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Master’s Thesis
Expected Shortfall and the Cross-Section of Stock Returns
Kevin Kleijer
Student number: 11811927 Date of final version: July 15, 2022 Master’s programme: Econometrics
Specialisation: Financial Econometrics Supervisor: S.C. S. Barendse
Second reader: dr. B. J. L. Keijsers
Statement of Originality
This document is written by Student Kevin Kleijer who declares to take full responsibility for the contents of this document.
I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it.
The Faculty of Economics and Business is responsible solely for the supervision of completion of the work, not for the contents.
Abstract
This thesis studies the additional factor, Expected Shortfall (ES), in explaining the variation in cross-sectional expected stock returns instead of Value-at-Risk (VaR). ES has proven to solve the problems arising using VaR. ES, volatility, and size showed explaining power in the cross-sectional variation of expected stock returns, while beta did not show significant power.
Furthermore, the positive relation of ES with average stock returns is shown to be robust for different probability levels. Additionally to cross-sectional testing, time-series regressions showed that ES at a portfolio level has additional explanatory power after the market return, size, book-to-market ratio, and VaR factors are controlled for. The results were based on 10 1%ES sorted and six/25 size/book-to-market portfolios for the CRSP data containing the NYSE, AMEX, and NASDAQ stock exchange markets from January 1958 through March 2022.
Contents
1 Introduction 5
2 Literature Review 7
2.1 Explaining power in cross-sectional variation in expected returns . . . 7
2.2 Expected Shortfall in addition to Value-at-Risk . . . 9
3 Data 11 4 The Model 14 4.1 Value-at-Risk and Expected Shortfall . . . 14
4.2 Comparing methods . . . 15
4.2.1 Volatility . . . 16
4.2.2 Backtesting VaR and ES . . . 18
4.3 ES and Expected Stock Returns . . . 19
5 Results 22 5.1 ES and Cross-sectional Returns . . . 22
5.2 ES and Portfolio Selection . . . 24
6 Conclusion 30
A Programs 32
B Figures & Tables 36
C Volatility Comparison 49
Bibliography 51
Chapter 1
Introduction
Individual stock and portfolio returns have positive and negative movements in expected returns.
The Capital Asset Pricing Model of Sharpe (1964), Lintner (1965), and Black (1972) imply that the cross-section in expected stock/portfolio returns is related to the market beta and therefore beta expressed individual risk. With variation in cross-sectional stock returns, we mean the varying returns across different stocks at one point in time (Cochrane, 2005). Additionally, it was found that the variation in movements of expected stock and portfolio returns can be explained by several characteristics such as the size, historic beta, liquidity, and the Book-to- Market (B/M) ratio of an asset. Fama and French (1993) proposed the three-factor model where they used portfolio grouping based on a market, size, and value factor and argued that this model explained the cross-sectional expected returns well. Then, using these factors, each factor’s beta is estimated via time-series regression, where the beta is the factor’s coefficient/sensitivity.
Naturally, investing in stocks and portfolios in the financial markets is paired with risk. The three main categories of financial risk are credit risk, operational risk, and market risk (Tsay, 2010). According to Tsay (2010), Value-at-Risk (VaR) has become the standard measure of market risk, where VaR is defined as the estimate of a maximum loss of an individual due to market movements given a holding period under a certain probability. To explain the cross- section in expected returns, Bali and Cakici (2004) used VaR as an additional explaining factor.
According to Tsay (2010), a problem with VaR is that in practice, the actual loss may be more significant than the estimated VaR leading to an underestimation of the actual loss. Ad- ditionally, Tasche (2002) & Acerbi and Tasche (2002) both have argued why Expected Shortfall (ES) is a better-performing risk measure than VaR. Expected Shortfall is the expected value of the loss function if the VaR is exceeded under a certain probability, where the loss function fol- lows a particular distribution. ES is more sensitive to the shape of the tails of the loss function’s distribution (Tsay, 2010). This paper uses ES as a risk factor since it has not been considered as an additional explanation in the cross-sectional expected returns. Therefore this leads to the main objective of the paper: to test whether Expected Shortfall can explain the variation in cross-sectional expected stock returns.
The models belonged to panel data models with large cross-sections and time series. Panel
data are observations for a sequence of periods with multiple variables (Cameron and Trivedi, 2005). First, ES tried to predict the risk premium using Fama-MacBeth cross-sectional regres- sions based on the panel data from the NYSE, AMEX, and NASDAQ stock exchange markets.
Additionally, size and beta were also included in the cross-sectional regressions. Then, the vari- ation of the cross-section in portfolio returns was tried to be explained by a four- and five-factor model that corrected for the three factors by Fama and French (1993), VaR and an additional ES factor. These models showed whether ES has additionally explaining power in cross-sectional portfolio returns. At last, the three-factor model by Fama and French (1993) was expanded with volatility and ES, where we used six and 25 size-B/M portfolios showing if ES had additional explaining power in expected stock returns by calculating the factor’s risk premiums.
The cross-sectional regressions by Fama and MacBeth (1973) showed that ES and size ex- plained the variation in cross-sectional expected stock returns while beta does not, contradicting the Capital Asset Pricing Model. In the portfolio setting, ES explained additionally after con- trolling for the VaR factor. However, not when the models were controlled for VaR and ES factors. The three- and four-factor models showed the explaining power from ES and volatility by having significant risk premiums in expected portfolio returns after correcting for market return, size, and the B/M ratio.
The remainder of this thesis is organized into five chapters—chapter 2 reviews past literature on the importance of explaining expected returns and variation in risk measures. Chapter 3 presents the dataset. The fourth chapter presents the model and framework for estimating VaR, ES, and cross-sectional regressions. In Chapter 5, the results for the methods are shown. At last, Chapter 6 concludes and suggests further research.
Chapter 2
Literature Review
2.1 Explaining power in cross-sectional variation in expected returns
The original model for understanding the trade-off between risk and return was followed by the Capital Asset Pricing Model (CAPM) by Sharpe (1964), Lintner (1965), and Black (1972) (SLB). The CAPM only takes one factor into account and is written as:
Rt= α + βRMt + ϵt (2.1)
where Rtis the asset’s return, RMt is the market portfolio return, β is the market risk/exposure, α is the excess return adjusted for market risk, and ϵtare idiosyncratic movements. In particular, Luenberger (2009) stated that when α = 0, the market portfolio corresponded with market risk explaining that the relation between the expected return and the individual asset risk can be expressed using the CAPM. Additionally, Frazzini and Pedersen (2014) stated that a high β was associated with a low α, confirming the relationship between risk and return. For asset pricing, the CAPM is the most famous and widely used (Cochrane, 2005). However, Jagannathan and Wang (1996) wrote that it could not explain the cross-sectional difference in stock returns.
Alternatively to the CAPM, Ross (1976) developed the arbitrage pricing theory (APT).
Contrary to the CAPM, the APT assumes that markets misprice assets. The APT is a multiple factor model that relates risk variables to prices of financial support. Hence, the APT expressed that a linear relationship between the asset’s expected returns and their market beta (risk) exists.
