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Citation Gerrard R J G Guilleacuten M Nielsen J P and Peacuterez-Mariacuten A M (2014) Long-run savings and investment strategy optimization The Scientific World Journal 2014 510531 - doi 1011552014510531

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Research ArticleLong-Run Savings and Investment Strategy Optimization

Russell Gerrard1 Montserrat Guilleacuten2

Jens Perch Nielsen1 and Ana M Peacuterez-Mariacuten2

1 Cass Business School City University London 106 Bunhill Row London EC1Y 8TZ UK2Department of Econometrics Riskcenter-IREA University of Barcelona Avenue Diagonal 690 08034 Barcelona Spain

Correspondence should be addressed to Montserrat Guillen mguillenubedu

Received 10 November 2013 Accepted 8 January 2014 Published 23 February 2014

Academic Editors J M Calvin C-T Chang and M Pal

Copyright copy 2014 Russell Gerrard et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We focus on automatic strategies to optimize life cycle savings and investment Classical optimal savings theory establishes thatgiven the level of risk aversion a saver would keep the same relative amount invested in risky assets at any given time We showthat when optimizing lifecycle investment performance and risk assessment have to take into account the investorrsquos risk aversionand the maximum amount the investor could lose simultaneously When risk aversion and maximum possible loss are consideredjointly an optimal savings strategy is obtained which follows from constant rather than relative absolute risk aversionThis result isfundamental to prove that if risk aversion and themaximumpossible loss are both high then holding a constant amount invested inthe risky asset is optimal for a standard lifetime savingpension process and outperforms some other simple strategies Performancecomparisons are based on downside risk-adjusted equivalence that is used in our illustration

1 Introduction

We study long term investment strategies characterised by afirst period of savings followed by a period of consumptionafter retirement This is the most common form of optimiza-tion problem that is faced by most citizens who seek a way tobuild up a pension payment stream that provides a balancebetween risk and reward Most investors request downsideprotection and desire upside potential

There are two competing magnitudes related to risk inclassical optimality theory when studying lifetime savingsand retirement investmentsOne is themaximumamount thesaver is allowing to lose during his life cycle The other is hisrisk aversion a parameter that is linked to the power of apower utility function However risk aversion andmaximumpossible loss have not been studied together Most articlesfocus on one and consider the other parameter as fixedWhenlooking at the distribution of terminal wealth our resultsshow that those two parameters should not be consideredseparately when optimizing long-term expected returnswhile controlling for downside risk

In this paper we use the risk adjusted ad hoc performancemeasure recently developed in [1 2] to evaluate the conse-quences of choosing a given combination of maximum loss

and risk aversion We study optimality of a constant absoluterisk aversion investment strategy that differs slightly from butis reminiscent of standard Constant Proportion PortfolioInsurance (CPPI) which was introduced by [3] (see also[4ndash6]) It is unsurprising that combining a high maximumpossible long-term loss with a high risk aversion that is alower power in the power utility function will be superior inthe long run to the risk-adjusted equivalent with a lower long-termmaximumpossible loss and a lower risk aversion that ishigher power in the utility function However the first com-bination is a good deal better than the second when studyingthe distribution of terminal wealth leading us to furtherinvestigations in this direction We take this point of view toits logical extreme and let the maximum loss go to infinity atthe same time as the power in the utility function goes to zerowhich means that we also raise risk aversion to infinity Ourlimit behaviour is carried out in such a way that the limit ofpower utility functions is also a utility function which hap-pens to be the exponential utility and which gives a fixed con-stant absolute risk aversion It turns out that both the optimalstrategy in the constant relative risk aversion and the perfor-mancemeasures converge smoothly to the equivalent entitiesderived directly from the exponential utility

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 510531 13 pageshttpdxdoiorg1011552014510531

2 The Scientific World Journal

Our contribution is to establish the link between riskaversion and maximum possible loss in some classical utilityfunction optimization We also show that performance anal-ysis in terms of downside protection and riskreturn trade-offenhances the understanding of lifetime investment strategies

In Section 2 we provide some background and notationIn Section 3 we introduce the market model and the opti-misation problem which will be used to illustrate the resultson power utility functionsThe optimal strategy prescribes aninitial investment in the risky asset proportional to the ratiobetween maximum possible loss and risk aversion which is aprincipal reason for keeping this quantity fixed when passingto the limit We demonstrate that the optimal strategy forthe generalized shifted utility that is defined when wealth canbe negative (as is permitted up to a certain maximum loss)converges to a limiting strategy and that is indeed theoptimal strategy under the limiting utility function Section 4discusses the risk and performance measures of lifetime in-vestment plans and the approach to the comparison of dif-ferent strategies We use risk measure tools that are generallyaccepted in this context

Our focus here is the expected shortfall risk measures[7 8] such as downside value at risk [9 10] and tail conditionalexpectation [11ndash13] which are widely accepted in financialeconomics The probability of a lifetime ruin is alreadystudied by [14] as well as optimal portfolio selection terminalwealth problems (see [15]) Lifetime ruin has receivedincreased attention (see [16])

We use simulations to evaluate the expected shortfall forldquobenchmarkrdquo strategies that is those which prescribe that aconstant proportion of the wealth should be invested in therisky asset We call these ldquobenchmarkrdquo strategies because thistype of strategies is frequently recommended by investmentadvisors [17ndash19] Similar questions have been investigated bymany authors [20ndash29] More recently reference [30] contri-buted to the analysis of investment strategies thatmeet certaintargets (see also [31]) In our presentation we fix a constantproportion and then we calculate the value of the maximumpossible loss such that the generalized power utility-optimalstrategy produces the same expected shortfall and proceed tocompare the two strategies in terms of the internal rate ofreturn (IRR)We also investigate whether it is possible to finda benchmark strategy which can give the same IRR as theshifted power utility-optimal strategy or special CPPI Thediscussion of the results which takes place in Section 5 isfollowed by Section 6 which concludes

2 Background and Preliminaries

It is well documented in the finance literature that under verygeneral conditions the power utility function leads to con-stant relative risk exposure that is a constant proportion ofthe wealth is invested in risky assets with the remainder inrisk-free assets An investment strategy that maximizes thepower utility function is called constant relative risk aversion(CRRA) See [32] for one of the more compact and generalintroductions So in practice an investor maximizing powerutility would invest a fixed proportion of his wealth in the

stock market and the remainder in bonds He would revisehis positions about once a year as the stock prices and returnsare known Then gains would be accumulated to previousyearrsquos wealth The outcomes are such that each yearrsquos wealthdistribution depends on the previous yearrsquos outcome Soyearly gains (or losses) matter a lot to the next period invest-ment balance between risk-free and risky assets

