Pension funds rules: Paradoxes in risk control

Accepted Manuscript

Pension funds rules: paradoxes in risk control Marinella Cadoni, Roberta Melis, Alessandro Trudda PII: DOI: Reference:

S1544-6123(16)30199-4 10.1016/j.frl.2017.05.003 FRL 707

To appear in:

Finance Research Letters

Received date: Revised date: Accepted date:

6 October 2016 11 April 2017 6 May 2017

Please cite this article as: Marinella Cadoni, Roberta Melis, Alessandro Trudda, Pension funds rules: paradoxes in risk control, Finance Research Letters (2017), doi: 10.1016/j.frl.2017.05.003

This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

ACCEPTED MANUSCRIPT

Highlights • Pension funds investment rules define assets risk according to typology or origin. • We define the risk of an asset as the roughness of series of returns. • We model the series with a multifractional Brownian motion with random exponent H(t). • We use H(t) to measure risk and model it with a combination of two beta distributions.

AC

CE

PT

ED

M

AN US

CR IP T

• We find that pension funds investment rules can lead to paradoxes.

1

ACCEPTED MANUSCRIPT

Pension funds rules: paradoxes in risk control Marinella Cadonia , Roberta Melisb,∗, Alessandro Truddac of Sassari, Department of Political Science, Communication, Engineering and Information Technologies, Sassari, Italy b University of Sassari, Department of Economics and Business, Sassari, Italy c University of Sassari, Department of Economics and Business, Sassari, Italy

CR IP T

a University

Abstract

AN US

Pension funds are financial institutions that invest retirement savings from workers to provide pension benefits. Due to this social security function, each country enforces laws to regulate investments. Usually regulations identify pension portfolio’s risk level based on the nature of its financial products. After the latest financial crisis, it became evident that such approach may not be sufficient to control the risk. In this paper we measure risk level with a multifractional Brownian motion with random exponent. We show how current rules can lead to paradoxes, where portfolios which comply with the laws are riskier than those that do not. Keywords: Pension Funds, investment risk, risk control, multifractional Brownian motion. JEL: C22; G11; G23.

M

1. Introduction and motivation

AC

CE

PT

ED

Security and the uncertainty of retirement incomes are extremely topical issues for young workers and future generations especially in European countries, where a reduction in the ratio of workers over retirees has occurred. In several countries, public pension schemes are progressively adopting reforms to contain public pension expenditure (e.g. by increasing pensionable age or shifting from defined benefit to defined contribution), at the cost of a reduction of replacement rates. At the same time, fully funded pension fund systems have been developed, to provide individuals with a supplementary pension benefit compared to the first pillar, which, unlike mutual funds, must focus on the social security function than speculative one. The demographic structure of developed countries and the related labour market has meant that pension funds have accumulated large capitals over the years so they play an important role as institutional investors [22]. Qualitative rules are often used to classify the risks of individual financial products according to their typology: liquidity, bonds, stocks, derivatives, commodities and others. For example, bonds are always considered less risky than stocks. Otherwise, limits may be related to geographical areas of origin, where usually non-OECD countries are considered more risky than the others. Following the financial crisis it has been shown how this classification can often be misleading, either leading to paradoxical situations (e.g. Greek government bonds and the sovereign debt crisis of 2010 in Europe) or to moral hazard policies. There are some restrictions about investments in equity and bonds traded in the over the counter markets and/or in non OECD countries. OECD (2014) [16] describes the main quantitative regulation applied to OECD and IOPS countries updated to December 2013, showing also the main changes occurred in the period 2002-2013. The Directive 2011/61/EU, in force from July 2013 was enacted to harmonise the laws that first appeared fragmented among the states of the Union. The severe financial crisis of 2008-2009 has led the EU to create

∗ Corresponding author: University of Sassari, Department of Economics and Business, Via Muroni 25, 07100 Sassari Italy Tel.+39 079 213030 - Fax +39 079 213012 Email addresses: [email protected] (Marinella Cadoni), [email protected] (Roberta Melis ), [email protected] (Alessandro Trudda)

