Determining equivalent charges on flow and balance in individual account pension systems

Transcription

1 Journal of Eonomis, Finane and Administrative Siene 21 (2016) 2 7 Journal of Eonomis, Finane and Administrative Siene Artile Determining equivalent harges on flow and balane in individual aount pension systems Luis Chávez Bedoya Esan Graduate Shool of Business, Lima, Peru artile info abstrat Artile history: Reeived 17 July 2014 Aepted 14 Marh 2016 JEL lassifiation: G23 Keywords: Pension fund Defined benefit Individual aount Charge on balane Charge on flow In this artile, we determine a harge on balane that is equivalent to a ertain fixed harge on flow for a partiular utility maximizer affiliate partiipating in a defined-ontribution pension fund under the system of individual aounts. We also prove, under market ompleteness, that the equivalent harge on balane depends only on the urrent level of the harge on flow, the length of the aumulation period and the risk free rate of return Universidad ESAN. Published by Elsevier España, S.L.U. This is an open aess artile under the CC BY-NC-ND liense ( Determinaión de exaiones de efeto equivalente en el flujo y el balane de los sistemas de pensiones de uentas individuales resumen Códigos JEL: G23 Palabras lave: Fondo de pensiones Benefiio definido Cuenta individual Exaión en el balane Exaión en el flujo En este artíulo se determina una exaión en el balane, que es equivalente a ierta tasa fija en el flujo de una empresa asoiada partiular maximizadora, que partiipa en un fondo de pensiones de aportaión definida en el sistema de uentas individuales. También se prueba, en la integridad del merado, que la exaión de tipo equivalente en el balane depende solo del nivel atual de la tasa en el flujo, la duraión del período de aumulaión y un tipo de rentabilidad sin riesgo Universidad ESAN. Publiado por Elsevier España, S.L.U. Este es un artíulo Open Aess bajo la CC BY-NC-ND lienia ( 1. Introdution Two important harateristis of a defined ontribution (DC) pension fund are that affiliates borne the risk derived from flutuations in the value of assets and that imposed administrative harges have a diret and signifiant impat on the terminal wealth of the orresponding individual aount (IA). For example, Murthi, Orszag, and Orszag (2001) estimate that in the U.K. over 40% of the address: lhavezbedoya@esan.edu.pe IA s value is dissipated through fees and harges while Whitehouse (2001) determines that a levy of one per ent of assets adds up to nearly 20% of the final pension value. Administrative harges have also reeived a great deal of attention from the pension supervisory agenies, poliy-makers and researhers, espeially in ountries that have partially or totally transformed their publi defined-benefit pension systems into individual apitalization ones. The most familiar and doumented example is Chile and the reader an find main aspets of suh reform in Arrau, Valdés-Prieto, and Shmidt-Hebbel (1993), Diamond and Valdes-Prieto (1994), Edwards (1998), Arenas de Mesa and Mesa-Lago (2006). Also, Queisser (1998), Sinha (2000), Kay and Kritzer (2001), Mesa-Lago / 2016 Universidad ESAN. Published by Elsevier España, S.L.U. This is an open aess artile under the CC BY-NC-ND liense ( by-n-nd/4.0/).

