Chapter 9 financial risk in pension funds; application of value at risk methodology

9.2.2 Value at Risk Measure Algorithm 1899.2.2.1 Portfolio Exposure Measure Procedure 1899.2.2.2 Uncertainty Measure Procedure 1919.3 Market Risk in Banking and Pension Fund Sectors 1929

9.2.2 Value at Risk Measure (Algorithm) 1899.2.2.1 Portfolio Exposure Measure Procedure 1899.2.2.2 Uncertainty Measure Procedure 191

9.3 Market Risk in Banking and Pension Fund Sectors 1929.3.1 Market Risk in Banks and Economic Capital

Measurement with VaR Techniques as Internal

9.3.2 Market Risk and Its Regulatory Approaches to Capital

9.4 VaR Measure in Pension Funds Business 1999.4.1 M odifi cation of Portfolio Exposure Measure

Procedure: Taking into Account Changes

9.4.2 Sensibility of Uncertainty Measure Procedure:

Importance of Assumptions Characterizing

9.4.3 Selection of Transformation Procedure: Choice

of Appropriate VaR Model due to Data Sample

nan-cial risk It is also used for regulatory capital requirement purposes

in banking and insurance sectors VaR methodology has been developed mainly for banks to control their short-term market risk Although, the VaR is already widespread in fi nancial industry, this method has not yet become a standard tool in pension funds However, like other fi nancial institutions, pension funds recognize the importance of measuring their

VaR could be a good measure of long-term market risk Aft er a description

of a general VaR algorithm and the three main VaR methods, we present

propose necessary adaptations of VaR measures for pension funds ness specifi cations

busi-9.1 INTRODUCTION

Financial institutions’ activities entail a variety of risks One of the most important categories of hazard is market risk, defi ned as the risk that the value of a n i nvestment may decl ine due to economic cha nges or other events that impact market factors (e.g., stock prices, interest rates, or for-eign exchange rates) Market risk is typically measured using the value-at-Risk (VaR) methodology

In order to provide evidence of safety, fi rms have to maintain a mum amount of capital as a buff er against potential losses from their busi-ness activities, or potential market losses Th e literature distinguishes the economic capital from the regulatory capital Th e fi rst is based on calcula-tions that are specifi c to the company’s risk, while regulatory formulas are based on industry averages that may or may not be su itable to any par-ticular company Moreover, the economic capital can be used for internal corporate risk-management goals as well as for regulatory purposes Th is

mini-chapter focuses on economic capital estimations for market risk in two main fi nancial institutions: banks and pension funds.

For banks, the new Basel accord has provided increased incentives for developing a nd ma naging t heir i nternal c apital on a n economic ba sis Basel II encourages bankers to use the VaR for both internal management and regulatory capital requirements Th is framework has also been copied

by the new European prudential system in insurance (Solvency II) While pension funds are not subject to banks’ capital adequacy requirements,

a n umber o f si milar re strictions g overn de fi ned-benefi t p lans ( Jorion, 2001) Moreover, even if they are not enclosed by bank (and insurance) rules, in the future, pension funds might adopt and adjust banking regu-lations in the aim to harmonize with general fi nancial framework.Today, m ost o f pens ion f unds c an c alculate t heir own ma rket r isks just in in ternal aim s T raditionally, t hey h ave e mphasized m easuring and rewarding investment performance by their portfolio managers In the past decade, however, many of them have signifi cantly increased the complexity of portfolios by broadening the menu of acceptable invest-ments Th ese i nvestments c an i nclude foreign sec urities, commodities, futures, swaps, options, and collateralized mortgage obligations At the same time, well-publicized losses among pension funds have underlined the i mportance of r isk ma nagement a nd measuring per formance on a risk-adjusted b asis Th e VaR a ppears t o off er c onsiderable p romise f or them in this area

Moreover, in the future, pension funds could extend the use of the VaR into regulatory capital requirement purposes, because this k ind of pru-dential model is already applicable to a number of fi nancial sectors Th is could provide strong support for the integration of fi nancial system and supervision Th e VaR is likely to continue to gain acceptance because it provides a forward-looking approach around which the supervision in all

framework to assess the relative risks that function in a p rudent person investment r egime I t a lso i mposes n ew tech nical r equirements a nd a higher level of sophistication

