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

CHAPTER Financial Risk in Pension Funds: Application of Value at Risk Methodology Marcin Fedor CONTENTS 9.1 I ntroduction 9.2 Value at Risk 9.2.1 Value at Risk Definition 9.2.2 Value at Risk Measure (Algorithm) 9.2.2.1 Portfolio Exposure Measure Procedure 9.2.2.2 Uncertainty Measure Procedure 9.2.2.3 T ransformation Procedure 9.3 Market Risk in Banking and Pension Fund Sectors 9.3.1 Market Risk in Banks and Economic Capital Measurement with VaR Techniques as Internal Model Tools: Basel II Experience 9.3.2 Market Risk and Its Regulatory Approaches to Capital in Pension Funds Sector 9.4 VaR Measure in Pension Funds Business 9.4.1 M odification of Portfolio Exposure Measure Procedure: Taking into Account Changes of Portfolio Composition 186 188 188 189 189 191 191 192 193 196 199 199 185 © 2010 by Taylor and Francis Group, LLC 186 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 9.4.2 Sensibility of Uncertainty Measure Procedure: Importance of Assumptions Characterizing Risk Factors’ Distributions 9.4.3 Selection of Transformation Procedure: Choice of Appropriate VaR Model due to Data Sample Characteristics 9.5 C onclusions Appendix References 200 02 203 04 08 V a l ue a t R isk (Va R) s become a po pular r isk measure of financial risk It is also used for regulatory capital requirement purposes in banking and insurance sectors VaR methodology has been developed mainly for ba nks to control t heir short-term market risk A lthough, t he VaR is already widespread in financial industry, this method has not yet become a standard tool in pension funds However, like other financial institutions, pension funds recognize the importance of measuring their financial risks The aim of this chapter is to specify conditions under which VaR could be a good measure of long-term market risk After a description of a general VaR algorithm and the three main VaR methods, we present different aspects of market risk in banks and pension funds Ther efore, we propose necessary adaptations of VaR measures for pension f unds business specifications 9.1 INTRODUCTION Financial institutions’ activities entail a va riety of risks One of the most important categories of hazard is market risk, defined as the risk that the value o f a n i nvestment ma y decl ine d ue t o eco nomic cha nges o r o ther events that impact market factors (e.g., stock prices, interest rates, or foreign exchange rates) Market risk is typically measured using the value-atRisk (VaR) methodology In order to provide evidence of safety, firms have to maintain a m inimum amount of capital as a buffer against potential losses from their business activities, or potential market losses The literature distinguishes the economic capital from the regulatory capital The first is based on calculations that are specific to the company’s risk, while regulatory formulas are based on i ndustry averages t hat may or may not be su itable to a ny pa rticular company Moreover, the economic capital can be used for internal corporate risk-management goals as well as for regulatory purposes This © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 187 chapter focuses on economic c apital e stimations for ma rket r isk i n t wo main financial 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 o n a n eco nomic ba sis Basel II encourages bankers to use the VaR for both internal management and regulatory capital requirements This framework has also been copied by the new European prudential system in insurance (Solvency II) While pension f unds are not subject to banks’ capital adequacy requirements, a n umber o f si milar re strictions g overn de fined-benefit 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 regulations in the aim to harmonize with general financial framework Today, m ost o f pens ion f unds c an c alculate t heir o wn 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 t heir portfolio managers In the past decade, however, many of them have significantly increased the complexity of portfolios by broadening the menu of acceptable investments These i nvestments c an i nclude f oreign sec urities, co mmodities, futures, swaps, options, and collateralized mortgage obligations At the same time, well-publicized losses among pension funds have underlined the i mportance o f r isk ma nagement a nd m easuring per formance o n a risk-adjusted b asis The VaR a ppears t o offer 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 c apital requirement purposes, bec ause t his k ind of prudential model is already applicable to a number of financial sectors This could provide strong support for the integration of financial system and supervision The VaR is likely to continue to gain acceptance because it provides a forward-looking approach around which the supervision in all financial areas begins to be organized It potentially provides a common framework to assess t he relative risks t hat f unction i n 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 t he banking sector The VaR originated on der ivatives t rading desks and then spread to other trading operations The implementations of VaR de veloped at t hese i nstitutions naturally reflected t he needs a nd © 2010 by Taylor and Francis Group, LLC 188 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling characteristics of their trading operations, such as very short time horizons, generally liquid securities, and market-neutral positions In contrast, pension funds generally stay i nvested i n t he ma rket, can have i lliquid securities i n their portfolios, and hold positions for a long time Therefore, 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 modifications for pension funds industry operating with a long-term horizon Therefore, the aim of this chapter 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 framework, we can name only a few articles treating for the VaR concept in the long-term vision, characteristic for pension funds Here the most important ones are Albert et al (1996), Ufer (1996), Panning (1999), Dowd et al (2001), and Fedor and Morel (2006) The rest of this chapter is organized as follows: Section 9.2 introduces the VaR definition and methodology, currently in use in the banking sector Section 9.3 discusses market risk concepts and its applications in the 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 business Finally, we conclude and give a brief outlook for future research 9.2 VALUE AT RISK In this section, we briefly review the VaR approach, as has been traditionally used in the banking sector First, we define the VaR concept; next, we discuss t he VaR a lgorithm; finally, w e de scribe t he t hree m ost po pular VaR models in the banking sector 9.2.1 Value at Risk Definition Let Vt + h be the future (random) value of a portfolio of financial positions at time t + h Let Vt denote the (known) value of the corresponding portfolio at t he d ate of e stimation The cha nge i n t he ma rket va lue of a po rtfolio over a time horizon h is given by ∆V = Vt + h − Vt © 2010 by Taylor and Francis Group, LLC (9.1) Financial Risk in Pension Funds ◾ 189 The VaR of a po rtfolio i s t he pos sible ma ximum loss, noted a s VaR h(q), over a g iven time horizon h with probability (1 − q) The well-known formal definition of a portfolio VaR is P ⎡⎣ ∆V ≤ VaR h (q )⎤⎦ = − q (9 2) and therefore VaR h (q) = Rh−1 (1 − q) (9.3) where R h−1 is the inverse of the distribution function of random variables ∆V, also called P&L distribution function.