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65Time-varying Term Structure of Risk premium, Estimated with Credit Default Swaps*
Victoria KOSTYUK
Financial University, Moscow vikostyuk@gmail.com
Abstract. The paper estimates the time-variation and term structure of forward-looking equity premium of Australia, Asia excluding Japan, and CEEMEA region. Methodology is based on Berg and Kaserer (2009) approach which employs structural models of default within Merton framework (1974) to convert credit spread from CDS into equity premium. The paper extends the Berg and Kaserer (2009) approach for equity risk premium (ERP) estimation in the following ways. Firstly, the forward-looking equity premium is calculated for developing markets, which to our best knowledge has not been done in the literature yet. Second, the use of monthly data allows observing time variation of equity premium. Finally, the availability of CDS data for 5-, and 10-year CDS maturities provides the term structure of equity premium for CEEMEA region since 2010. Term structure is downward sloping which implies that short-term risks are priced higher than long-term, and the slope becomes more angled during financial turmoil. Historical equity premium dynamics demonstrate apparent relationship with stock market behavior.
Аннотация. В статье оценивается статическая и динамическая структура ожидаемой риск-премии обыкновенных акций на рынках Австралии, Азии, Центральной и Восточной Европы, Ближнего Востока и Африки. Методология оценки основана на подходе Berg, Kaserer (2008), предполагающем использование структурной модели дефолта Мертона для конвертации кредитного спреда по CDS в риск-премию. Эта работа расширяет подход Berg, Kaserer в трех аспектах. Во-первых, премия рассчитывается для развивающихся рынков, что, насколько нам известно, не делалось в литературе до нас. Во-вторых, использование месячных наблюдений позволяет рассмотреть вариацию риск-премии во времени. Наконец, доступность данных по CDS с 5- и 10-летним сроком истечения дает возможность оценить временную структуру риск-премии. Результаты исследования показывают нисходящий уклон риск-премии: ближайшие риски оцениваются инвесторами сильнее, чем более далекие, и разность увеличивается во время кризиса. Временная динамика премии тесно связана с динамикой фондового рынка.
Key words: equity premium, credit default swap, developing markets, structural models of default.
introduction
The equity risk premium (ERP) is an extremely important concept of the modern theory of finance. Basically, it is the price that investors charge for bearing extra risk from holding risky assets compared to risk-free securities. At each point in time and for each asset of each maturity there exists a price for risk which is determined by the market participants. ERP has numerous important practical and theoretical implications in risk and return financial models, corporate finance valuation and analysis, portfolio management, investing activi-
ties, as well as in the composition of savings and spending plans, etc.
The part of the confusion about the equity risk premium is the entire meaning of this concept to different users. The literature and practical implications may assume one of the following meanings, such as historical, required, expected, and implied premium. In this study the equity premium is considered as fundamental judgment of the degree of risk that market participants see in the market and of the price that they assign to this risk. This, in turn, determines the value of the asset and its expected return. Consequently, it influences the capital dis-
* Временная динамика премии за риск, рассчитанной с использованием кредитных дефолтных свопов.
tribution among different asset classes as well as the choice of specific securities within the chosen asset class.
For a long time academics and practitioners used to think that the risk premium is constant over time leading to belief that historical returns best explain the future ones. Later the empirical studies revealed the time-variation of equity premium; this gave grounds for various theories of equity premium assessment. Despite the knowledge, access to information, developed analytical tools and techniques, neither academics nor practitioners have reached the consensus on equity premia valuation approach so far.
Recent literature suggests only few estimation methods that allow to obtain forward-looking risk premium: extracting it from bond spreads (Campel-lo et al., 2007), option prices (Bhar and Chiarella, 2004), and credit default swap (CDS) spreads (Berg and Kaserer, 2007). The crux of the Berg and Kaserer method is the link of credit valuation with the expected equity premium. The credit default swap valuation appears to be a proxy for credit valuation. The equity premiums are extracted from CDS spreads in the following manner: the structural models of default within the Merton (1974) framework specify that the difference between the risk-neutral and actual default probabilities depends on the assets' Sharpe ratio. The real world and risk-neutral default probabilities specified by the structural model of default in the Merton framework differ in terms of drift: the firm's assets have to fall by - r) *T compared to the risk-neutral world to reach the default (where ^ — is a real world drift, r — risk-neutral drift, T — time to maturity). The real world, or actual, default probability may be estimated from rating or taken from statistics, while the risk-neutral default probability is approximated from CDS spreads. This approximation is based on the intuition that CDS spread may be decomposed into two parts: price that compensates for expected loss and the risk premium demanded by investors. The difference between risk-neutral and actual default probabilities yields the asset's Sharpe ratio which in turn may be transformed into the market Sharpe ratio according to the continuous time CAPM. Finally, the equity risk premium may be easily derived from the general measure of risk aversion for each point in time.
The Sharpe ratio estimator of Berg and Kaserer is based on the observable parameters, which makes it simple to estimate and available for practitioners. The use of CDS data in the model allows for capturing time-variation of equity premium value. Moreover these credit instruments discover the term
structure of the risk premium due to availability of different maturities.
The majority of researches devoted to equity premium estimation represent results on broad US market owing to the availability of historical data for extensive time period (some data time series are known since 1870) while the rest ones describe premiums for developed European countries.