However, potential pension funds’ quest for a VaR-based risk-management system is hampered by several factors One is a lack of generally accepted standards that would apply to them Most works in the area of the VaR has been done in the banking sector Th e VaR originated on derivatives trad-ing desks and then spread to other trading operations Th e implementations

of VaR developed at these institutions naturally refl ected the needs and

characteristics of their trading operations, such as very short time horizons, generally liquid securities, and market-neutral positions In contrast, pension funds generally stay invested in the market, can have illiquid securities in their portfolios, and hold positions for a long time Th erefore, the application

of the VaR for pension funds—already done in Mexico where VaR is used for capital requirement purposes—remains controversial because this short-run risk measure should not be adopted without modifi cations for pension funds industry operating with a long-term horizon Th erefore, the aim of this chap-ter is to study the market risk measurement in the pension funds sector and its necessary adaptation for long-term business particularities Since the VaR modeling derives from the banking sector, this analysis will be thus naturally provided in comparison with Basel II amendments and solutions

In spite of the large quantity of the literature concerning various aspects

of the VaR, deeply studding its application in short-term trading work, we can name only a few articles treating for the VaR concept in the long-term vision, characteristic for pension funds Here the most impor-tant ones are Albert et al (1996), Ufer (1996), Panning (1999), Dowd et al (2001), and Fedor and Morel (2006)

the VaR defi nition and methodology, currently in use in the banking

banking and pension fund industries Next, we propose changes in VaR methodology that are necessary to adapt the concept as an internal tool for the economic capital market risk measurement in the pension fund busi-ness Finally, we conclude and give a brief outlook for future research.9.2 VALUE AT RISK

In this section, we briefl y review the VaR approach, as has been ally used in the banking sector First, we defi ne the VaR concept; next, we discuss t he VaR a lgorithm; fi nally, we de scribe t he t hree most popular VaR models in the banking sector

tradition-9.2.1 Value at Risk Defi nition

at the date of estimation Th e change in the market value of a po rtfolio

over a time horizon h is given by

+

Th e VaR of a po rtfolio is the possible maximum loss, noted as VaRh (q), over a given time horizon h with probability (1 − q) Th e well-known for-

mal defi nition of a portfolio VaR is

R is the inverse of the distribution function of random variables

∆V, also called P&L distribution function.* Th erefore, the VaR estimations

9.2.2 Value at Risk Measure (Algorithm)

Although the VaR is an easy and intuitive concept, its measurement is a challenging statistical problem In t his pa ragraph, we d iscuss a p rocess that is common to a ll VaR calculations Th is a lgorithm is composed of three procedures:

posi-•

tions present in the investment portfolio to risk factors

•

distribution of risk factor variations

•

9.2.2.1 Portfolio Exposure Measure Procedure

First, we describe portfolio exposure by a mapping procedure (the sentation of investment portfolio positions by risk factors) Assume that

repre-the investment portfolio is composed from m fi nancial positions Let us

t + h Let v m,t be a value of the corresponding position at the date of

of fi nancial positions, and thus of the whole portfolio Assume that we chose

n risk factors In general, the number n of risk factors we need to model is

substantially less than the number m of positions held by the portfolio Let

of risk factors Th us, there must exist pricing formulas (valuation

v m,t+h ; thus we can express V t and V t+h in terms of risk factors:

for the relationships (9.6) and (9.7) are

Relationships (9.8) and (9.9) are called portfolio mapping and functions

f and g are called the portfolio mapping functions Functions f and g can

be linear if the model of portfolio position price’s evaluation is linear (e.g., equities positions) However, the evaluation model is not linear for certain

categories of assets (e.g., options), therefore, neither function f nor g is

lin-ear any more

9.2.2.2 Uncertainty Measure Procedure

not contain any information relating to the market volatility We obtain the information about this uncertainty by risk factor distributions Let us defi ne

∆X i distribution explains market behavior We can characterize this

distri-bution by historical data related to all risk factors We must dispose of the data sample of the risk factor variations, called the window of observations

Let T be t he size of the window of observations (length expressed in

K returns over h days for each risk factor (i = 1,…,n) Generally, this fi

nan-cial data is formed on the basis of one-day variations of risk factors over a

past period; consequently, we have T-long time series of one-day returns

i j

distribution Th e choice of the window of observations is very important since we must have quotations for all risk factors throughout this time