* Therefore, the VaR estimations depend on the Rh distribution 9.2.2 Value at Risk Measure (Algorithm) Although the VaR is an easy and intuitive concept, its measurement is a challenging st atistical p roblem I n t his pa ragraph, w e d iscuss a p rocess that i s co mmon t o a ll VaR c alculations This a lgorithm is composed of three procedures: • The measure of portfolio exposure: the mapping of all financial positions present in the investment portfolio to risk factors • The measure of uncertainty: the characterization of the probability distribution of risk factor variations • The computation of the VaR for investment portfolio 9.2.2.1 Portfolio Exposure Measure Procedure First, we describe portfolio exposure by a mapping procedure (the representation of investment portfolio positions by risk factors) Assume t hat the investment portfolio is composed from m financial positions Let us define vm,t+h as t he f uture random va lue of t he financial position at t ime t + h Let vm,t be a value of the corresponding position at the date of estimation Suppose the portfolio has holdings ωi in m financial positions Then Vt = ω1v1,t + * Profit and loss distribution function © 2010 by Taylor and Francis Group, LLC + ω mvm,t (9 4) 190 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Vt + h = ω1v1,t + h + + ω mvm,t + h (9 5) In general, investment portfolios are complex, and their analysis becomes infeasible if we treat directly all financial positions Thus, a more manageable approach of modeling the portfolio’s behavior is to represent numerous individual positions by a limited number of specific risk factors They can be defined as fundamental variables of the market (e.g., equity prices, interest rates, or foreign exchange rates) that determine (by their modeling) prices of financial 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 us define Xi,t+h as the future random value of the risk factor at time t + h and Xi,t as the value of the corresponding risk factor at date t Each asset vm,t or vm,t+h held by the portfolio must be expressed in terms of risk factors Thus, t here must ex ist pricing formulas (valuation f unctions) F and G for each position vm,t or vm,t+h such that vm,t = Fm(X1,t,…, Xn,t) and vm,t+h = Gm(X1,t+h,…, Xn,t+h) According to (9.4) and (9.5), values of the portfolio Vt an d Vt+h a re l inear po lynomials o f pos ition va lues vm,t an d vm,t+h; thus we can express Vt and Vt+h in terms of risk factors: m m i =1 i =1 Vt = ∑ ωi vi ,t = ∑ ωi Fi ( X1,t ,…, Xn,t ) m m i =1 i =1 Vt + h = ∑ ωi vi ,t + h = ∑ ωi Gi ( X1,t + h ,…, Xn,t + h ) (9.6) (9.7) These are functional relationships that specify the portfolio’s market values Vt and Vt+h in terms of risk factors Xi,t and Xi,t+h Shorthand notations for the relationships (9.6) and (9.7) are Vt = f (X 1,t ,…, X n ,t ) (9 8) Vt + h = g (X 1,t + h ,…, X n ,t + h ) (9 9) Relationships (9.8) a nd (9.9) a re c alled portfolio mapping a nd f unctions 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 © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 191 categories of assets (e.g., options), therefore, neither function f nor g is linear any more 9.2.2.2 Uncertainty Measure Procedure Functions f and g not e stimate portfolio r isk bec ause Xi,t and Xi,t+h not contain any information relating to the market volatility We obtain the information about this uncertainty by risk factor distributions Let us define ∆Xi as the change of the risk factor’s price over a time horizon h Thus, the ∆Xi distribution explains market behavior We can characterize this distribution 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 t he w indow of observations ( length ex pressed i n ⎡T ⎤ trading days), K is defined as K = ⎢ ⎥ and {∆Xi }Kj =1 are the time series of ⎣h ⎦ K returns over h days for each risk factor (i = 1,…,n) Generally, this financial 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 for a ll r isk fac tors {∆Xi }Tj =1 These t ime series serve to est imate t he ∆Xi distribution The 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 The portfolio mapping functions map n-dimensional spaces of risk factors to the one-dimensional spaces of the portfolio’s market value Although we need to characterize the distribution of ∆V, mapping functions simply give the value of Vt and Vt+h Thus, we need to apply portfolio mapping functions to the entire joint distribution of risk factors, with the aim of obtaining ∆V distribution Consequently, we define ∆V as a function of risk factors ∆V = f (∆X1 ,…, ∆Xn ) (9 10) ∆V = g (∆X1 ,…, ∆Xn ) (9 11) where ∆X1, …, ∆Xn are variations of portfolio’s risk factors over the period h A t ransformation p rocedure co mbines t hus t he po rtfolio’s ex posure (composition) w ith t he cha racterization o f ∆Xi d istribution i n ord er t o describe the ∆V distribution Next, we find q-quantile of the portfolio distribution t hat is equal to t he VaR metric The t hird procedure estimates portfolio risk © 2010 by Taylor and Francis Group, LLC 192 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling In brief, we face two problems while calculating VaR First, we map portfolio positions to the risk factor by f and g functions, which reflect the portfolio’s composition On its own, however, it cannot estimate portfolio risk because (9.8) and (9.9) not contain any information relating to the market volatility We obtain t his information in t he risk factor distributions We characterize the ∆Xi distribution by historic data We use time series of risk factors {∆Xi }Kj =1 for this purpose However, on its own, the ∆Xi distribution cannot measure the portfolio risk because it is independent of the portfolio’s composition Thus, as soon as we have estimated the distribution of risk factors, we continue on to the third procedure by converting ∆Xi description 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 categorized according to t he t ransformation procedures t hey employ Even though t hey f ollow t he g eneral st ructure p resented abo ve, t hey em ploy different methodologies for the transformation procedure The presentation 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 defined as exposure to the uncertain market value of a portfolio Usually, the literature specifies four standard market r isk fac tors: eq uity r isk, o r t he r isk t hat st ock p rices w ould 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., g rains, metals, etc.) may modify This common definition of market risk in the financial sector differs between the bank business a nd t he pension f und i ndustry This sec tion presents t he disparity in the market risk vision between these two sectors.