2 L. Chávez Bedoya / Journal of Eonomis, Finane and Administrative Siene 21 (2016) (2006), Kritzer, Kay, and Sinha (2011) and Marthans, J. and Stok, J. (2013) provide good referenes for the reform, situation and perspetive of pension systems in Latin Ameria. As mentioned by Mithell et al. (1998), James, Smalhout, and Vittas (2001) and Whitehouse (2001) the high harges of IA systems is one of their main ritiisms sine they disourage partiipation (as people onsider ontributions as taxes instead of savings), damage the reputation of the system, redue future pensions, and inrease future osts for the government whether there is guaranteed minimum pension. Devesa-Carpio, Rodríguez-Barrera, and Vidal-Meliá (2003) onsider that the harge sheme adopted by the IA system is very important sine fund aumulation proess is exponential and targeted for long horizons. Following Kritzer et al. (2011), the most ommon administrative harges in IA pension systems are proportional on flow (or a perentage of the affiliate s salary), fixed on flow, proportional on assets (balane) and proportional over exess returns. Analysis and omparison of administrative harges aross different ountries an be found in James et al. (2001), Whitehouse (2001), Devesa-Carpio et al. (2003), Corvera, Lartigue, and Madero (2006), Gómez-Hernandez and Stewart (2008), Tapia and Yermo (2008). Moreover, Sinha (2001), Masias and Sánhez (2007) and Martínez and Muria (2008) analyze in detail the administrative harges in Mexio, Peru and Colombia, respetively. However, this artile will fous only on harges that are proportional on balane and flow sine they are by far the most popular and important in Latin Ameria 1. Queisser (1998) onsiders that the harge on flow is more advantageous for the Pension Fund Administrator (PFA) in the initial stages of the system, and although the harge on balane aligns the PFA s objetives in terms of inreasing the fund s profitability, it tends to be more expensive in the longrun as personal aounts grow in size. On the other hand, Shah (1997) mentions that the harge on flow generates distortions and undesirable tendenies like promoting high start-up osts for the PFAs, disouraging ompetition in the system and generating losses for older affiliates. Asset alloation, performane and risk of a DC pension plan during its aumulation and deumulation phases have reeived a onsiderable attention in the literature. Blake, Cairns, and Dowd (2001) using different models for asset returns and portfolio strategies estimate the value-at-risk of the pension ratio. Poterba, Rauh, and Venti (2005) alulate the expeted utility of retirement wealth for different investment strategies and assumptions. Devolder, Bosh Prinep, and Domínguez Fabián (2003) derive several optimal portfolio strategies for different types of utility funtions assuming the risky asset follows a geometri Brownian motion (GBM). Gao (2009) provides a similar analysis but under a onstant elastiity variane (CEV) proess for the risky assets. The effiieny of the mean-variane portfolio seletion in a DC pension plan is studied in Vigna (2014) when the risky asset follows a GBM. Haberman and Vigna (2001) onsider downside risk of an optimal asset alloation strategy derived from a disrete-time dynami programming approah. Salary risk and inflation risk were inorporated in Battohio and Menonin (2004) and Han and Hung (2012) while maximizing the expeted utility of terminal wealth. Battohio, Menonin, and Saillet (2004) and Yang and Huang (2009) inorporate longevity risk in the optimal asset alloation of a DC plan; the former using as objetive expeted utility, and the latter deviation of terminal wealth with respet to a predetermined target. Stohasti lifestyling under terminal utility with habit formation is found and ompared with other strategies in Cairns, Blake, and Dowd (2006). Finally, the reader interested in the analysis and optimal alloation during the deumulation phase an be referred, among others, to Blake et al. (2001), Gerrard, Haberman, and Vigna (2004), Horneff, Maurer, Mithell, and Dus (2006) and Gerrard, Haberman, and Vigna (2006). Nonetheless, methodologies to ompare administrative harges in DC pension fund with IA during its aumulation period have not reeived that level of attention in the literature, espeially in a ontinuous-time stohasti setting. Therefore, we fill suh gap by developing a methodology, in the aforementioned environment, to determine equivalent harges on flow and balane. We onsider a risk-averse