Agnes Joseph, Dirk de Jong and Antoon Pelsser. Policy Improvement via Inverse ALM. Discussion Paper 06/

Transcription

1 Agnes Joseph, Dirk de Jong and Anoon Pelsser Policy Improvemen via Inverse ALM Discussion Paper 06/

2 Policy Improvemen via Inverse ALM AGNES JOSEPH 1 Universiy of Amserdam, Synrus Achmea Asse Managemen DIRK DE JONG 2 Synrus Achmea Asse Managemen ANTOON PELSSER 3 Universiy of Maasrich, Nespar June 25, 2010 Absrac Tradiional ALM firs ses he policy parameers and hen assesses he impac on some sub-se of risk and reurn measures. We propose a mehod o inver he radiional ALM approach: firs formulae he desired level of risk and reurn measures and hen sysemaically search hrough he policy space o find he policy ha bes mees he objecives and consrains. The mehod is more open minded han radiional ALM as is shown in a numerical example using an ALM model for a sylized pension fund. JEL classificaion: Keywords: Asse Liabiliy Managemen 1 Universiy of Amserdam, Dep. of Quaniaive Economics, Roeerssraa 11, 1018 WB Amserdam, The Neherlands, A.Joseph@inerpolis.nl, el (0031) Synrus Achmea Asse Managemen, Rijnzahe 10, 3454 PV De Meern, The Neherlands, DA.de.Jong@inerpolis.nl 3 Universiy of Maasrich, Dep of Finance, Dep of Quaniaive Economics, P.O.Box 616, 6200 MD Maasrich, The Neherlands, A.Pelsser@maasrichuniversiy.nl Corresponding auhor

3 1. Inroducion Asse liabiliy managemen, ypically used by for example pension funds and insurance companies, is challenging. The siuaion is complicaed and he consequences are uncerain bu significan. One has o accoun for muliple risk and reurn measures, for example fuure profis, conribuions and probabiliies of insolvency, on differen ime horizons, for example 1 year, 5 years and/or 30 years. There are muliple goals which may differ for each of he sakeholders and here are many policy insrumens. Examples of he laer are he invesmen policy, profi-sharing policy and conribuion policy. To make i even more difficul here is uncerainy abou for example fuure economic variables and fuure regulaion. Ofen, Asse Liabiliy Managemen (ALM) models are used when making (and o suppor) policy decisions. Tradiional ALM firs ses he policy parameers and hen assesses he impac on some sub-se of risk and reurn measures. We propose a mehod o inver he radiional ALM approach: firs formulae he desired level for risk and reurn measures and hen find he policy parameers. The mehod is based on he idea ha he objecives should drive he policy choice and no he oher way around as is ofen seen in pracice, see e.g. Chapman (1999) and Slaer (2002) for ineresing discussions on his issue. The remainder of his paper is organized as follows: we firs describe he mehod, which we refer o as Inverse ALM. Then we show a numerical example for a sylized pension fund and propose some exensions of he mehod. We end wih a summary and conclusions. 2. Inverse ALM using WLS In radiional ALM one chooses differen policy parameers and he ALM model (a non-linear mapping) shows he effec of he new policies on differen risk and reurn measures. Typically he number of policy- or conrol variables in he ALM model is limied: he conribuion policy, profi-sharing policy and he invesmen policy for example. There are far more oucomes: risk and reurn measures such as (he perceniles of) solvency measures and profis a differen ime horizons. Figure 1 on he lef shows he radiional ALM process schemaically, depiced as a box wih a few buons and many oupus. Policy Parameers Policy Parameers ALM Model Risk Measures Inverse ALM Model Risk Measures Figure 1 Tradiional ALM en inverse ALM Wih inverse ALM we propose o inver he radiional ALM approach. Firs formulae he desired levels of he risk measures and hen find he police parameers o ge here. Since he number of conrol variables is limied wih respec o he number of risk measures in his