* * We consider that banking conventions are well known in the finance industry because they were widely discussed in the literature and studied by research during Basel II implications Insurance particularities of t he market risk v ision is presented in a more e xhaustive manner 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 These questions are nowadays very important In consequence, we pay more attention to the insurance rules and their particularities © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 193 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 financial p osition at or ne ar it s m arket value because of a lack o f depth or disruption in the market for that instrument This uncertainty is one of the most important category of risk facing banks Consequently, it has been the principal focus of preoccupation a mong t he sec tor’s regulators The new Ba sel accord* defined market risk as the risk of losses in on- and off-balance-sheet positions arising from movements in market prices, in particular, risks pertaining to interest rate-related instruments and equities in the trading book; and foreign exchange r isk a nd co mmodities r isk t hroughout t he ba nk Ba nks ve to retain a specific amount of capital to protect themselves against these risks This capital charge may be e stimated by standardized methods† or by internal models.‡ This is why banks should have internal methodologies that enable them to measure and manage market risks Basel II enumerates the VaR as one of the most important internal tools (with stress tests a nd other appropriate r isk-management tech niques) i n monitoring market risk exposures and provides a common metric for comparing the risk being run by different desks and business lines The VaR techniques should be integrated, as an internal model, into the bank’s economic capital assessment, with the goal to serve as a regulatory capital measurement approach for ma rket r isk G eneral ma rket r isk i s t hus a d irect f unction of t he o utput f rom t he i nternal VaR m odel i nitially de veloped b y a nd for ba nks The Ba sel committee on ba nking supervision i n a mendment to the capital accord to incorporate market risks states rules for market risk * The a mendment to t he c apital a ccord to i ncorporate m arket r isks, B ank for I nternational Settlements, updated November 2005 † The s tandardized appro ach to m arket r isk me asurement w as prop osed by t he B asel c ommittee in April 1993 a nd updated in Ja nuary 1996 The European Commission in its capital adequacy d irective (CAD) adopted very si milar s olutions k nown a s t he bu ilding blo ck approach The m ain d ifference b etween t he B asel c ommittee’s a nd t he Eu ropean U nion’s approaches is in the weights for specific risk The capital charge is 8% (Basel) or 4% (EU) for equities, reduced to 4% (Basel) or 2% (EU) for well-diversified portfolios The overall capital charge for market risk is simply the sum of capital charges for each of the exposures ‡ The 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 p ortfolio d iversification effects The go al w as to more c losely a lign t he re gulatory assessment of risk capital with the risks faced by the bank © 2010 by Taylor and Francis Group, LLC 194 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling measurement Part B* presents principles for the use of internal models to measure market risk in the banking sector The document specifies a number of qualitative criteria that banks have to meet before they are permitted to use internal models for capital requirement purposes (models-based approach) These criteria concern among others the specification of market risk factors, the quantitative standards, and the external validation The specification o f r isk fac tors co ncerns s eparately: i nterest r ates, exchange r ates, eq uity prices, a nd co mmodity prices F or i nterest r ates, 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 number of generally accepted approaches, for example, by estimating forward rates of zero coupon yields The yield curve should be d ivided into various maturity segments in order to capture t he variation in t he volatility of rates along the yield curve; there will typically be o ne risk factor corresponding 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 The risk measurement system must incorporate separate r isk f actors (difference be tween y ield c urve movements, for example, government bonds and swaps) to capture spread risk In the case of equity prices, three risk factor specifications are possible The first co ncerns c apturing t he m onitoring o f ma rket i ndex, ex pressing 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 The second treats risk factors in a similar way, using more detailed risk factors corresponding to various sectors of the overall equity market The third, the most extensive approach, would be to have risk factors corresponding to the volatility of individual equity issues Commodity prices’ risk factors, being specified i n t he ex tensive approach should t ake account of the variation in the “convenience yield” between derivative positions such as forwards, swaps, and cash positions in the commodity * The document splits i nto parts A a nd B Pa rt A of t he a mendment describes t he standard framework for measuring different market risk components The minimum capital requirement is expressed i n terms of t wo separately c alculated charges (expressed a s percentage): one applying to the “specific 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; t he ot her to t he interest rate risk in t he portfolio (termed “general market risk”) w here lon g a nd s hort p ositions i n d ifferent s ecurities or i nstruments c an b e o ffset Capital charges are applied appropriately to the risk level of each category of assets © 2010 by Taylor and Francis Group, LLC 196 