The interest in emerging countries has grown substantially over the two recent decades from the direction of the international and domestic investors. Nevertheless the academic literature doesn't provide us with estimations of expected risk premium in developing countries. There are only few papers that estimate ex post risk premiums for broad stock index of emerging economies or the Eastern countries (e.g. Cohen, 2010; Donadelli and Prosperi, 2011).
The purpose of the study is to test Berg and Kaserer equity premium estimator to the least examined in the literature markets, such as CEEMEA (Central Eastern Europe, Middle East, Africa), Asia (excluding Japan), and Australia. Although we are interested in forward-looking premium, it is important to assess it on historical data in order to prove adequacy of results and to capture equity premium properties — time-variation and term structure (where possible). To our best knowledge, this is the first research that employs this approach to determine equity premium term structure for the mentioned markets.
The study is naturally limited by the data availability. CDS instruments are young and poorly introduced in developing markets, such as CEEMEA and Asia excluding Japan. Only investment rated firms may have CDS on their foreign debt. The paper employs MARKIT® CDS indices, which combine the most liquid corporate CDSs and cover considered regions. Among them only CEEMEA market CDS data let us examine term structure properties as there are two maturities for index available.
The need to determine the actual default probability in the model involves another limitation that may influence results. Berg and Kaserer estimate real world default probabilities out of Moody's EDF data, which are of limited access. In this paper the average historical default probabilities from Moody's ratings are used as a proxy for the expected ones. This approach reflects the means of most investors that use historical probabilities for forecasting.
Some other limitations are connected to the theoretical framework of the model, which is built on Merton (1974); they are described in the second section.
The paper is structured as follows. The first section discusses the model of forward-looking equity premium estimation. The second section describes dataset of the research and model parameters estimation methods. The third section discusses results, their properties and implications, and limitations of the study. Finally, the conclusion summarizes empirical findings and applications of results, proposes developments for future research.
MODELSETup
This section describes the methodology for extracting equity premium from CDS spread. The crux of the approach is that default and equity risk premiums are intrinsically linked because equity and debt are both contingent claims on the same assets (Merton, 1974). Building on this argument, the debt valuation is linked to equity valuation through structural model of default, which allows deriving the difference between the risk-neutral and actual default probabilities, which in turn produces the dynamics of the asset value process, the asset Sharpe ratio; given the correlation it may be transformed into market Sharpe ratio.
The market Sharpe ratio estimator, developed by Berg and Kaserer (2009), is calculated only from observable parameters: risk-neutral and actual default probabilities, maturity, correlation, and volatility. The maturity and the risk-neutral default probability is extracted from CDS spreads, the actual default probabilities from Moody's data, the correlation from equity prices, and the market volatility from market benchmark indices data. The important advantage of this estimator is that it doesn't require calibration process to evaluate actual and risk-neutral default probabilities. The fact that only difference between them is required simplifies the model significantly; neither the asset value process nor the default barrier should be calibrated as well as expected dividends and earnings growth.
The derivation of the market Sharpe ratio starts with the determining of the real world default probability. Merton (1974) framework defines firm's debt as one zero-coupon bond D, which defaults at maturity time T, if the asset value A falls below the nominal debt value. The asset value process At is modeled as a geometric Brownian motion with volatility a: dAp = ]iAtdt+a AtdBt for actual drift and dAf = rAtdt+a AtdBt for risk-neutral drift r, where Bt denotes a standard Wiener process. According to this the real world default probability Pd for the period between t and Tmay be presented as follows:
Pd (t;T) = P [ < D] = P
Ate
i ■
U.--G*
2
(T-t)+G • (BT -B, )
< D
Pd (t; T) = N
f-|V 1-21(t -1)
a
yfr—t
(1)
N stands for cumulative standard normal distribution function. Accordingly, the risk-neutral default probability Qd can be calculated as:
Qd (t;T) = N
, D ( 1 2 ln--1 r —a
A I 2
(T -1)
a
(2)
The next step is to express the risk-neutral default probability through the actual one:
Qd (t; T) = N
'n- (Pd (t, T))
r
a
• 4¥-t
(3)
From formula (3) one can easily derive the asset's Sharpe ratio which can be simply estimated from risk-neutral and actual default probabilities and maturity:
- r N-■ { (,,T)}-N-■ {{ (,T}
SRj 4T=< ( )
The asset's Sharpe ratio may be modified into market Sharpe ratio in accordance with the continuous time CAPM, taking into account the correlation of the assets with the market portfolio, assumed to be nonzero pA>M ^ 0 :
Llu - r LlM - r LI. - r 1
M'A = r + --Pa,m-°a = ^------(5)
aM aM <5a Pa,m
For correlation estimation the following approximation holds: pAM ~ pB M, which assumes that the correlation between equity and market returns is taken rather than assets and market.