Preprint submitted to Finance Research Letters

May 8, 2017

ACCEPTED MANUSCRIPT

PT

ED

M

AN US

CR IP T

a framework that would harmonise national rules to prevent risks to investors, their counterparts, the other operators in the financial markets and, more generally, to the financial stability of Europe. For example, Italy revised the rules of pension fund investments, implementing the Directive 2011/61/EU, that came in force at the end of 2014. It puts the emphasis firmly on the risk management of pension funds and the responsibility of their administrators. The new DM 166/2014 innovates the previous regulatory framework introducing new rules that aim to ensure more flexibility in the management of pension funds, while respecting a prudent management policy. The fund will have to define the horizons of the most appropriate risk-return objectives of the fund and indicate the tolerable upper limit with respect to individual investments and overall the entire portfolio. In this context, defining appropriate models to measure the risk levels of pension fund portfolios: in this paper we develop a model to control the volatility of investments portfolios and we further show how a simple qualitative analysis not only is insufficient but even deceptive. After the financial crisis of 2008, a number of quantitative risk analysis methods have been proposed to model and control the volatility of investment portfolios of pension funds. Otranto and Trudda (2008) [17] propose a statistical procedure to classify pension funds in different risk classes, in order to monitor the funds based on the study of the dynamics of GARCH volatility associated with the returns of the funds. Bianchi and Trudda (2008)[6] analyse the investment risk in pension funds and they provide a technique for rebalancing pension fund portfolios in function of their pointwise level of risk. Impavido and Tower (2009) [12] study the key sources of vulnerabilities for pension plans and insurance companies in the light of the global financial crisis of 2008 and they also discuss how the institutional investors transfer shocks to the rest of financial sector and economy. Halim et al. (2010) ([11]) analyse how funds manage investments and show that funds that manage both active and surplus risk have generated better risk-reward trade offs. It has been debated that regulation and supervision of the financial sector, an in particular for pensions, needs a risk-based approach. In particular Berstein and Chumacero (2010) [4] evaluate the effect of value at risk limits and quantitative restrictions on portfolio choices in the context of a risk-based supervision for a defined benefit pension fund. Bohl et al. (2011) [7] study the performance of pension funds in Poland and Hungary where governments impose regulation on the investment of the pension funds which differs from that of Eastern countries (strongly regulated ) finding that investment limits and performance regulations influence the investment decisions of pension funds in both countries. Lippi (2014) [14] explores the investment lines choices of occupational pension funds subscribers in Italy from 2007 to 2011. He finds that many subscribers opt for the middle options between the different risk investment lines and he suggests to the regulators to assign this line to people who do not express any preference instead of the so called no risky line. Thomas et al. (2014) [22] analyse the stock market volatility in the OECD countries and they observe a significant reduction in volatility of stock prices with the increase of the investment of pension funds.

AC

CE

Classic financial theory on stock market uses Brownian motion to model asset return. Brownian motion is an elegant representation of the stock market, but one of its drawback is the unrealistic hypothesis that successive returns are independent and normally distributed. It is assumed that the price has no memory and a sharp decline one day does not influence the price of future day. For this reason, long run memory models (See [23]) were introduced. The fractional Brownian motion fbm is an extension of the standard Brownian motion and it was presented in the pioneer work of Mandelbrot and Van Ness in [15]. In the fractional Brownian motion the increments are serially correlated, new information has an enduring influence on the process and this involves a certain level of predictability (On this topic see Rostek and Sch¨obel [20]). It is characterized by a slowly decay autocorrelation function depending on the Hurst exponent. One of the problem of fractional Brownian motion is the possibility of arbitrage. An extension of the fractional Brownian motion is the multifractional Brownian motion, in which the Hurst parameter is a function of the time H(t). On this topic see [18], [13], [1], [5]. By allowing H to be a stochastic process or a r.v, the mBm can be further generalized to the Multifractional Process with Random Exponent mpre (see [2]). Cadoni et al. (2015) [8] model the Hurst exponent with a mixture of beta distributions and apply it to different financial instruments and portfolios. 3

ACCEPTED MANUSCRIPT

CR IP T

In this paper, using a mpre model [8] to measure the volatility, we show that, investments in financial products considered high risk by current legislation (non OECD stocks and bonds) are in fact less turbulent than products defined as low risk (e.g. OECD bonds). In the experimental evaluation, different investment portfolios are simulated: the results show that portfolios constructed with bonds or stocks of non OECD countries, i.e. assets that are classified risky according to the regulations, have a lower degree of (quantitatively measured) risk than that of OECD countries, which in turn are considered less risky by current pension funds investment regulations. Current legislations can therefore lead to paradoxes, where portfolios composed according to investment rules are quantitatively more risky than portfolios that do not conform to regulations. The remainder of the paper is organised as follows: Section 2 presents the methodology used, in Section 3 the experimental evaluation is performed, in Section 4 the results are discussed and conclusions are drown up. 2. Methodology