Alternatively another investor may prefer to fix anamount of his wealth to be invested in the stock market Thatwould correspond to a strategy called constant absolute riskaversion (CARA) In other words this investor would takea constant amount and would invest that in the risky assetskeeping the rest of his wealth in risk-free investment Gen-erally after one year no matter what the gains are he wouldagain invest the same constant amount in stocks In a pureCPPI strategy the investor would have defined a floor and amultiplier and then his exposure to the risky assets would be aconstantmultiple of the cushion which is the excess of wealthover the floor Our investor however does not establish a floornor a cushion and does not need to look into yearly gains (orlosses) to tune the next periodrsquos investment balance Thisinvestor does not have a downside protection however wewill require for him an immense risk aversion Then we willfind out that he can outperform the risk-adjusted equivalentstrategy followed by the first investor

Investment strategies resulting from a constant relativerisk exposure and from a constant absolute risk exposure cor-respond to the maximization of utility functions that arerelated We provide the theory to show that investors choos-ing a constant absolute risk exposure can outperform thosethat use a constant relative risk exposure approach Indeedthose two strategies correspond to the optimization of twodifferent utility functions that can be written in such a waythat the constant absolute risk exposure is obtained as theoptimumof the utility functionwhich is the limit of the utilitythat has the optimum in the constant relative risk exposure

The practical consequences of this general theory when itcomes to analysing the distribution of finalwealth andoptimallong-term savings have not received much coverage in theliterature but recently some papers have started to look intostochastic dominance downside risk and performance eval-uation of portfolio insurance strategies Unfortunately thesecontributions look closely into most popular products likestop-loss synthetic put and constant proportion portfolioinsurance techniques (see [33ndash36])These strategies producewealth processes that depend on the previous yearrsquos wealthwhich are known to be suboptimal in the usual markethypothesis [37ndash39]

21 Time Framework and Terminal Wealth Distribution Inour paper we study long-term investment because we thinkthatmany savings investors consider optimal investment stra-tegies on the basis of having a long period of savings and thena retirement phase when they need a stream of income forconsumption or a pension So we focus on long periods of atleast sixty years

Lifetime savings are difficult to study because each personhas a particular cash flow pattern We will consider a simple

The Scientific World Journal 3

payment stream since we will not assume inflation then wecan assume that savingconsumption payments are constantand periodical during a life cycle Typically a pension planestablishes a period of many years of savings for instancefrom age 35 to age 65 (retirement age)Then after retirementthe saver needs to withdraw money from his investmentaccount on a regular basis for about another period of 30years until time 119879 This setting is similar to the time frame-work chosen for other authors who address retirement in-come (see [40]) Moreover to simplify our setting we do notconsider mortality uncertainty (A recent discussion on mor-tality risk transfer can be found in [41]) as our time horizon119879 is fixed

Anobvious question is to find an optimalway to invest theinitial stream of payments that will accumulate wealth in thesavings period and also to keep the remaining investmentduring the withdrawal time Any investor would prefer astrategy that will ultimately provide him with the highestreturn However the uncertainty about the evolution of thereturns makes prediction impossible We can only work withassumptions on the returns process and study the wealth dis-tribution at time 119879 The wealth distribution is the set of pos-sible returns together with the odds to achieve them In thispaper instead of limiting our study to fixed terminal wealthdistribution we will rather fix its downside risk level There isa risk that the investor does not have enough resources to beable towithdraw a pension during retirementMany investorscould be enormously averse to the risk of ruin or even to therisk that wealth at time 119879 falls below a certain level if they areinterested in leaving a bequest to their heirs The utility func-tion captures the preference from the investorrsquos point of viewof the final wealth outcome (see eg [6 42 43])

Investment planning is a central topic for pension funds[44 45] formutual funds and also for individualsThemodelby [44] uses a multiperiod stochastic linear programmingframework with a flexible number of time periods of varyinglength Not many researchers deal with the whole lifespaninvestment horizon The work by [46] looks at similar prob-lems but they rather focus on the fluctuations of incomestreams and on portfolio choice [47] rather than on propor-tion versus constant wealth in the bond versus stock combi-nation Our study uses a timing for lifetime savings similar tothe one proposed by [48] or the one used by [49] for illus-trative purposes Both establish about thirty years of savingsfollowed by about thirty more years of retirement

Performance evaluation of portfolio insurance strategiesusing stochastic dominance criteria was studied by [33] whofound that in typical CPPI strategies a higher CPPI multipleenhances the upward potential but harms the protection levelThey also indicated that choosing a different floor value orrelaxing the rebalancing discipline substantially harms thestrategiesrsquo performance Our approach is similar to theirs butcompares strategies that vary in terms of the combination ofrisk aversion and maximum possible loss

22 The Power Utility Function and the Optimisation of Life-long Savings Utility functions are widely used by financialeconomists because they measure the ldquovaluerdquo of wealth to

investors Concave utility functions are preferred becausethey reflect risk-aversion that is the utility of potential wealthincrease does not compensate the risk or the utility of wealthloss

In the simplest possible case we define the power utilityfunction by

119906120574 (119888) = 120574minus1119888120574 (1)

where 119888 and 1minus120574 are positive real numbersThe value of 1minus120574is known as relative risk aversion or simply risk aversionThelarger it is the larger is risk aversion and the flatter is theutility shape as 119888 increases Central features of utility func-tions are their relative and absolute risk aversions (see [50])Note that this utility is only defined for positive wealth andthus losses or negative wealth are not allowed

In this case the relative risk aversion is given by

119877 (119888) = minus119888

11990610158401015840

120574(119888)

1199061015840120574(119888)

= 1 minus 120574 (2)

and the absolute risk aversion is

119860 (119888) = minus

11990610158401015840

120574(119888)

1199061015840120574(119888)

=1 minus 120574

119888 (3)

In a power utility function there is a constant relative riskaversion (CRRA) while absolute risk aversion decreaseswhen 119888 increases

