Mateusz Pipien

English

Research Papers in Economics

© Copyright by The Nicolaus Copernicus University Scientific Publishing House
DYNAMIC ECONOMETRIC MODELS 
Vol. 8 – Nicolaus Copernicus University – Toruń – 2008 
Mateusz Pipień 
Cracow University of Economics 
On the Use of the Family of Beta Distributions in Testing 
Tradeoff Between Risk and Return. Bayesian Analysis 
for WIG Excess Returns 
1. Introduction 
  The main goal of this paper is an application of Bayesian inference in test-
ing the relation between risk and return of the financial time series. On the basis 
of the Intertemporal CAPM model, proposed by Merton (1973), we built a gen-
eral sampling model suitable in analysing such relationship. The most important 
feature of our model assumptions is that the possible skewness of conditional 
distribution of returns is used as an alternative source of relation between risk 
and return. Thus, pure statistical feature of the sampling model is equipped with 
economic interpretation. This general specification relates to GARCH-In-Mean 
model proposed by Osiewalski and Pipień (2000). 
  In order to make conditional distribution of financial returns skewed we 
considered a constructive approach based on the inverse probability integral 
transformation. In particular, we apply Beta distribution transformation with 
two free parameters; see Jones and Faddy (2003).  
  Based on the daily excess returns on the Warsaw Stock Exchange Index we 
checked the total impact of conditional skewness assumption on the relation be-
tween return and risk on the Warsaw Stock Market. Posterior inference about 
skewness mechanism confirmed positive and decisively significant relationship 
between expected return and risk.  
2. An Approach to Creating Asymmetric Distributions 
  The unified representation of the univariate skewed distributions that we 
study in the paper is based on the inverse probability integral transformation; © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Mateusz Pipień  62
see Ferreira and Steel (2006) for details. The family IP={εs, εs:Ω→R}, with the 
representative density s(.|θ,ηp), is called the skewed version of the symmetric 
family I (of random variables with unimodal symmetric density f(.|θ) and distri-
bution function F, such that the only one modal value is localised at x=0) if s is 
given by the form: 
s(x |θ,ηp)=f(x|θ)⋅p(F(x|θ) |ηp), for x∈R. (1) 
  The asymmetric distribution s(.|θ,ηp) is obtained from f(.|θ) by applying the 
density  p(.|ηp) as a weighting function. Within the general form (1) several 
classes of distributions P have been imposed on some specific families of sym-
metric random variables; see Pipień (2006) for a review. The most important 
feature of our approach is, that uniform density, namely p(y|ηp)=1 for y∈R, re-
trieves symmetry in (1); i.e. s=f if p(.|ηp)=I[0,1]. In the empirical part of the pa-
per, when defining skewing density p, we make use of the family of Beta distri-
butions. The random variable εp∈(0,1) has a Beta distribution with parameters 
a>0 and b>0 if the density, denoted by Be(.|a,b), is given by the formula: 
Be(y|a,b)=y
a-1(1-y)
b-1/B(a,b), for y∈R, 
where normalising constant B(a,b) denotes the value of the Euler beta function. 
3. Introducing Skewness into GARCH-in-Mean Model 
  Let denote by xj the value of a stock or a market index at day j. The excess 
return on xj, denoted by yj, is defined as the difference between the logarithmic 
daily return on xj in percentage points (rj=100ln(xj/xj-1)) and the risk free short 
term interest rate (denoted by rj
f), namely yj=rj–rj
f. Following Engle, Lilien and 
Robins (1987), and Osiewalski and Pipień (2000) we consider for yj a simple 
GARCH-In-Mean process, defined as follows 
 yj=[α+E(zj)] hj
0.5+uj, j=1,2,..., (2) 
where uj=[zj–E(zj)]hj
0.5, and zj are independently and identically distributed ran-
dom variables with E(zj)<+∞. The scaling factor hj is given by the GARCH(1,1) 
equation; see Bollerslev (1986): 
hj=α0+α1uj-1
2+β1hj-1. 
The specific form of the conditional distribution of yj strictly depends on the 
type of the distribution of zj. Initially, in model denoted by M0, we assumed for 
zj the Student-t density with unknown degrees of freedom ν>1, zero mode and 
unit inverse precision. We denote the value of this density as ft(z|0,1,ν). Given 
model M0, E(zj)=0, uj=zjhj
0.5, and hence (2) reduces to the simpler form yj=α 
hj
0.5+uj. Let denote by θ=(α,α0,α1,β1,ν) the vector of all parameters in model 
M0. The following density represents conditional distribution of the excess re-
turn at time j: 
p(yj |ψj-1,θ, M0)= hj
-0.5⋅ft(hj
-0.5⋅(yj–αhj
0.5)|0,1,ν), j=1,2,... . © Copyright by The Nicolaus Copernicus University Scientific Publishing House
On the Use of the Family of Beta Distributions…  63