The market Sharpe ratio derived in the Merton framework is determined as the following:
^ " - „ N" { ('>7)}-N" { «>7>} 1 (6)
srm =-~-------(6)
V7 -1 Pe,M
The last parameter required for the equity risk premium estimator is the volatility of the market portfolio. Putting all together yields an estimator for the equity premium:
N -1 \Qd (t, T)}- N'1 fPd(t, T)} a
ERP =-^ V 7 1 V (7)
VT -1 Pe,M
Let's discuss all required parameters in particular. The market volatility is estimated from market equity index for concerned regions. The correlation coefficient is defined as the weighted average of median correlations for most highly capitalized firms from each industry with the market index. Median industry correlations are taken for robustness reasons. Since the correlation appears in the denominator of the formula (7), estimation errors result in an upward biased equity premium. Industry medians have lower standard errors than each company separately. The real world default probability is taken from Moody's ratings. The following approximation is used to obtain the risk-neutral default probability: CDS = ■ LGD — is the risk-neutral default intensity1, LGD — loss given default — is the percentage amount of loss due to default over the total exposure. LGD may be rewritten as 1-RR, where RR — the recovery rate, the percentage amount of total exposure which may be recovered during bankruptcy procedures in event of default. According to the default intensity model the risk-neutral cumulative default probability is derived from default intensity through the following relationship:
Qd (T, t) = 1 - e ^(T-t> (8)
Rewriting the formula 8 for CDS spread input parameter yields:
Qd (T, t) = 1 - e^l-RR J (9)
Putting this relation into the equity premium estimator produces the final formula for equity premium, which includes all required observable parameters:
1 The risk-neutral default intensity is the risk-neutral probability of default per unit time.
Figure 1. Monthly average Markit iTraxx CDS indices spreads (in percent) for CEEMEA, Australia, and Asia (excluding Japan)
N 1
ERP =
1 - e
CDS 1-RR
(T-t )
-N-1 {Pd (t, T )}
a
M
yfr-t
(10)
CDS spreads are taken from the historical data in Bloomberg while the recovery rates — from Markit statistics for each particular CDS index.
Although the derived in the Merton framework estimator for the market Sharpe ratio generates adequate results, it is the subject for criticism concerning default timing and the assumption about complete information. Berg and Kaserer (2009) expand the simple Merton framework to more advanced first passage time models, which allow for default before maturity, for the observable and unobservable asset values. Comparison of three approaches discovers immaterial results differences. They conclude that their market Sharpe ratio estimator is robust with respect to model changes because it comprises the relation between actual and risk-neutral probabilities, which itself is not submitted to the influence due to model changes. This study employs simple Merton Sharpe ratio estimator for deriving equity premium. The following sub-section describes model parameters estimation and descriptive statistics.
DATA AND pARAMETERS ESTIMATION
Current section describes data and statistics employed by the research. The basis of the research are the CDS spreads, which have 5- and 10-year maturities. Such periods are too enduring for developing markets; moreover hardly someone could be interested in estimating 5-year forward-looking equity premium. Market conditions change much more often: high volatility and uncertainty force investors to rebalance their portfolios often. Taking into consideration crisis years 2008-2009, when investor sentiments varied frequently, we decide to estimate 1-month forward-looking equity premium. The calendar month is considered as a period. This idea is supported by the assumptions that investors are sensitive to calendar timing due to necessity of periodic reporting, quarterly and annual returns calculation.
The empirical study requires a number of input parameters, which compose the equity premium estimator and are mentioned in the previous section. They may be divided into two types: the raw data and parameters to be calculated.
The raw data include corporate CDS spreads, actual default probabilities and recovery rates. Due to the fact that credit markets, particularly CDS, are underdeveloped in emerging countries, there is a lack of liquid CDS indices traded. As the only developed country in this research, Australia nevertheless has single 5-year maturity CDS index although it has the longest CDS index history.
Table 1. Average Cumulative Issuer-Weighted Global Default Rates by Alphanumeric Rating, 1998-2012*
Rating/ Year 1 2 3 4 5 10
Aaa 0.000 0.037 0.037 0.037 0.037 0.037
Aa1 0.000 0.000 0.000 0.000 0.000 0.000
Aa2 0.000 0.014 0.224 0.472 0.644 1.539
Aa3 0.055 0.164 0.233 0.344 0.525 1.499
A1 0.153 0.359 0.617 0.951 1.342 3.361
A2 0.116 0.336 0.614 0.829 1.113 4.441
A3 0.086 0.278 0.570 0.864 1.307 3.742
Baal 0.188 0.436 0.663 0.874 1.102 2.662
Baa2 0.216 0.528 0.900 1.369 1.747 3.928
Baa3 0.307 0.821 1.415 1.985 2.720 7.354
Bal 0.411 1.576 2.998 4.321 5.791 13.71
Ba2 0.708 1.692 3.036 4.586 5.843 13.36
Ba3 1.101 3.304 5.938 8.794 10.6 22.68
B1 1.592 5.247 9.373 12.94 15.9 31.1
B2 3.196 8.365 13.76 18.85 22.67 38.43
B3 4.739 11.81 19.39 25.39 30.48 49.41
Caal 8.278 18.32 27.42 35.05 42.20 64.43
Caa2 17.63 30.30 39.91 46.93 52.62 69.95
Caa3 28.39 44.48 54.31 61.34 66.89 69.37
AH rated 2.072 4.255 6.283 7.934 9.240 14.16
* Data in percent
The study employs Markit iTraxx type of CDS indices that appear in tradable credit default swap family of indices and cover the following regions: Asia excluding Japan, Australia, and CEEMEA. The Markit iTraxx indices are traded 5-year maturities (including 10-year maturity for CEEMEA) by series. A new series is determined on the basis of liquidity every 6 months — if there is enough liquidity the next series is rolled for the following half-year.