MH(t) (t)

AN US

We describe the log price evolution by a multifractional process with random exponent mpre. In the following, we outline the mathematical instruments the method is based on, for a thorough treatment the reader is referred to [8]. The multifractional Brownian motion (mBm, see [18], [13], [1] ) is a generalisation of the fBm obtained allowing that the path regularity can vary with time (See [9]). The process has the following representation: = C{πK(2H(t))}1/2

Z

ft (s)dB(s)

(1)

R

=

n o 1 H(t)− 12 H(t)− 21
|t − s| 1 (s) − |s| 1 (s) ]−∞,t] ]−∞,0] Γ H(t) + 21

ED

ft (s)

M

with

CE

PT

where H : [0, ∞) → (0, 1] is required to be a H¨older function of order 0 < η ≤ 1 to ensure the continuity of the motion. Since H(t) is the punctual H¨older exponent of the mBm at point t, the process is locally asymptotically self-similar with index H(t) (see, e.g., Benassi et al.[3])) in the sense that, denoted by Z(t, au) := MH(t+au) (t + au) − MH(t) (t) the increment process of the mBm at time t and lag au, it holds d

lim a−H(t) Z(t, au) = BH(t) (u), u ∈ R.

a→0+

(2)

AC

The above distributional equality implies that at any point t there exists an fBm with parameter H(t) tangent to the mBm. Furthermore, since BH(t) (u) ∼ N (0, C 2 u2H(t) ), the infinitesimal increment of the mBm at time t, normalised by aH(t) , normally distributes with mean 0 and variance C 2 u2H(t) (u ∈ R , a → 0+ ). The increments of the mBm are no longer stationary nor self-similar; despite this, the process is extremely versatile since the time dependency of H is useful to model phenomena whose punctual regularity is time changing. In our context, H(t) can represent the degree of confidence the investors have in the past. High values of H(t) (small roughness) correspond to trends (or low volatility phases), i.e. to periods in which the past information weighs in the investors’ trading decisions; low values of H(t) (great roughness) are associated to high volatility periods, in which prices display an anti-persistent or mean reverting behaviour because of the quick buy-and-sell activity that is typically induced by uncertainty. Standard financial theory is recovered when H = 21 , case in which the mBm reduces to the Brownian motion. 4

ACCEPTED MANUSCRIPT

By allowing H to be a stochastic process or a r.v, the mBm can be further generalised to the Multifractional Process with Random Exponent. Ayache and Taqqu (see [2]) define it by replacing the deterministic parameter H(t) by a random variable or a stochastic process. Let: H : [0, 1] → [a, b] ⊂ [0, 1] be a random or stochastic process, BH : [0, 1] × [a, b] ⊂ (0, 1) a Gaussian field and f1 : [0, 1] → [0, 1] × [a, b], t → (t, H(t, w)) and f2 : [0, 1] × [a, b] → R, (t, H) → BH (t, w). Then the process is defined by: Z(t, w) = f2 (f1 (t)) = BH(t,w) (t, w)

CR IP T

The process has the following properties:

1 (Ayache and Taqqu [2] ) - The pointwise H¨older exponent of the mpre is determined by H(t, w). 2 (Ayache, Taqqu [2]) - If H is a random variable independent of the Brownian motion then the process is stationary. 3 (Bianchi [5]) - H(t, w) can be interpreted as the confidence level investors have in past information.

