, qual-ity and momentum. The portfolios are constructed both corrected for regional and industrial effects (within industries) and not corrected for industrial effects (across industries). For these factor mimicking portfolios monthly data from the US stock market for the time period January 1997 to December 2019 is consid-ered. The portfolios are constructed by the cross-sectional approach specified in the methodology section of this thesis and are provided by PGGM. Descriptive statistics of the returns of the portfolios are given in table 5.
Table 5: Descriptive statistics factor mimicking portfolio returns US market January 1997 - December 2019
equity factor mean stdev auto corr min med max Size wi. 0.10 0.82 0.027 -3.47 0.043 3.28 Value wi. 0.0044 0.80 0.20 -2.22 -0.0017 2.95 Qual. wi. 0.069 0.45 0.13 -0.92 0.055 1.87 Mom. wi. 0.088 0.90 0.00 -4.84 0.010 3.93 Size acc. 0.17 3.61 0.069 -11.94 0.057 17.79 Value acc. -0.039 1.87 -0.059 -8.73 0.019 8.76 Qual. acc 0.20 1.61 0.035 -5.07 0.21 6.47 Mom. acc. 0.15 2.51 -0.051 -8.74 0.29 8.87
The table reports descriptive statistics for the monthly returns of the constructed factor mimicking portfolios considered in this thesis for the US market for the time period between January 1997 and December 2019. The statistics given are mean (mean), standard deviation (stdev), first-order autocorrelation (auto corr), minimum (min), median (med) and
maximum (max). Apart from the autocorrelation the statistics are given in hundreds. Source: PGGM SES calculations.
the portfolios corrected for regional and industrial effects (within industries). The factor mimicking portfolios constructed across industries also experienced higher maximum returns and lower minimum returns. Comparing the results in table 5 with the results in table 3, the historical unconditional performance of the factor mimicking portfolios constructed across industries is much more comparable to the performance of the standard equity factors. All in all, the results of the factor mimicking portfolios suggest that correcting for regional and industrial effects leads to portfolios with lower return standard deviations and less total return spreads, possibly at the expense of slightly lower mean returns.
4.1
Constructing macroeconomic regimes
From an economical point of view, investors bear risk from unexpected shocks in particular macroeconomic state variables. However, the relevance of the risks from unexpected changes in each state variable will vary among investors, due to personal differences in allocation of wealth and risk attitude for example. There-fore, more general measures of the macroeconomic environment are needed to capture macroeconomic risks for an average investor in a more appropriate way. Since it is not immediately clear from empirical work or theory how to con-struct a macroeconomic regime that captures the risks of an average investor in the most suitable way, we consider four different regime specifications. This also allows us to compare dependencies of equity factor portfolios on macroeconomic regimes obtained in different ways and to identify possible misspecifications of constructed regimes.
The created macroeconomic regimes are used to analyse the effect of simul-taneous shocks in multiple state variables. The regimes should be an accurate reflection of the general state of the (macro) economy. Since a single macroeco-nomic regime could be misspecified, we use four different regimes specifications in the spirit of Amenc et al.(2019). In the next part of this section the four constructed macroeconomic regimes are described in detail.
4.1.1 Risk Tolerance Regime
The first macroeconomic regime we create in this thesis is the Risk Tolerance composite variable. In order to construct this composite variable, the relation-ship between the future equity premium and the innovations in all macroeco-nomic state variables is considered. Monthly time series observations are used and the prediction horizon is chosen to be twelve months. Defining EMtas the
monthly equity return at time t and RFt as the yearly risk-free rate at time t,
we define EPt,t+12=Q 12
i=1(1 + EMt+i) − 1 − RFtas the future yearly equity risk
equation for the time period from April 1990 to March 2019:
EPt,t+12= α+βSDIVt+γSSIRt+δST ERMt+ηSDEFt+θSSVt+κSABASt
+ ξSAP It+ t, (5)
t = 1, ..., 349,
where, as indicated, EPt,t+12 is defined as the future yearly equity premium
at time t and the shocks in the macroeconomic state variables are defined as before. The estimated coefficients of the model can be found in table A1 in the appendix.
Since we expect that the required yearly equity premium by investors is higher during times of lower risk tolerance of the market, we expect a negative relation-ship between the yearly equity premium and the risk tolerance of the market. Therefore, after estimating the model, we obtain the negative of the fitted val-ues for the yearly equity premium. Subsequently, these valval-ues are standardized to form the composite variable.
