Approximating Risk in A Large Portfolio and Empirical
Applications
Ba M Chu∗
Carleton University
(Research in progress, comments are welcome.)
∗
E-mail address: ba chu@carleton.ca
1
Abstract
This paper provides an alternative behavioral approach for the portfolio selection problem of an investor who desires to ensure himself from loosing more than a specified amount.
This approach is essentially based on the foundation proposed by Stutzer (2002, 2003), which is
grounded on the conjecture that the investor’s aspiration to minimize the objective probability
that the invested wealth will not exceed an investor-selected target is equivalent to his desire
to maximize an expected utility function. This can be shown with an application of Large
Deviations principle. In addition, it is well-known that systematic risks inherent in financial
investments can be avoided through diversification provided that assets have low correlations
(see, inter alia, Sharpe (1964)). Unfortunately, our knowledge of the particular risk components which have relevant influences on the investment decision is very limited. In this paper,
we sheds some light on the relationship between the Value-at-Risk of a well-diversified portfolio (or the probability of loosing more than a specified amount), and an investment decision
based on the expected utility maximization. As a result, an optimal portfolio is determined
by either minimization of the Value-at-Risk or maximization of an expected utility. Moreover,
the expected utility is dependent on both moments and co-moments of asset returns, which
thus determine the optimal portfolio and Value-at-Risk. We apply our approach to derive
optimal asset allocation strategies investing in (1) U.K. industrial subindices and (2) funds of
hedge funds – this problem is of interest because hedge funds are construed as well-diversified
portfolios of low correlated assets and their hedging instruments, and often yield better performance than many market benchmarks and their ‘style’ classifications in terms of risk-adjusted
returns.
JEL classification: C4;D8;
Keywords: Portfolio theory; Large deviation; Skewness preference; Value-at-Risk; Portfolio
insurance; Decay function;
2
1.
Introduction
Such rare, extreme events as ”the bank may lose more than a specified amount of capital”..etc..
are referred to as large deviations or extreme values which might fall within the ends or ”tails”
of a probability function characterizing the probabilistic law of those events. Hence, it is obvious
that the use of under-performance probability1 as a standard measure of risk has hitherto become
prevalent. William (Fall 1997), Stutzer (2003), and Browne (1999b) have pointed out that the
investment criteria of the minimization of under-performance probability is best appropriate to
achieve a particular investment goal. Hence, William (Fall 1997) has advocated the use of the
under-performance probability rather than the classical volatility
...popular measures of risk include the expected volatility of a portfolio’s value as measured by its standard deviation, or beta, which measures the portfolio’s volatility relative
to an index such as the S&P. These common measures of risk, however, cannot be used
to determine objectively which specific efficient portfolio is best suited to achieve a
particular investment goal...
...conventional analysis often concludes that investors will choose portfolios that maximize various utility functions that typically aim to characterize investor attitudes toward
risk and return. But this conclusion assumes away the problem...
Stutzer (2003) have successfully unravelled this puzzle of William (Fall 1997). In Stutzer (2003)’s
paper, Large Deviations principle is applied to show that an investor’s desire to minimize the underperformance probability is also equivalent to his objective to maximize a power utility function. This
behavioral foundation sheds a new light on the optimal investment theory.
So far, many portfolio insurance theories involves minimization of various risk measures for a
given investment goal, but have not been quite successful (good references are Giorgi (August 2002),
Pedersen and Satchell (2002), and Malevergne and Sornette (2002) and references therein). Malevergne and Sornette (2002); Sornette et al. (1998) advocates the use of the higher-order cumulants
1
Probability of not achieving investment goals
3
as the measures of larger risks of a portfolio. Since the higher order expectation E[X n ] is simply
represented geometrically as equal to the area below the curve xn P (x), the curves illustrate the
fact that the main contributions to the moment E[X n ] of order n come from the values of X in the
neighborhood of the maximum of xn P (x) which increases fast with the order n of the moment we
consider. Thus, increasing the order of the cumulants allows one to sample larger fluctuations of
asset prices. However, the question of the coherence of the cumulants as risk measures may arise.
For instance, If an investor assesses the fluctuation of his invested assets based on C4 = µ4 − 3.µ22 ,
he/she presents aversion to the large fluctuations as measured by µ4 whilst is attracted by the small
fluctuations as measured by µ2 . Indeed, C4 depicts an investor who tries to avoid the larger risks
but is willing to accept the smallest ones. Apparently, the question of the coherence of risk measures can only be answered by taking the whole ”tails” of pdf into consideration since the ”tails” of
pdf encompass all the higher risk dimensions pertaining to larger, extreme events associated with
assets. Nevertheless, the implemental question of the portfolio selection by minimizing the ”tails”
might arise when it deviates from the Gaussian.
To overcome caveats of the methodology of cumulants, the methodology of Value-at-Risk2 was introduced as a new distribution-based tool for measuring risk (see Linsmeier and Pearson (1996), Fallon (1996), and Danielsson and Vries (1997) for a comprehensive review of this method). Two basic
approaches to implement VaR have hitherto been proposed: variance-covariance and extreme value
methods. In terms of the variance-covariance method, the wealth gained from the invested portfolio
can be a nonlinear function of time horizon and state variables which can be approximated linearly
up to second or third order3 . Furthermore, the vector of state variables are assumed to be multivari
ate Gaussian, thus VaR is simply equal to 1.65 standard deviation of change in the portfolio wealth .
On the contrary, the extreme value method does not use very restrictive Gaussian assumption as
used in the variance-covariance method, it assumes three major extreme value distributions such
as Pareto, Gumbel, Weibull to capture the probabilistic law of extreme values of state variables
2
Classical VaR is defined as inf[r : P rob{X(ω) > r} = α], where α is a specified probability of losing more than
VaR. Obviously, the under-performance probability is defined as an inversion of VaR or inf −1 [r : P rob{X(ω) > r} =
α]
3
These approximations are referred to as the Delta and Gamma method respectively
4
instead. The natural idea behind the extreme value approach is that VaR estimation is highly
dependent on good predictions of uncommon events, or catastropic risk, since the VaR is computed
from the lowest portfolio returns. As a result, according to McNeil and Frey (2000), any statistical
method used for VaR has to have the prediction of tail events as its primary goal.
Many authors such as Giorgi (August 2002) or Sengupta (1985, chapter 3) have hitherto attempted to interpret/apply VaR or its inverse function (under-performance probability) as a risk
criterion to solve for the optimal investment. In their studies, either the portfolio of two assets or
the Gaussian distribution are assumed. Obviously the extension to either the portfolio of more than
two assets or the Non-Gaussian distribution is analytically very complicated. The large deviations
approach is an alternative, but efficient approach to the variance-covariance and extreme value
methods. Similar to the extreme value theory, the large deviations method is used to modelled tail
events occurring at the ”ends” of a distribution. Nonetheless, the large deviation method differs
from the extreme value theory in that it provides with an asymptotic, simple functional form of
the tail of a distribution (or a decay function). In other words, the decay function converges to the
tail probability function under some regular assumptions as the number of risky assets/the time
horizon is large enough.
However, the problem of formulation of the decay function may arise when investors have specific, short to long term valuations for their respective investment horizons. In this case, the optimal
asset allocations also depend on the time horizon of investments. Indeed, as the time scale increases,
the distribution of asset returns converges to the Gaussian distribution, so that only the variance
remains relevant for very long horizons, thus portfolio weights might be independent of the investment horizon. For instance, Stutzer (2003) proposes the long-term optimal asset allocation
using a power utility function of the logarithmic cumulative period wealth of two assets portfolio.
He proves that investor’s desire to minimize the probability of the logarithmic cumulative period
wealth under-performing a benchmark is equivalent to his/her to maximize a power utility function
when the investment horizon is infinite. The optimal asset allocations obtained by maximizing the
power utility function is dependent on both benchmark and relative risk aversion parameter which
5
depends on the set of investment opportunities. Pham (2003) extends Stutzer’s idea to a dynamic
benchmark. His analysis which is based on the assumption that both the dynamics of risky asset
prices and benchmark depend on the dynamics of an ergodic economic indicator process yields an
optimal portfolio which is dependent on both the initial value of the benchmark and an investment
opportunities set dependent relative risk aversion4 .