9.2.2.3 Transformation Procedure

to the one-dimensional spaces of the portfolio’s market value Although we

need to characterize the distribution of ∆V, mapping functions simply give

to the entire joint distribution of risk factors, with the aim of obtaining ∆V distribution Consequently, we defi ne ∆V as a function of risk factors

A t ransformation p rocedure co mbines t hus t he po rtfolio’s ex posure

describe the ∆V distribution Next, we fi nd q-quantile of the portfolio

dis-tribution that is equal to the VaR metric Th e third procedure estimates portfolio risk

In brief, we face two problems while calculating VaR First, we map

folio positions to the risk factor by f and g functions, which refl ect the

port-folio’s composition On its own, however, it cannot estimate portfolio risk because (9.8) and (9.9) do not contain any information relating to the market volatility We obtain this information in the risk factor distributions We

i j

cannot measure the portfolio risk because it is independent of the portfolio’s composition Th us, as soon as we have estimated the distribution of risk fac-

into a characterization of the ∆V distribution by mapping functions.

We c an spec ify t hree ba sic f orms o f t ransformation p rocedures: variance–covariance, M onte C arlo, an d h istorical tr ansformations Traditionally, VaR models—the computation of a VaR measure providing

an output of those calculations (which is the VaR metric)—have been egorized according to the transformation procedures they employ Even though t hey follow t he general st ructure presented above, t hey employ

presenta-tion of three broad approaches to calculating VaR is beyond the scope of this chapter and can be found in Fedor and Morel (2006)

9.3 MARKET RISK IN BANKING AND PENSION

FUND SECTORS

Conventionally, market risk is defi ned as exposure to the uncertain ket value of a portfolio Usually, the literature specifi es four standard mar-ket risk factors: equity risk, or t he risk t hat stock prices would cha nge; interest rate risk, potential variations of interest rates; currency risk, the possibility of foreign exchange rate changes; and commodity risk, the risk that commodity prices (i.e., grains, metals, etc.) may modify Th is com-mon defi nition of market risk in the fi nancial sector diff ers between the bank business and the pension fund industry Th is section presents the disparity in the market risk vision between these two sectors.*

mar-* We consider that banking conventions are well known in the fi nance industry because they were widely discussed in the literature and studied by research during Basel II implications Insurance particularities of t he market risk vision is presented in a more e xhaustive man- ner b ecause t he v ision of m arket r isk me asurement i s to day i n e volution More over, ne w European prudential system preparations demand the research on market risk solutions in the insurance sector Th ese questions are nowadays very important In consequence, we pay more attention to the insurance rules and their particularities.

9.3.1 Market Risk in Banks and Economic Capital Measurement with VaR Techniques as Internal Model Tools: Basel II Experience

In the banking industry, the market risk is generally combined with “asset liquidity r isk,” wh ich r epresents t he r isk t hat ba nks ma y be u nable t o unwind a pos ition i n a pa rticular fi nancial p osition at or ne ar it s mar-ket value because of a lack o f depth or disruption in the market for that instrument Th is uncertainty is one of the most important category of risk facing banks Consequently, it has been the principal focus of preoccupa-tion among the sector’s regulators Th e new Basel accord* defi ned mar-ket risk as the risk of losses in on- and off -balance-sheet positions arising from movements in market prices, in particular, risks pertaining to inter-est rate-related instruments and equities in the trading book; and foreign exchange r isk a nd commodities r isk t hroughout t he ba nk Ba nks have

to retain a specifi c amount of capital to protect themselves against these

methodolo-gies that enable them to measure and manage market risks Basel II merates the VaR as one of the most important internal tools (with stress tests and other appropriate risk-management techniques) in monitoring market risk exposures and provides a common metric for comparing the risk being run by diff erent desks and business lines Th e VaR techniques should be integrated, as an internal model, into the bank’s economic capi-tal assessment, with the goal to serve as a regulatory capital measurement approach for market risk General market risk is thus a d irect function

enu-of t he o utput f rom t he i nternal VaR m odel i nitially de veloped b y a nd for banks Th e Basel committee on banking supervision in amendment

to the capital accord to incorporate market risks states rules for market risk

* Th e amendment to t he capital accord to i ncorporate market risks, Bank for I nternational Settlements, updated November 2005.