4.1.2 Macro Outlook Regime
The second regime created in this thesis is the Macro Outlook composite vari-able. The construction of this composite variable is similar to the construction of the Risk Tolerance composite variable. Instead of a predictive time series regression with the future yearly equity premium as dependent variable, we now estimate the regression relation between future yearly industrial produc-tion growth and the innovaproduc-tions in all macroeconomic state variables. In a similar way as before, defining IP GMt as the monthly industrial production
growth at time t, we define IP Gt,t+12=Q 12
i=1(1 + IP GMt+i) − 1 as the future
yearly industrial production growth at time t. Subsequently, we estimate the following equation for the time period from April 1990 to March 2019:
IP Gt,t+12= α+βSDIVt+γSSIRt+δST ERMt+ηSDEFt+θSSVt+κSABASt
+ ξSAP It+ t, (6)
t = 1, ..., 349,
where, as indicated, IP Gt,t+12 is defined as the future yearly industrial
are defined as before. The estimated coefficients of this model can also be found in table A1 in the appendix of this thesis.
To form the composite variable, the fitted values for the future yearly industrial production growth are obtained from the model and standardized. Since we ex-pect that a higher future industrial production growth leads to a more positive view of the macro outlook for investors, we do not take the negative of the fitted values in the construction of the Macro Outlook Regime.
4.1.3 Macro Stability Regime
As a third regime, the Macro Stability composite variable is created. To form this composite variable, we use all macroeconomic state variables except sys-tematic volatility. We omit the state variable syssys-tematic volatility because we are now interested in the conditional volatility of unforecastable disturbarces. The innovations of the macroeconomic state variables are again estimated using the VAR(1) model specified in equation (1). However, we now omit the state variable systematic volatility in the model and define
Xt= (T Bt, T ERMt, DEFt, DIVt, BASt, AM Ht)0.
After estimating the model, we extract the innovations from the model. Using similar notation as before we define
Yt= (ST Bt, ST ERMt, SDEFt, SDIVt, SBASt, SAM Ht)0.
Subsequently, we estimate a multivariate DCC-GARCH(1,1) model on the ob-tained innovations. Defining April 1990 as t = 1, the model is specified as follows:
Yt= Σt0.5Zt (7)
Σt= ∆tPt∆t (8)
Pt= P ((1 − α1− β1)Pc+ α1Σ−1t−1Xt−1(Σ−1t−1Xt−1)0+ β1Pt−1), (9)
t = 1, ..., 361, where Σt is a diagonal volatility matrix, Pc is a constant
cor-relation matrix, P is a specified function, Ztis SWN(0,Id). Diagonal elements
of Σt are given by σ2t,k = αk,0+ αk,1Xt−1,k2+ βk,1σ2t−1,k.
Figure 1: Monthly values of the composite variables for the Risk Tol-erance Regime, Macro Outlook Regime and Macro Stability Regime April 1990 - March 2020
The figure displays the monthly values of the composite variables for the Risk Tolerance Regime, Macro Outlook Regime and Macro Stability Regime for the time period from April 1990 to March 2020.
Regime. Statistics of the principal component analysis performed can be found in table A2 in the appendix.
Monthly values of the composite variables for the Risk Tolerance Regime, Macro Outlook Regime and Macro Stability Regime for the time period from April 1990 to March 2020 are displayed in figure 1. As can be seen in the figure, the values of the Risk Tolerance Regime and Macro Outlook Regime vary considerably over time, whereas the value of the Macro Stability Regime is relatively stable over time. The big spikes observable in the figure correspond to the second half of the year 2008, approximately the time period when the financial crisis hit the US market the most. So, it seems like our macroeconomic regime specifications have well captured the singularity of this event.
4.1.4 Risk-On Conditions Regime
Unlike the other composite variables, the fourth regime created in this the-sis is not based on a particular choice of model specification. Instead, we sort innovations in the macroeconomic state variables dividend yield and sys-tematic volatility into four different groups. We estimate equation (1) with Xt= (DIVt, V OLt) and extract again the innovations of the estimated model.
Based on their signs we divide the obtained innovations into four groups. Obser-vations from moments in time with negative shocks in both aggregate dividend yield and systematic volatility form a group, just as observations from moments with positive shocks in both state variables. The other two groups are formed by moments with a negative shock in aggregate dividend yield and a positive shock in systematic volatility and moments with a positive shock in aggregate dividend yield and a negative shock in systematic volatility.
The created groups are supposed to represent environments. Negative shocks in both dividend yield and systematic volatility can be considered as favorable macroeconomic conditions. On the contrary, positive shocks in both dividend yield and systematic volatility possibly indicate a more unstable market with less investment opportunities and therefore unfavorable macroeconomic condi-tions. The other two groups are in between these groups and give ambiguous signals about the state of the macro economy.
Similar to the minimum dependency portfolios constructed for single macroe-conomic state variables, we use April 1990 to November 2016 as the estima-tion period for the construcestima-tion of the minimum dependency portfolios for the macroeconomic regimes specified. We choose the out of sample period to be De-cember 2016 to March 2020 again. We evaluate the performance of the portfolios obtained in both periods using the performance measures specified before.