4 conex, we will on overall no be able o achieve all desired oucomes of risk and reurn measures. We can come as close as possible by projecing he desired oucomes on he number of conrol variables dimensional subspace of achievable oucomes. Figure 2 shows he inverse ALM process schemaically, where he desired change in risk measures is R, he achievable change in risk measures is ˆR, and he cosine of he angle beween R and ˆR, cos, is a measure of realism of he desired R (see Davidson and MacKinnon, 2004). Desired change R Sub-space of Achievable Oucomes Achievable change ˆR Angle beween R and ˆR Figure 2 Sub-space of achievable oucomes Firs we look a radiional ALM. For small changes in policy parameers B (ΔB) he radiional ALM is approximaely linear 4. If we look a risk measures R, he change in hese risk measures caused by a small change in policy insrumens B equals: R XB (2.1) where R is a N-vecor, B is a M-vecor (we assume he ypical case ha N M ) and where X is a non-square ( N M )-marix of parial derivaives X nm Rn / Bm. Inverse ALM is also linear: B X R (2.2) where R is he desired change in risk measures, and X is he pseudo-inverse of he marix X of parial derivaives. If ( X ' X ) is well-defined and non-singular (2.2) can be wrien as: B 1 X ' X X ' R (2.3) X ' X 1 X ' X is also called he Moore-Penrose pseudo-inverse (see for example Greene, 2004, pp 45-46). One can recognize (2.2) and (2.3) as an applicaion of leas squares regression. When R is he desired change in he oucomes of risk measures R, equaion (2.3) gives he change in he policy parameers o ge as close as possible o he desired oucomes. The achievable change in risk measures is given by: 4 Acually for small changes in parameers any model is approximaely linear.

5 Rˆ XB XX R X 1 X ' X X ' R (2.4) 1 X where X X ' X ' is he leas squares projecion marix. One could use he singular value decomposiion (SVD, see e.g. Brescher 1997) of he ( N M )-marix X o obain X. The marix X can be decomposed as X UV ' so ha X V 1 U '. Here U and V are (column) orhogonal N M and M M marices respecively, is a M M diagonal marix conaining he singular values i i 1,.., M of 1 X, and is a diagonal marix wih elemens 1 / i, for i 0, zero oherwise. An advanage of SVD is ha i can also be used o approximae he inverse of he marix of parial derivaives in case ( X ' X ) is ill-defined or singular. In he formulas above all risk measures are aken ino accoun as equally imporan. When some risk measures are considered more imporan hen ohers, we can use weighed leas squares (WLS), which resuls in formula (2.5) Rˆ X 1 X ' WX X ' WR (2.5) where W is a marix wih elemens (he weighs ) W nn n { 1,..., N} on he diagonal, and oher elemens equal o zero. Using ieraion of he inverse ALM process i should be checked ha he change risk measures by a change in policy conrols is approximaely linear. The appropriae sep-size may depend on he risk measures under consideraion. For example, order saisics such as value a risk measures will be highly non-linear when he sep-size is oo large. Sochasic flow analysis works locally bu no any more if scenario s cross each oher, because hen he assumpion of local differeniabiliy will no hold any more for order saisics. Many exensions of his simple inverse ALM are possible. We will propose some exensions in secion 4, bu we firs illusrae he resuls of inverse ALM using he WLS mehod proposed above in a numerical example for a sylized pension fund.