  √ P   k  k/2 VH − log δπ t−1 log 2k/2 Γ k+1 j=t−δ Xj+1,N − Xj,N 2

=

k log (N − 1)

PT

k Hδ,N (t)

ED

M

AN US

Property 1 means that we can evaluate the roughness of the series by using H(t, w). Property 2 claims that stationarity is a necessary condition for H to be a random variable independent of the Brownian motion. A consequence of property 3 is that H is not symmetrical with respect to its central value 1/2 as it is easier (quicker) to loose confidence than to build it. It therefore follows that, provided we are dealing with stationary processes we can model H with a random variable and this will give us a measure of the roughness and so the risk of a price series. Given the asymmetry of H w.r.t. its central value of 1/2 and its range in the interval [0, 1], we choose a random variable with a mixture of beta distribution. Indeed, as shown in [8] the beta distribution presents some characteristics that mime very well those of H. We define the risk as the roughness of the series of returns and model the series of returns by a mpre and the roughness by the H¨older exponent. At this stage, we need to estimate the H¨older exponent from real data. To do so, we adopt a family of ”moving-window” estimators of H(t) based on the k-th absolute moment of a Gaussian random variable of mean zero and given variance VH (the variance of the unit lag increment of a mBm) as defined in [5]. Given a series of length N and a window of length δ, the estimator has the form

(3)

for j = t − δ, ..., t − 1; t = δ + 1, ...,  N1; k ≥ 1. −1 The quick rate of convergence O δ − 2 (log N ) ), the estimator is reliable even for very short δ 0 s. It is

CE

demonstrated that when H =

1 2

AC

k V ar(Hδ,N (t)) =

the variance of the estimator reduces to √

π

h  i2 · δk2 ln2 (N − 1) Γ k+1 2

Γ



2k + 1 2



   ! 1 k+1 2 −√ Γ π 2

(4)

and the optimal value of k is deduced by minimizing the last relation. Equation (4) reaches a minimum for k = 2, so this is the value we set in our experiments. Once we have estimated H, we model it with a random variable with a mixture of beta distributions. The Beta distribution, being defined in the interval [0, 1] and being dependent on two parameters, has the potential to well approximate the H¨older exponent. The probability density function of the beta distribution is given by: y = f (x, a, b) = where B(a, b) =

Γ(a+b) Γ(a)Γ(b)

1 xa−1 (1 − x)b−1 B(a, b)

is the beta function. 5

ACCEPTED MANUSCRIPT

Its mean and variance are given by: µ=

a a+b

Var =

ab (a + b + 1)(a + b)2

To better model possible bimodal characters of the H¨older exponent we adopt a linear combination of two Beta distributions: α1 f (x, a1 , b1 ) + α2 f (x, a2 , b2 )

CR IP T

with α1 + α2 = 1. 3. Numerical Applications and Results

AC

CE

PT

ED

M

AN US

We test the model outlined in the previous section on a variety of financial products, classified according to their typology and geographical area. In our context, H(t) represents the degree of volatility the asset shows at time t based on the chosen lag time. As discussed in section 2, we define the risk as the roughness of the series of returns. A high level of roughness of a return series is the sign of an intense buy and sell activity, the higher the peaks and falls, the higher the uncertainty the operators express on the asset. Conversely, a stable behaviour of the price series, with a low roughness level, means that, over time, the market has deemed the asset to be low risk, resulting in stable prices. Therefore, when the series of returns has a low level of roughness (mean H near to one) means that prices are not turbulent and the market considers the asset to be low risk. On the other side, a high level of roughness (lower levels of H) implies that the asset is perceived as risky and its price is highly volatile. To show the effectiveness of H(t) at measuring the market volatility, in the appendix we report the results of the computation of H(t) for a variety of financial assets and we compare H(t) with other classical risk measures. The assets in the appendix consist of daily closing prices from January 2006 to December 2011, a period that includes the 2007-2008 market crisis, as well as for seven sample portfolios composed with these assets. As it can be seen from the Tables 3 and 5, the mean value of H(t) varies from 0.40 to 0.86, an ample range that enables us to distinguish between risky assets from non risky ones. Here, we want to see if H(t) can detect potentially risky assets or portfolios that current regulations would permit. The time series considered span from July 05 2008 until 19/09/2015, which includes the sovereign debt crisis in Europe of 2010, and are relative to OECD and non OECD bonds and stocks. For each asset, we estimated H(t) with a moving window of 20 days lag. The results of the experiments are summarised in Table 1, where the time series are grouped into non OECD bonds, non OECD stocks, OECD bonds and OECD stocks. As we can see from the table, in the period that follows the crisis, the non OECD bonds are noticeably less volatile than all OECD assets (bonds and stocks). Also the non OECD stocks prove to be, on average, less risky than the OECD ones. Figure 1(a) shows how H(t) of a non OECD bond and H(t) of an OECD bond vary in the considered time period, while Figure 1(b) compares a non OECD bond to a OECD stock. By observing the graph 1(a) it is evident that the non OECD bond has an H(t) almost always higher than the OECD one and 1(b) that the non OECD bond has an H(t) almost always higher than the OECD stock.