affiliate who maximizes her expeted utility of terminal wealth in a omplete Blak-Sholes market model 2. Then, we determine the equivalent harges by equating the maximum terminal ertainty equivalent that an be ahieved under both kinds of harges. Moreover, under ertain assumptions, we prove that the equivalent harges on balane and flow depend only on the length of the aumulation period and the risk-free rate of return; and, to the best of our knowledge this relationship between harges is new in the literature. This result is independent on the risky asset s growth rate and volatility, as well as, the affiliate s risk-aversion sine the omparison of administrative harges an be performed by simple terminal wealth expetations under a risk-neutral probability measure. The rest of the artile proeeds as follows: Setion 2 introdues a methodology to mathematially represent and ompare harges on balane and flow. Setion 3 disusses an appliation of the methodology to the Peruvian Private Pension System. Finally, Setion 4 draws onlusions. 2. Methodology { Throughout this paper (, F, P, F t ) represents a filtered }t 0 { } and omplete probability spae on whih a standard F t t 0 adapted one-dimensional Brownian motion B(t) is defined. We denote by L 2 (0,T,R) the set of all R -valued, measurable stohas- F T { ti proesses g(t) adapted to F t, suh that E[ }t 0 g(t) 2 dt] <. 0 For any t [0,T], we assume that the PFA an invest the affiliate s ontributions in only two assets whih satisfy: dp 0 (t) = rp 0 (t) dt, P 0 (0) = P 0 > 0, (1) dp 1 (t) = P 1 (t) dt + P 1 (t) db (t), P 1 (0) = P 1 > 0. (2) It is lear that r is the risk-free rate of return, and are the risky asset s growth rate and volatility, respetively. The stohasti differential equation (SDE) in (2) generates a geometri Brownian motion (GBM) whih is a ommon speifiation to model asset values and it is heavily utilized in stohasti ontrol of DC pension funds as mentioned in the introdution. But most important, assets (1) and (2) generate a omplete finanial market and therefore it guarantees the existene of a risk-neutral probability measure. This property will be extremely useful to verify our theoretial results of Setion The affiliate s problem Consider a partiular PFA s affiliate who has T>0 months before retirement, i.e., T represents the length of her aumulation phase. She already has W 0 > 0 ready to be invested in her individual 1 On the one hand, Bolivia, Colombia, Chile, El Salvador, Peru, and Uruguay have harges on flow. On the other hand, Mexio, Bolivia, Costa Ria, and Uruguay have harges on assets. Notie that in Bolivia and Peru both type of harges oexist. 2 This market onsists on a risky asset following a geometri Brownian motion and a risk-free asset. Both assets an be traded ontinuously and fritionless.

3 4 L. Chávez Bedoya / Journal of Eonomis, Finane and Administrative Siene 21 (2016) 2 7 aount, and after that initial deposit she will ontribute at a onstant rate >0 per month for the next T months. Also, for any tε[0,t] let x (t) L 2 F (0,T,R) be the proportion of her IA that is invested in the risky asset. We also assume that the adjustments are performed instantly and free of harge. Let W(t) be the affiliate s wealth in her IA at time t [0,T]. If the PFA does not harge any administrative fees to the affiliate, then W(t) satisfies dw (t) = [W (t) [ x (t) + (1 x (t))r] + ] dt + W (t) x (t) db (t), with W(0) = W 0. It is in the affiliate s interest that the PFA will maximize her expeted utility of terminal wealth, E[U(W (T))], by determining an optimal proportion x (t). We assume that U is stritly inreasing, differentiable and onave in its domain. Therefore, the affiliate wants the PFA to solve problem (P) given by Max E[U(W (T))] St. x(t) L 2 [ (0,T; R) F ] dw (t) = W (t)[x(t)( r) + r] + dt + W (t) x (t) db (t) W (0) = W 0. Introduing V, the value funtion of the problem, we have V (t, W) = max {x} E[U(W (T)) W (t) = W]. (4) Following Vigna (2014), it is possible to find a losed-form expression for the value funtion and the optimal ontrol under a general hyperboli absolute risk aversion (HARA) lass of utility funtions. Beause the main result of the paper regarding equivalent harges will be independent of risk-aversion, we hoose for simpliity a partiular ase of the HARA lass given by the exponential utility funtion. Then: U (W) = 1 e W,>0. (5) The utility in (5) exhibits onstant absolute risk aversion sine U (W) U = and it allows an expliit solution for (P). Following (W) Devolder et al. (2003), the optimal proportion to be invested in the risky asset is x (t) = e r(t t) W (3) r 2, (6) and the orresponding value funtion is V (t, W) = 1 exp { ( ( e e r(t t) r(t t) 1 r ) )} (T t). 