The sample includes generic daily mid-day spreads time-series for the corporate CDS indices: MARKIT iTRAXX CEEMEA 5- and 10-year maturity index, MARKIT iTRAXX ASIA excluding JAPAN IG (investment grade) 5-year maturity, and MARKIT iTRAXX AUSTRALIA 5-year maturity.
Daily roll-adjusted mid-quotes are obtained from Bloomberg (Figure 1).
The use of aggregate CDS index is stipulated by the fact that index is more liquid, covers the concerned region, and provides us with the equity premium for these regions rather than particular companies which is consistent with the purpose of
the study. The employed iTraxx Markit indices represent standardized public financial instruments which are priced in U.S. dollars, significantly liquid, and accepted as a key benchmark of the overall market credit risk. The presence of such characteristics contributes a reliable unified platform for assessing risk premium to the study.
The CEEMEA index is composed from 25 most liquid corporate underlying debt securities (bonds or LPN2) of mainly Russian, United Arab Emirates, and Turkish firms. Spreads are available since January 2010 for 5- and 10-year maturities. 10-year CDS index is rather illiquid, it is lacking for trades in May-September 2012 and May-July 2011. Regardless of this fact it is used to provide insight into term structure of equity premium. The Asian index excludes Japan and consists of 37 investment grade companies' 5-year maturity CDS3; it starts in Octo-
2 Loan participation note, or LPN, is common juridical form of Eurobond issue.
3 The total amount is 40, three of which are CDS on sovereign debt and are not considered in this study.
figure 2. Time-variation of equity premium estimations for Australia, Asia, 5-year CEEMEA, and 10-year CEEMEA CDS.
Figure 3. Historical daily chart of MSCI Australia index, MSCI CEEMEA index, and MSCI Asia excluding Japan,
January 2005 - April 2013. Source: Bloomberg.
ber 2007. The Australian index has the history since January 2005 and includes 25 names CDS of various Australian firms' debts for 5-year maturities.
As all underlying debt securities which comprise indices are rated by Moody's rating agency4, the average default rate for the each region is used
4 There are only three exceptions when debts are rated by S&P, the relevant Moody's rating is taken for those.
as a proxy for actual default probability. Since the CDS indices have 5-year maturity, only the 5-year default rates are considered; except for CEEMEA region, there are two CDS of 5- and 10-year maturities for which 5-year and 10-year default rates are taken accordingly.
The average cumulative issuer-weighted global default rates for 1998-2012 (Table 1), estimated by Moody's for each alphanumeric rating and adjusted
Table 2. Australian model parameters and estimates.
Table of average yearly parameters for Australia
Year CDS Spread 0 P market a mean p ERP p.a. Asset Sharpe Ratio Market Sharpe Ratio
10 2013 1.1% 9.2% 1.0% 4.2% 0.41 5.7% 40.5% 99.4%
2012 1.6% 12.5% 1.0% 9.7% 0.35 16.1% 48.5% 137.3%
2011 1.4% 10.3% 1.0% 17.0% 0.36 25.5% 43.0% 129.2%
2010 1.1% 8.4% 1.0% 16.7% 0.47 14.6% 38.1% 81.8%
2009 2.1% 17.2% 1.0% 17.8% 0.43 39.7% 56.0% 140.5%
2008 1.7% 11.3% 1.0% 22.9% 0.41 34.3% 44.5% 107.3%
2007 0.4% 2.8% 1.0% 11.7% 0.53 3.2% 13.7% 26.2%
2006 0.3% 2.3% 1.0% 8.4% 0.56 1.5% 10.2% 18.7%
2005 0.3% 2.3% 1.0% 8.2% 0.40 2.4% 10.8% 28.4%
for rating withdrawals, are taken for proxy for actual default probabilities. In their study Berg and Kaserer estimate actual default probabilities from Moody's KMV EDF rates which are available only for Moody's subscribers and are chargeable.
The recovery rates are taken from ISDA CDS standard contract terms5 for the each region: 0.25 for CEEMEA, 0.4 for Australia, 0.4 for Asia excluding Japan. This approach provides consistency of data and seems to be more practical and applicable for market participants who don't have access to Moody's EDF.
The parameters that require calculations include the correlation of the equity returns with market portfolio and the market volatility. For each region the relevant MSCI Index is chosen as a proxy for market portfolio — EM EMEA, Asia excluding Japan, and Australia. The estimation of correlation involves several steps. First, since the CDS indices engage companies of different industries, each industry weight in particular CDS index is defined. The top-10 stocks by market capitalization for each industry sector are selected in Bloomberg. Then, for each stock and each MSCI index 20-day rolling returns are calculated which constitute the basis for calculation of 20-day rolling correlation for each cross-section of stock and MSCI index. The averaging daily data for 20-day rolling correlations yields the monthly average values. Finally, the sum of weighted mean monthly correlations for industry sectors represents the final correlation figure for each calendar month of the sample. Market volatility is estimated as aver-
5 The recovery rate is specified in ISDA CDS standard contract terms, http://www.cdsmodel.com/cdsmodel/fee-computations. page.
age monthly standard deviation of 20-day rolling MSCI index returns.
Correlation and volatility calculations are made in RStudio and Microsoft Excel. Mean industry correlation is more objective and smoothed than that of particular company, it has lower errors, and mirrors market more accurate. The industry sector classification is based on Global Industry Classification Standard (GICS).
The CDS spreads, stocks prices, MSCI data are taken from Bloomberg in U.S. dollars for data uniformity.