Fama and French (1992) examined the effect of beta and size and found that high returns, small size, and high beta were all correlated. However, when the stocks were ordered by size, the explaining power of beta disappeared. Fama and French (1992) found that beta seemed to have a positive relationship with returns but that it was obscured by noise. They stated that there is an absence of the relation between beta and expected returns in contradiction to the SLB model in (2.1). Fama and French (1993) later followed by adjusting more than market and
size returns by implying the three-factor model:
Rt= α + βMRMt + βHM LRHM Lt + βSM BRSM Bt + ϵt (2.2) The now two added parameters were HML, which abbreviates for high minus low based on the book-to-market ratio (B/M) that corresponded with a value effect, and SMB, which shortens for small minus big based on a size effect. Bali and Cakici (2004) stated that these two effects were contradictions of the CAPM.
Cohen et al. (2003) showed that a high book-to-market ratio shows poor earnings and profitability, and a low book-to-market ratio shows the opposite. The stock with a high B/M currently has a low price and will have positive expected future returns (Pedersen, 2019). The
”value” effect came from the relation between the returns and the B/M ratio. The value strategy is obtained by having ”long” positions in stock with a high B/M and ”short” positions with a low B/M. Cohen et al. (2003) and Asness et al. (2013) stated that this was a preliminary cross-sectional result and contradicted the CAPM.
According to Pedersen (2019), a size strategy can be profitable: therefore, the size of stocks showed cross-sectional variation in expected returns. A ”size” strategy is to have ”long” po- sitions in small stocks and ”short” positions in more extensive stocks. Bali and Cakici (2004) concluded that there is a clear negative relationship between size and average return, hence the reason this strategy worked on generating positive expected returns and therefore contradicted the CAPM.
One of the most studied capital market processes is the relation between the asset’s return and its past performance alleged ”momentum” effect (Jegadeesh and Titman, 1993). A ”mo- mentum” strategy is to compute past returns of stocks for the last months and then to have
”long” positions in the top x% stocks (winners) and ”short” positions in the bottom x% stocks (losers) (Pedersen, 2019). A ”momentum” strategy proved to work; therefore, the past stock returns have a relationship with the expected future returns (Asness et al., 2013).
In addition to these explanations of cross-sectional variation in expected returns, Bali and Cakici (2004) concluded that Value-at-Risk (VaR) had additional explaining power on the ex- pected returns. Bali and Cakici (2004) constructed 1%, 5%, and 10% VaR and then used the regressions by Fama and MacBeth (1973) to explain the variation in cross-sectional expected stock returns. Bali and Cakici (2004) then followed the same method used by Fama and French (1992). Further, they constructed portfolios based on VaR and used these in the setting of Fama and French (1993), adding to (2.2). Bali and Cakici (2004) concluded that when constructing common stock portfolios, the average stock returns were positively correlated with VaR meaning lower risk led to lower expected returns. In the words of Bali and Cakici (2004): the stock with the lowest maximum likely loss under a certain probability had the lowest average returns.
2.2 Expected Shortfall in addition to Value-at-Risk
Financial institutes’ most prevalent risk measure is the Value-at-Risk (VAR) (Nadarajah et al., 2014). VaR measures the maximum loss, with confidence q = 1 − p, over horizon l (Tsay, 2010):
P[Xi,t≤ −V aRqi,t(l)] = p (2.3)
where Xi,t = −∆log(Vi,t) = −log(Vi,t+l− Vi,t) is the negative log return of some asset i with stock price Vi,t at time t, where Vi,t are expected stock prices of an asset i. VaR in this paper is written as:
V aRqi,t(l) = xqi,t(l) (2.4)
where V aRqi,t(l) is defined for an asset/company i at time t, given the holding period of an asset’s stock l. The calculation of VaR depends on the quantile q and the calculation method.
It is well known that, in general, Value-at-Risk is not a coherent risk measure (Acerbi and Tasche, 2002). Acerbi and Tasche (2002) stated that a risk measure ρ(X) is defined to be coherent if it satisfies axioms:
1. Translation invariance: ρ(X − a) = ρ(X) − a ∀ constant a;
2. Subadditivity: ρ(X1+ X2) ≤ ρ(X1) + ρ(X2);
3. Positive homogeneity: ρ(λX) = λρ(X) ∀ λ ≥ 0;
4. Monotonicity: if X ≤ Y with probability one, then ρ(X) ≤ ρ(Y ).
VaR violates the subadditivity property (2.) proved in the paper of Tasche (2002) and hence is incoherent. The violation of subadditivity means that the risk of a portfolio X1+ X2 can be larger than the sum of the individual risks: X1 and X2 (Acerbi and Tasche, 2002), where the sign-in bullet two is flipped. For example, in certain circumstances, it might be better to split a large firm into two smaller firms (Tasche, 2002). The incoherence of VaR then might discourage diversification, making the split no longer possible.
VaR is a law invariant risk measure (Tasche, 2002). Law invariance means that taking the risk measure VaR and X1 and X2 have identical distribution between the variables (X1 ∼ X2):
X1 ∼ X2 → V aR(X1) = V aR(X2) (2.5) Tasche (2002) stated that VaR is law invariant in a powerful sense: the distributions of X1
and X2 do not need to be identical to imply V aR(X1) = V aR(X2). There is criticism against this feature because random variables X1 with light tails probabilities and X2 with heavy tails probabilities may have the same VaR.
At last, VaR is based on the upper tail of a loss distribution. Hence there is criticism that VaR only measures the percentile of profit loss distributions and is not informative of the losses in upper percentiles Tsay (2010). Furthermore, the loss function’s Cumulative Distribution Function (CDF) is unknown. According to Tsay (2010), on page 327, studies are concerned
with the estimation of the upper tail behavior of the loss CDF. Because VaR as a risk measure suffers from several drawbacks, an alternative for VaR to measure risk is conditional Value-at- Risk (CVaR) known as Expected Shortfall (ES).
The alternative risk measure, Expected Shortfall, is computed using VaR. ES calculation is based on a conditional value if the VaR is exceeded. Following the notation of Tsay (2010) we would write:
E[Xi,t|Xi,t> V aRqi,t(l)] = ESi,tq (l) = Si,tq (l) (2.6) Here it is assumed that Xi,t follows a loss distribution. ESi,tq (l) is again defined for a company i at time t, given the holding period of a an asset’s stock l. The calculation of ES also depends on the quantile q and the calculation method.
Nadarajah et al. (2014) stated that ES is a similar risk measure to VaR but does not suffer from certain drawbacks. The main advantage of the alternative risk measure, Expected Shortfall, is that the tails of the distribution of the returns are taken into account Tsay (2010).
On the other hand, Acerbi and Tasche (2002) stated that the most critical property of ES is its coherence, which was proven in their paper by satisfying the four axioms. Hence ES is a subadditive risk measure contrary to VaR. In addition, Tasche (2002) stated that ES is law invariant and an additive risk measure as well as VaR.
Because ES follows the same methods as VaR, there should be reasons to believe that ES will have additional explaining power. However, Nadarajah et al. (2014) stated some shortcomings of the ES risk measure in addition to VaR. For example, ES is inconsistent with the right tail risk measured by convex order. Additionally, ES needs a larger sample size than VaR for the same level of accuracy (Nadarajah et al., 2014). Because of the larger sample size for ES, problems in additional explanatory power in expected returns could occur (McNeil et al., 2015).
Chapter 3
Data
Following Bali and Cakici (2004), the data were derived from the Center for Research in Se- curity Prices (CRSP) for all non-financial companies. The CRSP data was available using the Wharton Research Data Services (wrds). The CRSP contained NYSE, AMEX, and NASDAQ stock exchange markets with monthly data from January 1958 through March 2022. Not every company had shares listed on the stock exchange through the selected period because they would go insolvent or not even exist at the starting date. However, the stock data available for these companies were still very relevant to the report.