Now let us introduce a second parameter119870 so that

119906120574119870 (119888) = 120574minus1(119888 + 119870)

120574 (4)

where 119888 gt minus119870 and 1 minus 120574 gt 0 Now we can accept havingnegative wealth and the utility is defined in the interval(minus119870 0] aswell as in the positive real line as the standard powerutility function Then we have that relative risk aversion is

119877 (119888) = minus119888

11990610158401015840

120574119870(119888)

1199061015840

120574119870(119888)= (1 minus 120574)

119888

119888 + 119870 (5)

and absolute risk aversion is

119860 (119888) = minus

11990610158401015840

120574119870(119888)

1199061015840

120574119870(119888)= (1 minus 120574)

1

119888 + 119870 (6)

For this extended power utility neither CRRA nor CARAhold Under this setting that allows for the existence of lossesrisk aversion depends on the two parameters 120574 and119870

23 Limiting Utility Function One of the features of interestof the present paper is the investigation of the limiting casewhere119870 rarr infin and 120574 rarr minusinfin in such a way that119870(1minus120574) isequal to a constant Λ This corresponds to a balanced situa-tion where an increasing maximum possible allowed loss 119870and an increasing risk aversion (1minus120574) tend to an equilibriumin the limit

Although the utility functions 119906120574119870 do not converge in aformal sense we may make use of the fact that if one utility


function is a linear function of another then the maximiza-tion of the two utility functions will give rise to identicalstrategies Let us then renormalize 119906120574119870(119888) in such a way that1199061015840

120574119870(0) = 1 This produces a new utility function

119906120574119870 (119888) =1

120574119870120574minus1(119888 + 119870)

120574 (7)

Now we set 119870 = Λ(1 minus 120574) to obtain

119906120574119870 (119888) =Λ (1 minus 120574)

120574(1 +

119888

Λ(1 minus 120574))

120574

(8)

As 120574 rarr minusinfin the limit of this is

119906 (119888) = minusΛ119890minus119888Λ

(9)

Theutility function119906produced by this limiting procedure hasconstant absolute risk aversion 119860(119888) = minus11990610158401015840(119888)1199061015840(119888) = Λminus1So an investment strategy that maximizes this utility has theconstant absolute risk aversion property

This result shows that there is a continuous and smoothtransition from the power utility function with constant rel-ative risk aversion to the constant absolute risk aversion caseBelow we will investigate this transition by employing a riskmeasure to determine the behaviour of the process underoptimal control when utility function 119906120574119870 is used and to takethe limit of this as 120574 rarr minusinfin We will finally compare thelimiting results with those derived directly in the constantabsolute risk aversion case

We will see that having an adequate combination of hugerisk aversion and enormous maximum loss under the powerutility function leads to the constant absolute utility functionof the exponential utility in (9)

3 The Optimisation Problem

In this paper we address the problem of choosing a strategyfor the selection of an asset mix in such a way as to maximizethe expected utility of terminal value at time 119879 of an invest-ment consisting of a sequence of annual premiums of size 119886paid into an account at times 0 1 1198792 minus 1 followed by asequence of annual withdrawals of benefits also of size 119886 attimes 1198792 1198792 + 1 119879 minus 1 This is a continuous timeinvestigation in the sense that the investment portfolio can beadjusted at any time not just at the times when paymentsoccur

The setting corresponds to a lifecycle investment schemeaimed at planning for retirement in the absence of longevityrisk which means that we have a fixed time horizon 119879 whichdoes not depend on the investorrsquos survival Therefore nomortality projection needs to be assumed We also have thehypothesis of no inflation or financial interest rate We willconsider the risk-free rate to be the basis and therefore wewillfix it equal to zero This allows us to concentrate on the purestructure of the investment strategy and savings manage-ment

We assume that themarket consists of a risky asset whosevalue evolves as a geometric Brownian motion with drift

(excess return) equal to 120572 and diffusion coefficient (volatility)equal to 120590 as well as a riskless asset which has a constantreturn of zero In other words we are carrying out the analysisnet of risk-free interest rates This ensures that any positiveamount of money left in the fund at time 119879 represents a gainin comparison with investing solely in the riskless assetthroughout the time period whereas a negative amount attime 119879 represents a loss

Define 119862(119905) to be the net sum of all payment andwithdrawal transactions by time 119905 so that119862(119905) changes only bymeans of jumps equal to

Δ119862 (119905) = +119886 for 119905 = 0 1 1198792 minus 1minus119886 for 119905 = 1198792 1198792 + 1 119879 minus 1

(10)

The dynamics of 119883(119905) which is the total wealth accumulatedat time 119905 are given by

119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905) (11)

where 119882(119905) is a standard Brownian motion and 120587(119905) is theproportion of119883(119905) invested in stocks

The investor can at any time 119905 isin [0 119879) choose the pro-portion120587(119905)of the totalwealth119883(119905) to invest in the risky assetThis is the more general setting where we believe in a con-tinuous rebalancing assumption meaning that at any time 119905the desired proportion of wealth invested in the risky assetcan be chosen and is available in themarketThis assumptionis also known as dynamic allocation

We concentrate on 119883(119879) which is the so-called termi-nal wealth or the resulting outcome of a lifelong savingconsumption plan (The model is pretty restrictive and styl-ized the incomeconsumption streams as well as the timehorizon are deterministic However the main optimalityresult is not dependent on the nature of the payment streamas long as the timings of the payments are deterministic Theonly reason for specifying a particular form for the paymentstream is so that the simulations for comparisons can becarried out in the performance illustrations)

In this setting it is easily shown (see Appendix A) that theproblem of maximizing the expected utility of terminalwealth for the utility function 119906120574119870

max120587

E [1

120574(119883 (119879) + 119870)

120574] (12)

is solved by a strategy which invests an amount in the riskyasset at time 119905 equal to

120587 (119905)119883 (119905) = 119860 (119870 + 119883 (119905) + 119892 (119905)) (13)

where 119892(119905) = sum119879119897=119905Δ119862(119897) indicates the future payment stream

that for sure must be taken into account to measure wealth atany time 119905 because it is a compromised paymentrefundstructure In a continuous time payment stream we wouldhave119892(119905) = int119879

119905119889119862(119904) A simple version of this particular opti-

misation problem was already solved by [6] with no consum-ption process 119862(119905)