By looking at results of Table 1 and Figure 1 we can see that the single financial products considered risky and limited by pension funds investment rules show a low risk using a volatility measure: this demonstrates that qualitative rules coupled with limits in investments are insufficient and in some cases paradoxical. A quantitative analysis proves to be necessary to reliably estimate the risk of the assets. After considering single financial products, to simulate a pension fund investment scenario, we composed six portfolios out of the assets in Table 1. Four portfolios are composed of assets of the same type (homogeneous portfolios): P1 is made of all non OECD bonds, P2 is made of all non OECD stocks, P3 is made of all OECD bonds and P4 of all OECD stocks. Two portfolios are composed of assets of different types (mixed portfolios): P5 is made of two non OECD bonds, namely Philip 10, China, and a non OECD stock (Public 6

ACCEPTED MANUSCRIPT

Table 1: Mean H of single assets and portfolios

NON OECD BOND

OECD BOND

NON OECD STOCK

Mean H 0.7730 0.7195 0.8116 0.7908 0.5704 0.7317 0.7298 0.6291 0.6819 0.5854 0.5686 0.6076 0.6115 0,5581 0.5885

ED

M

AN US

OECD STOCK

Asset Philip 10 5/8 03/16/25 Philip 7 3/4 01/14/31 Turkey 7 09/26/16 China 4 3/4 10/29/13 Greece Ireland Switzerland Malay Banking BhD Public Bank Bhd Link/REIT/The SABMiler PLC Coca-Cola Co McDonalds Corp Iberdrola British American Tobacco

CR IP T

Asset type

PT

Figure 1: Estimated H for a non OECD bond vs a OECD bond (a) and for a non OECD bond vs an OECD stock (b)

AC

CE

Bank), while P6 is made of two OECD bonds (Greece and Switzerland) and a OECD stock (Iberdrola). All portfolios are composed by taking the assets in equal parts. To estimate the volatility of the portfolios, we considered the return series of the composing assets and then we calculated the value of H(t) of the portfolio returns. In table 4, for each portfolio, we can see its composition and its H average value. Figure 2 (a) shows the results for the homogeneous portfolios, while Figure 2 (b) shows the results for the mixed portfolios. The results of the analysis of the risk of portfolios lead to the same considerations we make for the single assets: the portfolios that show a lower volatility are P1, P5 and P2, which are all composed by non OECD financial assets, bonds, stocks and mixed. Paradoxically, the portfolios P3 (OECD bonds) and P6 (OECD mixed) appear to be more risky than the latest. Notice how the diversification effect in the portfolios can cause the portfolio volatility to be lower of that of each of its composing assets (see OECD Stocks in table 1 and the portfolio P4 in table 4). By modelling H(t) with a mixture of beta distributions we can analyse the risk level of the assets and portfolios more deeply. In figure 3, the mixture of two beta distributions are fitted to the probability density functions of H(t) of nine of the assets in table 1. These are four non OECD stocks, three OECD stocks, one non OECD bond and one OECD bond. As it can be seen, the fittings can model H(t) accurately 7

ACCEPTED MANUSCRIPT

Table 2: Portfolios’ composition

Mean H

P5(Non-OECD) Philip 10 5/8 03/16/25 China Public Bank

Mean H

0.7931

Mean H 0.6492

Mean H 0.6653

Mean H 0.6329

Mean H 0.6191

M

AN US

0.7734

P2(Non-OECD Stocks) Malay Banking Bhd Public Bank Bhd Link /REIT/The SABMiller PLC P4(OECD Stocks) Coca-Cola Co McDonalds Corp Iberdrola British American Tobacco PLC P6(OECD) Greece Switzerland Iberdrola

CR IP T

P1(Non-OECD Bonds) Philip 10 5/8 03/16/25 Philip 7 3/4 01/14/31 Turkey 7 09/26/16 China 4 3/4 10/29/13 P3(OECD Bonds) Switzerland Greece Ireland

ED

(a) H of Homogeneous Portfolios (b) H of Mixed Portfolio

PT

Figure 2: Estimated H for the six portfolios

CE

in most cases. In figure 4, the H(t) function is modelled by a mixture of beta distributions for each of the six portfolios and the same distributions are plotted on the same plane in figure 5. By looking at the distributions, it is clear the portfolios P1 and P5 (non OECD bonds and non OECD mixed, respectively) are sensibly less volatile than P3 and P6 (OECD bonds and OECD mixed). The non OECD stock portfolios also proves to be less risky than the OECD one.