4.2
Equity factor returns regressions
Table 6: Estimated coefficients time series regressions standard equity factor returns on unexpected shocks macroeconomic state variables April 1990 - March 2020
equity factor Size Value High prof. Low inv. Mom. intercept 0.00093 0.00075 0.0033** 0.0018* 0.0054** SDIVt -2.92 1.52 2.44 3.05* 0.58 SSIRt -7.06 1.42 2.12 -2.28 -0.03 STERMt 0.79 1.39* 0.60 -0.27 -1.39 SDEFt 1.29 2.23 1.09 0.37 3.12 SSVt 0.04 -0.07 -0.06 -0.11*** 0.02 SABASt 1.71** -0.04 -0.87 -1.10** -0.20 SAPIt -0.12 0.18 0.12 0.27* 0.83**
The table reports the estimated coefficients for the time series regressions of the five single standard equity factors considered in this thesis on the derived unexpected shocks in the seven constructed macroeconomic state variables specified in equation (2). Monthly data in the time period from April 1990 to March 2020 is used for the analysis. Significance levels of 10,5,1 and 0.1 percent are indicated with *,**,*** and ****, respectively.
In table 6 we can see that the Value factor has an economically significant posi-tive dependence on shocks in the state variables aggregate dividend yield, short interest rate,term spread and default spread. The effects of shocks in the other variables on the value factor are close to zero in magnitude. The size factor has an economically significant negative dependence on shocks in dividend yield and short interest rate. This suggest that combining the Value factor and Size factor in an investment portfolio gives an opportunity to immunize the effect of shocks in the state variables short interest rate and term spread. Even more impor-tantly, the size factor is the only factor with an economically significant negative dependence on shocks in dividend yield. This characteristic should imply that this factor plays an important roll in the minimum dependency portfolio con-structed for the state variable aggregate dividend yield. As we will see later in this paper, with a portfolio weight of 42.7 percent in this portfolio this is indeed the case. Another interesting observation is that the effect of shocks in the state variable systematic volatility is close to zero for all equity factors. So, changes in the expectation of this state variable do not seem to influence the performance of all equity factors considered much.
Table 7: Estimated coefficients time series regressions within factor mimicking portfolio returns on unexpected shocks macroeconomic state variables January 1997 - December 2019
equity factor Size wi. Value wi Qual. wi. Mom. wi. intercept 0.0010** 0 0.00075*** 0.00090* SDIVt 0.98 -0.84 0.67 2.19** SSIRt 1.51 0.65 0.39 -0.31 STERMt -0.10 0.30 0.021 -0.82*** SDEFt -0.64 0.95** 0.53** 0.24 SSVt -0.02 -0.020 0.00049 -0.0059 SABASt 0.36 0.37 -0.25* 0.067 SAPIt 0.01 0.16 0.016 -0.060
The table reports the estimated coefficients for the time series regressions of the four within factor mimicking portfolios considered in this thesis on the derived unexpected shocks in the seven constructed macroeconomic state variables specified in equation (2). Monthly data in the time period from January 1997 to December 2019 is used for the analysis. Significance levels of 10,5,1 and 0.1 percent are indicated with ∗,∗∗,∗ ∗ ∗ and ∗ ∗ ∗∗, respectively.
Table 8: Estimated coefficients time series regressions across factor mimicking portfolio returns on unexpected shocks macroeconomic state variables January 1997 - December 2019
equity factor Size acc. Value acc. Qual. acc. Mom. acc. intercept 0.0017 -0.00046 0.0021** 0.0016**** SDIVt -6.77* -3.07 -0.0026 -0.063 SSIRt 3.09 -6.09 1.72 -6.03 STERMt -0.60 -0.12 0.45 -2.58 SDEFt 1.83 1.16 0.060 1.47 SSVt 0.04 0.0052 0.049 -0.079 SABASt 0.49 -0.02 -0.31 -1.43* SAPIt 0.56 0.16 0.17 0.36
The table reports the estimated coefficients for the time series regressions of the four across factor mimicking portfolios considered in this thesis on the derived unexpected shocks in the seven constructed macroeconomic state variables specified in equation (2). Monthly data in the time period from January 1997 to December 2019 is used for the analysis. Significance levels of 10,5,1 and 0.1 percent are indicated with ∗,∗∗,∗ ∗ ∗ and ∗ ∗ ∗∗, respectively.
portfolios not corrected for regional and industrial effects (across industries) on these shocks. An explanation for this could be that the factor mimicking portfolios corrected for industrial effects capture more the pure effects of the unexpected shocks in the macroeconomic state variables due to the neutrality of the portfolios to regional and industrial effects, which leads to less uncertainty in the parameters estimated (lower standard deviations).
5
Results
5.1
Minimum dependency portfolios constructed for
sin-gle macroeconomic state variables
We first construct portfolios with a minimum dependency on a single macroe-conomic state variable using the standard equity factors size, value, high prof-itability, low investment and momentum. We allow short-selling, in the next parts of this section we will restrict the equity factor weights of the constructed portfolios to be non-negative. We first proceed as in the example given in the methodology section. So, we define states of the single macroeconomic state variable specified as quintiles (we choose N=5) of the unexpected shocks in the respective state variable. We use monthly observations and innovations of the constructed macroeconomic state variables and monthly equity factor returns from the time period April 1990 to November 2016 as input for the construction of the minimum dependency portfolios. As a next step, we evaluate the per-formance of the portfolios constructed using monthly data for the time period December 2016 to March 2020.