Note that strategy (13) consist in investing 119860 timesthe accumulated wealth where the leverage corresponds to


119860 = (1205721205902(1 minus 120574)) which is proportional to the inverse of

(1minus120574) a parameter associatedwith risk aversionThis optimalstrategy is similar to a constant proportion portfolio insur-ance (CPPI) strategy with multiplier 119860 The CPPI strategyguarantees a minimum level of wealth at some specifiedhorizon minus119870 lt 119883(119879) and the exposure to the risky asset is amultiplier119860 times the cushion In our problem the cushionis the difference between wealth at time 119905 which is equal to119883(119905)+119892(119905) and a floor which for us is not a given proportionof wealth as usual but it is rather equal to minus119870

This strategy gives rise to the identity

119883 (119905) + 119892 (119905) + 119870 = 119885 (119905) (14)

where 119885(119905) is a geometric Brownian motion of the form

119885 (119905) = 119885 (0) exp (120572119860 minus 1212059021198602) 119905 + 119860120590119882(119905) (15)

with119882 being a standardBrownianmotion and119860 = 120572(1205902(1minus120574)) Further details can be found in [51] The positivity ofthe geometric Brownian motion together with the fact that119892(119879) = 0 proves that our lower bound for119883(119879) is effectivelyminus119870

31 The Limiting Strategy As we allow both 119870 rarr infin and120574 rarr minusinfin in such a way that119870 = Λ(1minus120574) the amount inves-ted in the risky asset according to (13) converges to 120572Λ1205902In other words when taking the limit in (13) we see that thelimiting optimal strategy is one in which a constant amount isinvested in the risky asset Additionally we can prove thislimiting optimal result from a direct optimisation of theexponential utility

Proposition 1 The optimal strategy under the exponentialutility 119906 is to invest a constant amount 120572Λ1205902 in the risky asset

The proof of the CARA optimality can be found inAppendix B

The consequence of Proposition 1 is that the optimal strat-egy for the limiting utility function is the limit of the optimalstrategies something which is not guaranteed to be the casein all optimisation problems

If the limiting strategy is used we have

119889119883 (119905) =1205722Λ

1205902119889119905 +

120572Λ

120590119889119882(119905) (16)

resulting in the identity

119883(119905) =1205722Λ

1205902119905 +120572Λ

120590119882(119905) (17)

The limiting strategy has many interesting properties Forinstance the distribution of terminal wealth is easily obtainedusing the fact that 119882 is a standard Brownian motion As aconsequence typical downside risk measures of the terminalwealth distribution can be obtained analytically This partic-ular feature makes the limiting strategy very interesting fromthe point of view of investigating its performance compared

to other standard product insurance investment strategiesMoreover since we will concentrate on the yearly returns thatprovide exactly the median terminal wealth value for thisstrategy or internal return rates wewill explore how to appro-ximate IRRs in a one-step formula This will indeed speed upcomputations that are usually rather time consuming whenaddressing the performance of sophisticated investmentstrategies especially if outcomes are path dependent

4 Performance Measurement Methodology

In this section we illustrate the limiting strategy and compareit to other popular schemes such as holding a constant pro-portion invested in stocks We follow the performance mea-surementmethodology of [1] when analysing life cycle invest-ments and in particular we study the stochastic behaviour ofterminal wealth distribution

The performance measurement methodology is a toolwhich allows to compare products with different risks Toevaluate an investment plan we search for its equivalent ben-chmark strategy We consider two strategies equivalent if theyhave the same downside risk

We have chosen to work with the expected shortfall asdownside risk measure Sometimes it is referred to as condi-tional tail expectation or tail value at riskWe denote expectedshortfall (See more information on risk measures at [52])with tolerance 120579 as ES120579 and define

ES120579 = E [119883 (119879) | 119883 (119879) lt V120579] (18)

where V120579 is the value at risk with the same tolerance that isthe (1 minus 120579) quantile of the distribution of119883(119879)

P [119883 (119879) gt V120579] = 120579 (19)

The tolerance level is fixed at 95 The benchmark strategy isoften called the trivial strategy with a constant stock propor-tion 120587119887 for all 119905 isin [0 119879]

For any investment plan and for its equivalent strategythe internal interest rate (119903int) is calculated that is

119879minus1

sum

119905=0

Δ119862 (119905) (1 + 119903int)119879minus119905minus 119883119898 (119879) = 0 (20)

where 119883119898(119879) is the median of the final wealth distributionthat is P(119883(119879) gt 119883119898(119879)) = 05 Intuitively a higher medianis associated with a higher internal interest rate The internalinterest rate is the exact yearly rate of return that would leadto the median final wealth 119883119898(119879) given the structure ofpayments and withdrawals from initial wealth

The difference between the internal interest rate of a CPPI(or any other) strategy 119903119901int and that of the benchmark strategy119903119901

int is called the yearly financial gain (whenever positive) orloss (whenever negative)The difference indicates whether ornot the plan beats its risk-equivalent benchmark

41 Calculation of the Expected Shortfall for the CPPI StrategyFrom (14) and 119892(119879) = 0 it can be proved (see Appendix C)


that the value at risk at the 120579th percentile for theCPPI strategyis given by

V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ

minus1(1 minus 120579)119860120590radic119879) minus 1

(21)

where Φ represents the standard Normal distribution func-tion

The expected shortfall is given by

ES120579 = minus119870 +1

1 minus 120579119870119890120572119860119879Φ(Φminus1(1 minus 120579) minus 119860120590radic119879) (22)

as proved in Appendix CThis means that if ES119887

120579for the benchmark strategy 120587119887 is

known then the value of 119870119887 which gives an equivalentexpected shortfall is

119870119887 =ES119887120579

minus1 + (1 minus 120579)minus1119890120572119860119879Φ(Φminus1 (1 minus 120579) minus 119860120590radic119879)

(23)

We will call 119870119887ES119887

120579the factor that relates the maximum

possible loss in the CPPI strategy to the expected shortfall ofthe benchmark This means that one can compare the max-imum possible loss to the expected loss of the benchmarkproduct beyond some value at risk with a given tolerance

42 Median Terminal Value for the Limiting Strategy Themedian value of 119883(119879) corresponds to the median value of119885(119879) which in turn corresponds to themedian value of119882(119879)which is 0 by symmetry We therefore see that the medianvalue of119883(119879) is