AC

4. Conclusions

In 2008 the financial crisis caused huge losses of major pension funds worldwide. Several analysis show that before the crises, pension funds increased their portfolio risk in order to obtain higher values of the expected global asset return, although pension funds should maintain a prudent profile in view of their social function (in particular for the first pillar): this kind of moral hazard policy, with the aim of obtaining high returns, has led to the failure of some funds. Some authors identified a weakness in the financial laws which imposed limits to contain the portfolio’s risk based on qualitative rather than quantitative measures. In this paper, we propose to use a framework to quantitatively asses the risk of pension fund investments with fast response time which could be used to monitor investments in real time. The methods consists of modelling the return series with a mpre and the risk with the H¨older exponent H(t) of the process. In order to quantify 8

CE

PT

ED

M

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

Figure 3: Mixture of beta distribution fit to the H¨ older exponent of different types of assets. First row: Link Reit, Sab Miller and Malayan; second row: Philip 7, Public Bank and Swiss; third row: British American Tobacco, Mac Donald and Coca Cola.

the risk, we model H(t) with a random variable with beta distribution, which depends on two parameters that can vary to fit very well the asymmetry of H(t), at the same time providing us with different possible measure of the risk. Compared to classical risk measures such as standard deviation of returns and mod Var 95, the advantages of H are that it is a punctual risk measure, so it can be estimated in real time, can be modelled as a random variable. The empirical applications unveil several paradoxes: securities or portfolios considered to be low risk by current legislations, such as portfolios consisting of OECD bonds, may be riskier even than a portfolio made up of non OECD stocks and, more generally, the considered portfolios composed by assets of non OECD countries are less risky than portfolios made up of OECD assets. The proposed approach, as well as providing a quantitative risk measure, can also be efficiently used to monitor the risk 9

AN US

CR IP T

ACCEPTED MANUSCRIPT

AC

CE

PT

ED

M

Figure 4: Mixture of beta distribution fit to the H¨ older exponent of different portfolios. First row: mixture of beta fitting of H of non OECD bond (P1), non OECD stock (P2) and non OECD bond and stock mixed (P5) portfolios. Second row: mixture of beta fitting of H of OECD bond (P3), OECD stock (P4) and OECD mixed (P6) portfolios.

Figure 5: Mixture of beta distributions of all portfolios.

10

ACCEPTED MANUSCRIPT

of investment over time, and various techniques could be introduced to rebalance the portfolio when the measured risk level is too high. Quantitative measures such as the one proposed in this paper, could be adopted by pension fund regulations to effectively limit the risk in their investment portfolios. Appendix

AN US

CR IP T

In this section, we report the computations of H(t) for a variety of financial assets, and we compare its values with classical risk measures. First, we analyse time series of daily closing prices from January 2006 to December 2011, which includes the 20072008 market crisis. Assets include market indexes (Bovespa, Athen Index Compos, Hang Seng Index (HSI), Ibex, Dow Jones, NASDAQ, FTSE), some shares listed in those indexes (Cheung Kong Holdings Ltd, Bank of China, Petroleo Brasileiro, Telecom), bonds of emerging countries (Arca Bonds Emergenti) and EU bonds: Belgium Kingdom-BGB, Bundesrepub. Deutschlan-DBR, Irish TSY-Irish, Intesa San Paolo SPA-ISPIM, Netherlands Government-Nether, Obrigacoes do TesouroPGB, Republic of Austria-RAGB, Telecom Italia FIN SA-TITIM. In table 3, the mean values of H for the assets are shown. As it can be noticed, bonds are generally less risky than stocks, with a portfolio of non OECD bonds (Arca Bonds Emergenti) less risky than all OECD bonds. With the assets of table 3 we composed seven sample portfolios, the first three are made of market indexes and stocks, while the last four are made out of bonds. The mean values of H(t) are reported in the first column of table 5. Table 3: Mean H of single assets of the first group