Our present approach uses a single time scale, over which the returns are computed, and thus is
restricted to portfolio selection with a fixed investment horizon. Nevertheless, our approach extends
Stutzer (2003) and Pham (2003)’s approaches in a sense that the portfolio contains a large number
of risky assets. Therefore, the optimal asset allocations are obviously dependent on both RRA and
the number of assets in the portfolio. The advantage of our approach is its possibility to analyze
explicitly the impact of skewness on risk taking behaviour of investors who hold a well-diversified
portfolio and expect all idiosyncratic risks to be eliminated5 as conjectured by Kane (1982). Kane
(1982) proves that investors require a higher risk premium for a negative skewness whilst a lower risk
premium for a positive skewness, thus, skewness preference helps to explain expected asset returns.
Just as in CAPM, the investor aversion to variance makes an asset return’s covariance with the
market return a risk factor, Friend and Westerfield (1980), Patton (2002) and Knight and Satchell
(2001, chapter 4) argue that investor preference for a positively skewed wealth distribution should
make market co-skewness or nonlinear correlation an additional factor, that should ”be negatively
related to an asset expected return”. Intuitively, The impact of skewness of the portfolio return or
the nonlinear correlations between assets, given the optimal asset allocation, is shown to increase,
thus the under-performance probability increases as the target rate increases.
This paper is organized as follows:
Section 2 presents a general theorem on an asymptotic approximation of the average logarithmic
probability of the return of the optimal multiple assets portfolio under-performing a fixed positive
benchmark r: limn−→∞
1
n
inf α∈A log P rob{Rnα (ω) ≤ r}. In other words, the theorem provides an
4
In this situation, the choice of relative risk aversion parameter (RRA) is very limited because it depends on the
investment opportunities set. High values of RRA might give rise to ”Equity premium puzzle” (see Siegel and Thaler
(1997) for a comprehensive review)
5
This is a basic principle underlying CAPM and APT
6
asymptotic approximation of the average logarithmic probability of the multiple assets portfolio
being ensured from losing more than a specified amount |r| when r is negative. This asymptotic
approximation of the minimal under-performance probability is proven to be equal to the minus
maximal decay/growth function of the portfolio return.
Section 3 applies the theorem presented in Section 2 to derive the asymptotic under-performance/loss
probability in two cases:
1. Assets are Gaussian distributed
2. Assets are Non-Gaussian and skewed distributed
Section 4 uses UK equity data to examine the impact of skewness on the risk taking/investing
behaviour of investors with a homogeneous exponential utility function. We also derive generalized
empirical efficient frontiers manifesting the trade-offs between target rates and risk or expected
returns.
Section 5 concludes the paper and proposes some directions for future research.
2.
General portfolio analysis: a large deviations theorem
Following Malevergne and Sornette (2002), the final gain/loss resulting from a fixed term investment
P
in a portfolio is Snα (ω) = W0 ni=1 αi Xi (ω), where W0 is the initial capital, αi is the proportion held
in asset i returning a gain/loss Xi (ω) in a state of the world ω, and n is the number of risky assets in
the portfolio. The common problem facing an investor is how to select proportions to invest (αi ) so as
to ensure that he will not lose more than a specified amount |r|, i.e, α = arg minα∈A P rob{Snα (ω) ≤
r}, where A is a set of feasible portfolios, r < 0 is a loss target6 . The use of P rob{Snα (ω) ≤ r}
as a risk criterion has supported by Bawa (June 1978) and Harlow and Rao (1989). This underperformance probability (or Roy’s First-Safety rule) is a special case of the general LPM proposed
6
The event {ω : Snα (ω) ≤ r} implies:
• If r > 0, the investor might gain less than a target r
• If r ≤ 0, the investor might lose more than a target |r|
7
by Harlow and Rao (1989): minα LP Ms (r; α) = minα
Rr
−∞
(r − α, X(ω))s dP (r) when s = 0. Roy’s
first safety rule is shown by Bawa (June 1978) to be consistent with stochastic dominance rules,
thus, it is equivalent to an expected utility maximization for a very special utility function as
shown in Bawa (June 1978, proposition 1). Following Stutzer (2002) who shows that Roy’s first
safety rule is equivalent to maximization of a power utility function of final cumulative wealth
P
WT = W0 exp{ Tt=1 log Rpt /T }T as T −→ ∞ where T is the time horizon of the investment, we
will derive an asymptotic equivalence of Roy’s first safety rule which is the maximization of an
exponential utility function of portfolio return defined as Snα (ω) when n −→ ∞.
1. Lower bound: if θ ∈ (∞, 0) then an application of Tchebyshev’s inequality yields
inf
α∈A
P rob{Snα (ω)
≤ r} = inf
α∈A
E1[−∞,r] (Snα (ω))
≤ − sup sup
α∈A θ∈[−∞,0]
E[exp{θSnα (ω)}]
−
exp{θr}
{z
}
|
=
exponential utility function
exp{− sup sup [θr − log{E[exp{θSnα (ω)}]}]} = exp[− sup sup Λ∗ (θ, α)],
α∈A θ∈[−∞,0]
(2.1)
α∈A θ∈[−∞,0]
where Λ(θ, α) = θr − log E exp[θSnα (ω)] denotes the deviation function (or decay function).
2. Upper bound: if θ ∈ (0, ∞) then an application of Tchebyshev’s inequality yields
E[exp{θSnα (ω)}]
=
exp{θr}
α∈A θ∈[0,∞]
sup P rob{Snα (ω) ≥ r} = sup E1[r,∞] (Snα (ω)) ≤ sup inf
α∈A
α∈A
exp{− inf sup [θr − log{E[exp{θSnα (ω)}]}]} = exp[− inf sup Λ∗ (θ, α)],
α∈A θ∈[0,∞]
α∈A θ∈[0,∞]
(2.2)
Risk management literature, for instance Linsmeier and Pearson (1996) often uses the lower tail
probability (or the under-performance probability) as defined in Equation 2.1 as a risk measure.
In addition, as shown in Equation 2.1, This is equivalent to the exponential utility maximization,
thus, consistent with Bawa (June 1978)’s LPM theory. Therefore, our analysis is hereafter based
on the under-performance probability in Equation 2.1 instead of the over-performance probability
in Equation 2.2
8
The exponential utility utilized in Equation 2.1 has positive third derivative, thus expresses
skewness preference. The exponential utility function might be better than the power utility function
used in Stutzer (2003)’s approach in the sense that investors utilizing the exponential utility function
do not invest as excessively in risky assets as the investors utilizing the power utility function.
Meanwhile, the power utility function often results in excessive risky asset holding, which gives rise
to the equity premiums which are higher than what the empirical evidences document. Thus, high
risk aversion parameters are needed to explain empirical evidences. This phenomenon is named by
Mehra and Prescott (1985) as the equity premium puzzle.
The absolute risk aversion parameter in Equation 2.1 is determined by maximization, thus
dependent on both investment opportunity set and target rates. As stated by Stutzer (2003):
...conventional portfolio theory assumes that the risk aversion parameter θ is a preference
parameter that is independent of the investment opportunity set. Investors could use different investment opportunity sets, either because of differential regulatory constraints,
such as hedge funds’s greater ability to short sell, or because of different opinions about
the parameters of portfolios’ log return processes...
In Stutzer (2003)’s paper, the relative degree of risk aversion (θ) must be set in [−1, 0] so that the
power utility function is increasing and concave. Meanwhile, in our approach, it is not necessary
to set the absolute degree of risk aversion (θ) in [−1, 0] because the exponential utility function is
concave, increasing for any θ ∈ [−∞, 0].
Since the optimal proportions to invest (α) is determined by maximization of the exponential
utility, thus dependent on the absolute risk aversion parameter θ, target rate r, and the number
of risky assets (n). Hence, Cramér’s theorem and Gärtner-Ellis’s theorem in Dembo and Zeitouni
(1998, page 19,45), and Bucklew (1990, page 15) utilized by Stutzer (2003) to approximate the
P
under-performance probability of long-term gross return T1 Tt=1 log Rαt , where Rα,t = αRt +(1−α)r
are not straight-forward to be applied7 .