119883119898 (119879) = minus119870 + 119870 exp (120572119860 minus 1212059021198602)119879 (24)

From (17) it clearly follows that the median value of 119883(119879)when we use the limiting strategy of investing a constantamount in the risky asset is equal to

119883119898 (119879) =1205722Λ

1205902119879 (25)

This result indicates that in the limit the only factor that altersthe long-term median returns given the market parametersis the difference between initial and terminal time 119879 So theobvious way to increase terminal wealth median returns is towiden life cycle asmuch as possible thusmaking119879 largeThismeans that investors aiming at an optimal lifelong investmentstrategy should not delay the decision to start the savingsphase

43 Internal Rate of Return The internal rate of return orinternal interest rate is the value 119903 which makes the dis-counted value of the payment stream 119862(119905) equal to 0 In thissimple situation it means that

1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0 (26)

which is equivalent to

119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (27)

as proved in Appendix DThis equation defines 119883(119879)119886 in terms of 119903 and the pay-

ment stream mechanism considered in the previous sectionits inverse is the definition of 119903 as a function of119883(119879)119886

Proposition 2 For any given terminal wealth distribution119883(119879) and consumption as in (10) 119883(119879)119886 is an increasingfunction of 119903 over the range 119903 gt 0

The proof of Proposition 2 can be found in Appendix DMoreover it immediately follows from Proposition 2 that themedian value of 119903 corresponds to the median value of119883(119879)

Finally as shown in Appendix E the internal interest ratecan be approximated by

119903 asymp1

119879(minus1 + radic1 +

8119883119898 (119879)

119888119879) (28)

44 A Benchmark Strategy Providing an Equivalent InternalRate of Return All our performance evaluations will be con-nected to a benchmark strategy We fix a simple straightfor-ward benchmark strategy We have chosen to work with theinvestment that implies constant relative risk aversion whilewealth can fall below zero This benchmark strategy impliesthat the investment is designed so that a constant proportion120587119887 of the wealth is exposed to the risky assets at any time 119905We

can thenwrite an expression for terminal wealth so that119883(119879)is given by

119883 (119879) = 119886

1198792minus1

sum

119904=0

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

minus 119886

119879minus1

sum

119904=1198792

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

(29)

This enables us to calculate its expectation

E [119883 (119879)] = 119886

1198792minus1

sum

119904=0

119890120572120587119887(119879minus119904)

minus 119886

119879minus1

sum

119904=1198792

119890120572120587119887(119879minus119904)

= 119886119890120572120587119887(1198901205721205871198871198792minus 1)

2

119890120572120587119887

minus 1

(30)

This would correspond to an internal rate of return of 119903 =119890120572120587119887

minus1 Looking at it in another way the value of 120587119887 requiredto achieve an internal rate of return equal to 119903 would be 120587119887 =120572minus1 log(1 + 119903)The positive skewness of the distribution of 119883(119879) means

that the value of 120587119887 calculated by this formula will always bean underestimate of the true value of 120587119887 required to achievethis return Indeed it may be the case that there is no value of120587119887 which gives an equivalent median return at this level


5 Illustration

In our illustration we are calculating the median of the finalwealth rather than the mean and the corresponding internalinterest rate based on the simulated distribution of the finalwealth

51 The Strategies and the Parameters We set a time horizonof 119879 = 60 years We use 119886 = 10 in our paymentconsumptionstream process The risk-free rate of interest is set equal tozero because we are only interested in seeing the return thatexceeds the risk-free rate This obviously affects the choice ofthe excess interest rate return 120572

The constant proportion portfolio insurance (CPPI) strat-egywith a certainmultiplier119860 guarantees aminimum level ofwealth at some specified horizonThe investmentmechanismis to invest 119860 times 119883(119905) + 119892(119905) + 119870 in risky assets where119883(119905) is wealth at time 119905 119892(119905) is the discounted future paymentstream119870 is the guaranteed maximum possible loss and 119879 isthe ultimate time horizon In this model setup the amountinvested in risky assets at time 119905 is equal to

120572

(1 minus 120574) 1205902(119883 (119905) + 119892 (119905) + 119870) (31)

and the terminal wealth after 119879 = 60 years never falls belowthe guaranteed level minus119870

In the benchmark strategy a proportion of wealth isinvested in stocks every period 119905 and the wealth 119883(119905) is apath-dependent process that we need to simulate in order toexamine its terminal distribution after 119879 = 60 years

Setting the risk-free rate equal to zero has also been donein [1] The choice of the risk-free rate does not affect theresults as has been investigated in [51] With regard to theremaining parameters we have estimated the yearly excessstock return to be equal to 120572 = 343 and the volatility 120590 =1544 Similar levels have recently been used by [1 33 53]

52 Values of the CPPI Leverage Tables 1 2 and 3 presentthe equivalent CPPI plan product for a variety of benchmarkstrategies corresponding to leverage values of 119860 equal to 051 and 2 respectively Together with the value of 119860 in eachtable we also indicate the value of the corresponding factorfrom expression (23) that allows us to calculate themaximumpossible loss 119870 as a function of the downside risk measurethat is the expected shortfall at the 95 level of the bench-mark strategy

In each case the first column indicates the percentageinvested in stocks in the benchmark strategyThe second col-umn corresponds to its expected shortfall at the 95 toler-ance level which is the left tail conditional expectation ofwealth at 119879 The third column shows the value of 119870 whichwould be required in our CPPI strategy to achieve the samemedian risk derived from (23) The rest of columns areobtained by Monte Carlo simulation Columns four and fiveare the internal interest rates for the benchmark and the CPPIstrategies respectively Column six is the difference betweeninternal interest rate returns A positive sign indicates that theCPPI is better in comparison with the benchmark strategy

with exactly the same expected shortfall The last columnshows the percentage which needs to be invested in stocks ina benchmark strategy plan in order to reach the same internalinterest rate as theCPPI in the same row If the sixth column ispositive then the final column is larger than the first columnand the final column benchmark strategy would have morerisk than the corresponding CPPI