The stock data can be designated using company codes assigned by the CRSP. The company data were distinguished using the PERMNO. The PERMNO is a unique 5-digit permanent security identification integer posted by CRSP varying from -999989 to -100 and 100 to 99989.
The PERMNO neither changes during the trading history of an asset nor is reassigned after an asset ceases trading.
The data from the three exchange markets are each assigned with an exchange code. The NYSE has exchange code 1, the AMEX has exchange code 2, and the NASDAQ has exchange code 3. Some companies changed exchange markets in the past, so they contained information in both data sets. The overlapping stocks were put together and treated as one complete dataset (CRSP).
From the CRSP data, the following company data were extracted: the closing price (V ), the share volume (V OL), the number of shares outstanding (SHROU T ), and the value-weighted return (vwretx) excluding dividends. The vwretx was multiplied by 100 to gain percentages.
The closing price contained negative signs to indicate that the price is the bid/ask average and not the actual closing price. Therefore, the negative signs were removed from the price column.
If neither the closing price nor bid/ask average was available, the price was set to 0 or a NaN value. Companies with only 0 or NaN values and with less than 24 months of data on prices were dropped from the dataset. Prices that contained a zero or NaN value at the start or end of the period of an individual stock were dropped. When the 0 or NaN value occurred in the middle, that value was replaced by linearly interpolating the previous and next month’s price.
The volume is the sum of trading volumes during that month expressed as hundred shares. The
SHROU T measures the number of publicly held shares in thousands.
The monthly rate of return was calculated using the continuously compounded logarithmic returns. The computation of the logarithmic expected return for a company i at time t was as follows:
Ri,t = 100 × [∆log(Vi,t)] = 100 × [log(Vi,t+1) − log(Vi,t)] (3.1) Vi,t was multiplied by 100 so that the returns are in percentages, and the time horizon l was set to one month. This paper takes the logarithm (log) as the natural logarithm. The returns contained outliers due to extreme historical events. Hotta and Tsay (1998) stated that over- looking outliers may lead to substantial over/underestimation of asset volatility. Charles (2008) used a detection method for outliers that came close to the 99th percentile of the empirical distribution for the returns. When taking the 99th percentile of returns out of the CRSP data set, it corresponded with a |60|% return. Therefore the individual stock returns above 60% or below minus 60% were linearly interpolated, resulting in 1% of returns changing.
Table 3.1: The number of unique stocks within the three exchange markets: NYSE, AMEX, and NASDAQ
NYSE (1) AMEX (2) NASDAQ (3)
NYSE (1) 7827 873 1202
AMEX (2) 4471 945
NASDAQ (3) 17326
Table (3.1) contains the number of unique companies selected by the PERMNO after pre- processing. The top left value is a cross-section of NYSE (1), meaning that there were 7827 individual company stocks on the NYSE exchange market. The diagonal of Table (3.1) follows the same reasoning. The off-diagonal values are the number of overlapping companies, so for the cross-section of NYSE (1) and AMEX (2), there were 873 companies listed in history on both exchanges. So in total there were (7827 + 4471 + 17326 − 873 − 945 − 1202 =) 26604 unique company stocks in the CRSP data set.
For the CRSP data, we calculated size, beta, VaR, and ES for each company i at month t.
VaR and ES were calculated with different methods that required a comparison, discussed in Section (4.1) and (4.2). Following Bali and Cakici (2004), the size variable was computed for each of the individual stocks by:
sizei,t = log(M Ei,t) = log(Vi,t× SHROU Ti,t) (3.2) where M Ei,t stands for the market value of equity for a company i at time t.
Following Fama and French (1992), the beta variable was calculated using the value-weighted returns from the CRSP index and sets these as a proxy for the market returns. Then beta was obtained using a rolling time-series regression with a window of 24 months:
Ri,t= β0,i+ β1,iRi,tM+ β2,iRi,t−1M + ϵi,t (3.3)
where RMi,t is the market return for the company i at time t, and ϵi,t are the residuals from the time regression. The sum of beta given by betai,t = β0,i+ β1,i+ β2,i is beta given by the primary CAPM model from Sharpe (1964), Lintner (1965), and Black (1972).
At last, for testing explaining power in the risk factors, the factors and the six and 25 size-B/M value-weighted portfolio returns from the Fama and French (1993) factor model were obtained from the wrds. First, Fama and French (1993) created six portfolios by splitting the data on the median from size into a small and big group. Then the two size portfolios were split into three groups using the B/M ratio. The B/M ratio splits into 30 percent low, 40 percent middle, and 30 percent high portfolios. Next, the 25 portfolios were constructed similarly but now split into five groups twice based on size and the B/M ratio. The six portfolios are small-low (smlo), small-medium (smme), small-high (smhi), big-low (bilo), big-medium (bime), and big- high (bihi). The factors that follow from the six portfolios are SMB and HML. In addition to the factors, there is the the value-weighted return on the market (MKTRF), and the momentum return (UMD) for the monthly periods January 1958 through March 2022. They follow the NYSE, AMEX, and NASDAQ stock exchanges.
The computation of the SMB factor using the six portfolioss was as follows:
SM B = 1
3(smlo + smme + smhi) − 1
3(bilo + bime + bihi) (3.4) HML was the average return of two value portfolios (that is, high B/M ratios) minus the average return of the two growth portfolios (low B/M ratios):
HM L = 1
2(smhi + bihi) −1
2(smlo + bilo) (3.5)
UMD was created using past returns and split the data into two portfolios based on the median of returns. The ’winners’ portfolio is the 50 percent stock with the highest past returns, and the
’losers’ portfolio is the 50 percent with the lowest past returns. To acquire the UMD portfolio, subtract the ’losers’ from the ’winners’ portfolio.
Chapter 4
The Model
4.1 Value-at-Risk and Expected Shortfall
Value-at-Risk and Expected Shortfall were calculated using unconditional VaR/ES, historical simulation for VaR/ES following the paper of Bali and Cakici (2004), conditional VaR/ES, and RiskMetrics VaR/ES by Morgan (1996). The horizon l was set to one month, and the confidence level q was taken at 99%, 95%, and 90%. That means VaR and ES represented 1%, 5%, and 10% VaR/ES for each company i at month t.
For VaR and ES, we follow the notation of McNeil and Frey (2000). Let Xi,t = −Ri,t be a strictly stationary time series defined as the negative log-return of a company i at time t. The dynamics of X are assumed to be:
Xi,t= µi,t+ σi,tZi,t (4.1)
where Zi,t is a strict white noise process that is independent and identically distributed (i.i.d.) with mean zero and variance one (0,1), and marginal distribution FZi(zi). It is assumed that µi,t+1= E[Xi,t|Gi,t], σi,t+12 = Var(Xi,t|Gi,t) where Gi,t= {Xi,t, Xi,t−1, ...} is data for a company i available up to time t.
Let FXi(xi) be the marginal distribution of Xi,t. Then the unconditional method for VaR and ES required the calculation of a quantile from the marginal distribution given by:
xqi,t= inf{xi∈ R : FXi(xi) ≥ q} (4.2) Si,tq = E[Xi|Xi > xqi,t] (4.3) Here the deviation from the notation in McNeil and Frey (2000) followed because we wanted to calculate the VaR/ES for each company i at month t. The conditional method forecasts for the following months; therefore, a rolling window of 24 months was used to fit Xi,t to a normal distribution assuming FXi(xi) to be i.i.d. normal distributed with mean µi,t, and variance σ2i,t to obtain monthly VaR and ES estimates. Then VaR/ES at time t was obtained using the first 24 values of Gi,t = {Xi,t, ..., Xi,t−23}. Therefore for each company, this left out the first 24 months.