6 3. Numerical example using WLS 3.1. The ALM model We demonsrae he mehod using an ALM model for a sylized pension fund which is inspired by Grosen and Jørgensen (2000). We choose his relaively simple seing o illusrae he idea wihou he necessiy o focus on echnical deails of a full-version pracical ALM mehod. We firs inroduce he following balance shee of he pension fund a ime : Asses A Liabiliies L E where A are he asses, L he liabiliies and E is he equiy or buffer of he pension fund a ime. Given his balance shee we define he funding raio a ime as: A FR L (3.1) Wih respec o he liabiliies we make he following simplifying assumpions. There is a single paymen afer T years o he average cohor of all paricipans. The value of he liabiliies a ime 1 is deermined by he liabiliies a ime where he liabiliies will only grow by indexaion and ineres unil he conrac expires. Oher facors, such as moraliy are assumed o be deerminisic and already accouned for in he liabiliies a saring ime =0, L. 0 The ineres and indexaion depend on each year s marke reurn. The ineres rae, denoed by r, is assumed o be consan and equal o he economy s coninuously compounded riskless rae of ineres. The yearly indexaion is denoed by i. The dynamics of he liabiliies L, { 1,2,..., T}, of he pension fund are hen given by L L (1 r)(1 i ) 1. For he indexaion policy, which resuls in i, we assume here is some minimum guaraneed level of indexaion denoed by i g. Addiional indexaion depends on he solvency posiion of he pension fund. We assume he pension fund has some arge funding raio, denoed by FR arge. If he funding raio is above his level, hen some percenage α of he buffer above he arge level is used for indexaion. We will, as in Grosen and Jørgensen (2000), refer o α as he disribuion raio. The liabiliies L, { 1,2,..., T}, can be wrien as: L, 1 L ( 1 r)1 max FR FR arg e ig (3.2) We now urn o he dynamics on he asse side of he balance shee. The pension fund has a yearly rebalancing porfolio wih some percenage, denoed by w s, invesed in risky asses and (1- w s ) is invesed in riskless bonds. We assume he process of he risky asses o evolve according o a geomeric Brownian moion on a finie ime inerval [0,T] as: ds S d S db (3.3) S S

7 where S 0, S and S are consans. The sandard Brownian moion B is defined on he probabiliy space, H, P wih filraion ( H ) 0 T saisfying he usual condiions, and P denoes he real world probabiliy measure. The annual (log) reurns of he risky asses are condiionally Gaussian disribued: lns / S H ~ ( S S, ) (3.4) 1 2 S Lasly, we need inflaion in he economy, denoed i, which is assumed o be consan Policy insrumens, risk measures, economic parameers There are five policy parameers in his simple model. They are relaed o he indexaion policy, he invesmen policy and conribuion or iniial value of he asses. One can decide on he guaraneed minimum rae of indexaion i g, he arge funding raio FR arge, disribuion raio α, he percenage of risky asses w s and he funding raio (L 0 =1) or asses a sar A 0 respecively. In his example he pension fund s iniial policy is given by i g =0, FR arge =105%, α=25%, w s =40% and A 0 =125%. We defined he following risk and reurn measures: he expeced value, he 5 h percenile, he 10 h percenile and he 25 h percenile of 1) he value of he liabiliies and 2) he buffer (asses minus liabiliies). All risk measures are calculaed on four ime horizons: 5, 10, 20 and 40 years, and will be considered in oday s currency o make hem comparable. The parameer se o generae 1000 economic scenarios are for he risky asses S =8%, 2 S =10%, for he ineres rae r =4% and for inflaion i =1% Resuls The ALM resuls under he iniial policy are given in figure 1. I shows he risk measures we defined (expecaion and perceniles of he liabiliies and he funding raio) on he four differen ime horizons: 5, 10, 20 and 40 years. Liabiliies Funding raio 2,00 1,40 1,80 1,30 1,60 1,40 Exp 5% 1,20 1,10 Exp 5% 1,20 1,00 10% 25% 1,00 0,90 10% 25% 0, , Figure 1 ALM oucomes under iniial policy From figure 1 we conclude ha he averages are good: he expeced value of he liabiliies (in oday s currency) is 1,86 afer 40 years, he expeced funding raio is 130% on he same ime horizon. Bu he lower perceniles are bad, especially hose of he funding raio: he 10% percenile is 93%, and he 5% percenile 85% a he 40 year horizon. Can we improve hese oucomes?