When all parameters are defined, we put them into the model as of formula (10) and estimate the expected equity premium for the next month over the whole period of each CDS index. The last calculated value in the sample (timestamped March 2013) represents the expected equity premium for April 2013.
RESULTS AND INTERPRETATION
Based on the Sharpe ratio estimator, as of equation (6), derived by Berg and Kaserer in the Merton framework, we estimate implied equity premium, as of equation (7), from CDS spreads of generic iTraxx CDS indices of Australia, Asia excluding Japan, and CEEMEA region. Appendix 1 provides the results for monthly asset Sharpe ratio, market Sharpe ratio, and equity premium, denoted as per annum percents, for Australia, Asia excluding Japan, CEEMEA 5-year, and CEEMEA 10-year.
Figure 2 demonstrates graphically the time-variation of estimated equity premium.
Obtained results are consistent with empirical market observations. For the period 2005-2007
Table 3. CEEMEA model parameters and estimates for 5-year and 10-year CDS.
Table of average yearly parameters for CEEMEA (5-year)
Year CDS Spread 0 P market a mean p ERP p.a. Asset Sharpe Ratio Market Sharpe Ratio
10 2013 1.7% 10.9% 4.1% 5.1% 0.27 5.6% 22.9% 87.5%
2012 2.5% 15.9% 4.1% 11.2% 0.37 13.9% 33.1% 98.2%
2011 2.4% 14.1% 4.1% 15.4% 0.40 14.3% 29.7% 78.6%
2010 2.2% 13.5% 4.1% 13.0% 0.29 15.3% 28.7% 109.1%
Table of average yearly parameters for CEEMEA (10-year)
10 2013 1.9% 22.7% 9.7% 5.1% 0.27 4.3% 17.5% 66.8%
2012 2.4% 28.5% 9.7% 11.2% 0.37 8.8% 23.4% 67.0%
2011 2.5% 27.6% 9.7% 15.4% 0.40 11.8% 22.4% 62.5%
2010 2.2% 25.5% 9.7% 13.0% 0.29 10.8% 20.5% 78.3%
APrE)P 10Y
figure 4. Term structure dynamics of CEEMEA equity premium.
Australian premium curve is flat and close to zero. Volatility starts growing in 2008. From October 2008 to February 2009 Australian and Asian equity premiums significantly exceed 100%. Australian premium reaches its peak of 9% per month or 180% per annum in December 2008 when MSCI index for Australia drops 58% during four months to 377 on November 20, 2008 from its maximum in July 2008-890; and when the largest monthly slump of 38% takes place in October 2008. Asian equity premium reaches 7.3% per month or 133% per annum in December 2008 after monthly stock fall by 38% during October 2008.
In spite of the fact that CEEMEA data starts considerably later and unfortunately does not cover the crisis, its patterns since 2010 are very similar to Australian and Asian ones.
Figure 3 provides an insight of stock market behavior since 2005. It may be clearly noticed that equity premium movements correspond to stock indices raises and downturns.
In order to assess performance of stock index we should estimate its returns relative to the first value of the corresponding period: the base value for Australia is of 03.01.2005, for Asia — of 01.10.2007, and for CEEMEA — 01.01.2010.
Table 4. Asian model parameters and estimates.
Table of average yearly parameters for Asia excluding Japan
Year CDS Spread 0 P market a mean p ERP p.a. Asset Sharpe Ratio Market Sharpe Ratio
10 2013 1.1% 8.6% 1.0% 4.2% 0.35 6.3% 43.5% 124.8%
2012 1.6% 12.5% 1.0% 8.6% 0.38 14.3% 53.0% 141.2%
2011 1.4% 10.5% 1.0% 13.2% 0.45 15.6% 47.9% 105.6%
2010 1.1% 8.9% 1.0% 10.5% 0.57 8.9% 44.3% 80.1%
2009 2.2% 17.4% 1.0% 14.4% 0.59 20.1% 60.9% 101.5%
2008 2.0% 13.1% 1.0% 20.8% 0.53 28.1% 52.4% 100.3%
40 2007 0.6% 4.0% 1.0% 14.3% 0.52 7.6% 25.4% 49.1%
AuSTRALIA
The evidence of high-volatile Australian equity premium is supported by illustration of MSCI Australia index behavior: Australian stocks are more volatile than that of other regions. The standard deviation of Australian equity premium is 15% for the whole period and 18% since 2008, while the volatility of Asian equity premium is 10%, of CEEMEA 5-year — 8%, of CEEMEA 10-year — 6%.
The average annual estimations and input parameters for Australia are displayed in Table 2.
The estimations yield the average equity premium for the whole period around 15% for Australia, average market Sharpe Ratio is 85%, average yearly market volatility is 13%. CDS spreads are rather
low, compared to other regions, the mean since 2008 is 1.5%, including per-crisis period — 1.1%. Based on CDS spreads estimations of risk-neutral default probabilities vary significantly: from 2.3% before crisis to 17.2% during crisis. Correlations are rather low, approximately 0.44 on average, in contrast to the consideration that during turmoil correlations should increase. In general, Australian stock market shows itself as highly-volatile when sharp rises replace large slumps. The evidence of this is demonstrated in the Appendix 2. The graphical view displays clear inverse relation between stock market performance and estimated equity premium. This relation implies that during downturns investors demand much higher price for increased risk and therefore premiums jump;
Figure 5. Monthly correlation estimates for Asia excluding Japan.