Table 9: Statistics minimum dependency portfolios for single macroe-conomic state variables using standard equity factors
Panel A: Statistics estimation period state variable mean stdev tot. spr.
VaR95 ES95 macr. spr.
sq. dev. Agg. div. yield 0.23 1.99 20.5 -2.51 -3.53 0 0 Short interest rate 0.29 1.87 17.7 -2.33 -3.74 0.159 0 Term spread 0.20 3.83 25.3 -5.55 -7.59 0 0 Default spread 0.35 7.73 59.0 -11.3 -19.0 0 0 Systemic volatility 0.23 10.6 77.3 -15.7 -22.8 0 0 Agg. eff. b-a spread 0.28 3.44 29.4 -5.86 -8.04 0 0 Agg. price impact 0.15 3.54 29.1 -4.61 -7.97 0 0 Panel B: Statistics out of sample period
state variable mean stdev tot. spr.
VaR95 ES95 macr. spr.
sq. dev. Agg. div. yield -0.55 1.65 8.7 -2.6 -4.3 1.68 0.022 Short interest rate -0.56 1.41 7.8 -2.6 -3.7 1.38 0.0099 Term spread -1.6 3.46 17.6 -5.89 -8.58 1.09 0.0087 Default spread -2.1 10.1 63.4 -11.8 -31.0 6.34 0.27 Systemic volatility 0.55 1.12 65.7 -11.9 -18.4 15.1 1.45 Agg. eff. b-a spread -0.25 3.21 17.4 -4.6 -9.0 3.21 0.083 Agg. price impact -0.33 2.95 14.1 -5.0 -6.1 3.02 0.048
The table reports the main statistics for the returns of the constructed minimum
dependency portfolios for all single macroeconomic state variables using five standard equity factors. The statistics are given for the estimation period (Panel A) and the out of sample period (Panel B). The estimation period ranges from April 1990 to November 2016, the out of sample period ranges from December 2016 to March 2020. The statistics presented are mean value (µ), standard deviation (σ), difference between maximum return and minimum return (total spread), the 95 percent Value at Risk (VAR95), the 95 percent Expected Shortfall (ES95), the maximum difference between conditional expected returns for states of the macroeconomic state variable (macro spread) and the value of the minimized sum of squared deviations (squared deviation). The 95 percent Value at Risk and 95 percent Expected Shortfall are calculated using the historical simulation method.No restrictions were placed on the portfolio weights during the optimization. All statistics are given in hundreds.
Table 10: Portfolio weights minimum dependency portfolios for single macroeconomic state variables using standard equity factors
state variable SMB HML RMW CMA MOM Agg. div. yield 0.620 0.0459 0.0171 0.273 0.0444 Short int. rate 0.0377 0.369 0.154 0.340 0.0993 Term spread 0.0213 0.962 -0.800 0.715 0.103 Default spread 1.43 2.54 0.661 -4.17 0.534 Syst. vol. -1.23 -1.70 -2.14 5.79 0.275 Agg. eff. b-a spread 0.674 0.182 1.30 -0.957 -0.203
Agg. price impact -0.0118 -0.447 0.328 1.63 -0.503
The table reports the portfolio weights of the minimum dependency portfolios constructed for all single macroeconomic state variables using five standard equity factors. No restrictions were placed on the portfolio weights during the optimization. The following abbrevations for the equity factors are used: SMB (size), HML (value), RMW (high profitability), CMA (low investment) and MOM (momentum).
Looking at the results given in table 9, we observe that the macro spreads of the constructed portfolios are much higher in the out of sample period compared to the estimation period. So, the portfolios constructed to have the theoretical lowest possible dependency on the single macroeconomic state variables in the estimation period do not seem to hold similarly low dependencies on the respec-tive state variables out of sample.
In table 10 we can see that for all minimum dependency portfolios constructed, except the portfolios constructed for aggregate dividend yield and short interest rate, some equity factors are shorted. So, the non-negative restriction on the portfolio weights imposed in the next part will have an influence on most port-folios constructed, resulting in higher macro spreads, at least in the estimation period, for the respective portfolios.
We suspect that allowing short-selling when constructing minimum dependency portfolios does not benefit the out of sample performance of the portfolios con-structed, since the minimization problems solved here are very sensitive to input. Moreover, in many practical applications short-selling is not considered. There-fore, in the remainder of this thesis we will restrict the portfolio weights of the minimum dependency portfolios constructed to be non-negative.
Table 11: Statistics minimum dependency portfolios for single macroe-conomic state variables using standard equity factors
Panel A: Statistics estimation period state variable mean stdev tot. spr.