8 Horizon Risk measure oday Weigh Targe linear new parameers 40 Exp Liabiliies % Liabiliies % Liabiliies % Liabiliies Exp Buffer % Buffer % Buffer % Buffer Table 1 Resuls and goal differences for 40 year horizon in inverse ALM, oher horizons are given in he appendix To illusrae he inverse ALM mehod able 1 shows he resuls for he ime horizon of 40 years, he whole able is given in he appendix. The column Today gives he risk measures under he iniial policy. Using he inverse ALM mehod we firs need o define he arge values of hese risk measures (we choose o weigh all risk measures equally). The Targe Δ column says by how much he value of he risk measure should be adjused according o he pension fund objecives and consrains. For example, he 5% percenile of he liabiliies 40 years from now is esimaed a The arge is ha he 5% percenile is 1 ha is 0.94( Today)+0.06(Targe Δ), so ha he liabiliies do no lose any purchasing power in he 5% percenile. The 5% percenile of he buffer is now -0.15, he arge is ha he 5% percenile is 0 ha is -0.15( Today)+0.15(Targe Δ), so ha buffer is nonnegaive. Noice ha 1) all values are in real euro s, and 2) we can see wheher he acual arge values can be achieved by adjusing he policy parameers. Table 1, column Δ is he weighed leas squares resul of Targe Δ when policy parameers are adjused in order o ge as close o he arge as possible. For example, for he 5% percenile of he liabiliies he goal was o ge his risk measure equal o 1, so we where 0.06 shor. By adjusing he policy parameers, we an adjusmen of jus Wih he 5% percenile of he buffer, iniially equal o -0.15, he arge was o adjus he policy parameers such ha he risk measure equals zero, achieved is he value of The firs conclusion is ha improvemen can be reached, bu no all arge values can be achieved, he measure of realism of he desired change in oucomes, cos, equals 74%. As a final check, he las column in able 1 gives he resuls of he risk measures by re-simulaing he ALM model using he new policy parameers. Table 2 shows he iniial values of he policy parameers and hose suggesed by he inverse ALM mehod. Policy parameer iniial policy new policy Iniial asses, cash in/ou Minimum guaraneed indexaion 0.0% -0.2% Disribuion raio 25% 35% Targe funding raio 5% 11% Percenage of risky asses 40% 33% Table 2 Iniial policy parameers and policy parameers suggesed by he inverse ALM mehod In order o achieve he goals of he pension fund he inverse ALM mehod suggess o add 12% o he pension fund s asses via a single premium paymen; when he buffers are inadequae (< 11%) he liabiliies are cu wih 0.2% every year; condiional indexaion is

9 given when he funding raio is higher han 111%; he percenage of he buffer above 11% ha will be given as indexaion is adjused o 35%; he percenage socks is reduced o 33%. Given he example resuls we can conclude ha inverse ALM is an open minded mehod. Policy varians ha a forehand would be unmenionable (negaive guaraneed indexaion percenage) can show up o be he bes way o mee he objecives of he pension fund. Figure 2 shows all resuls given he iniial policy and he new policy. Almos all risk measures have improved on all ime horizons. The new policy, alhough surprising, seems o be a good alernaive for he iniial policy. Bu remember he policy suggesed includes a single premium equal o 12% of he iniial asses. Liabiliies iniial & new policy Funding Raio iniial & new policy 2 1,4 1,8 1,3 1,6 Exp 5% 1,2 Exp 5% 10% 10% 1,4 25% 1,1 25% Exp new Exp new 1,2 5% new 10%new 1 5% new 10%new 25%new 25%new 1 0,9 0, , Figure 2 ALM oucomes under new policy (sold lines) and under iniial policy (doed lines) 4. Inverse ALM using LP In his secion we sugges several improvemens of he inverse ALM mehod as proposed in secion 2. Firs of all, in he numerical example we added 12% by a single premium paymen o he asses of he pension fund, which could be done wihou concern or punishmen. The mehod could easily be adjused by adding quadraic coss for changes in he policy parameers: min( R B Rˆ )' W ( R Rˆ ) B' CB (4.1) where Rˆ XB Bu in pracice, here may also be sric limiaions o he change in conrol variables. Can we really expec a single premium paymen as high as x% of he iniial asses even if i seems o be he bes policy despie using a model ha includes quadraic coss as in (4.1)? Therefore one could also add resricions on B o he model, see for example Judge e al (1966) and Liew (1976) on resricions in regression analysis. Anoher possible improvemen over he WLS-mehod for inverse ALM is o incorporae asymmeric penalies. In he numerical example above he real value of he expeced liabiliies