For the historical simulation method, we selected a rolling window of 24 months again to predict VaR/ES at time t. The prediction was based on the empirical quantile of the negative log returns. For each sample period of 24 months, Xi,t was ordered in increasing order. Then VaR at time t is selected as the qth quantile of the ordered Xi,t for the past 24 months. ES can also be directly estimated from the sample returns Tsay (2010). ES is calculated by taking the average over all Xi,t that exceed VaR at the given confidence level q. Again, in this method, the first 24 months were left out.
Following McNeil and Frey (2000), the conditional method would simplify to write VaR and ES as:
xqi,t = µi,t+1+ σi,t+1zqi (4.4)
Si,tq = µi,t+1+ σi,t+1E[Zi|Zi ≥ ziq] (4.5) where zqi is the upper qth quantile for FZi(zi) for a company i, assuming not to depend on t given by inf{zi ∈ R : FZi(z) ≥ q}. Calculating the conditional mean and volatility requires a model for (4.1). In Section (4.2), we compared different models for volatility dynamics. It is commonly used that for General Autoregressive-Conditional Heteroskedasticity (GARCH) models, Zi,t is standard normal so that ziq= Φ−1(q), where Φ(z) is the standard normal cumulative distribution function (CDF).
At last, the RiskMetrics method by Morgan (1996) was calculated using an Exponentially Weighted Moving-Average (EWMA) volatility model for calculating σi,t+1. VaR and ES by the RiskMetrics method were then calculated as in (4.4) & (4.5), assuming the mean µi,t+1 to be zero and Zi,t to be standard normal.
4.2 Comparing methods
Table 4.1: Summary of the statistics for the representative sample
NYSE (1) AMEX (2) NASDAQ (3)
return size return size return size
PERMNO, i 10302 26622 84362 91379 10568 91616
mean return 0.00 -0.14 -0.11 0.28 -0.25 -3.01
mean size 13.99 12.77 10.85 10.23 9.46 10.57
start date 1988-11 1958-12 1997-01 2006-08 1982-07 2006-12 end date 2009-10 1989-06 2003-04 2022-03 2011-07 2018-08
Note. The return and size columns represent the stock selected on the median stock’s return and size, respectively. The return row represents the mean over time as percentages and the size row represents the size in (3.2) over time.
We wanted to compare methods for calculating VaR and ES by testing on a smaller sample. So first, for each stock, the mean over time was taken. Then a representative sample was selected
from each of the three exchange markets based on the median stock from these returns and size, which left six stocks that were treated separately. Table (4.1) describes the representative sample. The start and end dates represent the first and last month where the stock was reported.
The methods that were compared are split into two sections. First, Section (4.2.1) explains the order selection for a Exponential GARCH (EGARCH) volatility model used in the condi- tional method. Volatility was calculated using a fitted normal distribution’s standard deviation, the standard deviation of a sample, a GARCH model with an Autoregressive-Moving-Average (ARMA) mean model, an exponentially weighted moving average (EWMA) model, and the volatility model in the paper of Fleming et al. (2001). The comparison and explanation for selecting ARMA-GARCH are given in Appendix C.
Section (4.2.2) presents the backtesting results for the calculated VaR and ES according to the unconditional, historical simulation, conditional, and RiskMetrics by Morgan (1996) methods.
4.2.1 Volatility
Let ai,t= Xi,t− µi,t be the innovation for a company i at time t. Because of the leverage effect (Takahashi et al., 2013), an EGARCH model for volatility was used where we would write the conditional variance of ai,t in the alternative form of (Tsay, 2010):
ai,t = σi,tϵi,t log(σ2i,t) = α0+
s
X
k=1
αk|ai,t−k| + γkai,t−k
σi,t−k +
m
X
j=1
βjlog(σ2i,t−j) (4.6)
so that ai,t follows an EGARCH(m, s) model, where the {ϵi,t} are i.i.d. standard normal, and α0 is a constant. The conditional mean, µi,t, followed from an ARMA model of order p, q:
Xi,t= ϕ0+
p
X
k=1
ϕkXi,t−k+ ai,t−
q
X
k=1
θkai,t−k (4.7)
where p and q are non-negative integers. If q = 0, we would be left with an Autoregressive (AR) model of order p given by:
Xi,t= ϕ0+ ϕ1Xi,t−1+ ... + ϕpXi,t−p+ ai,t (4.8) If p = 0, we would have a Moving-Average (MA) model with an extension of the ai,t series written by:
Xi,t = c0+ ai,t− θ1ai,t−1− ... − θqai,t−q (4.9) where c0 is a constant term.
The parameters of both the EGARCH and ARMA model were selected using the Bayesian information (BIC) criterion, given by: BIC = k × log(T ) − 2 × log( ˆL), where k is the number of parameters, T the number of observations, and ˆL the maximized value of the likelihood function of the given model. The BIC supports smaller models, and since some of the the individual stock’s monthly data only consisted of a small number of observations, this criterion was used.
The optimization code for the orders of ARMA/EGARCH and lags in EGARCH that were used are in Appendix A.1-3.
The optimal order/lags/distribution for the CRSP data was based on test results and the most common orders in the representative sample. First, the ARMA-EGARCH model was tested using an ARCH-Lagrange Multiplier (LM) test that assessed the significance of the EGARCH effects. The residuals were defined as: ei,t = Xi,t− ˆµi,t, and the auxiliary regression was given by: e2i,t= α0+ α1e2i,t−1+ αme2i,t−m+ ϵi,t, where {ϵi,t} is defined as before, and m is the number of lags. The null hypothesis states that the standardized residuals are homoskedastic given by:
H0: α0 = ... = αm= 0 (4.10)
Secondly, the Jarque-Bera test was used to test the goodness of the fit, where the null hypothesis states that the given model is normally distributed. The test statistic is given by:
J B = n
6(S2+1
4(K − 3)2) ∼ χ(2) (4.11)
where S is the skewness, and K is the kurtosis of the sample. The test statistic is chi-squared distributed with 2 degrees of freedom. At last, a Wald/F-test was used for serial correlation in the volatility with volatility clustering (Appendix A.4). The Wald test has a null hypothesis:
no serial correlation.
Table 4.2: Sample outcomes for the optimization of an ARMA-EGARCH model and testing results within the representative sample
NYSE (1) AMEX (2) NASDAQ (3)
PERMNO, i 10302 26622 84362 91379 10568 91616 A. optimization
ARMA (1,1) (1,1) (1,1) (1,1) (1,1) (1,1)
EGARCH (0,1,1) (1,1,1) (1,1,1) (1,1,1) (0,1,1) (1,0,1)
lags 0 1 0 0 1 0
B. test
ARCH-LM 0.778 0.755 0.941 0.787 0.104 0.678
Jarque-Bera 0.539 0.000 0.577 0.124 0.000 0.294
Wald 0.987 0.697 0.948 0.995 0.895 0.999
Note. Panel A. shows the selected orders for ARMA/GARCH/lags for each company according to the minimal BIC. The ARMA row states the optimal order of (p, q), the EGARCH row states the optimal order (m, o, s), where o is the lag order of the asymmetric ai,t, and the lag row is the number of optimal lags for estimating EGARCH. Panel B. shows p-values for the corresponding tests.