ED

Market Indexes

Asset Bovespa DJI FTSE HSI NASDAQ Athens Ibex Telecom Cheung Kong Holdings Ltd Bank of China Petroleo Brasileiro Arca Bonds Emergenti Belgium Kingdom-BGB Bundesrepub. Deutschlan-DBR Irish TSY-Irish Intesa San Paolo SPA-ISPIM Obrigacoes do Tesouro-PGB Republic of Austria-RAGB Telecom Italia FIN SA-TITIM Netherlands Government-Nether

M

Asset type

AC

CE

PT

Shares

Bonds

Mean H 0.5713 0.6304 0.5917 0.5792 0.5967 0.5360 0.5855 0.5534 0.5472 0.5521 0.5695 0.8565 0.8153 0.8064 0.7582 0.7935 0.7186 0.8188 0.7090 0.8193

We want to see how the H¨older exponent compares to other well known risk measures, namely the standard deviation of the returns, Value at risk and modified Value at risk. In classical portfolio theory the standard deviation of the asset returns (std ret) is used to measure the risk of financial assets or portfolios. Being the standard deviation a measure used to quantify the amount of variation or dispersion of returns from their expected value, large standard deviations indicate larger degrees of risk. The Value at Risk (VaR) is a general measure of risk developed to equate risk across products and to aggregate risk on a portfolio basis. VaR measures the worst expected loss over a given horizon (for example, 11

ACCEPTED MANUSCRIPT

Table 4: Sample Portfolios

SP2 SP3 SP4 SP5 SP6 SP7

Assets in portfolio Athen, Bovespa, Ibex, HSI, Cheung Kong Holdings Ltd, Bank of China, Brasil Petroleo, Telecom CPFE, HSI, Natu3 Ibex, Telecom, Bovespa, HSI 90% Arca Bond Emergenti, 10% Cheung Kong Holdings Ltd Irish, ISPIM, TITIM BGB, DBR, RAGB BGB, DBR, Irish, ISPIM, Nether, PGB, RAGB, TITIM.

CR IP T

Portfolio SP1

AN US

1 day) under normal market conditions at a given confidence level (for example, 95%). The VaR is returned as a positive percentage even it represents a loss and assumes a normal distributions of returns. The Modified Value at Risk MVaR is calculated as the VaR, but it does not assume a normal distribution of the return as it is calculated by using the skewness and the kurtosis of returns. We compared the estimated portfolio risk (mean H) with the other risk measure, std ret, VaR at 95% and MVaR at 95%. Table 5 summarises the risk measures for the sample portfolios. Table 5: Risk measures of sample portfolios.

std ret 0.00466 0.00263 0.00327 0.00409 0.01297 0.01366 0.01372

M

mean H 0.8556 0.8214 0.8070 0.7716 0.5902 0.4342 0.4047

VaR 95 0.00787 0.00440 0.00533 0.00664 0.02134 0.02249 0.02276

mod VaR 95 0.00396 0.00427 0.00535 0.00662 0.02048 0.02116 0.01419

PT

ED

Portfolio SP4 SP6 SP7 SP5 SP1 SP3 SP2

AC

CE

In table 5, the portfolios (rows) are ordered according to the proposed measure, the mean of H(t) (shown in the first column). In the first row is SP4, the lowest risky portfolio according to H(t), which has a mean value of 0.8556, while in the last row is the highest risk portfolio, with a mean value of 0.4047. Columns three to five show that the of risk measured by H is in line with the other risk measures, Standard Deviation, VaR at 95% and MVaR at 95%. Table 6 reports the results of the same analysis for the six portfolios that we used in section 3 to show the paradoxes. Acknowledgement The authors acknowledge the financial support provided by the Banco di Sardegna Foundation (prot. n. U1371.2013).