10 on he 40-year horizon was 1.86 (see able 1), which is a good resul so ha we se he arge value a If we search for new policy parameer using WLS a downside deviaion from 1.86 is equally reaed as an upside deviaion, while an upside deviaion in his case is more aracive. One can use linear programming (LP, see e.g. Danzig and Thapa, 2003) o incorporae asymmeric penalies and add resricions on he change in policy parameers a he same ime. Use for example B N 2 min I w ( R Rˆ ) s.. n1 Rn Rn n n n d u B B B (4.2) Here I oherwise n { N is an indicaor funcion, equal o one if Rn Rn 1,..., } R n R n and zero 4.1. Numerical examples using LP Using again he ALM model of he sylized pension fund as in secion 3, we now demonsrae inverse ALM using linear programming o obain new policy parameers (4.2). Firs, we calculaed he resuls as if here are sill no resricions on he change in policy parameers. We jus incorporaed asymmeric penalies. Table 3 gives he risk and reurn measures concerning he liabiliies and he buffer on he 40 year horizon. Table 4 gives he newly proposed policy parameers. Horizon Risk measure oday Weigh Targe LP unresriced LP unresriced 40 Exp Liabiliies % Liabiliies % Liabiliies % Liabiliies Exp Buffer % Buffer % Buffer % Buffer Table 3 Resuls and goal differences for 40 year horizon in inverse ALM incorporaing asymmeric penalies, oher horizons are given in he appendix Policy parameer iniial policy new policy Iniial asses, cash in/ou Minimum guaraneed indexaion 0.0% 0.2% Disribuion raio 25% 12% Targe funding raio 5% 18% Percenage of risky asses 40% 43% Table 4 Iniial and adjused values of he policy parameers using Linear Programming As we can see, comparing able 3 wih able 1, he expeced liabiliies and perceniles of he liabiliies are more or less he same using he new parameers suggesed by inverse ALM using WLS respecively LP. Bu by using he policy parameers suggesed by linear

11 programming (incorporae asymmeric penalies), he expeced buffer and perceniles of he buffer are much higher. Bu, comparing he policy parameers we see ha he policy suggesed by LP adds 20% o he iniial asses, compared o 12% in he case of WLS. If addiional paymen is a problem, hen nor WLS or he LP wihou policy resricions will no give a proper soluion. Therefore, we calculaed a hird example where we resric he change in he value of he iniial asses A0 A0. We also increased he weighs for he liabiliies over he expeced buffer, since high buffers as in able 4 are aracive, bu we would raher like o achieve higher liabiliies. The resuls using LP, now wih resricions, are given in able 5 and able 6. Horizon Risk measure oday Weigh Targe LP resriced LP resriced 40 Exp Liabiliies % Liabiliies % Liabiliies % Liabiliies Exp Buffer % Buffer % Buffer % Buffer Table 5 Resuls and goal differences for 40 year horizon in inverse ALM, oher horizons are given in he appendix Policy parameer iniial policy new policy Iniial asses, cash in/ou Minimum guaraneed indexaion 0.0% 0.3% Disribuion raio 25% 11% Targe funding raio 5% 3% Percenage of risky asses 40% 43% Table 6 Iniial and adjused values of he policy parameers using Linear Programming wih resricions on he adjusmen of he iniial asses Adding resricions, here now is no single premium any more. The invesmen policy is similar o he policy suggesed by LP wihou resricions and oher weighs (able 6 compared wih able 4). However, he parameers concerning he indexaion policy are slighly differen, due o he exra weigh on liabiliy resuls. The guaraneed minimum rae of indexaion is slighly higher (0.3% compared o 0.2%). The arge funding raio is only 103%, so ha condiional indexaion is given earlier (arge funding raio was 118%), he disribuion raio is 11% (was 12% in LP wihou resricions). Saring wih lower iniial asses does resul in lower expeced liabiliies and buffers (able 5 compared o able 3), given he resricions inverse ALM shows which goals w.r.. he risk and reurn measures are realisic. By saring o define he goals of he pension fund, inverse ALM sysemaically searches hrough he policy space. The numerical examples show ha he policies suggesed by inverse ALM clearly improve he ALM-resuls on he differen risk and reurn measures.