† Th e standardized approach to m arket r isk measurement was proposed by t he Basel mittee in April 1993 and updated in January 1996 Th e European Commission in its capi- tal adequacy directive (CAD) adopted very similar solutions known as the building block approach Th e m ain d iff erence b etween t he B asel c ommittee’s a nd t he Eu ropean Union’s approaches is in the weights for specifi c risk Th e capital charge is 8% (Basel) or 4% (EU) for equities, reduced to 4% (Basel) or 2% (EU) for well-diversifi ed portfolios Th e overall capital charge for market risk is simply the sum of capital charges for each of the exposures.

com-‡ Th e 1996 amendment to the capital accord provided for the supervised use of internal models

to establish capital charges Regulators considered that an internal model approach is able

to address more comprehensively and dynamically the portfolio of risks and is able to fully capture portfolio diversifi cation eff ects Th e goal was to more c losely a lign the regulatory assessment of risk capital with the risks faced by the bank.

measurement Part B* presents principles for the use of internal models to measure market risk in the banking sector Th e document specifi es a num-ber of qualitative criteria that banks have to meet before they are permit-ted to use internal models for capital requirement purposes (models-based approach) Th ese criteria concern among others the specifi cation of mar-ket risk factors, the quantitative standards, and the external validation.

exchange rates, equity prices, a nd commodity prices For interest rates, there must be a se t of risk factors corresponding to interest rates in each currency in which the bank has interest-rate-sensitive on- or off -balance sheet positions Banks should model the yield curve using one of a num-ber of generally accepted approaches, for example, by estimating forward rates of zero coupon yields Th e yield curve should be d ivided into vari-ous maturity segments in order to capture the variation in the volatility

of rates along the yield curve; there will typically be one risk factor responding to each maturity segment Banks must model the yield curve using a minimum of six risk factors; in general one risk factor is related to each segment of the yield curve Th e risk measurement system must incor-porate separate risk factors (diff erence between y ield curve movements, for example, government bonds and swaps) to capture spread risk

cor-In the case of equity prices, three risk factor specifi cations are possible

market-wide movements in equity prices Positions in individual securities

or in sector indices could be expressed in “beta-equivalents” relative to this market-wide index Th e second treats risk factors in a similar way, using more detailed risk factors corresponding to various sectors of the overall equity market Th e third, the most extensive approach, would be to have risk fac-tors corresponding to the volatility of individual equity issues Commodity prices’ risk factors, being specifi ed in the extensive approach should take account of the variation in the “convenience yield” between derivative posi-tions such as forwards, swaps, and cash positions in the commodity

* Th e document splits into parts A a nd B Part A of t he amendment describes the standard framework for measuring diff erent market risk components Th e minimum capital require- ment is expressed in terms of t wo separately calculated charges (expressed as percentage): one applying to the “specifi c risk” of each security (an adverse movement in the price of an individual security owing to factors related to the individual issuer), whether it is a short or

a long position; the other to t he interest rate risk in the portfolio (termed “general market risk”) where long a nd short positions i n d iff erent securities or i nstruments can be off set Capital charges are applied appropriately to the risk level of each category of assets.

Th e Ba sel co mmittee o n ba nking su pervision a pplied m inimum quantitative standards for the purpose of calculating market risk capital charge No particular type of model is prescribed by Basel II accord; for example, banks are free to use: variance–covariance matrices, historical simulations, or Monte Carlo simulations models Th e window of observa-tions should not be shorter than 1 year and data sets should be updated no less frequently than once every 3 months Th e VaR must be computed on

a daily basis with a 99th percentile confi dence level, for a 10 days horizon Banks may scale up 1 day VaR to 10 days by the square root of time, com-

From a t heoretical point of v iew, t he sc aling r ule needs to be l ed i n

i j

X ,

which serve to estimate ∆V distribution, must be not only i.i.d (as stated

in Fedor and Morel, 2006) but also normally distributed Th is additional

restriction for the h rule can be ex plained by the subsequent

reason-ing F ollowing F edor a nd M orel ( 2006), va riations o f l og r isk fac tors must be i dentically a nd i ndependently d istributed [i.i.d.] It means t hat