VaR95 ES95 macr. spr.
sq. dev. Agg. div. yield 0.23 1.99 20.5 -2.51 -3.53 0 0 Short int. rate 0.29 1.87 17.7 -2.33 -3.74 0.159 0 Term spread 0.26 1.74 14.0 -2.50 -3.23 0.100 0 Default spread 0.28 1.45 12.7 -1.74 -2.67 0.153 0 Syst. vol. 0.28 1.62 14.4 -1.89 -3.09 0.121 0 Agg. eff. b-a spread 0.26 1.43 11.2 -1.66 -2.47 0.110 0 Agg. price impact 0.30 1.56 13.6 -1.98 -3.07 0.103 0 Panel B: Statistics out of sample period
state variable mean stdev tot. spr.
VaR95 ES95 macr. spr.
sq. dev. Agg. div. yield -0.55 1.65 8.7 -2.6 -4.3 1.68 0.0222 Short int. rate -0.56 1.41 7.8 -2.6 -3.7 1.38 0.00986 Term spread -0.69 1.58 8.6 -2.6 -4.4 0.72 0.00306 Default spread -0.52 1.47 8.9 -2.0 -5.0 0.84 0.00477 Syst. vol. -0.33 1.02 4.0 -1.8 -2.0 1.3 0.0122 Agg. eff. b-a spread -0.48 1.51 8.6 -2.3 -4.9 1.76 0.016 Agg. price impact -0.15 1.09 5.7 -1.5 -3.1 1.30 0.0118
The table reports the main statistics for the returns of the constructed minimum dependency portfolios for all single macroeconomic state variables using five standard equity factors. The statistics are given for the estimation period (Panel A) and the out of sample period (Panel B). The estimation period ranges from April 1990 to November 2016, the out of sample period ranges from December 2016 to March 2020.The statistics presented are mean value (µ), standard deviation (σ), difference between maximum return and minimum return (total spread), the 95 percent Value at Risk (VAR95), the 95 percent Expected Shortfall (ES95), the maximum difference between conditional expected returns for states of the
macroeconomic state variable (macro spread) and the value of the minimized sum of squared deviations (squared deviation). The 95 percent Value at Risk and 95 percent Expected Shortfall are calculated using the historical simulation method. The statistics are given for the estimation period ranging from April 1990 to November 2016. The portfolio weights were restricted to be non-negative during the optimization. All statistics are given in hundreds.
Table 12: Portfolio weights minimum dependency portfolios for single macroeconomic state variables using standard equity factors
state variable SMB HML RMW CMA MOM Agg. div. yield 0.620 0.0459 0.0171 0.273 0.0444 Short int. rate 0.0377 0.369 0.154 0.340 0.0993 Term spread 0.205 0.325 0 0.412 0.0586 Default spread 0.331 0.299 0.293 0 0.0775 Syst. vol. 0.221 0 0.0177 0.642 0.120 Agg. eff. b-a spread 0.429 0.167 0.335 0.0696 0
Agg. price impact 0.358 0 0.642 0 0
The table reports the portfolio weights of the minimum dependency portfolios constructed for all single macroeconomic state variables using five standard equity factors. The portfolio weights were restricted to be non-negative during the optimization. The following
abbrevations for the equity factors are used: SMB (size), HML (value), RMW (high profitability), CMA (low investment) and MOM (momentum).
As expected, except the portfolios constructed for aggregate dividend yield and short interest rate, the portfolios constructed under the restriction of non-negative portfolio weights exhibit higher macro spreads and squared deviations than the portfolios constructed without portfolio weights restrictions in the es-timation period (the portfolios constructed for aggregate dividend yield and short interest rate are the same portfolios as the ones constructed without the restrictions imposed). Interestingly, however, the macro spreads and squared de-viations of these five portfolios in the out of sample period are all lower than the macro spreads and squared deviations of the respective portfolios constructed with short-selling allowed in this period. So, in hindsight these portfolios were better able to immunize shocks in the respective macroeconomic state variable in future times, whereas their historical performance in this aspect was worse. Comparing the unconditional performance of the portfolios constructed under the portfolio weights restriction and the portfolios constructed with no portfo-lio weights restricted, we see that the mean returns in the estimation period are comparable. However, the standard deviations of the returns of the port-folios constructed under the restriction of non-negative weights are much lower and the 95 percent Value at Risks and 95 Expected shortfalls are on average much higher (less negative) in this period. So, we conclude that historically the portfolios constructed without allowance of short-selling provide a better unconditional risk-return trade-off.
minimum dependency portfolio constructed for the state variable aggregate ef-fective bid-ask spread has the lowest standard deviation, lowest total spread and highest (least negative) 95 percent Value at Risk and 95 percent Expected Shortfall. Thus, historically this portfolio seems to have the lowest risk of all portfolios constructed. However, the portfolio has a relatively low historical monthly return compared to the other portfolios constructed (0.26 percent), indicating that in the past the lower risk of the portfolio went together with lower returns. Looking at the out of sample performance of the portfolio, we see that the unconditional and conditional performance of this portfolio in the out of sample period is comparable with the performance of the other portfolios constructed. So, the relatively low risk of the portfolio in the estimation period does not seem to translate to a lower risk in the out of sample period.