Table (4.2) shows that the EGARCH(1,1,1) with an ARMA(1,1) mean model was the most prevailed. The tests also showed the most significant p-values for this combination within the
representative sample. Since the Jarque-Bera test for the sample led to significant outcomes, a normal distribution for the errors was assumed. The comparison continues using backtest- ing methods for VaR and ES, where the conditional method used volatility-mean calculations according to the EGARCH(1,1,1)-ARMA(1,1) model for the sample.
4.2.2 Backtesting VaR and ES
The selected method for the complete dataset followed from backtesting 1% VaR and ES (q = 0.99) for the representative sample. Backtesting Value at Risk counts the number of exceptions.
Following the notation of Tsay (2010), we would define the number of hits as:
Ii,t= 1{Xi,t+1>xq
i,t} (4.12)
If the model is well specified, the number of hits follow a binomial distribution,PT
t=1Ii,t∼Bin(T, p) where p = 1 − q, and T the total number of months/observations for a company i. For the representative sample, the number of exceptions should not differ significantly from p = 0.01.
However, the Binomial test only checks for unconditional coverage. Misspecification in the model could also mean that the hits are serially dependent. Serial dependence could be due to hits occurring in clusters. The autocorrelation function (ACF) was used for testing written as Tsay (2010) (appendix A.5):
ρi,l= corr(Ii,t, Ii,t−l) = cov(Ii,t, Ii,t−l)
var(Ii,t) l = 1, 2, ... (4.13) where corr and cov stand for the correlation and covariance between two random variables, var stands for the variance of a random variable, and l is the number of lags. The ACF should not show a significant correlation at a 5% level.
Backtesting ES can be done following the paper of Acerbi and Szekely (2014) with formula:
Ki,t+1= Xi,t+1− sqi,t σi,t+1
(4.14) Then under the null hypothesis:
H0: E[Ii,t+1× Ki,t+1|Gi,t] = 0 (4.15) where Gi,t= {Xi,t, Xi,t−1, ...} is data from the past for a company i up to time t. The expected value in (4.15) should not differ significantly from zero for the representative sample. Under H0, the sample ˆKi,t+1 should be i.i.d. with mean zero tested using bootstrapping.
Table (4.3) shows that VaR’s conditional and unconditional method performed best when testing the hits since the results show that the mean does not differ significantly from 0.01. For ES, the unconditional method performed worse since the values are not close to zero except for PERMNO 84362. The conditional method, in most cases, still performed well, but some values did differ significantly from zero. The Empirical method for ES also performed well since most values are close to zero. The autocorrelation test results for l = 24 are given in Table (B.1-3).
For VaR hits, most of the models were significantly uncorrelated for a large number of lags.
However, some showed autocorrelation with several lags at a trim level but were still significant.
Out of Table (4.3), the conditional method performed the best based on VaR and ES back- testing results. Therefore for each stock in the CRSP data, VaR/ES was calculated according to the conditional method using an EGARCH(1,1,1) volatility with an ARMA(1,1) mean model.
Some of the company stocks violated the boundary constraints in volatility estimation; hence these stocks were dropped from the data. In addition, some volatilities were estimated very high, which were dropped and replaced by linear interpolation.
Table 4.3: VaR and ES backtesting results within the representative sample
NYSE (1) AMEX (2) NASDAQ (3)
PERMNO, i 10302 26622 84362 91379 10568 91616
A. mean hits
VaR Conditional 0.0161 0.0082 0.0133 0.0137 0.0230 0.0074 VaR RiskMetrics 0.0267 0.0175 0.0000 0.0000 0.0402 0.0181 VaR Unconditional 0.0133 0.0146 0.0000 0.0082 0.0402 0.0181 VaR Empirical 0.0491 0.0322 0.0413 0.0277 0.0433 0.0272 B. expectation sample mean
ES Conditional -0.0990 0.5076 -0.0037 -0.0033 0.4293 -0.2221 ES RiskMetrics 0.0856 0.5613 0.0000 0.0000 0.9469 0.0964 ES Unconditional 0.2576 0.6350 0.0000 -0.1694 0.9504 0.1596 ES Empirical 0.1100 0.1399 0.0640 0.0730 0.2814 0.1392 Note. Panel A shows the mean number of hits: T1 PT
t=1Ii,t, where T is the total number of months. Panel B shows the estimated sample mean from (4.15) under H0. VaR/ES conditional represents the conditional method. VaR/ES RiskMetrics represent the RiskMetrics model by Morgan (1996). VaR/ES unconditional used the fitted normal distribution’s standard deviation for calculating Var/ES using the unconditional method. Finally, VaR/ES Empirical represents VaR/ES calculated using the historical simulation method. For the empirical method of cal- culating the VaR/ES, the volatility in (4.14) was taken as the standard deviation of a rolling window through 24 months.
4.3 ES and Expected Stock Returns
In the asset-pricing framework of Fama and French (1992), a month-by-month Fama-Macbeth procedure was used to test for explaining power in cross-sectional returns (Fama and MacBeth, 1973). The Fama-MacBeth procedure works by first regressing the returns (Ri,t) for each company i against the risk factor ki,t to obtain each variable’s beta:
Ri,t = αi+ βi,k∗ ki,t+ ui,t ∀ i = 1, ..., n (4.16)
where ui,t are the regression residuals, n is the number of stock, and ki,t is the risk factor that stands for the sizei,t, betai,t, and Si,tq (ES) at a given q = 0.99, 0.95, 0.90. Then the estimated βˆi,k, where k is the random variable from (4.16), was used to determine the risk premium by regressing over time:
Ri,t= ωt,k+ γt,k∗ ˆβi,k+ ϵi,t ∀ t = 1, ..., T (4.17) where ϵi,t are the cross-sectional residuals series from the regression. Since the stock starts and stops on the exchanges within the period, the panel is unbalanced over time. So over time, there are T estimates for each risk factor k: ˆγt,k ∀ t = 1, ..., T . Averages of the estimates were used to obtain the slope coefficient: ˆγk = T1 PT
t=1ˆγt,k. The same method was used to obtain the average estimate, ˆωk, for each k. Therefore, the regression results show which variables, on average, have explaining power by checking whether the expected premiums are nonzero (Fama and French, 1992). The null hypothesis for the time-series average is that the monthly regression estimate ˆγt,k is zero, tested using a standard t-test. The t-statistic was computed as follows:
t = σ(ˆγˆγk
k), where σ is the standard deviation and t is student t-distributed. The R-squared in the regression follows the average of the overtime R-squared following these regressions.
In the paper of Fama and French (1993) and Bali and Cakici (2004), stocks were sorted based on size and the B/M ratio into 25 portfolios. These are then used to evaluate the model’s performance according to the asset pricing models. In this case, the 25 portfolios based on the CRSP data were sorted using the risk factors as in (4.16) and (4.17): size, beta, and Sq (ES) for the three different critical levels q = 0.99, 0.95, 0.90 (Appendix A.6). Then following Bali and Cakici (2004), an Ordinary Least-Squares (OLS) regression was used where the 25 equally- weighted portfolio returns were regressed on the 25 equal-weighted portfolio’s risk factor to determine the relationships between the cross-section of portfolio returns and the risk factors.