7
Classical Gärtner-Ellis theorem
Pn is used to approximate the tail probability of the arithmetic average of random
variables limn−→∞ n1 log P rob{ i=1 Xi (ω)/n ∈ [a, b]}, where weights are equal to n1 which is obviously independent
of θ and (a,b)
9
Now, we state a theorem saying that the upper bounds in Equations 2.1 and 2.2 can be tightened
under some regular conditions.
Theorem 1. Given a set of n risky assets {Xi (ω)}ni=1 with returns as random elements in the
n
probability space (Ω, ⊗ni=1 Fi=1
, P(⊗ni=1 dXi (ω))).
Λn (nθ, α) = log
Z
Ω
exp{nθSnα (ω)}P(⊗ni=1 dXi (ω))
• AS1a
lim
n−→∞
1
sup Λn (nθ, α) = Λ(θ, r) exists,
n α∈A
′
′
and there exists an unique exposing point θ(r) ∈ [Λ (0, r), Λ (θ, r)] (where θ = {θ : Λ(θ, r) =
+∞}) to the exposed point of the hyperplane
∂Λ(θ,r)
∂θ
= r. Let us denote
1
inf [nθr − Λn (nθ, α)] = Λ∗ (θ, r)
n−→∞ n α∈A
lim
• AS1b
lim
n−→∞
1
inf Λn (nθ, α) = Λ(θ, r) exists,
n α∈A
′
′
and there exists an unique exposing point θ(r) ∈ [Λ (θ, r), Λ (0, r)] (where θ = {θ : Λ(θ, r) =
−∞}) to the exposed point of the hyperplane
∂Λ(θ,r)
∂θ
= r. Let us denote
1
sup[nθr − Λn (nθ, α)] = Λ∗ (θ, r),
n−→∞ n α∈A
lim
where the notations Λ(θ, r), Λ∗ (θ, r) which have been used in AS1a are used here for the sake
of parsimony.
• AS2 Set of feasible portfolio A(α) is a compact convex n-1 dimentional set symmetric around
10
the origin with meas(A) = 2.
• AS3 Set of portfolio return probability measures P(dSnα (ω)) is complete and separable.
If AS1a,AS2,AS3 hold, then
1
sup log P{Snα ≥ r} = −Λ∗ (θ, r)
n−→∞ n α∈A
lim
(2.3)
And if AS1b,AS2,AS3 hold, then
1
inf log P{Snα ≤ r} = −Λ∗ (θ, r),
n−→∞ n α∈A
lim
(2.4)
Proof. See Appendix A
Remark 2.1. Equation 2.4 implies an asymptotic equivalence between the average of logarithmic
minimum under-performance probability and the average of maximum decay rate. In another word,
assuming that all idiosyncratic risks inherent in risky assets are eliminated through diversification,
investor’s preference to maximize the expected utility is transformed into his/her desire to minimize VaR. Furthermore, by Assumption AS1b, target rate r must be chosen to be smaller than
limn−→∞ E[Snα (ω)] so that the exposing point θ(r) (or absolute risk aversion parameter) to the decay
function exists.
3.
Multivariate normal case
Stutzer (2003) proposes a time-varying portfolio choice which requires a proportion invested in a
single risky asset, whose price dynamics is a covariance-stationary geometric Brownian motion, with
the rest invested in a riskless asset with constant return r. He derives the decay rate function which
2
log S2α (ω)−log r
1 E
, where Covτ denotes its τ -lags covariance8 . The higher
is a squared Sharpe ratio 2 √Pτ =+∞
τ =−∞
Covτ
the target rate is the lower the under-performance probability is.
8
Given a portfolio, the decay rate is obviously a parabolic function of the target rate
11
The purpose of this section is to extend Stutzer (2003)’ approach to the multivariate normal
case. Let X(ω) denotes a vector of (n) risky assets whose multivariate distribution is normal.
An application of Theorem 1 yields a large deviations approximation of the average of minimum
logarithmic under-performance probability. Now, we are in position to state Corollary 2.
Corollary 2. Supposing that the multivariate return distribution is Gaussian (e.g Nn (µ, Σ)) then
Assumptions AS1b, AS2, AS3 as stated in Theorem 1 hold. We make some further assumptions
as follows
µT (Σ + ΣT )−1 1
−→ A
n−→∞
n
1T (Σ + ΣT )−1 1
lim
−→ B
n−→∞
n
1T (ΣT + Σ)−1 Σ(Σ + ΣT )−1 1
lim
−→ C
n−→∞
n
1T (ΣT + Σ)−1 Σ(Σ + ΣT )−1 µ
−→ D
lim
n−→∞
n
µT (ΣT + Σ)−1 µ
lim
−→ E
n−→∞
n
µT (Σ + ΣT )−1 Σ(Σ + ΣT )−1 µ
−→ F
lim
n−→∞
n
lim
(3.1)
then an asymptotic upper tail of the optimal portfolio return probability can be represented as
1
inf log P rob{Snα (ω) ≤ r} = θ∗ r − Λ(θ∗ )
n−→∞ n α∈A
lim
where θ∗ is defined in Equation B.4
Proof. See Appendix B
Remark 3.1. When X(ω) is multivariate normal, the optimal lower tail of the portfolio return can
be approximated with a decay function which as a close form. Given a dataset, the target rate r
∗
must be chosen so that θ∗ < 0 and r ≤ limn−→∞ E[Snα (ω)], where α∗ is the optimal portfolio. Thus,
the accuracy of the approximation of the lower tail probability is much dependent on the choice of
the target rate.
12
4.
Non-Gaussian case with skewness
The asymptotic link between the minimization of under-performance probability of the portfolio
return and the maximization of the exponential utility function with skewness preference characteristics, as defined in Equation 2.1, makes it interesting to study the impact of skewness on the
risk-taking behaviour of investors. The impact of skewness, thus, co-skewness or nonlinear correlations on the prediction of stock return which often results in higher order CAPM has been studied
extensively in the literature, for instance, Patton (2002), Kane (1982), and Friend and Westerfield
(1980).
An application of the third-order Taylor expansion yields the following approximation to the
logarithmic moment generating function
T
Λn (θ, α) = log E[exp(θα X(ω))] = θ
3
θ
E
6
X
n
i=1
n
X
i=1
2
n
θ2 X
αi [Xi − E[Xi ]] +
αi E[Xi ] + E
2
i=1
3
(1 − ǫ)3 4 ∂ 4 log E[exp(xαT X(ω))]
αi [Xi − E[Xi ]] +
θ
kx=ǫθ , where ǫ ∈ [0, 1]
6
∂x4
(4.1)
where the third term of Equation 4.1 is the third central moment of the portfolio return which
incorporates third central moments of individual assets and second order covariance between two
assets E[Xi − E[Xi ]][Xj − E[Xj ]]2 . This expansion can facilitate the study of the impact of skewnesses/nonlinear correlations of/between risky assets on the optimal investment since
it embeds the
E
skewness of portfolio return, where skewness of the portfolio return is defined as E
P
P
n
i=1
n
i=1
αi [Xi −E[Xi ]]
2 3/2 .
αi [Xi −E[Xi ]]
Moreover, The forth term of the expansion can be made as small as possible as θ is small enough.
Therefore, this results in the following approximation of the average of the minimum logarithmic
under-performance probability:
1
1
inf n log P rob{αT X(ω) ≤ r} = − lim
sup [nθr − Λn (nθ, α)]
n−→∞ n α∈R
n−→∞ n α∈Rn
T
lim
1 α=1
1T α=1
θ<0
13
3
(4.2)
The right hand side of Equation 4.2 can be calculated by using nonlinear optimization techniques
such as the Quasi-Newton or Nelder-Mead simplex method.
5.
Empirical evidences
[This section is in progress.]
In this empirical study, we use 5 years daily UK equities data (in the period from 1/1/1999 to
1/1/2004) retrieved from Datastream9 . We categorized UK firms in six industries: Aerospace&High
Tech, Banking service, Construction, Food stuffs, Information Technology, and Leisure&Hotels.