16% 14% 12% 10% 8% 6% 4% 2% 0%
i MSCI Asia monthly volatility
J
\
A
r-OOOOCOOWWOOOT-
OOOOOQOT-T-T-T-
IIIIIIIIIII
ac>.ac>.ac>.iic
®<B<c®(o<a©(C(o®<a
T- T- (<g CM N «
CL C ® IS M ->
Figure 6. Monthly volatility estimates for MSCI Asia excluding Japan.
when volatility is low and the market demonstrates steady growth — equity premiums are lower.
CEEMEA
Average annual CEEMEA model parameters and estimated values are presented in the Table 3 for two cases: estimations based on 5-year and 10-year default swaps.
Although CDS spreads on average seem to be very similar for both CEEMEA cases, the difference in implied risk-neutral probabilities is significant: they are twice as high for 10-year period (13.6% versus 26%). These estimations may be supported by the evidence that actual default probabilities differ for the same level approximately (4.1% versus 9.7%). Correlations for CEEMEA are 0.33 on average; mean annual market volatility is 11.2%. Five-year CDS-based estimations yield average 12.3% equity premium while 10-year — only 9%. The same effect is reflected in market Sharpe ratio figures: long-term ratios are 25% lower than medium-term ones (93% versus 69%). This leads to conclusion that the term structure of equity premium is downward sloping: long-term assets are considered to be less risky than medium-term. Figure 4 provides insight into variation of term structure of CEEMEA equity premiums estimated using 5-year and 10-year CDS.
10-year CDS index is rather illiquid: the data on trades during May-July 2011 and May-September 2012 is absent. Despite the lack of data the pattern of positive difference between 5-year and 10-year estimated is obvious.
Appendix 3 demonstrates inverse relation between movements of market (MSCI CEEMEA Index) and estimated equity premiums for CEEMEA.
ASIA
Along with Australian, the Asian data allows observing equity premium before, during, and after the crisis. Estimations for Asia and input model parameters are displayed in Table 4. The average CDS spread is 1.4% with its peak 2.2% in 2009. Risk-neutral probabilities of default vary significantly from year to year: from 4% in 2007 to 17.4% in 2009 with mean 10.7%. MSCI Asia excluding Japan Index volatility, or market volatility, follows financial markets behavior increasing in critical stages of economy, e.g. 20.8% in 2008, with mean 12.3%. During the peak of economic turmoil correlation increases to 0.59 while in 2012 it is 0.38 and continues to reduce in 2013.
Equity premiums reach the highest level in 2008-28.1% on average, in contrast to other examined regions with maximums in 2009. Appendix 4 displays inverse relation between movements of market (MSCI Asia excluding Japan Index) and estimated equity premiums.
discussion
The estimated values constitute the upper limits for equity premiums due to several reasons. First, due to assumption that 100% of CDS spread is attributable to credit risk. Another major issue that impacts results is parameters estimations.
First parameter that needs to be defined is real world default probability. Average cumulative issuer-weighted global default rates by alphanumeric rating for the period 1998-2012 as of Moody's were taken as a proxy for expected default probabilities due to lack of Moody's EDF data. Real world default probabilities are time-varying. The average values are underestimated in the periods of turmoil. As a result the Sharpe ratios and equity premiums are biased upwards, because the difference between risk-neutral probabilities of default, which are time varying, and actual default probabilities is overestimated. The literature on modeling expected real world default probabilities is developing. The time-varying cumulative default probabilities are required for further research in order to obtain more accurate data on Sharpe ratios and equity premiums.
Methods of correlation and market volatility estimation may also affect results due to their high variability from month to month. Historical monthly values of Asian correlations and stock market volatility are demonstrated on Figures 5 and 6 respectively.
Volatility and correlation parameters' estimates have direct crucial influence on equity premium results. More precise and robust calculation approach will lead to more accurate equity premium and Sharpe Ratio outcomes.
The use of constant recovery rates is implied by the default intensity model although recovery rates may vary over time. Current paper uses conservative rates implied in Markit iTraxx CDS indices which yields upper limit for risk-neutral default probability as well as Sharpe ratio and equity premium.
Although due to possible upward bias discussed above, the overall size and behavior of regarded economies equity premium is consistent with market data.
conclusion
This paper estimates expected equity premium for developing markets using the methodology of Berg and Kaserer (2009). The approach is based on the idea that risk premium may be extracted from CDS spread as it consists of expected loss of price and premium for bearing credit risk. Structural model of default within Merton framework is applied to define actual and risk-neutral probabilities. Basically, the only difference between them is the drift: for risk-neutral default probability it is risk-free rate, and for real world default probability it is return. Expressing risk-neutral default probability by the means of actual yields the relation which contains such parameter is asset Sharpe ratio. Then having estimated risk-neutral default probability from CDS spread according to default intensity model and received market default rates, we are able to derive asset Sharpe ratio, convert it into market Sharpe ratio and thus obtain equity premium. As a result, the simple estimator for equity premium is based on risk-neutral and actual default probabilities, the maturity, corresponding stock market volatility, and equity correlations. The major advantage of the model is that it is based only on observable parameters and doesn't require forecasts or calibrations. Either Sharpe ratio estimation dispenses from risk-free rate calculation which is rather arguable concern.