Given model M0 the expected excess return (conditional to the whole past ψj-1) 
is proportional to the square root of the inverse precision hj: 
 E(yj |ψj-1,θ, M0)= α hj
0.5. (3) 
The parameter α∈R captures the dependence between expected excess return 
and the level of risk both measured by E(yj|ψj-1,M0,θ) and the scale parameter 
hj
0.5 respectively. 
  Now, defining model M1, we introduce skewness into our GARCH-In-Mean 
model. The resulting asymmetric distribution is obtained by skewing the density 
of the random variable zj according to method presented in the previous section. 
Asymmetric density of zj takes the form related to the formula (1): 
p(z |M1)=ft(z|0,1,ν)⋅p[Ft(z) |η1,M1], for z∈R, (4) 
where p(.|η1,M1) defines the skewing mechanism parameterised by the vector 
η1, and Ft(.) is the distribution function of the Student-t random variable with 
ν>1 degrees of freedom parameter, zero mode and unit inverse precision. For 
the skewing density p in M1 we assumed the Beta distribution density with two 
free parameters a>0 and b>0: 
p(y|η1,M1)=Be(y|a,b), η1=(a,b), a>0, b>0. 
In model M1 the conditional distribution of yj is heteroscedastic, where time 
varying dispersion measure hj, defined by GARCH(1,1) specification, is a func-
tion of the whole past of the process. The degrees of freedom parameter ν>1 
enable for fat tails of p(yj|ψj-1,θ,η1,M1). It is also possible to test whether the 
dataset supports conditional distribution with Gaussian-type tails (for ν→∞). 
Asymmetry of the conditional distribution can be captured by the presence of a 
particular skewing mechanism. If a>b (a<b), then the conditional distribution is 
skewed to the left (right), while in case a=b=1 the skewed density in (4) reduces 
to the simple symmetric case. Consequently restriction a=b=1 nests M0 in M1. 
Additionally, skewness of the distribution of zj in M1 generates nonzero expecta-
tion E(zj)<+∞, hence: 
E(yj |ψj-1,θ,η1,M1)= [α+E(zj)]hj
0.5, for E(zj)≠0, 
and conditional skewness of excess returns yj in M1 can be interpreted as an ad-
ditional source of the relationship between risk and return. Our idea fully corre-
sponds to Harvey and Siddique (2000), who emphasize, that systematic skew-
ness is economically important and governs risk premium. 
4. Empirical Analysis 
  In this part we present an empirical example of Bayesian comparison of all 
competing specifications. We also discuss the results of the total impact of the 
conditional skewness assumption on the relationship between risk and return on 
the Warsaw Stock Exchange (WSE). Our dataset was constructed on the basis © Copyright by The Nicolaus Copernicus University Scientific Publishing House
Mateusz Pipień  64
of T=2144 observations of daily growth rates, rj, of the index of the WSE (WIG 
index) from January 06, 1998 till July 31, 2006. The risk free interest rate, rj
f, 
used in excess return yj, was approximated by the WIBOR overnight interest 
rate (WIBOR O/N instrument). Our empirical results remained practically un-
changed for rj
f calculated on the basis of the middle and long term WIBOR in-
terest rate and also in the case rj
f=0 for each j. 
  Table 1 presents decimal logarithms of the marginal data density value, as 
well as the posterior probabilities of M0 and M1. The substantial data support is 
attached to model M1, making the case of conditional symmetry f yj rather im-
probable in the view of the data. Comparing posterior properties of α+E(zj) in 
both models, we checked the empirical importance of conditional skewness 
(imposed in M1) on the relationship between risk and return. Initially, given M0, 
E(zj)=0, and the whole information about relative risk aversion coefficient is re-
flected in parameter α; see (3). Just like many other researchers we obtained 
positive, but rather weak relationship between expected excess return and the 
level of risk. The posterior probability P(α>0|M0,y) equal 0.92 leaves consider-
able level of uncertainty about the true strength and the sign of tested relation. 
However, in case of model M1, the posterior probability of the positive sign of 
α+E(zj) is greater than 0.99, leaving no doubt about the sign of the relationship. 
Hence, we restored the positive sign of the relationship, by imposing a particu-
lar skewing mechanism on the conditional distribution of excess return. Beta 
distribution transformation, applied in M1, seems to be a proper model compo-
nent, making conditional distribution of yj more sensitive to information about 
risk premium contained in the dataset. 
Table 1.  Decimal logarithms of the marginal data density values, posterior probabilities 
of competing specifications, and posterior analysis of the impact of the condi-
tional skewness assumption on the relation between risk and return 
 