Focusing on the macro spreads of the constructed portfolios in the out of sample period, we can see that the minimum dependency portfolio constructed for the interest rate term spread holds the lowest macro spread (0.72 percent), whereas the portfolios constructed for aggregate effective bid-ask spread and aggregate dividend yield hold the highest macro spreads (1.76 percent and 1.68 percent respectively). So, apparently in the out of sample period specified dependence of equity factor portfolio returns on unexpected shocks in the interest rate term spread is immunized much better than dependence of portfolio returns on unex-pected shocks in aggregate dividend yield or aggregate effective bid-ask spread for the portfolios constructed. This is confirmed by the squared deviations of the portfolios in this period, which are very high for the minimum dependency portfolios constructed for aggregate dividend yield and aggregate effective bid-ask spread in comparison to those for the other portfolios constructed. Both portfolios also exhibit relatively high downside risk as measured by the 95 per-cent Value at Risk and 95 perper-cent Expected Shortfall.
Looking at the portfolio weights given in table 12, we first observe that all five standard equity factors have a significant weight in at least one portfolio constructed. So, it seems like all standard equity factors considered in this paper contribute to creating portfolios with the lowest possible dependency on macroe-conomic conditions. The size factor has a substantial weight in the minimum dependency portfolios constructed for all single macroeconomic state variables except short interest rate. Since we observed earlier that the size factor expe-rienced on average a relatively low return in the past compared to the other equity factors, this seems not to be particularly good news for the historical performance and future performance of the constructed portfolios. However, the volatility of the return of the size factor is similar to the volatilities of the other factor returns and the correlations between the size factor and all four other equity factors are negative, indicating that the volatilities of the portfo-lios obtained should be relatively low in the estimation period and out of sample period.
Value and Momentum or Size and High Profitability in equity portfolios pro-vides good diversification opportunities due to the clearly negative correlations between these equity factors. We see in table 12 that in particular the minimum dependency portfolio constructed for default spread seems to utilize the negative correlation between Size and High Profitability by giving substantial weights to these factors. So, apparently both the unconditional performance and the con-ditional performance for the state variable default spread is improved by giving high portfolio weights to the equity factors Size and High Profitability in the portfolio constructed.
Comparing the results of table 6 and table 12, we observe that in most portfo-lios constructed substantial weights are given to equity factors with an econom-ically low (close to zero) dependence on shocks in the respective macroeconomic state variable. This does not come as a surprise, since that kind of equity fac-tors provides relatively stable conditional returns for the different states of the macroeconomic state variable and is therefore important in constructing portfo-lios with the least possible dependence on the respective single macroeconomic state variable.
5.2
Minimum dependency portfolios constructed for
macroe-conomic regimes
Besides dependencies of equity factors on shocks in macroeconomic state vari-ables, we also analyse dependencies of these equity factors on constructed macroe-conomic regimes. In the next parts of this section the results of the minimum dependency portfolios constructed for the four regimes specified earlier in this paper are presented.
5.2.1 Risk Tolerance regime and Macro Outlook regime
Table 13: Statistics returns minimum dependency portfolios for Risk Tolerance regime and Macro Outlook regime using standard equity factors
Panel A: Statistics estimation period Regime spec. mean stdev tot.
spr.
VaR95 ES95 macr. spr.
sq. dev. Risk Tolerance 0.237 1.79 11.7 -2.36 -3.29 0.869 0.0901 Macro Outlook 0.325 1.86 18.5 -2.11 -3.76 0.236 0 Panel B: Statistics out of sample period
Regime spec. mean stdev tot. spr.
VaR95 ES95 Risk Tolerance -0.53 1.83 10.4 -2.62 -5.53 Macro Outlook -0.08 0.882 4.06 -1.49 -1.73
The table reports the main statistics for the returns of the constructed minimum dependency portfolios for the Risk Tolerance regime and the Macro Outlook regime using five standard equity factors. The statistics are given for the estimation period (Panel A) and the out of sample period (Panel B). The estimation period ranges from April 1990 to November 2016, the out of sample period ranges from December 2016 to March 2020. The statistics given are defined similarly as before. All statistics are given in hundreds.
The weights of all equity factors in the minimum dependence portfolios con-structed for the Risk Tolerance regime and the Macro Outlook regime are given in table 14.