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling where M is a regulatory capital multiplier that equals 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 experience, 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 amendment, 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) The Basel committee on banking supervision requires banks to perform backtesting on a quarterly basis using year (about 250 t rading days) of data This process simply counts the actual number of times in the past year that the loss on the profit and loss account (P&L) exceeded VaR The formula (9.12) shows t he i mportance of t he predictive nature of VaR models that influence the final amount of capital requirements As regulators not define the technique of the modeling approach to be used for capital requirement purposes, it is important that the VaR model works as a good predictor It encourages ba nks t o a pply good q uality VaR m odels because more sophisticated techniques lower capital requirement amounts The crucial role of backtesting for VaR estimation purposes in pension funds 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 t he pension f und primarily relates to t he risk of investment performance, deriving from market value fluctuations or movements in interest rates, as well as an inappropriate mix of investments, an overvaluation 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 flows—from investments differing f rom t hose estimated, or f rom a l oss of va lue i f t he 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 The market risk, just as in the banking sector, is thus defined as the risk introduced i nto p ension f und op erations t hrough v ariations i n financial markets These 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 financial wealth differ from negative results in the banking sector The effects of these variations on a pension fund can be q uite co mplex a nd c an a rise s imultaneously f rom se veral so urces, f or example, company’s ability to realize sufficient value from its investments © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 197 to a llow it to satisfy liability expectations Subsequently, t hese approaches demand that the asset–liability matching (ALM) risk also be considered To understand well the market risk nature in the pension fund business, its financial policy particularities must be b riefly presented In our aim t o r emain g eneral, w e d o n ot a nalyze t he financial st rategies sepa rately f or defined benefit a nd defined co ntribution p lans We c an t hus stipulate t hat t he market risk v ision in t he pension f und sector depends on the asset allocation character: its long-term goals shorted however by regulation rules and provoking fi xed income instruments purchasing, the importance of liabilities, and “buy and hold” character The o bjective o f t he financial ma nagement i n pens ion f unds sec tor is t he portfolio’s return optimization, by respecting t he regulatory constraints and the engagements represented in liabilities Pensions are subjected to a d ouble requirement: the preservation of the nominal value of their short-term capital and the protection of the real value of this capital at long horizon The conflict be tween t he short-term r isk (evaluated 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 longest duration These liabilities a re a lso similar i n nature, a lthough t here will be q ualitative differences (e.g., life policies provide for such f eatures as policy loan, and early surrender options in a way that pension funds not, and defined benefit schemes have options on the invested assets in a way t hat l ife policies not) The g reatest systematic r isk faced by both sets of institutions arises from any mismatch of maturities between assets and liabilities To minimize the risks associated with maturity mismatching, the two sets of institutions will tend to hold a substantial proportion of long-term a ssets, such a s eq uities, property, a nd long-term bonds, i n their portfolios Although given the specific 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 percentage o f i nterest r ate pos itions i n t heir i nvestment po rtfolios Holding a n important portfolio of fi xed income assets (bonds) instead of equity positions changes a market risk perception in the pension business The pension f und financial policy follows generally a “ buy a nd hold” rule The pension investor thus buys financial 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 © 2010 by Taylor and Francis Group, LLC 198 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling in the banking sector The financial portfolio is more stable than the banking environment—pension f unds not have t he same reactivity compared to trading desks Indeed, a trader can easily release his positions, which justifies the estimate of short-term market risk Pensions cannot As underlined above, pension funds are principally facing liability risks while banks are mostly confronted with asset risks This distinction is very general because banks also have liability risks for many reasons, for example, interest rate or foreign currency risk; they can be exposed because of their own debt Pension f unds a re a lso confronted w ith a ssets r isks, for example, t heir per formance i s a ffected by financial ma rket fluctuations However, liability risks are the most important hazards they have to face Taking into consideration the correlation between assets and liabilities is thus crucial in pension funds The ex ample o f a sset–liability m ismatch r isk i n pens ion f und i s a n interest rate risk The impact of interest rate risk on an investment portfolio cannot be co nsidered in isolation to the effect on the valuation of liabilities and guarantees Therefore, it is critical for risk interaction to be properly reflected in the models This is a significant difference, as banking models tend to focus sepa rately on t he key risk a reas Commercial banks also face asset and liability mismatch risk Deposits constitute liabilities that may be due in a short term These financial sources are transformed in loans with longer maturities that make assets hard to recover in a sh ort term For that reason, liquidity cries can provoke insolvency and lead to failures in mechanism in a short term In the pension fund, a subst antial pa rt o f a ssets co uld be e asily r ealized a nd a n i mportant part of liabilities is not due in a short term (has a long-term “maturity”) Thus, l iquidity cries c annot precede i nsolvency i n pensions sec tor a nd asset liquidity risk is not, as stressed before, a primary preoccupation of control authorities Pension funds supervision