The empirical analysis of most liquid CDS indices spreads for Australia, Asia excluding Japan, and CEEMEA region results in quite homogenous time series of equity premiums with similar properties. Australia has the longest CDS history which allows estimating equity premium from 2005. Before crisis in 2008 average yearly equity premium value is 2.3% per annum, during crisis 2008-2009 premium rise sharply to 37% (reaching 9% for December 2008 or 180% in annual terms), from 2010 by the present it is very volatile, around 15% yearly on average. Other regions' results demonstrate behavior identical to Australian and respond to the same economic and financial events, but are less volatile. Asian CDS index starts from October 2007. It reaches its maximum of 7.3% for month, or 133% annually, in December 2008. Since 2010 the average monthly premium in annual terms per year is approximately 11%. Data of CEEMEA CDS indices for 5-, and 10-year maturities are available since 2010 although the 10-year CDS is rather illiquid (it has no trades for May-September 2012 and May-July 2011). The dynamics is the same for both maturities. On average yearly for 2010-2012 equity
premium is 14.5% for 5-year CDS and 10.5% for 10-year CDS. The term structure curve is downward sloping, or inverse, that may indicate that financial turmoil is still in place when short-term risks are higher than long-term.
This paper's contribution is threefold. Firstly, we estimate expected equity premiums for Australia and emerging markets of Asia excluding Japan, and CEEMEA, which to our best knowledge has not been done in the literature yet. Secondly, it estimates equity premiums for each calendar month during the life of CDS index used as a proxy for CDS spread, whereas Berg and Kaserer estimate yearly premiums. As a result we get historical time-variation of equity premium, which allows for its behavior assessing in different economic and financial conditions including pre- and post-crisis 2008. Finally, the paper provides an insight into term structure of equity premium by the example of CEEMEA market.
Empirical observations prove adequacy of estimated results of the study. The following conclusions regarding the equity premium are to be made:
1) Equity premium for Australia and two emerging markets (CEEMEA and Asia) are countercyclical;
2) Term structure of equity premium is inverse — which implies that investors price medium-term risks higher than long-term ones;
3) Term structure varies over time — curve slope changes; negative slope increases when stock market declines.
The numerical results appear to be the upper limit for equity premiums. To evaluate equity premiums more accurately there are several possibilities of approach improvement. The major one is to determine time-varying expected real world default probabilities. In addition, robust estimations of equity correlations and stock market volatility that comply with default probabilities and premiums timing will produce more reliable estimations. Nevertheless, one should bear in mind that "the complexity is often traded off against parsimony for practical reasons of implementation" (Balduzzi, Das, Foresi, and Sunda-ram 1996, p.43.) Estimated equity premiums dynamics reflects financial markets events and behavior rather well and may be used for trading purposes.
The brief look on relation between stock market movements and equity premium behavior does not lead to agreement upon the concern what affects what. Establishing the relationship between stock market behavior and estimated equity premium in order to understand whether the model may be used for returns prediction may expand practical implementation.
Appendix 1. Monthly asset Sharpe ratio (ASR), market Sharpe ratio (MSR), and equity premium (EP) estimations, denoted as per annum, for Australia, Asia excluding Japan, and CEEMEA.
Australia Asia excluding Japan CEEMEA 5Y CEEMEA 10Y
Month ASR MSR EP ASR MSR EP ASR MSR EP ASR MSR EP
ApriL-13 40% 110% 6% 44% 138% 6% 25% 104% 4% 18% 78% 3%
March-13 40% 95% 3% 44% 137% 6% 23% 74% 4% 18% 58% 3%
February-13 40% 88% 4% 43% 107% 4% 23% 88% 6% 17% 64% 4%
January-13 42% 105% 10% 44% 118% 9% 21% 83% 9% 17% 67% 7%
December-12 45% 130% 9% 45% 113% 5% 22% 40% 3% 18% 32% 2%
November-12 46% 124% 6% 47% 130% 12% 23% 42% 4% 19% 35% 3%
October-12 47% 130% 14% 48% 141% 15% 28% 61% 7% 23% 49% 5%
September-12 48% 152% 11% 52% 122% 9% 31% 91% 8%