2 2 (7) We an observe from (6) that the optimal ontrol does not depend on the ontribution rate. If we apply the optimal strategy x (t) stated in (6), then W (T) = W(T) and the maximum expeted utility of terminal wealth is [ ] E U W (T) = V (0,W 0 ) ( )} = 1 e { exp e rt rt 1 W o + r 2 2 T. (8) Moreover, the ertainty equivalent of W (T),CE W (T), is given by e CE W (T) = e rt rt 1 r 2 2 T. (9) As we an notie from (9), CE ( W (T) ) is the sum of the future value of W 0, the future value of a ontinuous annuity with rate (both using the risk-free rate r) and a term depending on the market prie of risk and the risk aversion parameter but independent of any ontribution made to the IA. Next, we desribe in detail the harges that the PFA will apply either on the affiliate s IA or on her ontributions. We will use a struture similar to the one onsidered in Shah (1997), Diamond (2000), Blake and Board (2000), Whitehouse (2001), Devesa-Carpio et al. (2003) and Gómez-Hernández and Stewart (2008) Charge on balane Let ı>0 be the monthly harge on balane expressed in ontinuous time. It is also known as harge on assets or on stok and, in general, it is applied as a perentage of the value of assets under management. The affiliate wants to study the value of her IA under this type of harge. If we denote suh wealth as W s (t), it will satisfy the following SDE dw s (t) = [ W s (t) [ x s (t)( r) + r ı ] + ] dt + W s (t) x s (t) db (t), (10) with W s (0) = W 0. Notie that the harge on balane will diminish the monthly growth rates and r by a quantity equal to ı. We will use a ontrol x s (t) to indiate the fration of the IA invested in the risky asset under the harge on balane. In this ase, the affiliate wants to solve problem (Ps) given by Max E[U(W s (T))] St. x s (t) L 2 [ (0,T; R) F [ ] ] dw s (t) = W s (t) x s (t)( r) + r ı + dt W s (0) = W 0. +W s (t) x s (t) db (t) Based on the results and assumptions regarding the exponential utility funtion, the optimal strategy for (Ps) is xs (t) = e (r ı)(t t) r W s 2, (11) while the maximum ertainty equivalent is e (r ı)t CE W s (T) = e (r ı)t 1 r ı 2 2 T. (12) Notie that the last term of (12) does not depend on the initial ontribution (W 0 ), the harge on balane (ı), and the ontribution rate (). Next, we desribe the harge on flow Charge on flow Let >0 be the harge on flow and it ould be applied as a fration of the affiliate s salary or ontributions. Whether the affiliate makes a ontribution X in a partiular month, we assume the harge she will pay to the PFA (at the moment the ontribution is made) will be F = (1 e ) X. Considering that F ould have been invested in the fund, it is possible to express ontribution X as e X when adjusted for the opportunity ost of F. In the ase of a onstant rate of ontribution,, the harge on flow will generate an adjusted ontribution rate of e. The affiliate wants to study the value of her individual aount under this type of harge. We denote suh wealth as W f (t) and it will satisfy the following SDE: dw f (t) = [ W f (t) [ x f (t)( r) + r ] + e ] dt + W f (t) x f (t) db (t), (13)

4 L. Chávez Bedoya / Journal of Eonomis, Finane and Administrative Siene 21 (2016) with W f (0) = e W 0. Reall that W f (T) does not represent the true wealth of the affiliate at the end of the aumulation phase but the final wealth adjusted by the opportunity ost of the harge on flow. Then, random variables W f (T) and W s (T) an be ompared. We will use a ontrol x f (t) to indiate the fration of the IA invested in the risky asset under the harge on flow. Thus, the affiliate wants to solve the problem (Pf) given by Max E[U(W f (T))] St. x f (t) L 2 (0,T; R) F [ [ ] dw f (t) = W f (t) x f (t)( r) + r + e ] dt W f (0) = e W 0. +W f (t) x f (t) db (t) Based on the previous results and assumptions regarding the exponential utility funtion, the optimal ontrol and the maximum ertainty equivalent of (Pf) are x e r(t t) f (t) = W f and r 2, (14) ] e CE W f (T) = e [e rt rt 1 r 2 2 T. (15) Similar to (12), only ( the ) last term of (15) depends only on the r market prie of risk, the risk aversion parameter () and the length of the aumulation period Comparing harges on balane and flow The affiliate wants to ompare her optimal expeted utility of adjusted terminal wealth under the two types of harges onsidered. Therefore, it