gradually tends toward risk-based supervision, originated in the supervision of banks and increasingly extended to other types of financial intermediaries This tendency is closely associated with movement toward the integration of pension supervision with banking and other financial services into a single authority However, the application of prudential rules to pension f unds requires some modifications (e.g., see Brunner et al 2008 for an overview of the private pension system of t he four countries, which introduced t he risk-based supervision) The authors stress that one of the main objectives of risk-based supervision in banking a nd pens ion f unds i s t o ensu re t hat i nstitutions ad opt so und © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 199 risk-management p rocedures a nd h old a n a ppropriate l evel o f c apital Thus, pens ion supervisors w ill fac e cha llenges t hat a re i n ma ny a spects similar to those faced by bank (and insurance) supervisors In consequence, the adaptation of the VaR in pension funds seems to be one of the first steps into building the integrated supervision framework 9.4 VAR MEASURE IN PENSION FUNDS BUSINESS The VaR wa s i ntroduced f or t he first t ime i nto pens ion f und su pervisory framework in Mexico where it is used to estimate the volatility risk Mexico requires t he pens ion f unds t o r etain a sset fluctuations w ithin a prescribed level, set by t he VaR However, Mexico’s f ramework presents many inconveniences, for example, the VaR is calculated by the supervisor on a daily basis, because the VaR model was taken directly from banking sector w ithout significant modifications for pension f und specifications We challenge this problem in this chapter where we propose a necessary adaptation of the VaR measure for pension fund characteristics We discuss all the three procedures presented in Section 9.2.2 9.4.1 Modification of Portfolio Exposure Measure Procedure: Taking into Account Changes of Portfolio Composition Throughout the rest of the chapter, we consider that the investor does not change any portfolio’s positions over h horizon (the investor does not sell or buy any assets between date t and date t + h) The first procedure of the VaR measure (Section 9.2.2.1) maps the portfolio position to the risk factor by the mapping function that reflects the portfolio composition In the banking sector, due to the short-term character of estimations (one day), the composition of Vt and Vt+h are identical (assuming that the investor does not change positions), in consequence, functions f and g remain equal for (9.8) and (9.9) In pension fund context, where portfolios include an important percentage of interest rate instruments and VaR estimations have a long-term character, therefore Vt and Vt+h cannot be supposed to be similar Proposition 9.1: The n ominal price a nd q uantities o f pos itions i n t he investment portfolio including interest rate instruments change over VaR estimation h orizon ( even t he ma rket co nditions r emain u nchanged a t times t and t + h) The difference between Vt and Vt+h a re c aused b y t he sum of cash flows generated by interest rate positions (in the case of bonds, coupons, and face values paid at maturities) as well as durations diminution, which changes the price of interest rate instruments over horizon h © 2010 by Taylor and Francis Group, LLC 200 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling It follows from Proposition 9.1 that Vt ≠ Vt+h (if the portfolio contains interest rate instruments) even if the investor does not change positions and the market conditions are similar at times t and t + h This particularity changes the risk profi le of the investment portfolio (f ≠ g) and has to be taken into consideration We propose to analyze the situation (the composition) of t he i nterest r ate portfolio a t t he en d of t he VaR horizon—at time t + h t hus, when we consider t he portfolio mapping f unction specified in (9.9) We are supposed to measure the real risk profi le of the portfolio Thus, the portfolio behavior will depend on its structure at date h, represented by (9.9) This statement also modifies VaR estimations by the square root of time rule Even though we should calculate one-day VaR, we should use portfolio representation at date t + h We c an e valuate i nterest r ate p osition p rices a t d ate t + h if we know their prices at time t (which is known because it is a current price), the zero coupon yield curve and all cash flows generated by bond in the portfolio Thus, we calculate asset values of each vm,t + h conditional from vm,t (information at time t) Next, we find Vt + h and function g specified in (9.9) 9.4.2 Sensibility of Uncertainty Measure Procedure: Importance of Assumptions Characterizing Risk Factors’ Distributions According to Section 9.2.2.2, the VaR measure depends on the characterization of the ∆Xi distribution Thus, VaR depends on the expected value of variations of risk factors over t he period h E[∆Xi] ex presses t rends (drifts) of risk factors’ prices in the future These expected returns must be assumed while VaR is estimated by pa rametric methods (variance– covariance and Monte Carlo approaches), and they are included in the historic data set distribution when VaR is measured with nonparametric models (historical simulations) The forecast of these trends are problematic and there does not exist one incontestable and unique methodology For short periods, the expected return is very weak Thus, in the banking sector, where the VaR is often calculated on one-day basis, the assumption o f a n ull ex pected r eturn i s bei ng made I n t he pens ion f und sec tor, VaR must be c alculated for longer horizons for which the portfolio’s expected return becomes significant While estimating VaR, we cannot take into account the expected return based on historical data (time series of {∆Xi }Kj =1) bec ause t hese expected returns change over time and depend on the length of the window of observations K The VaR a mount would become subjective © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 201 and biased Thus, expected returns must be supposed We propose two possibilities: • The ch oice o f n il ex pected r eturns for a ll r isk fac tors: t his n eutral assumption does not pass a ny judgment about f uture (favorable or adverse) changes of the market situation • The ch oice o f a ssumed ex pected r eturns f or r isk fac tors: ex pected returns should be forecasted by independent experts (this independence would g uarantee t he reliability of estimations), or should be based o n t he ma rket co nsensus a t t he d ate o f c alculation (e.g., b y using forward