The portfolio’s risk-factor was calculated by taking the mean over the portfolio constituents.
Finally, the regression results significance was tested again using a t-test computed in the same way as in the previous section.
Additionally, to investigate ES at a portfolio level, a risk factor ESHL (ES high minus low) was created for the relation between the risk factor’s returns and ES. The ESHL factor was constructed similar to the size portfolios from Fama and French (1993): each month, the stocks were ranked on 1% ES. Then the 1% ES median was used to split into two portfolios, where the top 50% portfolio was called the high ES (HES), and the lower half was the low ES (LES).
Then the equal-weighted returns for each portfolio were computed to obtain the monthly risk factor ESHL created by:
ESHL = HES − LES (4.18)
that is, subtracting the equal-weighted returns for the Lower half portfolio from the high ES portfolio to receive the 1% ESHL factor.
The VaRHL (VaR high minus low) factor was computed similarly to ESHL in (4.18) but is now based on VaR. The VaRHL portfolio was used to test whether ES has an additional explaining power after correcting for VaR. This was done in a similar setting as Fama and
French (1993) by producing a four-factor model based on VaRHL:
RESt = α + βM KT RFRtM KT RF + βHM LRHM Lt + βSM BRSM Bt + βV aRHLRV aRHLt + ϵt (4.19) where RESt are 10 sorted ES portfolio returns, and ϵt are the regression residuals.
Furthermore, the ESHL portfolio was then used to explain the cross-section of expected returns based on the six portfolios by the risk factors using the method of Fama and French (1993). Therefore additionally to (2.2), there is a four-factor model produced using:
Rs−B/Mt = α + βM KT RFRM KT RFt + βHM LRHM Lt + βSM BRSM Bt + βESHLRtESHL+ ϵt (4.20) where the dependent variable Rs−B/Mt are the six size-B/M portfolio returns given by smlo, smme, smhi, bilo, bime, and bihi, and ϵt are the regression residuals. In addition to the six portfolios, the 25 size and book-to-market portfolios based on Fama and French (1993) were also used as the dependent variable of regression (4.20).
To study the risk factors in stock returns, an additional risk factor was made based on volatility called VOLHL (volatility high minus low). VOLHL was constructed similarly to ESHL in (4.18) but is now based on volatility. The volatility used for creating this portfolio was calculated based on the ARMA(1,1)-EGARCH(1,1,1) model for the CRSP data.
Following Bali and Cakici (2004) & Fama and French (1993), one should concentrate on the intercepts from the time-series regressions to tell how well the explanation in the cross-section of returns is. According to Merton (1973) & Ross (1976), the simple test of whether the factors explain the cross-section of expected returns is that the intercepts of a time-series regression on given portfolios should not differ from zero.
Chapter 5
Results
5.1 ES and Cross-sectional Returns
1%, 5%, and 10% ES and VaR for q = 0.99, 0.95, 0.90 respectively were tested on a signifi- cance effect in cross-sectional returns. The CRSP data were sorted into 10 equal-weighted ES portfolios based on Expected Shortfall’s quantile (Appendix A.6).
Table 5.1: Average monthly portfolio returns, ES, and VaR, sorted by 1%, 5%, and 10% ES
1% 5% 10%
Portfolio ES VaR Return ES VaR Return ES VaR Return
1 10.06 8.74 -0.20 7.69 6.04 -0.31 6.47 5.64 -0.37
2 15.82 13.74 -0.43 12.12 9.54 -0.47 10.22 7.72 -0.49 3 19.36 16.83 -0.47 14.86 11.74 -0.50 12.56 9.19 -0.53 4 22.50 19.58 -0.43 17.31 13.71 -0.46 14.66 10.57 -0.48 5 25.67 22.36 -0.29 19.78 15.70 -0.30 16.78 11.94 -0.34 6 29.08 25.36 -0.21 22.46 17.86 -0.24 19.07 13.40 -0.25 7 32.87 28.69 -0.02 25.43 20.27 -0.05 21.63 14.97 -0.07 8 37.25 32.54 0.25 28.89 23.08 0.23 24.61 16.78 0.21 9 42.75 37.40 0.67 33.25 26.66 0.68 28.39 19.00 0.68 10 51.13 44.82 1.36 39.95 32.23 1.65 34.28 22.73 1.87 Note. Portfolio 1 (10) denotes the ES portfolio at critical level q with the smallest (largest) ES quantile. The 1%, 5%, and 10% column represent the sorting based on 1%, 5%, and 10% ES respectively. The ES, VaR, and Return columns represent the mean over time of the portfolio for each variable respectively and are defined as percentages for each portfolio.
Table (5.1) suggests that ES and VaR are increasing for the ascending ES portfolios. It is generally clear that the higher ES portfolios resulted in higher returns. However, it can be found that it was not the case for some of the mid-decile portfolios since they dropped in returns while ES increased. From the third portfolio, the returns start increasing. For q = 0.95 and
q = 0.90, the fifth portfolio had a higher return than the first portfolio, while for q = 0.99, this was only the case from the seventh portfolio. The table does not completely show a significant positive relation between ES and their portfolio return. Therefore we needed the cross-sectional regressions by the Fama-MacBeth procedure.
Table 5.2: Fama-MacBeth cross-sectional regressions & portfolio regressions of expected returns on size, beta, and 1%, 5%, 10% ES
Fama-MacBeth regression CRSP, 25 portfolios regression
Variable, k ωˆk ˆγk R2 αˆk βˆk R2
A. Monthly stock regressions B. Portfolio regressions size 4.3872* -0.3772* 0.012 5.2815* -0.4410* 0.746
(0.338) (0.022) (0.642) (0.054)
beta -0.3008 0.9778 0.056 -0.2579* 0.9932* 0.993
(0.169) (0.020) (0.036) (0.015)
ES 1% -1.1320* 0.0384** 0.030 -1.2396* 0.0449* 0.752
(0.090) (0.005) (0.172) (0.005)
ES 5% -1.3267* 0.0581* 0.030 -1.4321* 0.0665* 0.736
(0.090) (0.090) (0.206) (0.008)
ES 10% -1.4645* 0.0753* 0.031 -1.5676* 0.0851* 0.723
(0.091) (0.008) (0.233) (0.011)
Note. Panel A represents the cross-sectional average regression results for the Fama-MacBeth procedure. ˆωkis the average intercept estimate, ˆγk= T1 PT
t=1ˆγt,kis the average slope coefficient for risk factor k with their standard deviation between brackets, and R2is the average R-squared overtime from (4.17). Panel B shows the univariate OLS regression of 25 portfolio returns on the 25 portfolio’s risk factors for each k, where ˆαk is the estimated intercept and ˆβk the estimated coefficient.
* indicates p < 0.05. ** indicates p < 0.10.
The regressions in Table (5.2, Panel A.) were performed for the period ranging from February 1958 to March 2022, leaving a sample size of T = 770. Because beta calculations set the rolling window to 24 months, the data ranged from January 1960 through March 2022 with T = 747.
The R-squared is given as a decimal and was very low in all cases because these are stock data, and stock data tends to be very noisy. The computed returns were in percentages. For the variable size, this means that when the size increases by 1 point, the cross-sectional expected return increase/decrease by ˆγsize percentage point ceteris paribus. For beta, the regressions’
output was interpreted the same way as the size variable. ES was calculated as a percentage;
therefore, when the ES increases by 1 percent, the cross-sectional expected stock returns will increase/decrease by ˆγSq percent ceteris paribus.