is appropriate to ontrast both CE ( W s (T) ) and CE ( W f (T) ) given by (12) and (15), respetively. Moreover, we define the following ratio to establish suh omparison R sf = CE ( W s (T) ) CE ( W f (T) ) (16) If R sf > 1, the harge on balane will be preferred. If R sf < 1 the harge on flow will be preferred. Finally, when R sf = 1 the affiliate will be indifferent between both shemes. We will onsider W 0 = 0, i.e., the aumulation phase begins with an amount equal to zero in the affiliate s individual aount. Under this assumption and onsidering all the other variables fixed, we an express ratio R sf in (16) as a funtion of the harge on balane ı. For the exponential utility funtion we have: e (r ı)t ( r) 2 T r ı 2 2 R sf ı = e. (17) e rt 1 r ( r) 2 T 2 Given, let ı be the equivalent harge on balane, that is, the value ı suh that R sf ı in (17) is equal to one. Thus, ı satisfies e (r ı )T 1 r ı = e e rt 1. (18) r The left-hand side of (18) is the future value at T of a ontinuous annuity with unit rate and interest r ı. The right-hand side orresponds to the future value at T of a ontinuous annuity with rate e and interest r. If we denote the future values of suh annuities as s T(r ı ) and e s Tr, then (18) is equivalent to = ln( s Tr / s T(r ı ) ). Moreover, we an observe that ı will depend only on r, T and. Hene, it is independent of the parameters and of the risky asset, the ontribution rate and the risk aversion oeffiient. Finally, notie from (18) that if T and r inreases then ı dereases, improving the relative performane of the harge on flow with respet to the harge on balane. In the next setion we generalize equation (18) for any riskaverse affiliate as desribed in Setion Equivalent harges in a omplete market As mentioned before, the finanial market onsisting of the riskfree asset and the risky asset given by (1) and (2) is omplete. Also, given xs (t) and x (t), the optimal ontrols of problems (Ps) and (Pf), f we an determine both W s (T) and W f (T). The equivalent harges of flow and balane an be obtained by omparing the expeted present values of their orresponding IAs under the risk-neutral probability measure Q. For that purpose we define the ratio N sf as: [ ] N sf = E Q Ws (T) [ ] E Q Wf (T). (19) The ratio in (19) is equivalent to the expeted present values of W s (T) and W f (T) using probability measure Q sine the fator e rt appears in both numerator and denominator of (19). Additionally, if N sf > 1, the harge on balane will be preferred. If N sf < 1 the harge on flow will be preferred; and, when N sf = 1 the affiliate will be indifferent between both shemes. [ ] It is lear that both p s = e rt E Q Ws (T) and p f = [ ] e rt E Q Wf (T) represent the urrent pries of the affiliate s IA aounts under the orresponding harges. For example, p s is the amount of money that the affiliate will reeive today in exhange of giving her IA (entirely) at the end of the aumulation phase, in this ase it is assumed that the harge is on balane, and that she will ontinue to ontribute to the fund at a rate until T. Under Q both the risk free rate and the risky assets grow at a rate r, then for fixed harges and ı we have: [ ] e (r ı)t E Q Ws (T) = e (r ı)t 1 W o +, (20) r ı [ ] e E Q Wf (T) = e rt rt 1 W o +. (21) r Given N, let ı be the equivalent harge on balane, that is, the N value ı N suh that N sf ı N = 1. Then, it is easy to verify that ı N also satisfies equation (18) when W 0 = 0. This framework is far more general than the partiular ase of the exponential utility funtion beause the omparison of administrative harges performed through the ratio N sf will be valid for any risk-averse affiliate. 3. Numerial appliation In this setion, we present an appliation of the proposed methodology to the Peruvian Private Pension System. This appliation is relevant beause the PPS is going through an important reform exatly 20 years after its reation. Part of the reform onsists of replaing the harge on flow with a harge on balane, and this situation has partially motivated the present researh artile 3. We onsider a retirement age of 65 years and ignore the mandatory insurane fee. We will work with three harges on flow 3 Peruvian Law No ontains the main aspets of the reform. One is that affiliates will migrate to a mixed harge that has a 10-year transient flow omponent, and from year 10 onwards the harge will be only on balane. The reform also inludes a bidding mehanism on harges to alloate new affiliates and norms to inorporate independent workers.