Each category includes 10 equities issued by 10 firms respectively, see Table 1 for details. For
brevity, we tend not to report descriptive statistics of data here. Further descriptive statistics of
data such as means, variances, covariances, autocorrelations, and skewnesses of individual equities
are available upon request.
Given 10 equities in each industry, we apply Theorem 1 and Corollary 2 to calculate optimal
proportions invested in 10 equities, the under-performance/loss probabilities of invested portfolios
for various specified amounts of loss for six above-mentioned industries. Given a loss target, for
α
instance r = −0.05, P rob{S10
(ω) ≤ r} is the likelihood that we may lose more than 0.05 per-
centage of our wealth in the next period, thus, our insurance strategy is to select α so as to make
α
P rob{S10
(ω) ≤ r} as small as possible. The results that we should expect to obtain are as follows:
• Investors who set high loss target for the minimization of his VaR should have high absolute
risk aversion degree. This is because the VaR increases as the loss target decreases, thus, if
the investor is more concerned about risk he should select a high loss target. In doing so, he
hope that the magnitude of his loss probability is small.
• Supposing that investors investing in 6 industries try to minimize their loss probabilities, given
a loss target, the trade-off between the mean returns and VaRs of optimal portfolios, invested
in 6 industries, is expected.
9
Datastream is a product of Thomson Inc which provides financial/economic data services
14
• In a specific industry, as the loss target varies, the trade-off between the mean returns and
VaRs of optimal portfolios is expected.
• As it has been mentioned that the maximization of our exponential utility function enables
us to incorporate the impact of skewness preference on risk taking/investing behaviour. The
presence of skewness presumably adds an additional source of risk to the optimal investment.
Given a medium loss target which implies a moderate risk aversion degree10 , the VaR in
the presence of positive skewness is expected to be higher than the VaR with zero skewness.
Higher VaR in the presence of positive skewness is a trade-off for the skewness preference
which might result in a higher mean return.
Let Rit =
Pit −Pit−1
,
Pit−1
where Pt is the equity price at time t, denotes the return of an individual asset
i at t. A strong assumption that returns are not autocorrelated11 makes it possible to estimate
mean, covariance, and skewness with their corresponding sample ones. More specifically, the mean,
variance, and the third central moment in Equation 4.2 can be estimated as follows:
n
T
n
X
1X X
αi Ri ] =
E[
Rit
αi
T
t=1
i=1
i=1
n
T
T
n
i2
X
1X
1 XhX
2
αi [Rit −
Rit ]
αi [Ri − E(Ri )]] =
E[
T t=1 i=1
T t=1
i=1
"
#
X
3
10
T
T
T
1
1X
1X
1X
T
T
E
αi Ri ≈
1 [ R −
Rt ] α α [ R −
Rt ] [ R −
Rt ] α
T ×10
T 1×T T ×10 T t=1 1×10 10×1
T t=1 1×10 T ×10 T t=1 1×10 10×1
i=1
In the Gaussian case, a use of Corollary 2 yields an explicit solution to the maximization of the
decay function, i.e, sup α∈Rn [nθr − Λn (nθ, α)]. Meanwhile, in the Non-Gaussian case, we must
1T α=1
θ<0
use the Nelder-Mead simplex optimization algorithm12 to maximize the decay function in Equation
4.2 for the absolute risk aversion degree and optimal proportions invested in 10 equities in each
industry. Simulation results are presented in Tables 2, and 3.
10
This implies that the investor does not have strong aspiration to reduce his exposure to risk
This assumption is rather strict. We refer to Stutzer (2003) for a comprehensive discussion
12
SAS system has this algorithm as a built-in function
11
15
Results in Tables 2, and 3 are illustrated in Figures 1, and 2 respectively. It is easy to notice that
the higher the loss target is as the lower the VaR becomes in all six industries13 . The lowest/highest
VaR are obtained when r = −0.05 and r = −0.001 respectively. In the Gaussian case, Figure 1 shows
a seemingly clear picture of the hypothesis that the industry which yields higher optimal return also
has higher exposure to risk. In this picture, IN (Information technology) yields minimum return as
well as minimum VaR whilst Banking (BA) yields maximum return as well as maximum VaR. The
industry that attracts much our attention is Construction (CU) since it yields an optimal return
which is higher than that of BA, but, a VaR which is much less than that of BA. Are equities
of CU best investments?. To answer this question, we need to investigate the Non-Gaussian case.
Figure 2 shows that Construction industry (CU) is obviously best to invest since it yields highest
mean return at a VaR which is lower than other industries. A possible explanation of this fact is
that the skewness of the optimal return of the portfolio of CU equities is highly positive, thus, the
inclusion of skewness into the exponential utility function embedding the skewness preference results
in lower optimal VaR. Both Banking (BA) and Information Technology (IN) are still not attractive
to invest. The reason is that over the last five years, several economic recessions have brought about
closures of internet/e-commerce firms and mergers of banks due to their poor performance. This
phenomenon is sometimes called as the bubble in Banking/IN industries so that investors viewed
BA and IN as risky industries to invest.
Since Banking industry is not attractive to invest whilst Construction industry is very attractive
to invest, thus, we now study in depth the Banking (BA) and Construction (CU) data. We can
observe very clearly the trade-off between optimal VaRs and mean returns or loss targets in Figures
3 - 8 or Figures 10 - 15 in both Gaussian and Non-Gaussian cases. Additionally, Figure 9 shows a
negligible impact of skewnesses/nonlinear correlations of/between Banking equities on VaRs. This
implies the symmetry of the minimal probability of portfolio return (as we can see in Table 2, the
skewnesses of optimal portfolios are negligible), which explain its unusual high VaR. On the contrary,
Figure 16 illustrates the hypothesis that for medium loss targets, the optimal VaR of the portfolio
13
This is obvious since agents buy more insurance when they are more concerned about high VaR. Thus, they
expect to reduce VaR by buying more insurance
16
of CU equities in the presence of positive skewness is higher than the optimal VaR in the Gaussian
case. The discrepancy between VaRs in the Non-Gaussian and Gaussian cases, respectively is the
additional source of risk for skewness preference. Moreover, VaR in the Non-Gaussian case is more
sensitive to a change of r than in the Gaussian case.
6.
Conclusion
A large deviations theorem states that an investor desiring to minimizing the probability of losing
more than a specified amount should choose a portfolio that maximizes a endogenous expected
exponential utility function. Our result actually generalizes Stutzer (2003) to the multiple asset case
with fixed term investment, which facilitates the study of the impact of the dependence structure
on the portfolio choice and VaR.
UK equities data has been used to calibrate our theoretical results. It has been shown that our
theory helps to rank industries in term of their attractiveness to be invested. Empirical analysis
has confirmed that our method works, more specifically, the trade-off between the optimal VaR and
the optimal portfolio return or the loss target, and the illuminating impact of the skewness on the
optimal investment/VaR.
Future research might be focussed on the behavioural equivalence between VaR or other risk
measures and the myopic loss averse utility function, e.g Siegel and Thaler (1997). It would be
more interesting to extend our theory to incorporate the dynamics of asset prices14 . We believe
that those further generalization of our results would help to predict VaR/optimal investment, thus
create hedging strategies in different maturities.
14
The case of two assets has been handled by Pham (2003)
17
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20
7.
Appendix
A.
Proof of Theorem 1
We are now going to prove Equation 2.3, the proof of Equation 2.4 can be formulated in the same
way.