We see that the value of the factor is negative for the threetables and increases as 119860 increases This results in largervalues of119870 as119860 decreases for example the value of119870 for theCPPI strategy equivalent to investing 10 in stocks is 42 for119860 = 05 and 13 for 119860 = 2 We also observe that the CPPIstrategy is better than the benchmark strategy for119860 no higherthan 1 Moreover in several cases the CPPI strategy isconsiderably outperforming the benchmark strategy forexample the yearly financial gain in terms of internal interestrate is higher than 02 for 119860 = 05 and 119860 = 1 whencompared to the benchmark strategy and increases as thepercentage invested in stocks increases It reaches the value110 when 100 is invested in stocks and 119860 = 05 For119860 = 2 the CPPI plan underperforms the trivial strategywhen the constant percentage invested in stocks is lower than50 and performs equally at the level of 50 This meansthat for 119860 = 2 investing 50 in stocks in the benchmarkstrategy is equivalent to fixing the value of 119870 = 83 in theCPPI strategy both in terms of risk and performance asmeasured by the internal interest rate For 119860 lower than 2there is no such equivalent investment in stocks comparedto the CPPI plan Finally we also see that for 119860 = 2 and apercentage higher than 50 invested in stocks the CPPIplan again outperforms the trivial strategy but the financialgain is not so high as for lower values of 119860 In this sense itwould be interesting to find analytically a critical value of 119860for which the CPPI plan underperforms the trivial strategy

In conclusion in order to improve the typical fixedproportion invested in stocks that is a very common lifetimeinvestment strategy one should lower leverage 119860 which alsoimplies raising 119870 in the CPPI strategy Recall that as 119860 isproportional to the reciprocal of the coefficient of riskaversion lowering119860 is equivalent to increasing risk aversion

In summary our results show that whenever leverage 119860decreases and we find the risk-adjusted equivalent to thebenchmark strategy for a fixed 120587119887 then the maximum possi-ble loss 119870 increases This indicates that indeed there existsa balance between risk aversion and the CPPI multipleMoreover CPPI is better than the trivial strategy for 119860 le 1 ifwe look at the yearly internal rates of return difference butnotably the performance of CPPI becomes much better thanthe benchmark when119860 diminishes as we approach the limit-ing optimal strategy In several cases the CPPI is quite con-siderably outperforming the benchmark strategy

53 Graphical Display When Varying A Figure 1 shows theproportion invested in stocks that the benchmark strategyrequires in order to have the same median return as theequivalent CPPI plan that has the same risk as the benchmarkthat invests 20 in stocks We study variations of CPPI as afunction of 119860 We can see that it increases as 119860 gets close to


Table 1 Equivalence between lifetime benchmark investment strategy (a fixed percentage 120587119887 is invested in stocks) and the CPPI (fixed 119870)We set 119879 = 60 120583 = 343 and 120590 = 1544 Here we consider 119860 = 05 so the factor derived from (23) is equal to minus3255

120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887

int Equivalent 120587119887

10 minus1282 42 033 053 020 1620 minus2734 89 064 098 034 3230 minus4343 141 093 137 044 4840 minus6103 199 119 173 054 6450 minus8012 261 143 204 062 8260 minus10073 328 164 234 069 10770 minus12316 401 184 261 077 lowast

80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast

lowastThemaximum internal interest rate for the bechmarks is 25014 for 152 invested in stocks


120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887


10 minus1282 13 033 032 minus002 1020 minus2734 28 064 062 minus002 1930 minus4343 45 093 091 minus002 2940 minus6103 63 119 117 minus001 3950 minus8012 83 143 143 000 5060 minus10073 104 164 166 002 6170 minus12316 127 184 189 005 7380 minus14767 153 201 211 011 8790 minus17485 181 216 233 017 106100 minus20536 212 227 254 027 lowast



0

5

10

15

20

25

30

35

40

0 05 1 15 2

()

Trivial strategy with the equivalent returnA

Figure 1 For different values of 119860 (ranging from 0 to 2) dotsindicate the equivalent proportion of wealth invested in stocks forthe benchmark strategy which has the same median return as theCPPI which is exactly risk-equivalent to 120587119887 = 20

0

10

20

30

40

50

60

70

80

0 05 1 15 2

()

Trivial strategy with the equivalent return

A


zero and reaches a value around 35 We also observe that itdiminishes after 119860 = 1 Figure 2 shows the same graph butfor an initial level proportion of 40 in stocks Again we seethat it increases for small values of119860 and it is around 70 for119860 close to zero and again diminishes after 119860 = 1

Since 119860 is proportional to the reciprocal of risk aversionthese figures mean that as the multiplier goes to zero riskaversion increases and as risk aversion is very large the CPPIcan allow the maximum possible loss amount 119870 to increaseSo the CPPI strategy can produce returns that can be aboutthe same as those obtained with a constant proportionalstrategy but using 119860 = 2

6 Discussion and Conclusion

As a result of our investigation we conclude that an investorshould put at risk ldquowhatever he can afford to loserdquo 119870 andnot just a positive cushion as in standard CPPI products The

investment should be leveraged with some constant 119860 andthen in the limit the resulting constant amount should beinvested in the risky asset

If the investor decides that he does not want to loseanything (compared to the short interest rate) then he cannotgain either so that case is trivial So let us say that if a lifelonginvestor is not afraid of losing up to 119870 then the optimalstrategy is to invest 119860 times the accumulated wealth that is119860 lowast (119870 + gains) where 119860 is proportional to the inverse ofthe power of the utility function that is risk aversion This isexactly the principle of CPPI strategies

In one particular limit if we let 119860 and 119870 act togetherincreasing risk aversion and 119870 to infinity then we can reachthe limiting optimal strategy that corresponds unsurprisinglyto the constant absolute risk utility function

This limiting optimality is a beautiful result that has prob-ably escaped the analysis because of the difficulties arisingwhen studying lifetime investment strategies The literaturehas mostly overlooked the fact that the floor that is used totalk about the investment cushion could be fixed below zeroin order to accept losses

We have aimed at showing the inherent interaction bet-ween risk aversion and maximum possible loss a topic thathas not been treated in such away before and that can perhapsexplain why risk aversion is so difficult to communicate oreven measure in real life

We conclude that absolute relative risk aversion might beworth considering for long term savings

We claim that the optimal strategy for long-term investorsis to keep a constant amount invested in the risky assetsthroughout the whole investment horizon The constantamount is equal to the product of the maximum possible losstimes a multiple which we can also call a leverage Unlikeother popular products the optimal allocation does not needto be recalculated based on the performance of the marketsThus dynamic investment allocation is automatic and simplefor our proposed strategy Finally and most importantly itturns out that in the limit of our suitable combination ofmax-imum possible loss and infinite risk aversion wealth andaccumulated returns do not determine the definition of theconstant amount optimal strategy