5 6 L. Chávez Bedoya / Journal of Eonomis, Finane and Administrative Siene 21 (2016) 2 7 Table 1 Equivalent annualized harge on balane, ı, suh that R sf ı = 1 for different ages and harges on flow. We have onsidered r = 0.037%, onstant ontribution rate, and the following harges on flow: f min = 1.47%, f max = 1.69%, and f avg = 1.58% (the harges are based on salary and assume a mandatory ontribution of 10% of the affiliate s salary). Age (years) Equivalent harge on balane (in % and yearly) fmin = 1.47% favg = 1.58% fmax = 1.69% Author s elaboration. (expressed as perentages of the affiliate s salary): f min = 1.47%, f max = 1.69%, and f avg = 1.58% whih orresponds to the minimum, maximum and average PPS s harges on flow as in May Sine dependent workers in Peru have a mandatory ontribution of 10% of salary and f i are applied to it, we have i = ln(1 10f i ) and therefore min = , max = and avg = We will also assume that monthly risk-free is r = 0.037%, and it was estimated using the inflation-adjusted Peruvian bonds with maturity 30 days. The data series onsists on daily observations of the orresponding yearly rate from 20/12/2005 to 16/05/2014. Table 1 and Figure 1 show ı (annualized) for ertain ages 4 and three senarios for the harge on flow: f min, f max and f avg.as expeted from (18), we observe that ı is stritly inreasing in age (dereasing in T), and stritly dereasing in f for a fixed T. In the ase of a 40-year-old affiliate, or equivalently T = (65 40) 12 = 300 months, ı is 1.398% per year when f avg is the orresponding harge on flow. This implies that a harge on balane smaller than 1.398% makes suh sheme onvenient for the 40-year-old affiliate. The orresponding values for f min and f max are 1.289% and 1.510% per year, respetively. An important age to onsider is 37 years sine half of PPS s affiliates are younger than that age. The orresponding ı for f avg is 1.245% per year. Consequently, if the harge on flow is the system s average (or 1.58% of salary) and the harge on balane is greater than 1.245% per year, then the youngest half of the affiliates in the system will find the harge on 4 If E is the affiliate s age, then T = (65 E) 12 months will be the length of the aumulation phase. Charge on balane, % - δ * f min =1.47% f avg =1.58% f max =1.69% Equivalent harge on balane Age (years) Figure 1. Equivalent annualized harge on balane, ı, suh that R sf ( ı ) = 1 for different ages and harges on flow. We have onsidered r = 0.037%, onstant ontribution rate, and the following harges on flow: f min = 1.47%, f max = 1.69%, and f avg = 1.58% (the harges are based on salary and assume a mandatory ontribution of 10% of the affiliate s salary). Author s elaboration. balane undesirable. We an observe that ı 0.704% for all ases onsidered in the example, and suh level would make the harge on balane to be preferred for almost all PPS s affiliates. Reall that we are onsidering r = 0.037%; however, an inrement in r will make the values of ı smaller and therefore the harge on balane will beome less attrative. 4. Conlusions and further researh We have developed a methodology to determine equivalent harges on flow and balane for individual aount pension systems. We have onsidered a risk-averse affiliate who wants to maximize her expeted utility of adjusted terminal wealth in a omplete finanial market. Then, we need to solve the orresponding stohasti ontrol problems to find and ompare the maximum terminal ertainty equivalent (CE) whih an be ahieved under both harge shemes. Under a fixed ontribution rate, an exponential utility funtion and no initial amount in the IA, we are able to find the equivalent harges (those whih make both