Table 14: Portfolio weights minimum dependency portfolios for Risk Tolerance regime and Macro Outlook regime using standard equity factors
Regime specification SMB HML RMW CMA MOM Risk Tolerance 0.634 0.111 0.255 0 0 Macro Outlook 0 0.0115 0.498 0.424 0.0662
The table reports the portfolio weights of the minimum dependency portfolios constructed for the Risk Tolerance regime and the Macro Outlook regime using five standard equity factors. The following abbrevations for the equity factors are used: SMB (size), HML (value), RMW (high profitability), CMA (low investment) and MOM (momentum).
observe that the portfolio constructed for the Risk Tolerance regime experienced a relatively low mean return and volatility in the estimation period, whereas the portfolio constructed for the Macro Outlook regime experienced a relatively high mean return and slightly higher volatility in this period. The 95 percent Value at Risk and 95 percent Expected shortfall of both portfolios are similar and comparable to the ones of the portfolios constructed for the single macroeco-nomic state variables.
Remarkably, in the out of sample period the portfolio constructed for the Macro Outlook regime has a higher (less negative) average return, lower standard de-viation and higher (less negative) 95 percent Value at Risk and 95 percent Ex-pected shortfall than the portfolio constructed for the Risk Tolerance regime. So, the unconditional performance of the minimum dependency portfolio con-structed for the Macro Outlook regime can be considered to be superior to the unconditional performance of the minimum dependency portfolio constructed for the Risk Tolerance regime.
Focusing on the conditional performance of the constructed portfolios, we ob-serve that in the estimation period for both portfolios the macro spreads and squared deviations are relatively high compared to those of the portfolios con-structed for the single macroeconomic state variables. So, from this viewpoint it seems like constructing portfolios with the least possible dependency on macroe-conomic regimes is less successful than constructing portfolios with the least possible dependency on single macroeconomic state variables. Later on in this section we will see whether the results of the portfolios constructed for the Macro Stability regime and the Risk-on Conditions regime are in agreement with this. Note that in table 13 there are no macro spreads and squared devi-ations given for the out of sample period for the two portfolios. Since we have only constructed monthly values of the composite variables for both regimes until March 2019, we are not able to calculate these two statistics for the out of sample period ranging from November 2016 and March 2020.
The results of table 14 reveal that the minimum dependency portfolio con-structed for the Risk Tolerance regime consists for the majority out of the size factor (63.4 percent). Conversely, the portfolio constructed for Macro Outlook has significant weights in all but the size factor with the highest weight for the high profitability factor.
5.2.2 Macro Stability regime and Risk-on Conditions regime
obtained values of the composite variable and define the time period from April 1990 to November 2016 as the estimation period and the time period from De-cember 2016 to March 2020 as the out of sample period.
For the construction of the Risk-on Conditions regime we follow the procedure specified in section 4.1.4. We define the four groups specified in this section as states of the Risk-on Conditions regime. As before we define the time period from April 1990 to November 2016 as the estimation period and the time period from December 2016 to March 2020 as the out of sample period.
Descriptive statistics for the returns of the minimum dependency portfolios constructed for the Macro Stability regime and the Risk-on Conditions regime using standard equity factors for both the estimation period and the out of sample period are given in table 15.
Table 15: Statistics returns minimum dependency portfolios for Macro Stability regime and Risk-on Conditions regime using standard equity factors
Panel A: Statistics estimation period Regime spec. mean stdev tot.
spr.
VaR95 ES95 macr. spr.
sq. dev. Macro Stab. 0.271 1.73 13.0 -2.13 -3.04 0.308 0 Risk-on Cond. 0.342 1.95 21.3 -2.18 -4.42 0.640 0.00275 Panel B: Statistics out of sample period
Reg. spec. mean stdev tot. spr.
VaR95 ES95 macr. spr.
sq. dev. Macro Stab. -0.301 1.39 7.28 -2.52 -3.65 1.55 0.0169 Risk-on Cond. -0.076 1.14 6.16 -1.76 -2.49 0.589 0.00223
The table reports the main statistics for the returns of the minimum dependency portfolio constructed for the Macro Stability regime and Risk-on Conditions regime using five standard equity factors. The statistics are given for the estimation period (Panel A) and the out of sample period (Panel B). The estimation period ranges from April 1990 to November 2016, the out of sample period ranges from December 2016 to March 2020.The statistics given are defined similarly as before. All statistics are given in hundreds.
Table 16: Portfolio weights minimum dependency portfolios for Macro Stability regime and Risk-on Conditions regime using standard equity factors
Regime specification SMB HML RMW CMA MOM Macro Stability 0.601 0 0.308 0 0.0913 Risk-on Conditions 0 0 0.133 0.608 0.258
The table reports the standard equity factor portfolio weights of the minimum dependency portfolios constructed for the Macro Stability regime and the risk-on conditions regime using five standard equity factors. The following abbrevations for the equity factors are used: SMB (size), HML (value), RMW (high profitability), CMA (low investment) and MOM (momentum).
Table 15 shows that in the estimation period the macro spreads of the minimum dependency portfolios constructed for the Macro Stability regime and Risk-on Conditions regime are again relatively high. So, similar as for the Risk Toler-ance regime and Macro Outlook regime, it seems to be relatively difficult to construct an equity factor portfolio that is immune to shocks in the states of the macro economy as defined by the Macro Outlook regime and Risk-on Con-dition regime, the latter being a regime specification with no particular choice of modelling involved.
