Tijdsclirift voor Econoiuie ei1 Management Vol. XLVI, 4, 2001 How to Determine the Capita1 Requirement for a Portfolio of Annuity LiabiPities by J. DHAENE, M. J. GOOVAERTS, S. VANDUFFEL aiid D. VYNCICE ti.U.Leuven, Departiiient o[ Applied Economics, Actiiarial Sciences R.U.Leuven, Departinent of Applied Economics, Actuarial Sciences K.U.Leuven, Departinent of Applied Econoinics. Actuarial Sciences David Vyncke K.U.Leuven, Department of Applied Econornics, Actuarial Sciences ABSTRACT This paper illustrates an analytic inethod iliat caii be used to determine the total capita1 requirements necessary to properly provide for ilie fuhlre obligations of a portfolio of annuity liabilities and to protect the enterprise froin the related risks it iaces. This example is based on the work of Kaas, Dhaeiie and Goovaerts (2000). I. INTRODUCTION The deterinination of these requirements entails the aiialysis of the distribution function, more specifically the tail of the distribution function (os the catastrophic part) of tlie present value of tlie future cash flows. The projection of the future cash flows relating to the annuity obligations, and the subsequeiit deterniination of tlieir present value, may be at its simplest, when the aiiiount aiid timing of the asset aiid liability cash flows are iiisensitive to varying economic conditions. Eveii in these circumstances, tlie possible impact of inortality improvemeiits and the fliture course of the reiiivestinent assumptions are important. On the otlier hand, these coinputations are most complex, wlieii the timing and amount of the cash flows are affected by the economic scenario (e.g. annuity payinents whose aniount and tiining are solely, or partly, driven by economic perforiiiance of some type as wel1 as to the policyholder reaction to such performance). The use of inappropriately siinple (i.e. perliaps for computational ease os convenience) metliods for these computatioiis, is liltely to introduce hidden "surplus" (could be a deficit too). Capita1 req~iirements calculated under such a regime may iiot reflect the t-me capita1 req~iiredto support the insurer's busiiiess. In order to coinpute the liltelihood that an insurer wil1 not be able to meet its obligations when they fa11 due, knowledge is required of the nature of the cash flows and any underlying stocliastic process which drives tlieir timing and aiiio~int.The inethod illustrated in this exainple starts froin this knowledge. It enables the actuary to determine (approximately) tlie relevant probabilities of extreme events. Tt allows the actuary to deterinine the necessary provision, based on tlie level safety desired. IT. THE CASH FLOW OF THE FUTURE LIABILITIES Firstly, the cash flows depicting the filture paynlents of the annuity portfolio are projected int0 the fuh~re.The cash flows represent, for each year between 2002 and 2079, the expected payiiient of that year, adjusted with a safety niargin. The expectations of the paynients diie are calculated usiiig realistic technica1 bases concerning disability, n~orbidityetc. The additional inargin is a safety niargin against the possible negative deviations and also includes costs. 111. THE DISTRIBUTION OF THE CAPITAL REQUIREMENT We will detennine the present value at Januaiy 1, 2001, which we will consider as the time 0. The time unit is chosen to be one year. Let ai be the amount due at time i. The stochastic capital req~iirement, ?i is then given by: V = a,e-Y' + a,e-('i +Y" + ... + ane-(?+ K- +..