rates or informations deduced from options) The choice of E[∆Xi] is very important because it strongly influences the VaR estimations and, consequently, the portfolio risk It is thus advisable to make it with the greatest prudence The first option, when ∀i E[∆Xi] = 0, underestimates the VaR metric in a period of economic growth and, inversely, i t o verestimates VaR m etric wh en p rices dec rease The determination of expected returns for all risk factors, ∀i E[∆Xi] = ai, seems to be q uite p roblematic bec ause i t depen ds o n sub jective p references a nd demands solid analysis All these choices should be done with the greatest prudence because they impact directly final VaR calculations The seco nd st atement co rresponding t o E[∆Xi] co ncerns t he i nvestment portfolio including interest rate positions In these cases, the distribution of random variables ∆Xi for corresponding interest rate positions can be estimated either by variations of zero coupon bonds’ prices or by variations of zero coupon interest rates Th is choice is not equivalent Proposition 9.2: Measuring the risk of interest rate positions by interest rate variations or price variations is not equal: If we use variations of prices as risk factors for interest rate positions, we underestimate investment portfolio risk If we use va riations of i nterest rates as r isk fac tors for i nterest rate positions, we overestimate the risk of investment portfolio This choice should be taken with care Therefore, the selection of zero coupon interest rate variations seems to be the preferred solution © 2010 by Taylor and Francis Group, LLC 202 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling 9.4.3 Selection of Transformation Procedure: Choice of Appropriate VaR Model due to Data Sample Characteristics Basel II amendments not prescribe any particular type of VaR model for regulatory capital calculations and banks can use any method they prefer In the pension fund sector, many VaR models cannot be used for long-term estimations due to characteristics of historic data In consequence, many VaR techniques cannot be used for economic capital measurement, especially for capital charges estimation purposes, because results of the calculations are biased We propose a procedure for the choice of VaR method (among t he t hree tech niques presented i n Section 9.2.2) best ad apted to the long-term character of pension fund estimations The selection of the adequate model is based on data set proprieties Let us assume that we dispose of the one-day variations (daily returns) of all risk factors at the moment of VaR calculations As underlined in Fedor T and Morel (2006), all observed outcomes in the time series {∆ Xi } j =1 (w here i = 1,…,n) must be i.i.d If time series are not i.i.d., in general, in the banking sector w e u se a utoregressive co nditional h eteroscedasticity/generalized autoregressive conditional heteroscedasticity (ARCH/GARCH) processes that propose a spec ific pa rameterization f or t he beha vior o f r isk fac tors They a llow for t ime-varying conditional volatility: e ven t he u nconditional one-day returns are not i.i.d., suitably conditioned returns became normal ARCH/GARCH models make the assumption of i.i.d standardized residuals (the most generally used distribution is the standard normal) and specify the d istribution o f r esiduals.* H owever, t hese tech niques c annot be u sed for regulatory capital calculations in the pension fund sector According to Christoffersen et al (1998) and Christoffesen and Diebold (1997), if the shortterm application of GARCH models appears efficient, volatility is effectively not forecastable for horizons longer than 10 or 15 trading days (depending on the asset class) More detailed description is beyond the scope of this chapter The assumption that all time series are i.i.d allows estimating VaR with methods presented in Fedor and Morel (2006) We can use Monte Carlo simulations ( based on t he h istorical d istribution of r isk f actors) or t wo other methods based on {∆Xi }Kj =1, where (i = 1,…,n) In practice, variance– covariance and historical simulation approaches are inapplicable because the time series of 1-year returns, which allows the estimation of risk factor distributions, are too short (e.g., if we dispose of a 10 years data sample, * The definition and study of t emporal aggregations of G ARCH processes were presented by Drost and Nijman (1993) © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 203 we have 10 variables to estimate risk factor distributions) In these cases, the VaR is very sensitive to data set changes Sometimes, VaR metric cannot be calculated (e.g., in the historical simulation approach we cannot calculate VaR1 year(99.5%) if we have a 10 year-long window of observations) If we can formulate an additional assumption that all time series of risk factors are n.i.d., we can use Monte Carlo simulations based on the normal distribution or we can calculate 1-year VaR by scaling 1-day VaR with the h rule (as shown in Sections 9.3.1 and 9.4.1) 9.5 CONCLUSIONS This chapter discussed market risk measurement in the pension fund sector v ia VaR methodology A lthough VaR models have proved to be u seful to banking regulators for calculating the market risk, they cannot be directly applied in the pension fund service There are similarities in the risks to which banks and pension funds are exposed However, some notable differences between these two sectors are also important This chapter proposed t he necessary adaptation of t he VaR i n a n a im to reduce controversy remaining due to the limited linkages between such a short-term measure and the longer horizon of pensions In t he first step , w e d iscussed t he u se o f t he VaR f or i nternal ma rket risk management; but we also focus our interest on VaR adjustment for regulatory capital requirement purposes Since the VaR has already emerged f or ba nks a nd i nsurance co mpanies, i t w ill be t ransplanted to pension s ystems i n t he f uture This is bec ause t he t ypes of r isk a nd the associated method that focuses on solvency measurement are quite similar in all fi nancial sectors We already observed these tendencies in Mexico where the concept of VaR was applied as an attempt to contain downside losses in pension funds In the future, the VaR will be one of the most important tools of the risk-based su pervision f ramework i n pens ion f unds I ts i ntroduction will certainly have an impact on risk-management practices, regulatory approach, and even financial policies of pension funds (as shown in Brunner et a l 2008, pension f und ma nagers i n Mexico have made u se of the greater freedom, given