The cross-sectional size regression showed a significant negative relationship between size
and the monthly stock returns, supporting that smaller stocks lead to higher average returns.
Unlike the explanatory power of size, the regression showed that beta does not have significant explanatory power in the cross-sectional expected stock returns. This result was supported by Fama and French (1992), where the relationship between expected stock returns and beta is insignificant.
The Fama-MacBeth regressions showed that for different probability loss levels of ES, the positive relationship with expected stock returns was significant and nonzero, therefore explain- ing cross-sectional stock returns. The intercept ˆωk was negative for ES, indicating the expected stock return’s negative skewness to the left. The R-squared of the ES regressions was higher than the size regression and slightly lower than the insignificant beta regression. Therefore ES explained the variation in cross-sectional expected stock returns better than size, and beta did not capture explanation power in the returns.
Additionally, we tested the relationship between size, beta, and ES within the size, beta, and ES portfolios (Table (5.2), panel B). The positive relation between ES and expected returns was now significant in a portfolio setting, explaining the variation in cross-sectional expected portfolio returns. In this case, beta was also significantly positive with a high R-squared, supporting the CAPM. The R-squared values of all variables were high, indicating that the explanatory power in cross-sectional portfolio returns is substantial for size, beta, and ES at three critical levels. Therefore by the asset-pricing theory, beta and ES variables explained the variation in expected cross-sectional portfolio returns positively, while the size variable did in a negative relation.
5.2 ES and Portfolio Selection
Table 5.3: Correlation table between the three Fama and French (1993) factors, the momentum factor (UMD), the ES portfolio (ESHL), the VaR portfolio (VaRHL), and the volatility portfolio (VOLHL)
MKTRF SMB HML UMD ESHL VaRHL VOLHL
MKTRF 1 0.295 -0.22 -0.161 0.649 0.648 0.665
SMB 1 -0.170 -0.032 0.747 0.746 0.745
HML 1 -0.224 -0.195 -0.197 -0.199
UMD 1 -0.187 -0.186 -0.194
ESHL 1 0.999 0.998
VaRHL 1 0.997
VOLHL 1
We expanded the three-factor model from Fama and French (1993) to further explore ES’s em- pirical performance in a portfolio setting. The correlation between the ESHL portfolio and the three-factor model from Fama and French (1993) is given in Table (5.3). The Table indicates
the direction and extent of the relationship between the newly computed ESHL and the three factors. Additionally, the VOLHL, VaRHL, and UMD portfolios were added to find the rela- tionship between ESHL and these portfolios. Table (5.3) shows that the MKtRF and SMB had a large positive relation with the ESHL portfolio. The positive relationship between ESHL and SMB is larger than the negative one between HML and ESHL. The correlation between UMD and ESHL is negative but low, suggesting a possible approach to selecting a portfolio based on ES. VaRHL and VOLHL show an almost complete correlation with ESHL, indicating that ES has potentially no additional performance after VaR and volatility.
Table 5.4: Four-Factor model: Regression of 10 ES sorted portfolio returns on factors: MKTRF, SMB, HML, and VaRHL
Portfolio/Variable
Intercept MKTRF SMB HML VaRHL R2
1 -0.4128 0.5690 0.1906 0.1972 -0.2003 0.962 -5.979 31.500 6.262 9.843 -7.059
2 -0.3456 0.4728 0.2748 0.1519 -0.2290 0.693 -6.060 31.688 10.930 9.178 -9.139
3 -0.3246 0.7065 0.3317 0.2423 -0.1407 0.869 -6.442 53.606 14.935 16.574 -5.590
4 -0.3042 0.7900 0.3874 0.3003 0.3029 0.907 -6.010 59.672 17.368 20.451 10.647
5 -0.2759 0.8361 0.4175 0.3375 0.1282 0.921 -5.203 60.273 17.864 21.932 4.686
6 -0.3025 0.8504 0.4249 0.3416 -0.0203 0.931 -5.501 59.114 17.531 21.405 -0.769
7 -0.2637 0.6795 0.3559 0.2278 0.5539 0.969 -4.788 47.157 14.657 14.254 19.977
8 -0.2436 0.8429 0.4607 0.3493 1.1781 0.939 -4.260 56.335 18.274 21.047 42.996
9 -0.2116 0.7683 0.4128 0.2876 0.7963 0.963 -3.983 55.283 17.63 18.657 30.135
10 -0.2034 0.8101 0.4315 0.3244 1.7064 0.952 -3.648 55.548 17.563 20.055 49.689
Note. Portfolio 1 (10) denotes the 1%ES portfolio with the smallest (largest) 1%ES quantile.
Each row represents one regression of the 1%ES portfolio returns on the four factors (4.19) where the coefficients for the factors are given in the row with their t-statistic (t) below in italics.
|t| ≥ 1.96 indicates statistical significance at a 5% level.
Table 5.5: Five-Factor model: Regression of 10 ES sorted portfolio returns on factors: MKTRF, SMB, HML, VaRHL, and ESHL
Portfolio/Variable
Intercept MKTRF SMB HML VaRHL ESHL R2
1 -0.4536 0.5663 0.1878 0.1948 0.7691 0.9371 0.962 -5.373 30.896 6.132 9.624 0.689 0.841
2 -0.3481 0.4727 0.2746 0.1518 -0.2562 0.0559 0.693 -4.988 31.199 10.851 9.071 -0.278 0.061
3 -0.3270 0.7064 0.3315 0.2422 -0.2854 0.0564 0.869 -5.305 52.783 14.828 16.387 -0.350 0.069
4 -0.3377 0.7879 0.3851 0.2984 -0.9106 0.7697 0.907 -5.457 58.644 17.16 20.112 -1.114 0.942
5 -0.2951 0.8348 0.4162 0.3363 -0.4618 0.4414 0.921 -4.550 59.284 17.692 21.629 -0.539 0.515
6 -0.3260 0.8489 0.4233 0.3402 -0.4101 0.5382 0.931 -4.846 58.132 17.352 21.096 -0.461 0.606
7 -0.3160 0.6762 0.3523 0.2248 -0.0218 1.1996 0.969 -4.694 46.271 14.429 13.928 -0.024 1.350
8 -0.3522 0.8359 0.4532 0.3429 -2.1885 2.4909 0.940 -5.057 55.290 17.944 20.54 -2.378 2.709
9 -0.2544 0.7655 0.4099 0.2851 -0.1871 0.9832 0.963 -3.918 54.299 17.403 18.312 -0.218 1.147
10 -0.2579 0.8066 0.4278 0.3212 -0.6981 1.2518 0.952 -3.787 54.542 17.316 19.669 -0.776 1.392
Note. Portfolio 1 (10) denote the 1%ES portfolio with the smallest (largest) 1%ES quantile.
Each row represents one regression of the 1%ES portfolio returns on the four factors and the additional ESHL factor added to (4.19) where the coefficients for the factors are given in the row with their t-statistic (t) below in italics.
|t| ≥ 1.96 indicates statistical significance at a 5% level.
We used a time-series regression approach to find the relation between ES and VaR in the cross-section of portfolio returns. The regressions in Table (5.4) were performed on the 10 1%ES monthly sorted portfolio returns from Table (5.1). From Table (5.4), it can be seen that VaRHL showed significant coefficients except for the regression based on ES portfolio 6.