August-12 51% 131% 15% 54% 188% 21% 32% 94% 10%
JuLy-12 53% 161% 35% 57% 162% 27% 36% 148% 36%
June-12 51% 172% 23% 56% 169% 16% 42% 111% 14%
May-12 47% 116% 7% 53% 165% 9% 36% 94% 10%
ApriL-12 45% 143% 13% 51% 138% 12% 34% 129% 13% 25% 94% 9%
March-12 46% 135% 11% 53% 112% 7% 35% 88% 9% 25% 64% 6%
February-12 50% 128% 21% 59% 118% 16% 37% 155% 28% 27% 114% 20%
January-12 53% 124% 29% 60% 137% 23% 41% 126% 25% 27% 81% 15%
December-11 53% 122% 35% 60% 136% 36% 39% 140% 40% 27% 100% 27%
November-11 53% 149% 57% 61% 120% 35% 35% 115% 39% 28% 90% 30%
October-11 53% 205% 67% 57% 123% 29% 42% 101% 31% 26% 63% 18%
September-11 47% 126% 30% 50% 121% 25% 32% 56% 13% 24% 41% 9%
August-11 41% 175% 20% 45% 121% 10% 27% 51% 4% 21% 40% 3%
JuLy-11 40% 133% 11% 44% 131% 9% 25% 53% 7%
June-11 38% 115% 18% 43% 98% 10% 25% 51% 7%
May-11 38% 87% 16% 42% 86% 7% 24% 56% 4%
ApriL-11 38% 144% 18% 43% 100% 9% 25% 82% 6% 19% 60% 5%
March-11 38% 68% 6% 43% 76% 7% 27% 73% 6% 19% 52% 4%
February-11 38% 111% 12% 43% 82% 4% 27% 80% 7% 19% 56% 5%
January-11 38% 117% 16% 43% 73% 6% 27% 84% 8% 19% 60% 6%
December-10 38% 117% 9% 42% 88% 6% 30% 78% 6% 19% 49% 4%
November-10 40% 96% 9% 43% 95% 6% 26% 78% 5% 18% 55% 4%
0ctober-10 41% 64% 11% 46% 72% 7% 29% 74% 11% 21% 54% 8%
September-10 42% 78% 17% 47% 85% 8% 31% 89% 17% 22% 63% 11%
August-10 44% 118% 23% 48% 89% 10% 30% 134% 22% 22% 99% 16%
JuLy-10 44% 80% 27% 49% 74% 14% 31% 86% 19% 22% 61% 13%
June-10 38% 84% 21% 46% 73% 11% 28% 111% 23% 21% 84% 17%
May-10 32% 65% 4% 40% 65% 2% 25% 178% 10% 18% 126% 7%
ApriL-10 33% 59% 12% 41% 66% 10% 25% 122% 20% 18% 88% 14%
March-10 36% 76% 20% 45% 99% 18% 30% 119% 20% 23% 90% 15%
February-10 33% 69% 10% 40% 68% 6% 31% 131% 15% 21% 90% 10%
January-10 35% 78% 11% 43% 88% 7%
December-09 36% 123% 17% 44% 81% 5%
November-09 37% 141% 11% 43% 72% 5%
0ctober-09 41% 108% 12% 46% 80% 12%
September-09 46% 99% 22% 49% 100% 16%
August-09 52% 75% 17% 54% 96% 14%
JuLy-09 52% 96% 12% 56% 89% 12%
June-09 59% 138% 21% 60% 97% 11%
May-09 68% 169% 57% 71% 108% 25%
ApriL-09 74% 162% 49% 79% 131% 33%
March-09 69% 179% 53% 78% 116% 22%
February-09 67% 166% 53% 72% 109% 32%
Australia Asia excluding Japan CEEMEA 5Y CEEMEA 10Y
Month ASR MSR EP ASR MSR EP ASR MSR EP ASR MSR EP
January-09 72% 231% 153% 78% 138% 54%
December-08 63% 166% 180% 77% 167% 133%
November-08 61% 142% 81% 77% 158% 58%
October-08 49% 121% 18% 55% 90% 12%
September-08 46% 131% 13% 51% 95% 15%
August-08 46% 95% 8% 51% 96% 13%
JuLy-08 40% 79% 14% 48% 84% 10%
June-08 35% 64% 6% 41% 73% 8%
May-08 41% 100% 21% 47% 92% 20%
ApriL-08 50% 114% 21% 57% 108% 23%
March-08 42% 136% 31% 51% 101% 25%
February-08 35% 93% 13% 43% 81% 12%
January-08 26% 46% 7% 33% 59% 10%
December-07 22% 36% 6% 34% 67% 13%
November-07 14% 23% 3% 21% 41% 4%
0ctober-07 17% 32% 9% 21% 39% 7%
September-07 18% 33% 8%
August-07 13% 30% 2%
JuLy-07 9% 17% 1%
June-07 10% 20% 1%
May-07 12% 27% 3%
ApriL-07 12% 27% 3%
March-07 12% 22% 2%
February-07 13% 23% 1%
January-07 12% 24% 1%
December-06 8% 16% 1%
November-06 11% 18% 2%
0ctober-06 12% 18% 1%
September-06 11% 16% 1%
August-06 11% 16% 2%
JuLy-06 11% 16% 2%
June-06 10% 16% 2%
May-06 11% 19% 2%
ApriL-06 10% 25% 1%
March-06 9% 23% 2%
February-06 10% 24% 1%
January-06 9% 18% 1%
December-05 11% 20% 2%
November-05 11% 19% 2%
0ctober-05 11% 26% 2%
September-05 11% 31% 3%
August-05 12% 41% 4%
JuLy-05 14% 38% 3%
June-05 13% 44% 3%
May-05 11% 27% 3%
ApriL-05 8% 29% 2%
March-05 8% 19% 1%
February-05 8% 17% 1%
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Appendix 2. Estimated equity premium and stock market performance for Australia. Equity premiums are monthly data on a yearly basis. MSCI Australia index performance means indexed daily MSCI values with 03.01.2005=100 in percent.
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Appendix 3. Estimated equity premium (5-year CDS-based) and stock market performance for CEEMEA. Equity premiums are monthly data on a yearly basis. MSCI EM EMEA index performance means indexed daily MSCI values with 01.01.2010=100 in percent.
Appendix 4. Estimated equity premium and stock market performance for Asia excluding Japan. Equity premiums are monthly data on a yearly basis. MSCI Asia excluding Japan index performance means indexed daily MSCI values with 01.10.2007=100 in percent.
140%
120%
100%
80%
60%
40%
20%
Asia Equity Premium
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