12 5. Summary and conclusions Tradiional ALM firs ses he policy parameers and hen assesses he impac on some sub-se of risk and reurn measures. We propose a mehod o inver he radiional ALM approach: firs formulae he desired level of risk and reurn measures and hen find he policy parameers. We sared by using a weighed leas squares mehod o obain new policy parameers. An example showed ha 1) probably no all objecives and consrains se by he pension fund managemen can be realized, bu inverse ALM gives he policy ha bes mees he arges, and 2) he mehod is very open minded: policy varians ha a forehand would be unmenionable can show up o be he bes way o mee he objecives. We also proposed several exensions: o add quadraic coss, o incorporae asymmeric penalies using linear programming and o add resricions o he change in policy insrumens. We used a simple model o illusrae he ideas, bu he mehod also seems feasible for realisic ALM models, and is herefore ineresing for praciioners. In he examples we defined he arges of he company under consideraion as a whole. Anoher exension could be o use he ALM model o faciliae communicaion beween sakeholders of he company on risk/reurn measures, by defining arges for he sakeholders separaely, see Joseph e.al. (2010).

13 References BRETSCHER, O. (1997): Linear Algebra Wih Applicaions, 1 s ediion, Prenice Hall, Inc. CHAPMAN, R.J., GORDON, T.J. AND C.A. SPEED (2001): Pensions, funding and risk, Insiue of Acuaries and Faculy of Acuaries, Working Paper DANTZIG, G.B. AND M.N. THAPA (2003): Linear Programming: Theory and Exensions, 1 s ediion, Springer-Verlag DAVIDSON, R. AND J.G. MACKINNON (2004): Economeric Theory and Mehods, 1 s ediion, Oxford Universiy Press GREENE, W.H. (2004): Economeric Analysis, 4 h ediion, Prenice Hall Inernaional, Inc. GROSEN, A. AND P.L. JØRGENSEN (2000): Fair valuaion of life insurance liabiliies: The impac of ineres rae guaranees, surrender opions, and bonus policies Insurance: Mahemaics and Economics, 26, JOSEPH, A.S., JONG, D.A. AND A.A.J. PELSSER (2010): Policy choices based on risk profiles of sakeholders of a pension fund, SSRN working paper JUDGE, G.G. AND T. TAKAYAMA (1966): Resricions in Regression Analysis, Journal of he American Saisical Associaion, 61 (313), LIEW, C.K. (1976): Inequaliy Consrained Leas-Squares Esimaion, Journal of he American Saisical Associaion, 71 (355), SLATER, A. ET AL (2002): Monioring he effeciveness of asse allocaion decisions, Working paper presened a he Finance and Invesmen Conference, 27 June 2002

14 Appendix Horizon Risk measure oday Targe linear new parameers LP unresriced LP resriced 5 Exp Liabiliies % Liabiliies % Liabiliies % Liabiliies Exp Liabiliies % Liabiliies % Liabiliies % Liabiliies Exp Liabiliies % Liabiliies % Liabiliies % Liabiliies Exp Liabiliies % Liabiliies % Liabiliies % Liabiliies Exp Buffer % Buffer % Buffer % Buffer Exp Buffer % Buffer % Buffer % Buffer Exp Buffer % Buffer % Buffer % Buffer Exp Buffer % Buffer % Buffer % Buffer Table A Resuls examples inverse ALM