Proof. STEP 1: upper bound
By the equivalence of two events: {ω : Snα (ω) ≥ r} ∼ {ω : exp{θSnα (ω)} ≥ exp{θr}}, where
P
Snα (ω) = ni=1 αi Xi (ω) is the return obtained from investing in n risky assets we obtain
P rob{Snα (ω) ≥ r} = EP(⊗ni=1 dXi (ω)) 1[ω : Snα (ω) ≥ r] ≤
EP(⊗ni=1 dXi (ω)) [exp{nθSnα (ω)}]
exp{nθr}
∀θ ∈ [0, θ] (A.1)
(this is simply an application of Tchebyshev’s inequality). After taking the logarithm on both sides
of Equation A.1, we then obtain
1
1
log P rob{Snα (ω) ≥ r} ≤ −(θr − log{EP(⊗ni=1 dXi (ω)) [exp{nθSnα (ω)}]}),
n
n
(A.2)
i.e
lim sup
n−→∞
1
1
log P rob{Snα (ω) ≥ r} ≤ − lim sup(θr − log{EP(⊗ni=1 dXi (ω)) [exp{nθSnα (ω)}]})
n
n
n−→∞
(A.3)
From AS2 and Weierstrass’s theorem, which states that any function with its compact domain
always has a finite extreme values, we can deduce
lim sup
n−→∞
1
1
sup log P rob{Snα (ω) ≥ r} ≤ − lim sup inf (nθr − log{EP(⊗ni=1 dXi (ω)) [exp{nθSnα (ω)}]})
n α∈A
n−→∞ n α∈A
(A.4)
21
(since sup −f = − inf f )
STEP 2: lower bound
Now, we are going to show that
lim inf
n−→∞
1
1
∗
sup log P rob{Snα (ω) ≥ r} ≥ − lim inf inf nθr − log{EP(⊗ni=1 dXi (ω)) [exp{nθSnα (ω)}]}
n−→∞ n α∈A
n α∈A
Here we imitate the proof of Gärtner-Ellis theorem by defining another conjugate measure on the
n
joint probability space (Ω, ⊗ni=1 Fi=1
, P(⊗ni=1 dXi (ω))) as follows
P(⊗ni=1 dXi ) P(⊗ni=1 dXi ) Gn
∗
∗
= exp{nθSnα (ω) − log EP(⊗ni=1 dXi ) exp{nθSnα (ω)}},
(A.5)
(where G is smallest σ-algebra contained in the product σ-algebra ⊗n1 Fi ). Whence,
∗
∗
∗
1(ω : Snα (ω) ≥ r)P(⊗n1 dXi (ω)) = 1(ω : Snα (ω) ≥ r) exp{log EP(⊗n1 dXi ) exp{nθSnα (ω)}−
∗
nθSnα (ω)}P(⊗ni=1 dXi ),
(A.6)
where α∗ (θ) is the optimal portfolio found in Equation A.4, i.e,
∗
α∗ (θ) = arg inf [nθr − log EP(⊗ni=1 dXi ) exp{nθSnα (ω)}]
α∈A
We hereafter omit the suffix (∗ ) of α∗ for brevity.
Taking the expectation of Equation A.6, we obtain
P rob{Snα (ω)
≥ r} =
Z
Ω
1(Snα (ω) ≥ r) exp{log EP(⊗n1 dXi (ω)) exp{nθSnα (ω)} − nθSnα (ω)}
P(⊗n1 dXi (ω))
(A.7)
Let us perturb the event {ω : Snα ≥ r} by an infinitesimal value ǫ, thus, we obtain the following
22
equation:
P rob{Snα ≥ r + ǫ} ≥ P rob{r − ǫ ≤ Snα (ω) ≤ r + ǫ} ≥ exp{log EP(⊗n1 dXi (ω)) exp{nθSnα (ω)}
−nθSnα (ω)}P(r − ǫ ≤ Snα ≤ r + ǫ) ⇐⇒
lim inf
n−→∞
1
1
log P rob{Snα (ω) ≥ r + ǫ} ≥ − lim inf {nθr − log EP(⊗n1 dXi (ω)) exp{nθSnα }}+
n−→∞ n
n
1
(A.8)
lim inf P(r − ǫ ≤ Snα (ω) ≤ r + ǫ) − θǫ
n−→∞ n
To complete the proof, we need to show that
1
log Pn (Aǫ (ω)) = 0
n
lim inf
n−→∞
(A.9)
where Aǫ (ω) = {ω : r − ǫ ≤ Snα (ω) ≤ r + ǫ}.
Firstly, we need to show that:
lim sup
n−→∞
1
(1)
(2)
log Pn (Acǫ (ω)) < 0 =⇒ lim Pn {Aǫ (ω)} = 1 =⇒
n−→∞
n
1
lim inf log Pn (Aǫ (ω)) = 0.
n−→∞ n
(A.10)
(A.11)
Equation A.10 is obvious. In order to show Equation A.11, we first note that lim inf n−→∞
1
n
log Pn (Aǫ (ω)) ≤
0 is obvious. By Assumption AS3, the set of Pn is separable and complete, thus, its conjugate
measures Pn are also separable and complete. By Portmanteau theorem (e.g see Billingsley (1999)
for the definition), we have lim inf n−→∞
1
n
log Pn (Aǫ (ω)) =
Equation A.11 follows.
Secondly, we show that lim supn−→∞
1
n
log Pn (Acǫ (ω)) ≤ 0.
23
log lim inf n−→∞ Pn (Aǫ (ω))
∞
≥ 0. Therefore,
An application of Tchebyshev’s inequality yields the following equation
Pn {Snα (ω) ≥ r + ǫ} ≤ exp{− nλ∗ (r + ǫ) − log EP(⊗dXi (ω)) exp(nλ∗ Snǫ ) }
⇐⇒ lim sup
n−→∞
1
1
log P(Snα (ω) ≥ r + ǫ) ≤ −(r + ǫ)λ∗ − lim sup log EP(⊗n1 dXi ) exp{nλ∗ Snα (ω)}
n
n−→∞ n
e n (λ∗ , α(r, θ(r)))], with λ∗ ∈ [0, λ − θ(r)]
= −[(r + ǫ)λ∗ − lim sup Λ
(A.12)
n−→∞
e n (λ∗ , α(r, θ(r))) has the obvious meaning and θ(r) is an unique exposing point to the
where Λ
exposed point r of the hyperplane
∂Λ(θ,r)
∂θ
= r by Assumption AS1a.
On the other hand, by Assumption AS1, we can deduce
lim sup Λn (θ, α(r, θ)) = lim Λn (θ, α(r, θ)) = Λ(θ, r)
(A.13)
e n (λ∗ , α(r, θ)) = Λn (λ∗ + θ(r), α(r, θ)) − Λn (θ(r), α(r, θ))
Λ
(A.14)
n−→∞
n−→∞
By Equations A.13 and A.14, Equation A.12 is equivalent to
lim sup
n−→∞
1
∗
log Pn {Snα (ω) ≥ r + ǫ} ≤ −Λ∗ (λ∗ (r + ǫ), r) + Λ∗ (θ(r), r) + θ(r)ǫ
n
(A.15)
where Λ∗ is the Fenchel-Legendre transform of Λ


 rθ(r) − Λ(θ(r), r) if Λ′ (0) < θ(r) < Λ′ (θ)
∗
Λ (θ(r), r) = sup {θr − Λ(θ, r)} =

 0 if otherwise
θ∈(0,θ)
24
(A.16)
and
Λ∗ (λ∗ (r + ǫ), r) =
sup
{λ(r + ǫ) − Λ(λ + θ(r), r)}
λ∈(0,λ−θ(r+ǫ))

′
′


(r + ǫ)λ∗ (r + ǫ) − Λ(λ∗ (r + ǫ) + θ(r), r) if Λ (0) < λ∗ (r + ǫ) + θ(r) < Λ (θ)



=
(or λ∗ (r + ǫ) ∈ [0, θ − θ(r)])




 0 if otherwise
(A.17)
Similarly, we can obtain
lim sup
n−→∞
1
log P{Snr (ω) < r − ǫ} ≤ −Λ∗ (ζ ∗ (r − ǫ)) + Λ∗ (θ(r)) − θ(r)ǫ
n
(A.18)
def
where ζ ∗ (r − ǫ) ∈ (θ − θ(r), 0) and θ(r) ∈ (θ, 0) with θ = {θ : Λ(θ, r) = −∞}.