Our recommendation is not delaying or postponingdecisions on lifetime savingspension plans because arisingfrom the optimal limiting strategy of having a large constantamount invested in the risky assets combinedwith a large riskaversion it follows that the longer the investment period thelarger the expected returns at the terminal time

Appendices

A Optimal Strategy in the CRRA Case

This proof is an adapted version of the one presented by [6]and it is also been revised from [51]

The control problem of choosing a strategy 120587 = 120587(119905 119909) tomaximize for each 0 le 119905 lt 119879 and for each 119909

J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909

strategy 120587 is used] (A1)


is a familiar one and is treated by for example [54] Theingredients of the problem are the utility function 119906 and thedynamical process 119883(119905) which in our case evolve accordingto the stochastic differential equation

119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)

where 120587(119905) represents for each time 119905 the proportion of thetotal wealth which is invested in the risky asset

The optimal value function 119881(119905 119909) is defined as

119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)

The Hamilton-Jacobi-Bellman methodology (HJB) allows usto write the optimality equation in the form

sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)

for each 119905 such that Δ119862(119905) = 0 where 119881119905 119881119909 and 119881119909119909 denotepartial derivatives of 119881 If Δ119862(119905) = 0 the correspondingrequirement is

119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)

As a function of 120587 this is quadratic so the maximum isachieved at

= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)

as long as 119881119909119909 lt 0 (a condition which needs to be checkedseparately for each application of the theory) Substitutingthis into the optimality equation gives

119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)

where Δ119862(119905) = 0In this case the terminal utility function 119906 is 119906(119909) =

120574minus1(119909 + 119870)

120574 This prompts us to investigate solutions of theform

119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)

which satisfy the boundary condition

119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)

where 119892(119905) = int119879119905119889119862(119904) with corresponding strategy

(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)

It is elementary to verify that 119881119909119909 lt 0 when 120574 lt 1As we have found a strategy whose value function 119881

satisfies both the optimality equation and the boundary con-dition the verification theorem in [54] allows us to concludethat the strategy is optimal

The resulting optimal strategy characterised by the form(119905)119883(119905) = 119860(119870 + 119883(119905) + 119892(119905)) is known as the constant pro-portion portfolio insurance (CPPI) strategy with multiplier119860TheCPPI strategy guarantees aminimum level of wealth atsome specified horizon119883(119879) gt minus119870

B Optimal Strategy in the CARA Case

This investigation proceeds as in Appendix A except thatnow the utility function is 119906(119909) = minusΛ119890minus119909Λ

We seek a solution of the form

119881 (119905 119909) = minus119869 (119905) exp (minus 119909Λ) (B1)

Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)

For this to be a solution of (A7) at times when Δ119862(119905) = 0 wemust have

[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)

which has solution proportional to exp(minus(120572221205902)(119879minus119905))Thebehaviour at the times when Δ119862(119905) = 0 is accounted for by afactor exp(minus(1Λ) int119879

119905119889119862(119904)) = exp(minus119892(119905)Λ) Along with the

boundary condition 119869(119879) = Λ the solution is therefore

119881 (119905 119909) = minusΛ exp(minus 1Λ[119909 + 119892 (119905)] minus

1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)

Again the verification theorem (see eg [54]) enables us todeduce that this strategy is optimal


C Calculation of the Expected Shortfall forthe CPPI Strategy

The value at risk at the 120579th percentile for the CPPI strategy isthe value V120579 such that

(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879

times [log (V120579 + 119870) minus log (119885 (0)) minus (120572119860 minus1

211986021205902)119879]]

(C1)

Since 119882(119879)radic119879 is a standard Normal random variable wecan work out the value of V120579 explicitly in terms of the otherparameters noting that 119885(0) = 119883(0minus) + 119892(0minus) + 119870 = 119870

V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)


The conditional tail expectation corresponding to a left-tail probability of (1 minus 120579) is given by

E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)

In this expression119884 stands for the ratio119882(119879)radic119879 which weknow to have a standard Normal distribution

By definition P(119884 lt Φminus1(1 minus 120579)) = (1 minus 120579) We may

therefore write the expected shortfall as

ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int

Φminus1(1minus120579)

minusinfin

119890119860120590radic119879119910

times1

radic2120587exp(minus

1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1

radic2120587exp(minus1

2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1

1 minus 120579119870119890120572119860119879Φ(Φminus1(1 minus 120579) minus 119860120590radic119879)

(C4)

D Proof of Proposition 2

We have to prove that

1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)

is increasing in 119903 for 119903 gt 0 Proving that its logarithm isincreasing will be sufficient Differentiating the logarithmgives

1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)

which can also be written as

1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)

The denominator is positive for 119903 gt 0 as the first second andthird terms are positive As for the numerator it is anincreasing function of 119903 since it has derivative (12)119879(1 +119903)(1198792minus1)

(1 + (119879 + 2)119903) and its value at 119903 = 0 is 119879 gt 0

E Approximating the Internal Rate of Return

Theinternal rate of return or internal interest rate is the value119903 which makes the discounted value of the payment stream119862(119905) equal to 0 In this simple situation it means that

1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)


If119883(119879) = 0 it is clear that the internal rate of return is 0When119883(119879) is not too large we can use an asymptotic expansion toderive an approximate value for 119903

((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)

Multiplying this by (1 minus (1 + 119903)minus1) we obtain

4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)

This means that the value 4119883119898(119879)(1198881198792) will provide a good

approximation to themedian value of 119903 as long as 1199031198792 is quiteclose to 0 and in other cases we can use the quadratic derivedfrom this expression to find a closer approximation

119903 asymp1


8119883119898 (119879)

119888119879) (E6)

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

Support from the Spanish Ministry of ScienceFEDERECO2010-21787-C03-01 is gratefully acknowledgedMontser-rat Guillen thanks ICREA Academia

References

[1] M Guillen J P Nielsen A M Perez-Marın and K S PetersenldquoPerformance measurement of pension strategies a case studyof Danish life cycle productsrdquo Scandinavian Actuarial Journalvol 2012 no 4 pp 258ndash277 2012