However, if the rule of the square root of time is applicable to a percentile

of the distribution of h-day price variations (and the VaR is a q-quantile

of ∆V distribution), variations in prices need to be normally and

indepen-dently distributed (n.i.d.).*

Banks, using internal models, calculate capital requirements in dance to the following formula:

where M is a regulatory capital multiplier that equals 3 and m, depending

on the quality of internal model’s estimation (backtesting), varies between [0,1] To prove the predictive nature of the model from subsequent experi-ence, banks are supposed to use validation techniques Backtesting—the comparison of the VaR model’s outputs (forecasts) with actual outcomes (realizations)—is a regulatory requirement under the Basel market amend-ment, add itionally a sl iding sc ale o f add itional c apital r equirements i s imposed if the model fails to predict the exposure correctly (three zones approach) Th e Basel committee on banking supervision requires banks to perform backtesting on a quarterly basis using 1 year (about 250 trading days) of data Th is process simply counts the actual number of times in the past year that the loss on the profi t and loss account (P&L) exceeded VaR

models that infl uence the fi nal amount of capital requirements As tors do not defi ne the technique of the modeling approach to be used for capital requirement purposes, it is important that the VaR model works as

regula-a good predictor It encourregula-ages bregula-anks to regula-apply good quregula-ality Vregula-aR models because more sophisticated techniques lower capital requirement amounts

sector will be discussed in the following sections of this chapter

9.3.2 Market Risk and Its Regulatory Approaches

to Capital in Pension Funds Sector

A market risk for the pension fund primarily relates to the risk of ment performance, deriving from market value fl uctuations or movements

invest-in invest-interest rates, as well as an invest-inappropriate mix of invest-investments, an uation of assets, or an excessive concentration of any class of asset A market risk can also arise from the amount or timing of future cash fl ows—from investments diff ering from those estimated, or from a l oss of value if the investment becomes worth less than expected A particular and important example of investment risk is when liabilities (which cannot be reduced) are backed by assets, such as equities, where the market value can fall

introduced i nto p ension f und op erations t hrough v ariations i n fi nancial markets Th ese variations are usually measured by changes in interest rates,

in equity indices, or in prices of various derivative securities However, its consequences for pension fund’s fi nancial wealth diff er from negative results

in the banking sector Th e eff ects of these variations on a pension fund can

be quite complex a nd c an a rise simultaneously f rom se veral sources, for example, company’s ability to realize suffi cient value from its investments

to allow it to satisfy liability expectations Subsequently, these approaches demand that the asset–liability matching (ALM) risk also be considered.

To understand well the market risk nature in the pension fund ness, its fi nancial policy particularities must be briefl y presented In our aim to remain general, we do not a nalyze t he fi nancial st rategies sepa-rately f or defi ned benefi t a nd defi ned co ntribution p lans We c an t hus stipulate that the market risk vision in the pension fund sector depends

busi-on the asset allocatibusi-on character: its lbusi-ong-term goals shorted however by regulation rules and provoking fi xed income instruments purchasing, the importance of liabilities, and “buy and hold” character

is the portfolio’s return optimization, by respecting the regulatory straints and the engagements represented in liabilities Pensions are sub-jected to a double requirement: the preservation of the nominal value of their short-term capital and the protection of the real value of this capital

con-at long horizon Th e confl ict between the short-term risk (evalucon-ated for regulatory purposes) and the need for a long period management imposes, from the beginning, certain number choices of the asset allocation.Blake (1999) stresses that pension funds and life assurance companies—the principal long-term investing institutions—have liabilities of the lon-gest duration Th ese liabilities are also similar in nature, although there will be qualitative diff erences (e.g., life policies provide for such features

as policy loan, and early surrender options in a way that pension funds do not, and defi ned benefi t schemes have options on the invested assets in a way that life policies do not) Th e greatest systematic risk faced by both sets of institutions arises from any mismatch of maturities between assets and liabilities To minimize the risks associated with maturity mismatch-ing, the two sets of institutions will tend to hold a substantial proportion

of long-term assets, such a s equities, property, and long-term bonds, in their portfolios Although given the specifi c nature of the options attached

to life policies, life companies will hold a relatively larger proportion of more capital-certain assets, such as bonds, in their portfolios than pension funds However, pension funds also have a number of important percent-age of interest rate positions in t heir investment portfolios Holding a n important portfolio of fi xed income assets (bonds) instead of equity posi-tions changes a market risk perception in the pension business

rule Th e pension investor thus buys fi nancial instruments that guarantee

an output, enabling him to respect its engagements toward its customers and its shareholders Its goal is neither the speculation, nor the trading, as