From Equations A.15 and A.18, we obtain
lim sup
n−→∞
[ ∗
1
1
∗
∗
log Pn {Snα (ω) > r + ǫ Snα (ω) < r − ǫ} ≤ lim sup log Pn {Snα (ω) > r + ǫ}
n
n−→∞ n
1
∗
+ lim sup Pn {Snα (ω) < r − ǫ} ≤ max{−Λ∗ (λ∗ (r + ǫ), r) + Λ∗ (θ(r), r)
n−→∞ n
+ θ(r)ǫ, −Λ∗ (ζ ∗ (r − ǫ)) + Λ∗ (θ(r)) − θ(r)ǫ} (A.19)
To complete the proof, we need to show that the right-hand of Equation A.19 is less than zero.
By Definition 2.3.3, Dembo and Zeitouni (1998, page 44)
θr − Λ∗ (r, θ) ≥ θx − Λ∗ (x, θ)
(A.20)
for any x 6= r and r is a unique exposed point to the hyperplane specified by Assumption AS1a.
Now let us set x = r + ǫ, then from Equation A.20 we can derive that
−Λ∗ (λ∗ (r + ǫ), r) + Λ∗ (θ(r), r) + θ(r)ǫ ≤ 0
25
(A.21)
By a similar argument, we obtain
−Λ∗ (ζ ∗ (r − ǫ)) + Λ∗ (θ(r)) − θ(r)ǫ ≤ 0
(A.22)
Hereafter, we have proved that
lim sup
n−→∞
1
log Pn {Acǫ (ω)} ≤ 0
n
In addition, Assumption AS1 allows us to apply Lebesgue’s dominated convergence theorem by
sending the perturbing term ǫ to zero. Therefore, the lower bound has been proved.
Again by Assumption AS1, we obtain
lim sup
n−→∞
lim inf
n−→∞
1
inf (nθr − log{EP(⊗ni=1 dXi (ω)) [exp{nθSnα (ω)}]}) =
n α∈A
1
inf (nθr − log{EP(⊗ni=1 dXi (ω)) [exp{nθSnα (ω)}]}) = Λ∗ (θ, r)
n α∈A
Therefore, we have showed Equation 2.3.
B.
Proof of Corollary 2
Proof. Let X ∼ Nn (µ, Σ) where |Σ| =
6 0, Assumption AS3 in Theorem 1 holds. Then
n×1
P df (X) = (2π)−n/2 |Σ|−1/2 exp
(
)
1
− (X − µ)T Σ−1 (X − µ)
2
1
Λn (θ, α) = E exp{θSnα } = E exp{θαT X} = θ[αT µ + θαT Σα]
2
Since
∂ 2 Λn (θ,α)
∂α2
=
θ2
(Σ + ΣT )
2
(B.1)
is positive definite, α∗ = arg min Λn (θ, α) is a solution to the following
minimization problem
min Λn (θ, α) s.t 1T α = 1,
α∈Rn
26
i.e, the first part of Assumption AS1b in Theorem 1 holds while Assumption AS2 holds trivially.
Whereby, Lagrangian method yields
α = 2θ−2 (Σ + ΣT )−1 (λ1 − θµ)
"
#
1
1 2
where λ = T
θ + θ1T (Σ + ΣT )−1 µ
1 (Σ + ΣT )−1 1 2
(B.2)
Then we obtains
"
infn Λn (θ, α) = 2
α∈R
1T α=1
1 1T (ΣT + Σ)−1 µ µT (Σ + ΣT )−1 11T (ΣT + Σ)−1 µ
+
θ
2 1T (Σ + ΣT )−1 1
1T (Σ + ΣT )−1 1
1 1T (ΣT + Σ)−1 Σ(Σ + ΣT )−1 1
+
−µT (ΣT + Σ)−1 µ + θ2
4
[1T (Σ + ΣT )−1 1]2
1 1T (Σ + ΣT )−1 Σ(Σ + ΣT )−1 11T (Σ + ΣT )−1 µ 1 1T (ΣT + Σ)−1 Σ(Σ + ΣT )−1 µ
− θ
+
θ
2
2
1T (Σ + ΣT )−1 1
1T (Σ + ΣT )−1 1
1 µT (Σ + ΣT )−1 11T (ΣT + Σ)−1 Σ(Σ + ΣT )−1 1
+
θ
2
[1T (Σ + ΣT )−1 1]2
µT (Σ + ΣT )−1 11T (Σ + ΣT )−1 Σ(Σ + ΣT )−1 11T (Σ + ΣT )−1 µ
−
[1T (Σ + ΣT )−1 1]2
µT (Σ + ΣT )−1 11T (ΣT + Σ)−1 Σ(Σ + ΣT )−1 µ 1 µT (ΣT + Σ)−1 Σ(Σ + ΣT )−1 1
− θ
−
2
1T (Σ + ΣT )−1 1
1T (Σ + ΣT )−1 1
#
µT (ΣT + Σ)−1 Σ(Σ + ΣT )−1 11T (Σ + ΣT )µ
+ µT (ΣT + Σ)−1 Σ(Σ + Σ)−1 µ
1T (Σ + ΣT )−1 1
A use of assumption 3.1 yields the following limit
#
"
AC
A2 A2 C
1 2C 1 A
D
AD
1
inf Λn (nθ, α) = 2 θ 2 + θ( +2 2 −2 )+
+ 2 −2
+F −E
Λ(θ) = lim
n−→∞ n α∈Rn
4 B
2 B
B
B
B
B
B
T
1 α
(B.3)
Exposing point θ∗ which is the solution to
θ∗ =
2
"
∂Λ(θ)
∂θ
B
r−
C
= r is given by
A
AC
D
+2 2 −2
B
B
B
27
#
,
(B.4)
i.e, the second part of Assumption AS1b in Theorem 1 holds. Whereby, we obtain
1
inf log P rob{Snα ≤ r} = −(θ∗ r − Λ(θ∗ ))
n−→∞ n α∈Rn
T
lim
1 α=1
where Snα = αT X.
28
(B.5)
Table 1: List of firms categorized by industries
Industries
Aerospace&High tech (AH)
ARMOURGROUP
Banking (BA)
ABBEY NATIONAL
Construction (CU)
AGGREGATE INDUSTRIES
Foods (FO)
BIG FOOD GROUP
Information Technology (IN) ADVANCED TECHNOLOGY SUSP
Leisure and Hotels (LH)
CROWN SPORTS
29
HROWEN
EGG
ENSOR HOLDINGS
MORRISON(WM)SPMKTS
CONCURRENT TECHNOLOGIES
HOLIDAYBREAK
Firms
AVONRUBBER
ALLIANCE&LEICESTER
BELLWAY
COFFEE REPUBLIC
ARM HOLDINGS
HARDYS&HANSONS
Firms (to be continued)
LOOKERS
MID-STATES
HBOS
HSBC
GIBBS&DANDY
JOHNSTON GROUP
SAINSBURY (J)
SHOPRITE GP
FAYREWOOD
IMAGINATION TECHNOLOGIES
MANCHESTER UTD
RANK GROUP
Firms (to be continued)
ROLLS-ROYCEGROUP
VARDYREG
WILSHAW
NORTHERN ROCK
RYL.BK.OF SCTL
STD.CHARTERED
MORGAN SINDALL
RMC GROUP
TRAVIS PERKINS
TESCO
THORNTONS
WHITTARD OF CHELSEA
NETWORK TECHNOLOGY
PSION
SURFACE TECH.SYS
SPORTECH
WEMBLEY GP
YATES GROUP
GKN
BARCLAYS
BOVIS HOMES GROUP
GREGGS
CML MICROSYSTEMS
HILTON GROUP
PENDRAGON
LLOYDS TSB GP
KIER GROUP
SOMERFIELD
INTELEK
RESTAURANT GROUP
30
Table 2: Simulation results in the Gaussian case
r
Industries Risk aversion
α1
AH
-89.68864
0.0425835
BA
-23.70514
-0.062308
-0.05
CU
-126.0616
0.0400909
FO
-102.833
0.0357436
IN
-206.0717
0.4752185
LH
-120.8293
0.0096838
AH
-53.98941
0.0425392
BA
-14.25771
-0.135574
-0.03
CU
-76.17352
0.0407873
FO
-61.96095
0.0357924
IN
-123.8442
0.4914924
LH
-72.78555
0.0061952
AH
-36.13979
0.0424842
BA
-9.533987
-0.226658
-0.02
CU
-51.22949
0.041644
FO
-41.52494
0.0358527
IN
-82.7304
0.5117606
LH
-48.76369
0.001873
AH
-18.29017
0.0423219
BA
-4.810268
-0.496633
-0.01
CU
-26.28546
0.0441268
FO
-21.08893
0.0360302
IN
-41.61661
0.5720754
LH
-24.74183
-0.010842
AH
-2.225516
0.0399503
BA
-0.558922
-4.641289
-0.001
CU
-3.835832
0.0739699
FO
-2.696519
0.0384892
IN
-4.614208
1.5453475
LH
-3.122153
-0.189573
α2
0.0747094
0.3753904
0.0320837
0.0157913
0.224751
0.3044837
0.0646021
0.4324507
0.0300542
0.0159742
0.2234139
0.2997406
0.0520603
0.5033877
0.0275572
0.0162005
0.2217485
0.2938643
0.0150393
0.7136459
0.020321
0.0168657
0.2167925
0.2765773
-0.526021
3.9415391
-0.066658