The intercepts in the four-factor model were all significant therefore stating that the factors could not entirely explain their risk premium. The intercepts were increasing for the larger ES portfolios, explaining the positive relationship between ES and expected portfolio returns. The high R-squared and significant intercepts showed that ES explains the variation in the cross- sectional portfolio returns after controlling for the three factors of Fama and French (1993) and
VaR, suggesting that ES additionally explains the variation after controlling for VaR.
The four-factor model was expanded to a five-factor model where the ESHL portfolio was additionally taken as a factor. The five-factor model regression results are shown in Table (5.5), where the dependent variables were again the 10 ES sorted portfolio returns. Table (5.5) shows that the VaRHL and ESHL factors were only significant for the 8th ES portfolio. The insignificance of the factors may be caused by the high correlation between the two portfolios.
In this particular regression, the R-squared did increase compared to the four-factor regression.
This result may indicate an additional explaining power of the ES portfolio after VaR. However, the other regressions showed insignificant relationships. Therefore there is not enough evidence to conclude that the ES factor explained the cross-section in portfolio returns in addition to the VaR factor. The insignificance in the VaRHL and ESHL factors stated that these do not additionally explain the risk premium of the stock returns. The regressions’ intercepts remained significantly different from zero, and therefore ES still explained the variation in portfolio returns after controlling for the five factors.
Table 5.6: Correlations between the six portfolios and the three Fama and French (1993) factors, and the ES portfolio (ESHL)
size quantile Book-to-Market quantile MKTRF SMB HML ESHL small
low 0.8722 0.6895 -0.3338 0.8513
medium 0.8794 0.6531 -0.0613 0.7960
high 0.8376 0.6336 0.1112 0.7824
big
low 0.9705 0.2146 -0.3647 0.5777
medium 0.9327 0.1743 0.0050 0.5380
high 0.8653 0.2131 0.2457 0.5671
Note. Small - low (big-high) denotes the size-B/M portfolio with the smallest (largest) size quantile and the lowest (highest) B/M quantile. The values below each factor are correlations of the factor with the given size-B/M portfolio.
The correlations between the six/25 size-B/M portfolios and the three factors plus ESHL are shown in Table (5.6)/Table (B.8) respectively, for comparing the relative performance between the factors. The MKTRF factor has the largest explanation power in the portfolios since the correlation is, on average, the highest in both tables. The SMB and HML vary from high to lower correlation. The ESHL has, on average, a higher correlation with the six and 25 portfolios than SMB and HML, suggesting that the ESHL portfolio explains the cross-sectional returns better. In addition, the ESHL has an inverse relationship with the portfolio returns of size and B/M.
Table 5.7: Three- (& Four-Factor model): Regression of six size-B/M portfolio returns on MKTRF, SMB, HML (, and ESHL)
Three-Factor model Four-Factor model Book-to-Market quantile: Coefficient
Size quantile low medium high low medium high
Intercept
small 0.0019 0.0041 0.0041 0.0041 0.0028 0.0041 t-statistic 4.833 13.439 18.525 9.011 7.963 15.233 big 0.0046 0.003 0.0024 0.0042 0.0025 0.0042
17.462 6.967 5.912 13.256 4.852 8.800
MKTRF
small 0.0109 0.0096 0.0099 0.0102 0.0100 0.0100 115.975 134.047 188.671 84.303 105.972 140.056 big 0.0099 0.0097 0.0108 0.0101 0.0098 0.0103
157.337 94.363 110.824 118.575 71.221 80.069 SMB
small 0.0104 0.0082 0.0088 0.0091 0.0089 0.0088 75.755 77.903 113.629 44.662 56.243 73.539 big -0.0015 -0.0012 0.0001 -0.0013 -0.0009 -0.001 -16.26 -8.181 0.773 -8.812 -4.02 -4.618 HML
small -0.0023 0.0036 0.0071 -0.0023 0.0036 0.0071 -16.381 33.215 90.347 -17.344 34.143 90.27 big -0.0027 0.0031 0.0079 -0.0027 0.0031 0.0079
-28.618 20.482 54.416 -28.633 20.539 55.79 ESHL
small - - - -0.0020 0.0011 0.0000
- - - -8.616 6.163 0.211
big - - - 0.0004 0.0004 -0.0016
- - - 2.257 1.668 -6.724
R2
small 0.974 0.977 0.988 0.977 0.978 0.988
big 0.975 0.923 0.949 0.975 0.924 0.952
Note. Small - low (big-high) denotes the size-B/M portfolio with the smallest (largest) size quan- tile and the lowest (highest) B/M quantile, smlo (bihi). The three-factor (2.2) and four-factor (4.20) models show regressions based of the six size-B/M portfolio returns on the three/four factors respectively. The coefficients with their t-statistic below in italics are given for each factor together with the R-squared of the regression.
To continue testing how well the premiums for the three factors by Fama and French (1993) and the ES factor explained the cross-section of average stock returns, we need Table (5.7). The table shows the regression results of the Fama and French (1993) three-factor model (2.2) and the four-factor model, including ESHL (4.20). Additionally, Table B.4-7 shows the regression results for the three-factor and four-factor models that included VOLHL and ESHL for the 25 portfolios sorted by size and the B/M ratio.
Table (5.7) shows that the SMB coefficient for the bihi portfolio was the only insignificant estimate for the three-factor model. The coefficients for the market return were around 0.01.
The estimates of SMB and HML were related to the size and book-to-market ratio because the SMB coefficients decreased from small- to big-size portfolios. The HML coefficients increased from negative to positive values for the low to high book-to-market portfolios.
The coefficients for MKTRF, SMB, and HML were statistically significant in the four-factor model. For the added ESHL portfolio, 4 of the 6 estimates were statistically significant, where the coefficients of ESHL on smhi and the bime portfolio were insignificant. The estimates for ESHL tended not to have a natural pattern for the size and book-to-market portfolios. Within the four significant coefficients for ESHL, the R-squared from the 3 of the 4 four-factor models increased compared with the three-factor model. Therefore, ES has additional explaining power after controlling for MKTRF, SMB, and HML. The 25 portfolio regressions of the three- and four-factor model in Table B.5-7 supported the same findings. The volatility portfolio showed 20 of the 25 coefficients to be significant, and the ES portfolio showed the same 20 of 25 coefficients to be significant. From the 25 portfolio regressions, it can also be seen that the R-squared increased for the four-factor model when using volatility and ES. Therefore volatility and ES explained the variation in cross-sectional stock returns. The ES explained the variation slightly better because some of the R-squared increased when using ESHL instead of VOLHL.
From the intercepts, we can test whether the returns of SMB, HML, ESHL, and VOLHL controlled for size, B/M, ES, and volatility effects in the average returns. The intercepts from Table (5.7) for both models were close to zero, stating that ES had additional explaining power in the asset-pricing model. The three-factor model showed that MKTRF, SMB, and HML captured variation in the cross-section of stock returns. The intercepts for the three-factor of the 25 portfolio regressions in Table (B.4) showed that 5 of the 25 intercepts statistically differed from zero. In contrast, for both four-factor models, only 4 of the 25 intercepts were statistically different from zero. Therefore both the six and 25 portfolio regression resulting intercepts showed that VOLHL and ESHL additionally explained the variation in cross-sectional expected returns after the control for market return, size, and the B/M ratio. The difference between the four-factor models containing VOLHL and ESHL is small.