by VaR, by moving away from very basic portfolios and investing more in domestic and foreign equity, as well as foreign fixed income instruments) Th is chapter only opens the discussion on VaR adaptation in pension funds, which should be continued in the further research © 2010 by Taylor and Francis Group, LLC 204 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling APPENDIX: PROOFS Throughout the rest of the appendix, we make an assumption that we know financial market conditions (e.g., the yield curve or the discount rate) at time t and that these market conditions remain unchanged at times t and t + h We begin by proving three useful lemmas Lemma 9.1: If market conditions remain unchanged at times t and t +h, the nominal price of zero coupon bond changes with time Proof of Lemma 9.1: We analyze the evolution of the zero coupon bond over the time horizon h Let vtzc (C , τ) be t he time t price of a z ero coupon bond The value of this bond is a function of its coupon and maturity date τ (and t ≤ τ) Thus, vtzc (C , τ) = C [1 + rt (τ)]t −τ , where C is the face value and rt(τ) is the time t rate of interest applicable for period τ – t (e.g., instantaneous forward interest rate) The price vtzc (C , τ) changes over the time horizon h: • If t + h≥ τ, the zero coupon bond is not present any more in the portfolio at time t + h, but the holder of this bond has received a cash flow C at time t +τ • If t + h< τ, the zero coupon bond is still in the portfolio at time t + h, with new theoretical price vtzc+ h (C , τ) = C [1 + rt + h (τ)] t + h −τ Even if the yield curve does not change between times t and t + h, the time left until maturity and the yield to maturity rate (rt+h(τ) = rt(τ − h) ≠ rt(τ) if the yield curve is not flat) have varied The price of the zero coupon bond depends on different variables at times t and t + h Although we make the assumption of stable market conditions, the zero co upon bo nd’s p rice cha nges o ver t he t ime h orizon h H ence, vtzc (C , τ) ≠ vtzc+ h (C , τ) , which completes our proof Lemma 9.2: If the investor does not change any position in the portfolio between date t and date t +h and the market conditions remain unchanged at times t and t +h: Nominal prices and quantities of equities and similar positions are the same in time t and time t +h Nominal p rices a nd q uantities o f fi xed i ncome pos itions ( interest rate instruments) changes with time © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 205 Nominal p rice a nd co mposition o f i nvestment po rtfolio i ncluding interest r ate pos itions ( bonds, loans, a nd m ortgages) cha nges over time period h Proof of Lemma 9.2: First, we show (1) Assume vteq (Dk , rt ) as the value of the equity position at date t The theoretical price of this equity position is t he p resent va lue o f i ts f uture d ividends ( under t he co ndition o f c era tainty) Then, vteq (Dk , rt ) = ∑ k =1 Dk (1 + rt )− k , wh ere Dk re presents f uturedividends a nd rt i s a de sired r ate o f r eturn (discount r ate) a t d ate t L et a vteq+h(Dk , rt +h ) = ∑ k =1 Dk (1 + rt +h )− k be t he p rice o f eq uity pos ition a t t ime t + h I f portfolio positions a nd ma rket conditions a re equal at d ate t and t + h, t hen rt = rt + h, h ence vteq (Dk , rt ) = vteq+ h(Dk , rt + h ) , wh ich completes t he first part of the proof Second, we show (2) Let vtbd(C , τ) be the time t price of an interest rate (fi xed income) position Throughout the rest of the proof, we will consider that all interest rate positions (bonds and any other fi xed income instruments) in the pension fund’s investment portfolio can be represented as a portfolio of zero coupon bonds If the investor holds one bond generating in the future b cash flows (the investor receives a coupon C once per period and a face value at the bonds maturity), these cash flows can be regarded separately as b distinct zero coupon bonds with face values Ci and maturities t −τi b b τi (and i = 1, …,b) Then, vtbd(C , τ) = ∑ i =1 vizc,t (Ci , τi ) = ∑ i =1 Ci ⎡⎣1 + ri ,t (τi )⎤⎦ , where Ci i s t he fac e va lue a nd ri,t(τi) i s t he t ime t i nstantaneous te o f interest applicable for periods τi − t Using Lemma 9.1, we have • If t + h ≥ τb, t hen t he bond is not present a ny more in t he portfolio at the date t + h, but the investor has already received b cash flows of amounts Ci (representing coupons a nd t he fac e va lue of t he bond) since times t + τi (and i = 1, …,b) • If t + h< τb, t hen t he bo nd i s st ill co nsidered t o be i n t he po rtfolio a t t ime t + h Then, ∃ c ∈{0,…, b − 1} such a s τc ≤ t + h < τc+1 and τ0 =t, t he h older s r eceived c cas h flows Cj s ince t imes t +τj (a nd j = 1,…,c) The n ew ma rket p rice o f t he bo nd i s b b vtbd+ h (C , τ) = ∑ i =c +1 vizc,t + h (Ci , τi ) = ∑ i =c +1 Ci [1 + rt + h (τi )]t + h −τi The market price of the zero coupon bond does not depend on the same variables at dates t and t + h, even though the yield curve does not change Hence, b b if ∑ i =1 vizc,t (Ci , τi ) ≠ ∑ i =1 vizc,t + h (Ci , τi ), then vtbd (C , τ) ≠ vtbd+ h(C , τ), which completes the second part of our proof © 2010 by Taylor and Francis Group, LLC 206 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Third, we show (3) Let us denote Vt as the market price of an investment po rtfolio a t d ate t and Vt+h a s t he ma rket p rice o f t he i nvestment portfolio at date t + h If the portfolio contains interest rate positions, then vtbd (C , τ) ≠ vtbd+ h(C , τ) Hence, Vt ≠ Vt+h, which completes the third part of our proof Lemma 9.3: In the case of an interest rate position, we can measure the risk t hrough p rice cha nges o r i nterest r ate va riations I n a first option, we evaluate changes of prices using interest rate variations; in the second case, we c alculate i nterest r ate cha nges u sing price va riations However, the choice between two options is not equal: If we use interest rate changes to measure the variation of prices, we overestimate the expected value of prices’ variations, consequently, we underestimate the risk If we use price changes to measure the variation of interest rates, we overestimate the expected value of interest rates’ variations, in consequence, we overestimate the risk Proof of Lemma 9.3: First, we show (1) Let us study a zero coupon bond yielding w ith maturity τ Let us note vtzc(τ) and vtzc+ h(τ) as prices of this zero coupon bond at dates t an d t + h C onsequently, rtzc(τ) and rtzc+ h(τ) are i nterest rates of t he z ero coupon bond at d ates t and t+h By defini−τ −τ tion, w e ve vtzc (τ) = ⎡⎣1 + rtzc (τ)⎤⎦ and vtzc+ h(τ) = ⎡⎣1 + rtzc+ h(τ)⎤⎦ , t herefore − − rtzc(τ) = ⎡⎣vtzc(τ)⎤⎦ τ − an d rtzc+ h (τ) = ⎡⎣vtzc+ h (τ)⎤⎦ τ − L et u s define random va riables ∆vzc(τ) a nd ∆rzc(τ) a s f ollows ∆v zc (τ) = vtzc+ h (τ) − vtzc (τ) and ∆r zc (τ) = rtzc+ h (τ) − rtzc (τ) S uppose t hat i nterest r ates a nd prices of the zero coupon bond are always strictly positive Then ∆v zc (τ) = vtzc+ h (τ) − vtzc (τ) > −vtzc (τ) and ∆r zc (τ) = rtzc+ h (τ) − rtzc (τ) > −rtzc (τ) −τ Let us define the function δ on] − rtzc (τ); + ∞[, as δ( x ) = ⎣⎡1 + rtzc (τ) + x ⎦⎤ − −τ ⎡⎣1 + rtzc(τ)⎤⎦ W e t hen ve ∆vzc(τ) =δ[∆rzc(τ)] W hen w e der ive function δ at two times, we have the second derivative of d2 function δ as δ(x ) = (τ2 + τ)(1 + rtzc (τ) + x )−τ−2 > 0, wh ich m eans dx that the f unction δ is strictly convex Using Jensen inequality, we obtain © 2010 by Taylor and Francis Group, LLC Financial Risk in Pension Funds ◾ 207 nce, E[∆v zc (τ)] = E[δ(∆r zc (τ))] > δ(E[∆r zc (τ)]) He E[∆v zc (τ)] > δ(E[∆r zc (τ)]), which completes the first part of our proof Second, we show (2) We study the zero coupon bond from (1), and we use zc zc zc zc zc zc the s ame va riables: vt (τ), vl + h(τ), rt (τ), rt + h(τ), ∆v (τ), and ∆r (τ) , ] − vtzc (τ); + ∞[, as γ( x ) = and ∆rzc(τ) Sup pose f unctions γ on − − [vtzc (τ) + x] τ − [vtzc (τ)] τ Then, w e ve ∆rzc(τ) =γ[∆vzc(τ)] L et u s der ive function γ at two times Ther efore, − −2 d2 τ +1 γ( x ) = (vtzc (τ) + x ) τ > dx τ which m eans t hat f unction γ i s st rictly co nvex U sing J ensen inequality, w e ve E[∆r zc (τ)] = E[γ( ∆v zc (τ))] > γ( E[∆v zc (τ)]) H ence, E[∆r zc (τ)] > γ (E[∆v zc (τ)]), which completes the second part of our proof Proof of Proposition 9.1: Let us define Xi,t as risk factor’s value at time t and Xi,t+h as its value at date t + h(and i = 1, …, n) The change in value of a risk factor over period h is then ∆Xi = Xi,t+h –Xi,t Note that Vt and Vt+h are values of a portfolio of financial positions, respectively, at times t and t + h Using Equations (9.8) and (9.9), we have Vt = f(X1,t,…, Xn,t) and Vt+h = g(X1,t+h,…, Xn,t+h) U sing (3) i n L emma 9.2, w e ve Vt ≠ Vt+h if p ortfolio includes interest rate positions, thus f ≠ g Let N be t he value of the cash flows occ urring o ver t he per iod h No a ssumption i s m ade c oncerning the method of N estimation (e.g., N m ight b e s uccessively re invested i n bonds market or in monetary market) The change in the value of portfolio including interest rate positions between times t and t + his given by ∆V = Vt + h +N – Vt Ther efore, ∆V = g (X 1,t + h ,…, X n ,t + h ) + N − f (X 1,t ,…, X n ,t ) and ∆V = g ( X1,t + ∆X1 ,…, Xn,t + ∆Xn ) + N − f ( X1,t ,…, Xn,t ) A ssume t hat f and g are linear combinations of risk factors Then, Vt = f ( X1,t ,…, Xn,t ) = n n n ∑ i =1 di Xi,t and Vt +h = g ( X1,t +h,…, Xn,t +h ) = ∑ i =1 ei Xi,t +h = ∑ i =1 ei ( Xi,t + ∆Xi ) and so ∆V = g ( X1,t ,…, Xn,t ) + g (∆X1 ,…, ∆Xn ) + N − f ( X1,t ,…, Xn,t ) The expected va lue o f t he cha nge i n p rice o f a po rtfolio o f financial E[ ∆V ] = E[ g ( X1,t ,…, Xn,t )] + positions o ver a t ime h orizon h is E[ g (∆X1,…, ∆Xn )] + E[N ] − E[ f ( X1,t ,…, Xn,t )] an d E[ ∆V ] = g ( X1,t ,…, Xn,t ) + g (E[∆X1 ],…, E[∆Xn ]) + E[N ] − f ( X1,t ,…, Xn,t ) , wh ere g ( X1,t ,…, Xn,t ) + E[N ] − f ( X1,t ,…, Xn,t ) is the expected va lue of the portfolio price variations when the variations of the all portfolio’s risk factors have a null expected value © 2010 by Taylor and Francis Group, LLC 208 ◾ Pension Fund Risk Management: Financial and Actuarial Modeling Hence, we notice that a portfolio including bonds will never have the same value at times t and t + h even if market conditions are stable (unchanged), which completes our proof Proof o f Pr oposition 9.2: First, we show (1) Let us note that ∆Vbd is the change in the value of a portfolio of interest rate positions over a time horizon h We also define ∆X izc as variations of a portfolio’s risk factors (zero coupon bonds) over t he per iod h; ∆vzc(τ) a s v ariations of pr ices of zero coupon bonds over t he per iod h; and ∆rzc(τ) as variations of interest r ates o f z ero co upon bo nds o ver t he per iod h A ccording t o ( 9.11), ∆V bd = g (∆X 1zc ,…, ∆X nzc ) If we estimate risk factors through price variazc zc tions of zero coupon bonds, then ∆Xi = ∆vi (τ) and i = 1,…, n Thu s, we estimate ∆vizc (τ) b y z ero co upons’ i nterest r ate va riations Using (1) i n Lemma 9.3, note that we underestimate the risk of risk factors Hence, we underestimate portfolio risk, which completes the first part of our proof Second, we show (2) Let us consider the investment portfolio of interest rate positions from (1) If we use variations of zero coupon bonds’ interest rates to estimate risk factors, then ∆Xizc = ∆rizc (τ) and i = 1,…, n Thu s, we evaluate ∆rizc (τ) by variations in zero coupon prices Using (2) in Lemma 9.3, note that we overestimate the risk of risk factors Hence, we overestimate portfolio risk, which completes the second part of our proof REFERENCES Albert, P., Bahrle, H., and Konig, A., Value-at-risk: A risk theoretical perspective with focus on applications in the insurance industry, Contribution to the 6th AFIR International Colloquium, Nurnberg, Germany, 1996 Blake, D., Portfolio choice models of pension funds and life assurance companies: Similarities and differences, The Geneva Papers on Risk and Insurance, 24(3), 1999, 327–357 Brunner, G., Ro cha, R , a nd Hinz, R , Risk-Based S upervision o f Pension Funds: Emerging Practices and Challenges, World B ank Publications, Washington, D.C., 2008 Christoffesen, P a nd Dieb old, F., H ow r elevant is v olatility f 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long terme, Contribution to the 28th AFIR International Colloquium, Paris, France, 2006 Jorion, P., Value at Risk, 2nd edn., McGraw-Hill, New York, 2001 Panning, W.,H., The strategic uses of value at risk: Long-term capital management for property/casualty insurers, North American Actuarial Journal, 3(2), 1999, 84–105 Ufer, W., The “Value at risk” concept for insurance companies, Contribution to the 6th AFIR International Colloquium, Nurnberg, Germany, 1996 © 2010 by Taylor and Francis Group, LLC ... second part of our proof Proof of Proposition 9. 1: Let us define Xi,t as risk factor’s value at time t and Xi,t+h as its value at date t + h(and i = 1, …, n) The change in value of a risk factor... Section 9. 2 introduces the VaR definition and methodology, currently in use in the banking sector Section 9. 3 discusses market risk concepts and its applications in the banking and pension fund industries... random va lue of t he financial position at t ime t + h Let vm,t be a value of the corresponding position at the date of estimation Suppose the portfolio has holdings ωi in m financial positions