 
M1 
skewness imposed by Beta den-
sity with 2 parameters  
M0 
conditional symmetry and 
hence: E(zj)=0 
log⋅p(y|Mi)  -1558.41 -1559.06 
P(Mi|y), i=0,1 0.8171  0.1829 
α+E(zj) 
0.2148 
0.0852 
0.0483 
0.0337 
P(α+E(zj)>0|Mi,y)  0.9972 0.9201 
  Figure 1 presents the plot of the posterior mean of the skewing mechanism 
in model M1. The qualitative analysis of the shape of the density p(.|η1,M1) 
enables to identify the sources of skewness of the conditional distribution of yj 
supported by the dataset. As seen from the figure, the considerable amount of 
skewness is located in the tails of the density p(yj |ψj-1,θ,η1,M1). It is clear, that 
skewness is forced by subtle distinction between left and right tail of the condi-© Copyright by The Nicolaus Copernicus University Scientific Publishing House
On the Use of the Family of Beta Distributions…  65
tional distribution of excess return yj. The skewing mechanism p(.|η1) makes 
conditional distribution of excess returns more leptokurtic, as the function 
p(.|η1) has its extremes on the bounds of the interval (0,1). Since the global ex-
treme value of p(y|η1) is reached at y=0, the skewing mechanism forces left 
conditional asymmetry in M1. 
 
Figure 1. The plot of the posterior mean of skewing mechanism p(.|η1) (grey plot), to-
gether with the plot depicting the case of symmetry (black plot of the uniform 
density over the unit interval) 
References 
Bollerslev, T. (1986) Generalised Autoregressive Conditional Heteroscedasticity, Jour-
nal of Econometrics, 31, 307–327.  
Engle, R. F., Lilien, D. M., Robins, R. P. (1987) Estimating Time-varying Risk Premia 
in the Term Structure: The ARCH-M Model, Econometrica, 55, 391–408. 
Ferreira, H., Stell, M. F. J. (2006), A Constructive Representation of Univariate Skewed 
Distributions, Journal of the American Statistical Association, 101, 823–839. 
Harvey, C.  R., Siddique, A. (2000) Conditional Skewness in Asset Pricing Models, 
Journal of Finance, 55, 1263–1295. 
Jones, M. C., Faddy, M. J. (2003) A Skew Extension of the t-Distribution, with Appli-
cations, Journal of Royal Statistical Association B, 65, 159–174. 
Merton, R. C. (1973) An Intertemporal Capital Asset Pricing Model, Econometrica, 41, 
867–887. 
Osiewalski, J., Pipień, M., (2000) GARCH-In-Mean through Skewed t Conditional Dis-
tributions: Bayesian Inference for Exchange Rates, w: Welfe, W., Wdowiński, P. 
(red.), 26-th International Conference MACROMODELS'99, Łódź , 354–369.  
Pipień, M. (2006), Bayesian Comparison of GARCH Processes with Skewness Mecha-
nism in Conditional Ditributions, Acta Physica Polonica B, 37, 3105–3121. 
 
 
 
 
 
 
 
 
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9© Copyright by The Nicolaus Copernicus University Scientific Publishing House
 
 

A Constructive Representation of Univariate Skewed Distributions,

A Skew Extension of the t-Distribution, with Applications,

An Intertemporal Capital Asset Pricing Model,

Bayesian Comparison of GARCH Processes with Skewness Mechanism in Conditional Ditributions,

Conditional Skewness in Asset Pricing Models,

Estimating Time-varying Risk Premia in the Term Structure:

GARCH-In-Mean through Skewed t Conditional Distributions: Bayesian Inference for Exchange Rates, w:

Generalised Autoregressive Conditional Heteroscedasticity,

On the Use of the Family of Beta Distribution in Testing Tradeoff Between Risk and Return. Bayesian Analysis for WIG Excess Returns

http://www.dem.umk.pl/dem/archiwa/v8/8_Pipien.pdf

On the Use of the Family of Beta Distribution in Testing Tradeoff Between Risk and Return. Bayesian Analysis for WIG Excess Returns

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