0.0260841
0.136821
0.0335782
α3
0.0759304
-0.106226
0.0891923
0.5013352
0.0001919
0.0218956
0.0810562
-0.126395
0.0887323
0.5151735
0.0013488
0.0219387
0.0874165
-0.151469
0.0881663
0.5323082
0.0027897
0.0219921
0.106191
-0.225788
0.0865261
0.5826513
0.0070776
0.0221492
0.3805801
-1.366741
0.0668113
1.2803826
0.0762691
0.024358
α4
0.0597171
0.1227179
0.0376225
0.0687949
0.00098
0.2233119
0.0571805
0.0947398
0.0392317
0.073777
-0.000241
0.2277236
0.054033
0.0599575
0.0412117
0.0799459
-0.001761
0.2331893
0.044742
-0.043137
0.0469495
0.0980705
-0.006286
0.2492685
-0.091045
-1.625856
0.1159176
0.3492687
-0.079301
0.4752892
α5
0.095642
-0.022377
0.2724555
0.0471043
0.0017204
0.0713241
0.1012868
-0.020142
0.2933293
0.0384175
0.0008171
0.0753138
0.1082912
-0.017364
0.3190115
0.0276614
-0.000308
0.0802568
0.1289671
-0.009129
0.3934368
-0.003941
-0.003655
0.094798
0.4311441
0.1172988
1.2880266
-0.44193
-0.057674
0.2992
α6
0.0795974
0.2057034
0.1870443
0.0756734
0.0170291
0.0424469
0.0785331
0.1674844
0.1727331
0.0718505
0.0179619
0.0390864
0.0772125
0.1199708
0.1551251
0.067117
0.0191237
0.0349229
0.0733142
-0.02086
0.1040984
0.0532097
0.0225809
0.0226748
0.0163404
-2.182904
-0.509241
-0.139539
0.0783691
-0.149494
α7
0.1639596
-0.051694
0.1404173
0.0245336
-0.000476
0.0529117
0.1550418
-0.136819
0.1443526
0.0237271
-0.000827
0.0473445
0.1439761
-0.242646
0.1491945
0.0227284
-0.001264
0.0404472
0.111312
-0.556318
0.1632258
0.0197942
-0.002563
0.0201568
-0.366072
-5.371828
0.3318821
-0.020872
-0.023533
-0.265061
31
α8
0.0488782
0.4159247
0.0740377
0.0658551
0.2540302
0.0062457
0.0473931
0.4999731
0.0696595
0.0720906
0.2384694
0.0088673
0.0455505
0.6044613
0.0642727
0.0798113
0.2190894
0.0121152
0.0401112
0.9141659
0.0486621
0.1024956
0.1614177
0.0216699
-0.039384
5.6687642
-0.138976
0.4168887
-0.769205
0.1559785
α9
0.2492068
0.0575099
-0.000114
0.1288659
0.0031109
0.2323445
0.2607467
0.1547432
-0.008594
0.1174849
0.0026557
0.2444873
0.2750661
0.2756228
-0.019027
0.1033928
0.0020887
0.2595315
0.3173345
0.6339117
-0.049263
0.0619891
0.0004013
0.3037883
0.935085
6.1343781
-0.412694
-0.511846
-0.026826
0.9258954
α10
0.1097758
0.0653585
0.1271693
0.0363027
0.0234441
0.0353521
0.1116206
0.0695391
0.1297137
0.0357125
0.0249084
0.0293026
0.1139097
0.0747364
0.1328442
0.0349817
0.0267321
0.0218077
0.1206668
0.0901413
0.1419162
0.0328344
0.0321591
-0.000241
0.2194215
0.3266376
0.2509614
0.0030748
0.1197319
-0.310171
to be continued from Table 2
Portfolio return mean Portfolio return variance
0.0002634
0.0000562
0.000638
0.0002309
0.0005803
0.0000404
0.0003424
0.0000492
0.000149
0.0000245
0.0003383
0.0000419
0.0002744
0.0000565
0.0009393
0.0002647
0.0006082
0.000041
0.0003575
0.0000495
0.0001667
0.0000247
0.0003637
0.0000425
0.000288
0.0000572
0.0013139
0.0003303
0.0006425
0.0000421
0.0003762
0.0000503
0.0001888
0.0000251
0.0003952
0.0000436
0.0003281
0.0000605
0.0024243
0.0006776
0.0007418
0.0000479
0.0004312
0.0000542
0.0002545
0.0000275
0.0004879
0.0000492
0.0009149
0.0003562
0.0194708
0.0347203
0.0019358
0.0004046
0.0011929
0.0003728
0.0013149
0.0002828
0.0017912
0.0005193
α
P rob{S10
(ω) ≤ r}
1.64E-10
0.002597
1.46E-14
5.80E-12
3.72E-23
6.33E-14
0.0002842
0.1156592
8.86E-06
0.0000832
7.91E-09
0.0000162
0.02575
0.3800232
0.0051778
0.0146985
0.000242
0.0070709
0.391479
0.7785591
0.2496656
0.3364476
0.1213555
0.279013
0.9854888
0.9913409
0.9683375
0.981211
0.9717621
0.977636
32
Table 3: Simulation results in the Non-Gaussian
r
Industries Risk aversion
α1
AH
-22.18826
0.04498
BA
-19.38204
0.0349798
-0.05
CU
-24.77758
0.0420258
FO
-19.29847
0.0355648
IN
-14.76588
0.4771895
LH
-16.09262
0.0109281
AH
-18.39635
0.0404471
BA
-13.60896
0.0291983
-0.03
CU
-20.48225
0.0395649
FO
-16.19588
0.03447
IN
-12.82932
0.4981784
LH
-14.26498
0.0097314
AH
-18.39635
0.0404471
BA
-13.60896
0.0291983
-0.02
CU
-20.48225
0.0395649
FO
-16.19588
0.03447
IN
-12.82932
0.4981784
LH
-14.26498
0.0097314
AH
-11.24435
0.0423947
BA
-4.814226
-0.006185
-0.01
CU
-13.7861
0.0400021
FO
-11.3409
0.0350305
IN
-10.74216
0.4972038
LH
-10.71849
0.0090248
AH
-2.220753
0.0423398
BA
-0.667565
-0.282708
-0.001
CU
-6.840188
0.042657
FO
-7.39602
0.0357566
IN
-9.442187
0.5041507
LH
-7.993731
0.0070192
case
α2
0.084072
0.3007861
0.0354731
0.0137218
0.223345
0.2996543
0.0802267
0.3036723
0.0317874
0.0132326
0.2212311
0.3077229
0.0802267
0.3036723
0.0317874
0.0132326
0.2212311
0.3077229
0.0782063
0.3316497
0.0325715
0.016434
0.2228757
0.3052644
0.0282474
0.4935213
0.0304595
0.015982
0.2224747
0.3006462
α3
0.0717153
-0.079802
0.0827624
0.4905186
0.0028436
0.0226374
0.0734158
-0.080853
0.0901516
0.4985856
0.0012844
0.0209039
0.0734158
-0.080853
0.0901516
0.4985856
0.0012844
0.0209039
0.0735076
-0.090747
0.0869057
0.4969533
0.0017104
0.0220372
0.0994526
-0.122346
0.0844098
0.5068215
0.0021693
0.0220229
α4
0.0639175
0.1600993
0.0332278
0.069469
0.0000954
0.2270167
0.060997
0.1572449
0.0377236
0.0666701
0.0012878
0.226728
0.060997
0.1572449
0.0377236
0.0666701
0.0012878
0.226728
0.0624149
0.1441764
0.0382546
0.0687741
-0.000182
0.2242679
0.048117
0.0358316
0.0374255
0.0723068
-0.001184
0.2262773
α5
0.0894372
-0.025469
0.2505551
0.0518248
-0.000292
0.0678125