+Y>,) where Y, is the return on the investnients from 2001 to 2002, Y, is the rehirn from 2002 to 2003, etc. We will assume that t l ~ eyearly returns Y, are normally distributed with mean ,u and variance d.We also assume that the yearly returns are inutually independent. In the following Table, a number of possible investment strategies with values of the paraineters ,u and o are given (figures fictious, for illustrative purposes only). P 0 100% Shares 0.10 0.15 75% 125% 0.09 0.12295 5OYo / 50% 0.08 0.09922 p - 25% / 75% 0.07 100% Bonds 0.06 0.08173 0.075 . p It is clear that V is a sum of dependent random variables. Indeed, the different discount factors wil1 be strongly positive dependeilt. The computation of the distribution fùnction of V cannot be perforined exactly. One could try to deterniiile the distribution fuilction of V by siinulation, but this will lead to untsustable estimates for the tail probabilities. Moreover, siniulation will be very time-consuming for determiiiing the optiinal asset mix. Recent actuarial research results allow to coinpute lower and upper bounds for tlie distribution of V. It will be show11 that these lower and upper bounds are often close to each other, which of course illustrates the accurateness of the approxiinatioils. On the next pages one finds upper and lower bounds for the distribution of V for the different investinent strategies. These bounds are upper and lower bounds in the sense of convex order for the exact but unl<nown distribution íùnction. Tlie upper bound is denoted by CO (dotted line), the lower bound by ES (hl1 line). We also present approximations for the percentile, the expected shortfall and the conditional tail expectation at different levels. Lower bound 1 upper bouild I Exact Mean 4.225.360 1 4.225.360 / 4.225.360 Lower Percentile Expected Shortfall Percentile 1 Expected Shortfall I Conditional Tail I 1 / Exoectation 1 / 1 Conditional Tail Expectation bound I Upper / 1 bouild 1 1 Lower bound 4.413.501 Upper bound 4.413.501 4.413.501 StDev 1.898.995 2.267.604 1.908.791 Lower Percentile Expected Shortfall bound 1 Exact Mean Upper bound Conditional Tail Ex~ectation Percentile Expected Shortfall Conditional Tail Ex~ectation Lower bound 4.783.853 Upper bould 4.783.853 Exact Mean StDev l 660.798 1.966.5 17 1.665.608 Lower Percentile bouiid Upper bourid 4.783.853 Expectation Percentile Expected Shorlfall Conditional Tail Exoectation 1 Lower Percentile 1 Expected Sliortfall bound 80% 1 Conditional Tail Expectation 6.556.404 Expected Shortfall Conditional Tail Expectation 1 Lower bound Exact Upper bouild Mean 6.379.328 6.379.328 2.057.314 StDev 1.757.507 I ~ower / Percentile 1 Expected Shortfall 6.379.328 1.760.055 1 bound 80% Coilditioilal Tail / Expectation 276.730 7.690.022 1 9.073.672 1 Expectation I IV. COMPARISON BETWEEN THE DIFFERENT INVESTMENT STRATEGIES A. Upper Boz~nd(CO) B. Lower Bounds (ES) The expectations of tlie lower and upper bouild of V are always equal. This is beca~isethe bouiids have exact expectations. From numerical comparisons it follows tliat the best estimate for the distribution of the present value of the cash flow under consideration is tlie lower bouild. This is confirmed by the fact that tlie exact Standard Deviation aild tlie Staiidard Deviation of the lower bouiid are very close to each otlier. V. THE OPTIMAL ASSET ALLOCATION The optimal asset allocatioii is a coinbination of the different iiivestineiit strategies (also talcing int0 accouilt the dependenties between the yearly returns), that niini~njzesa certain criterion. Here we wil1 assuine that the optima1 asset allocation miniinizes the initia1 amount that has to be reserved sucli that, with probability of 99%, al1 future obligations can be met. Hence, the optiinal asset allocation is the one that minimizes the 99%-percentile of V. Considering the lower bouiids, the optiinal investment strategy tums out to iiivest 40,40% in shares aiid the remaining part in boiids, see the following figure. Considering the upper bouilds (which are less accurate) we find that the optiinal investment strategy is to iiivest 35,87% in shares. ESLM ----- , i I"" O#>" 3 O<##,IiUU '"O" ""O V B10 000 --, , _ s?OG% cotkprr ao,o(m 10814 Ooi >iu ~ i i a a Lower Lower bouiid Upper bo~uld 4.986.280 5.095.640 Percentile Expected Shoitfall bound 1 Uppei bound Conditional Tail Expectation / Perceniile Expected Shortfall Conditional Tail Exoectation K Kaas, J. Dhaene and M.J. Goovaerts, 2000, Upper aild Lower Bounds for Sums of Random Variables, Inszirntice Matl~emnt~cs & Econonzzcs 27, 151-168. R. Kaas, M. Goovaerts, J. Dhaene aiid M. Denuit, 2001, Modem Actuarial Risk Theory, (Kluwer), to appcar