12

ACCEPTED MANUSCRIPT

Table 6: Risk measures of portfolios of section 3

mean H 0.7931 0.7734 0.6653 0.6492 0.6329 0.6191

std ret 0.00643 0.00412 0.00802 0.00868 0.01366 0.0296

VaR 95 0.01072 0.00691 0.01362 0.01432 0.02285 0.03769

mod VaR 95 0.00245 0.01027 0.00889 0.00647 0.02088 0.05482

CR IP T

Portfolio P1 P5 P2 P3 P4 P6

References

AC

CE

PT

ED

M

AN US

[1] Ayache, A. and L´ evy V´ ehel, J. (2000) The Generalized Multifractional Brownian Motion, Statistical Inference for Stochastic Processes 3(1): 7-18. [2] Ayache, A. and Taqqu, M.S. (2005) Multifractional Processes with Random Exponent. Publicacions Matem` atiques, 49(2), 459-486. [3] Benassi, A., Jaffard, S. and Roux, D. (1997) Gaussian processes and Pseudodifferential Elliptic operators, Revista Mathematica Iberoamericana, 13(1): 1989. [4] Bernstein, S.M and Chumacero, R.A. (2012) VaR limits for pension funds: an evaluation, Quantitative Finance, 12(9): 1315-1324. [5] Bianchi, S. (2005) Pathwise Identification of the Memory Function of the Multifractional Brownian Motion with Application to Finance. International Journal of Theoretical and Applied Finance, 8(2): 255-281. [6] Bianchi, S. and Trudda A. (2008) Global Asset Return in Pension Funds: A Dynamical Risk Analysis. Mathematical Methods in Economics and Finance, 3(2): 1-16. [7] Bohl, M.T., Lischewski, J. and Voronkova, S. (2011) Pension Funds’ Performance in Strongly Regulated Industries in Central Europe: Evidence from Poland and Hungary. Emerging Markets Finance and Trade, 47(3): 80-94. [8] Cadoni, M., Melis, R. and Trudda A. (2015) Financial Crisis: A New Measure for Risk of Pension Fund Portfolios. Plos One, 10(6): 1-12, 10.1371/journal.pone.0129471. [9] Coeurjolly, J. (2005) Identification of the multifractional Brownian motion. Bernoulli, 11(6): 987-1008. [10] Embrechtz, P. and Maejima, M. (2002) Selfsimilar processes.Princeton University Press. [11] Halim S., Miller T. and Dupont D.C. (2010) How Pension Funds Manage Investment Risks: A Global Survey, Rotman International Journal of Pension Management, 3(2): 30-39. [12] Impavido, G. and Tower, I. (2009) How the financial crisis affects pensions and insurance and why the impacts matter, International Monetary Found, WP/09/151. [13] L´ evy V´ ehel, J. (1995) Fractal approaches in signal processing. Fractals, 3: 755-775. [14] Lippi, A. (2014) Does menu design influence retirement investment choices? Evidence fron Italian occupational pension funds. Judgment and Decision Making 9(1): 77-82. [15] Mandelbrot, B.B. and Van Ness, J.W. (1968) Fractional Brownian Motions, Fractional Noises and Applications, SIAM Review, 10(4): 422-437. [16] OECD (2014) Annual Survey of Investment Regulation of Pension Funds, OECD Publishing. [17] Otranto, E. and Trudda, A. (2008) Classifying the Italian pension funds via GARCH distance. In: Mathematical and Statistical Methods for Insurance and Finance, Springer: 189-197. [18] Peltier, R.F. and L´ evy V´ ehel, J. (1994) A New Method for Estimating the Parameter of Fractional Brownian Motion. Rapport de recherche INRIA n.2396, Le Chesnay Cedex. [19] Peltier, R.F. and Levy V´ ehel, J. (1995) Multifractional Brownian Motion: definition and preliminary results, Rapport de recherche INRIA n.2645, Le Chesnay Cedex. [20] Rostek, S., Sch¨ obel, R. (2013) A note on the use of fractional Brownian motion for financial modeling, Economic Modelling, 30(1): 30-35. [21] Samorodnitsky, G. and Taqqu, M.S. (1994) Stable non-Gaussian random processes, Chapman & Hall, London. [22] Thomas, A., Spataro, L., and Mathew, N. (2014) Pension funds and stock market volatility: An empirical analysis of OECD countries, Journal of Financial Stability, 11(1): 92-103. [23] Zai, F. (2013) Modelling: The power of long memory, The Actuary 02/2013, http://www.theactuary.com/features/2013/02/the-power-of-long-memory/

13