shemes indifferent in terms of terminal CE) by solving a nonlinear equation involving only the future values of two ontinuous annuities. Moreover, the results will hold for any utility funtion and investment strategy sine market ompleteness allows us to work in a risk-neutral environment. The methodology was applied to the Peruvian Private Pension System (PPS) in order to determine the equivalent harge on balane for different aumulation horizons and three senarios for the harge on flow. We found that a harge on balane lower than 0.704% per year would make suh sheme preferable to the one based on flow for almost all PPS s affiliates. However, suh threshold assumes a monthly risk-free rate of 0.037%, a onstant ontribution rate and a fixed harge on flow greater than 1.47% of the affiliate s salary. It is possible to extend this methodology to onsider a time-varying ontribution rate, risk-free rate and harge on balane, as well as other modifiations preserving market ompleteness. Finally, it will be worth to perform a omplete analysis in a general equilibrium environment onsidering a welfare target; but, this is beyond the sope of the paper. Referenes Arenas de Mesa, A., & Mesa-Lago, C. (2006). The strutural pension reform in Chile: Effets, omparisons with other Latin Amerian reforms, and lessons. Oxford Review of Eonomi Poliy, 22(1),

6 L. Chávez Bedoya / Journal of Eonomis, Finane and Administrative Siene 21 (2016) Arrau, P., Valdés-Prieto, S., & Shmidt-Hebbel, K. (1993). Privately managed pension systems: Design issues and the Chilean experiene. Washington DC: Tehnial report, World Bank. Battohio, P., Menonin, F., & Saillet, O. (2004). Optimal asset alloation for pension funds under mortality risk during the aumulation and deumulation phases. Annal of Operations Researh, 152(1), Blake, D., & Board, J. (2000). Measuring value added in the pensions industry. Geneva Papers on Risk and Insurane-Issues and Pratie, 25(4), Blake, D., Cairns, A. J. G., & Dowd, K. (2001). Pensionmetris: Stohasti pension plan design and value-at-risk during the aumulation phase. Insurane: Mathematis and Eonomis, 29(2), Cairns, A. J. G., Blake, D., & Dowd, K. (2006). Stohasti lifestyling: Optimal dynami asset alloation for defined ontribution pension plans. Journal of Eonomi Dynamis and Control, 30(5), Corvera, F., Lartigue, J., & Madero, D. (2006). Análisis omparativo de las omisiones por administraión de los fondos de pensiones en los países de Améria Latina. Mexio City: Mimeo. Devesa-Carpio, J., Rodríguez-Barrera, R., & Vidal-Meliá, C. (2003). Mediión y omparaión internaional de los ostes de administraión para el afiliado en las uentas individuales de apitalizaión. Revista Española de Finaniaión y Contabilidad. XXXII, 116, Devolder, P., Bosh Prinep, M., & Domínguez Fabián, I. (2003). Stohasti optimal ontrol of annuity ontrat. Insurane: Mathematis and Eonomis, 33(2), Diamond, P. (2000). Administrative osts and equilibrium harges with individual aounts. In J. B. Shoven (Ed.), Pages of: Administrative aspets of investment-based soial seurity reform (pp ). University of Chiago Press. Diamond, P., & Valdés-Prieto, S. (1994). Soial seurity reforms. In B. Bosworth, R. Dornbush, & R. Laban (Eds.), The Chilean eonomy: Poliy lessons and hallenges (pp ). Washington DC: Brookings Institution. Edwards, S. (1998). The Chilean pension reform: A pioneering program. In M. Feldstein (Ed.), Privatizing soial seurity (pp ). University of Chiago Press. Gao, J. (2009). Optimal portfolios for DC pension plans under a CEV model. Insurane: Mathematis and Eonomis, 44(3), Gerrard, R., Haberman, S., & Vigna, E. (2004). Optimal investment hoies post retirement in a defined