5.3
Comparison dependencies standard equity factor
port-folios and factor mimicking portport-folios
In order to compare the dependencies of both standard equity factors and factor mimicking portfolios on unexpected shocks in macroeconomic state variables in the most appropriate way, we now construct minimum dependency portfolios for all single macroeconomic state variables both using standard equity factors and using corresponding factor mimicking portfolios. We use the time period from January 1997 to August 2016 as the estimation period of all portfolios constructed. The performances of the constructed portfolios are evaluated for the time period from September 2016 to December 2019. As indicated earlier, the standard equity factors considered in this comparison are size, value, high profitability and momentum. The factor mimicking portfolios considered are the mimicking portfolios for the factors size, value, quality and momentum. Both factors corrected for region and industry effects (within industries) and factors not corrected for industry effects (across industries) are used in the construction of the minimum dependency portfolios consisting of factor mimicking portfolios.
The portfolio weights of the minimum dependency portfolios constructed for all single macroeconomic state variables using the four standard equity factors mentioned before are given in table 17.
Table 17: Portfolio weights minimum dependency portfolios for single macroeconomic state variables using four standard equity factors
state variable SMB HML RMW MOM Agg. div. yield 0.639 0.260 0 0.101
Short int. rate 0.180 0.473 0 0.347 Term spread 0.426 0 0.512 0.0616 Default spread 0.0280 0.614 0 0.358
Syst. vol. 0.383 0.347 0.158 0.112 Agg. eff. b-a spread 0.592 0.299 0 0.108
Agg. price impact 0.359 0 0.641 0
The table reports the portfolio weights of the minimum dependency portfolios constructed for all single macroeconomic state variables using the four standard equity factors mentioned.
Table 18: Portfolio weights minimum dependency portfolios for single macroeconomic state variables using eight factor mimicking portfolios state variable SMBw HMLw QUAw MOMwSMBa HMLa QUAa MOMa Agg. div.yield 0 0.348 0.341 0.0169 0.0722 0.0788 0 0.144 Short int. rate 0.429 0.192 0.379 0 0 0 0 0 Term spread 0.0149 0.282 0.179 0 0.176 0.0975 0.250 0 Default spread 0.537 0.0413 0.0718 0.350 0 0 0 0 Syst. vol. 0.328 0.271 0.136 0.182 0 0 0.0821 0 Agg. eff. b-a spr. 0.199 0.431 0.053 0.245 0 0.0284 0.0493 0 Agg. pr. impact 0.236 0.187 0.360 0.210 0 0 0 0
The table reports the portfolio weights of the minimum dependency portfolios constructed for all single macroeconomic state variables using the eight factor mimicking portfolios mentioned. Factor mimicking portfolios corrected for regional and industrial effects (within) and not corrected for industrial effects (across) are indicated with wi. and acc. added to the equity factor name, respectively. The abbreviation QUA is used for the quality factor mimicking portfolio.
Relevant statistics of the returns of the minimum dependency portfolios con-structed for all single macroeconomic state variables using the four standard equity factors mentioned before for both the estimation period and the out of sample period are given in table 19.
Table 19: Statistics minimum dependency portfolios for single macroe-conomic state variables using four standard equity factors
Panel A: Statistics estimation period state variable mean stdev tot.
spr.
VaR95 ES95 macr. spr.
sq. dev. Agg. div.yield 0.24 2.20 19.23 -3.05 -4.18 0.51 0.00136 Short int. rate 0.27 2.22 20.84 -2.93 -5.37 0.34 0 Term spread 0.30 1.56 10.96 -1.73 -2.87 0.18 0 Default spread 0.27 2.46 24.24 -3.51 -6.13 0.40 0.00148 Syst. vol. 0.26 1.67 13.13 -2.25 -3.14 0.0049 0 Agg. eff. b-a spr. 0.24 2.10 17.18 -2.89 -4.05 0.081 0 Agg. pr. impact 0.30 1.70 13.58 -2.36 -3.34 0.20 0 Panel B: Statistics out of sample period
state variable mean stdev tot. spr.
VaR95 ES95 macr. spr.
sq. dev. Agg. div. yield -0.23 1.81 9.19 -2.65 -2.94 1.01 0.0070 Short int. rate -0.18 1.28 6.46 -1.84 -2.31 1.49 0.012 Term spread 0.019 0.91 4.18 -1.26 -1.47 0.84 0.0043 Default spread -0.21 1.50 7.20 -2.26 -2.85 1.12 0.0085 Syst. vol. -0.18 1.35 7.38 -2.09 -2.25 0.66 0.0030 Agg. eff. b-a spr. -0.24 1.76 9.14 -2.63 -2.93 1.54 0.017 Agg. pr. impact 0.051 0.90 3.83 -1.06 -1.31 0.91 0.0067
The table reports the main statistics for the returns of the constructed minimum