0.0917688
-0.025265
0.2601095
0.0526556
0.0012052
0.0704
0.0917688
-0.025265
0.2601095
0.0526556
0.0012052
0.0704
0.0930857
-0.024102
0.2700462
0.0485871
0.0001823
0.0715698
0.1215875
0.0019734
0.3040033
0.0428858
0.0001318
0.0743161
α6
0.0799228
0.2560689
0.2002877
0.0799129
0.018438
0.0456786
0.0823838
0.2536696
0.1945663
0.0776489
0.017034
0.0425522
0.0823838
0.2536696
0.1945663
0.0776489
0.017034
0.0425522
0.0814081
0.2349489
0.1878724
0.0785809
0.0180704
0.0420906
0.0746695
0.0856686
0.1666117
0.0728006
0.0187372
0.0407595
α7
0.1758205
0.0602288
0.1359669
0.0240594
-0.001939
0.0536463
0.1739538
0.0545849
0.1379239
0.0260954
-0.001135
0.0536063
0.1739538
0.0545849
0.1379239
0.0260954
-0.001135
0.0536063
0.163595
0.0135668
0.1413901
0.0241694
-0.001486
0.0522017
0.1230881
-0.294128
0.1474465
0.0232826
-0.001193
0.0486332
33
α8
0.0482203
0.3039389
0.0758598
0.0623172
0.2504713
0.0043262
0.0495247
0.311548
0.075948
0.0635221
0.232607
0.0059442
0.0502007
0.3206863
0.074995
0.0629924
0.2382777
0.0052325
0.0485317
0.3514909
0.0748948
0.0654932
0.2331376
0.0069239
0.0420615
0.6674101
0.0703957
0.0696442
0.2263471
0.00829
α9
0.2352061
-0.071122
0.0115253
0.1357864
0.0025424
0.2279932
0.2384611
-0.063502
0.0070479
0.1303571
0.0043494
0.2290867
0.2425807
-0.052715
0.0032814
0.1330948
0.0004206
0.2344434
0.2468095
-0.016976
0.0003502
0.1305162
0.0025517
0.2325853
0.3022002
0.3395812
-0.01197
0.1227981
0.0022898
0.2413914
α10
0.1067083
0.0602912
0.132316
0.0368251
0.0211301
0.0403065
0.1088212
0.0597027
0.1251767
0.0367626
0.0239574
0.0333244
0.1056613
0.0606307
0.1244322
0.0371319
0.0256989
0.0348505
0.1100464
0.0621759
0.1277125
0.0354613
0.025936
0.0340344
0.1182364
0.0751961
0.1285614
0.0377219
0.026077
0.0306442
to be continued from Table 3
Portfolio return mean Portfolio return variance
0.0002517
0.0000561
0.0002388
0.000212
0.0005493
0.0000402
0.0003334
0.000049
0.0001587
0.0000243
0.0003251
0.0000418
0.0002556
0.0000561
0.0002626
0.0002123
0.0005631
0.0000402
0.0003371
0.0000491
0.0001673
0.0000247
0.000337
0.0000419
0.0002566
0.0000561
0.0002965
0.0002129
0.0005697
0.0000403
0.0003367
0.0000491
0.0001691
0.0000247
0.0003335
0.0000419
0.0002599
0.0000562
0.000407
0.0002163
0.0005784
0.0000404
0.0003393
0.0000491
0.0001723
0.0000248
0.0003426
0.000042
0.0003137
0.0000569
0.0015089
0.0003758
0.0006193
0.0000413
0.0003504
0.0000493
0.0001804
0.0000249
0.0003575
0.0000423
Portfolio return skewness
2.29E-14
7.45E-15
7.37E-16
-2.47E-14
1.01E-14
1.89E-14
2.26E-14
7.17E-15
1.26E-15
-2.44E-14
9.28E-15
1.89E-14
2.26E-14
6.77E-15
1.46E-15
-2.46E-14
9.80E-15
1.84E-14
2.21E-14
5.45E-15
1.77E-15
-2.48E-14
9.46E-15
1.89E-14
2.15E-14
-5.63E-15
3.03E-15
-2.51E-14
9.54E-15
1.87E-14
α
P rob{S10
(ω) ≤ r}
0.0000571
0.0031651
0.0000125
0.0001507
0.0007918
0.0005222
0.0098839
0.1161759
0.0044427
0.0139882
0.0255583
0.0202237
0.0872214
0.3800244
0.0458015
0.09096
0.1083842
0.1054071
0.4499025
0.7785593
0.3414889
0.4245275
0.3868374
0.4201405
0.9854889
0.9916604
0.9860524
1.0357037
0.9997037
1.027143
P rob{S10
34
α(r)
≤ r}
Figure 1: The trade-off between mean returns and VaRs of different industries in Gaussian case
α(r)
E[S10 (ω)]
α(r)
P rob{S10
35
≤ r}
Figure 2: The trade-off between mean returns and VaRs of different industries in Non-Gaussian case
α(r)
E[S10 (ω)]
P rob{S10
α(r)
≤ r}
Figure 3: Banking sector (BA): the trade-off between mean return and VaR in the Gaussian case
α(r)
E[S10 (ω)]
P rob{S10
α(r)
≤ r}
Figure 4: Banking sector (BA): the trade-off between target rate and VaR in the Gaussian case
r
36
P rob{S10
α(r )
≤ r}
Figure 5: Banking sector (BA): 3D plot of target rate, mean return, VaR in the Gaussian case
E [S α
10 (ω
)]
r
P rob{S10
α(r)
≤ r}
Figure 6: Banking sector (BA): the trade-off between mean return and VaR in the Non-Gaussian
case
α(r)
E[S10 (ω)]
37
P rob{S10
α(r)
≤ r}
Figure 7: Banking sector (BA): the trade-off between target rate and VaR in the Non-Gaussian
case
r
P rob{S10
α(r )
≤ r}
Figure 8: Banking sector (BA): 3D plot of target rate, mean return, VaR in the Non-Gaussian case
E[
Sα
10 (
ω
)]
r
38
P rob{S10
α(r)
≤ r} in Non-Gaussian case
Figure 9: Banking sector (BA): comparison between Gaussian VaR and Non-Gaussian VaR
α(r)
P rob{S10
≤ r} in Gaussian case
P rob{S10
α(r)
≤ r}
Figure 10: Construction sector (CU): the trade-off between mean return and VaR in the Gaussian
case
α(r)
E[S10 (ω)]
39
P rob{S10
α(r)
≤ r}
Figure 11: Construction sector (CU): the trade-off between target rate and VaR in the Gaussian
case
r
40
P rob{S10
α(r )
≤ r}
Figure 12: Construction sector (CU): 3D plot of target rate, mean return, VaR in the Gaussian
case
E [S α
10 (ω
)]
r
P rob{S10
α(r)
≤ r}
Figure 13: Construction sector (CU): the trade-off between mean return and VaR in the NonGaussian case
α(r)
E[S10 (ω)]
41
P rob{S10
α(r)
≤ r}
Figure 14: Construction sector (CU): the trade-off between target rate and VaR in the Non-Gaussian
case
r
P rob{S10
α(r )
≤ r}
Figure 15: Construction sector (CU): 3D plot of target rate, mean return, VaR in the Non-Gaussian
case
E[
Sα
10 (ω
)]
r
42
P rob{S10
α(r)
≤ r}
Figure 16: Construction sector (CU): comparison between Gaussian VaR and Non-Gaussian VaR
r
43