ontribution pension sheme. Insurane: Mathematis and Eonomis, 35(2), Gerrard, R., Haberman, S., & Vigna, E. (2006). The management of deumulation risk in a defined ontribution pension plan. North Amerian Atuarial Journal, 10(1), Gómez-Hernández, D., & Stewart, F. (2008). Comparison of osts and fees in ountries with private defined ontribution pension systems. Paris: Working paper 6. International Organisation of Pension Supervisors. Haberman, S., & Vigna, E. (2001). Optimal investment strategies and risk measures in defined ontribution pension shemes. Insurane: Mathematis and Eonomis, 31(1), Han, N., & Hung, M. (2012). Optimal asset alloation for DC pension plans under inflation. Insurane: Mathematis and Eonomis, 51(1), Horneff, J., Maurer, R., Mithell, S., & Dus, I. (2006). Optimizing the retirement portfolio: Asset alloation, annuitization, and risk aversion. Tehnial report National Bureau of Eonomi Researh. James, E., Smalhout, J., & Vittas, D. (2001). Administrative osts and the organization of individual retirement aount systems: a omparative perspetive. World Bank Poliy Researh Working Paper Kay, S., & Kritzer, B. (2001). Soial seurity in Latin Ameria: Reent reforms and hallenges. Federal Reserve Bank of Atlanta Eonomi Review, 86(1), Kritzer, B., Kay, S., & Sinha, T. (2011). Next generation of individual aount pension reforms in Latin Ameria. Soial Seurity Bulletin, 71(1), Marthans, J., & Stok, J. (2013). Una propuesta para reformar los sistemas privados de pensiones: el aso peruano. Doumento de trabajo, Universidad de Piura. Martínez, O., & Muria, A. (2008). Sistema de omisiones de las administradoras de fondos de pensiones en Colombia. Reporte de estabilidad finaniera, Bano de la Repúblia de Colombia. Masias, L., & Sánhez, E. (2007). Competenia y reduión de omisiones en el sistema privado de pensiones: el aso peruano. Doumento de Trabajo. Superintendenia de Bana, Seguros y Administradoras de Fondos de Pensiones (SBS). Mesa-Lago, C. (2006). Private and publi pension systems ompared: An evaluation of the Latin Amerian experiene. Review of Politial Eonomy, 18(3), Murthi, M., Orszag, J. M., & Orszag, P. R. (2001). Administrative osts under a deentralized approah to individual aounts: Lessons from the United Kingdom. In R. Holzmann, & J. Stiglitz (Eds.), New ideas about old age seurity (pp ). The World Bank. Poterba, J., Rauh, J., & Venti, S. (2005). Utility evaluation of risk in retirement saving aounts. In D. A. Wise (Ed.), Analyses in the eonomis of aging (pp ). University of Chiago Press. Queisser, M. (1998). Regulation and supervision of pension funds: Priniples and Praties. International Soial Seurity Review, 51(2), Shah, H. (1997). Towards better regulation of private pension funds. World Bank Paper Sinha, T. (2000). Pension reform in Latin Ameria and its lessons for international poliymakers. Kluwer Aademi Pub 23. Sinha, T. (2001). Analyzing management fees of pension funds: A ase study of Mexio. Journal of Atuarial Pratie, 9, Tapia, W., & Yermo, J. (2008). Fees in individual aount pension systems: A rossountry omparison OECD Working Papers on Insurane and Private Pensions 27. OECD Publishing. Vigna, E. (2014). On effiieny of mean-variane based portfolio seletion in defined ontribution pension shemes. Quantitative Finane, 14(2), Whitehouse, E. (2001). Administrative harges for funded pensions: omparison and assessment of 13 ountries. In Private pension systems: Administrative osts and reforms. pp OECD. Yang, S., & Huang, H. (2009). The impat of longevity risk on the optimal ontribution rate and asset alloation for defined-ontribution pension plans. The Geneva Papers on Risk and Insurane-Issues and Pratie, 34(4),