[2] M Guillen A K Konicz J P Nielsen and A M Perez-MarınldquoDo not pay for a Danish interest guarantee the law of the tripleblowrdquoThe Annals of Actuarial Science vol 7 no 2 pp 192ndash2092013

[3] A Perold ldquoConstant portfolio insurancerdquo Tech Rep 3 HarvardBusiness School 1986

[4] F Black and R Jones ldquoSimplifying portfolio insurancerdquo Journalof Portfolio Management vol 13 pp 48ndash51 1987

[5] A Perold and W Sharpe ldquoDynamic strategies for asset alloca-tionrdquo Financial Analyst Journal vol 51 pp 16ndash27 1988

[6] F Black andA Perold ldquoTheory of constant proportion portfolioinsurancerdquo Journal of Economic Dynamics and Control vol 16no 3-4 pp 403ndash426 1992

[7] J Belles-Sampera J M Merigo M Guillen and M SantolinoldquoThe connection between distortion risk measures and orderedweighted averaging operatorsrdquo InsuranceMathematics andEco-nomics vol 52 no 2 pp 411ndash420 2013

[8] B Abbasi andMGuillen ldquoBootstrap control charts inmonitor-ing value at risk in insurancerdquoExpert SystemsWith Applicationsvol 40 no 15 pp 6125ndash6135 2013

[9] C-D Fuh I Hu Y-H Hsu and R-H Wang ldquoEfficient simu-lation of value at risk with heavy-tailed risk factorsrdquoOperationsResearch vol 59 no 6 pp 1395ndash1406 2011

[10] R Alemany C Bolance and M Guillen ldquoA nonparametricapproach to calculating value-at-riskrdquo Insurance Mathematicsand Economics vol 52 no 2 pp 255ndash262 2013

[11] L Chen S He and S Zhang ldquoTight bounds for some risk mea-sures with applications to robust portfolio selectionrdquo Opera-tions Research vol 59 no 4 pp 847ndash865 2011

[12] C Bolance M Guillen E Pelican and R Vernic ldquoSkewed biv-ariate models and nonparametric estimation for the CTE riskmeasurerdquo Insurance Mathematics and Economics vol 43 no 3pp 386ndash393 2008

[13] J Belles-SamperaMGuillen andM Santolino ldquoBeyond value-at-risk in finance and insurance GlueVaR distortion risk mea-suresrdquo Risk Analysis vol 34 no 1 pp 121ndash134 2014

[14] K Van Weert J Dhaene and M Goovaerts ldquoComonotonicapproximations for the probability of lifetime ruinrdquo Journal ofPension Economics and Finance vol 11 no 2 pp 285ndash309 2012

[15] K VanWeert J Dhaene andM Goovaerts ldquoOptimal portfolioselection for general provisioning and terminal wealth prob-lemsrdquo Insurance Mathematics and Economics vol 47 no 1 pp90ndash97 2010

[16] H Huang and M A Milevsky ldquoLifetime ruin minimizationshould retirees hedge inflation or just worry about itrdquo Journal ofPension Economics and Finance vol 10 no 3 pp 363ndash387 2011

[17] R Josa-Fombellida and J P Rincon-Zapatero ldquoOptimal invest-ment decisions with a liability the case of defined benefit pen-sion plansrdquo InsuranceMathematics andEconomics vol 39 no 1pp 81ndash98 2006

[18] R Josa-Fombellida and J P Rincon-Zapatero ldquoOptimal assetallocation for aggregated defined benefit pension funds withstochastic interest ratesrdquo European Journal of Operational Re-search vol 201 no 1 pp 211ndash221 2010

[19] R Josa-Fombellida and J P Rincon-Zapatero ldquoStochastic pen-sion funding when the benefit and the risky asset follow jumpdiffusion processesrdquo European Journal of Operational Researchvol 220 no 2 pp 404ndash413 2012

[20] N Branger A Mahayni and J C Schneider ldquoOn the optimaldesign of insurance contracts with guaranteesrdquo InsuranceMathematics and Economics vol 46 no 3 pp 485ndash492 2010

[21] A Chen and M Suchanecki ldquoDefault risk bankruptcy proce-dures and the market value of life insurance liabilitiesrdquo Insur-ance Mathematics and Economics vol 40 no 2 pp 231ndash2552007

[22] N Cai and S Kou ldquoPricing Asian options under a hyper-exponential jump diffusion modelrdquo Operations Research vol60 no 1 pp 64ndash77 2012

[23] M Chronopoulos B De Reyck and A Siddiqui ldquoOptimalinvestment under operational flexibility risk aversion and un-certaintyrdquo European Journal of Operational Research vol 213no 1 pp 221ndash237 2011

[24] W S Buckley G O Brown and M Marshall ldquoA mispricingmodel of stocks under asymmetric informationrdquoEuropean Jour-nal of Operational Research vol 221 no 3 pp 584ndash592 2012

[25] T-Y Yu H-C Huang C-L Chen and Q-T Lin ldquoGeneratingeffective defined-contribution pension plan using simulationoptimization approachrdquo Expert Systems with Applications vol39 no 3 pp 2684ndash2689 2012


[26] T-J Chang S-C Yang and K-J Chang ldquoPortfolio optimiz-ation problems in different risk measures using genetic algo-rithmrdquo Expert Systems with Applications vol 36 no 7 pp10529ndash10537 2009

[27] M Decamps A De Schepper and M Goovaerts ldquoSpectraldecomposition of optimal asset-liability managementrdquo Journalof Economic Dynamics and Control vol 33 no 3 pp 710ndash7242009

[28] A Butt and Z Deng ldquoInvestment strategies in retirement in thepresence of a means-tested government pensionrdquo Journal ofPension Economics and Finance vol 11 no 2 pp 151ndash181 2012

[29] E Vigna and S Haberman ldquoOptimal investment strategy fordefined contribution pension schemesrdquo InsuranceMathematicsand Economics vol 28 no 2 pp 233ndash262 2001

[30] IOwadally SHaberman andDGomezHernandez ldquoA savingsplan with targeted contributionsrdquo Journal of Risk and Insurancevol 80 no 4 pp 975ndash1000 2013

[31] P Emms and S Haberman ldquoIncome drawdown schemes for adefined-contribution pension planrdquo Journal of Risk and Insur-ance vol 75 no 3 pp 739ndash761 2008

[32] I Karatzas and S E Shreve Methods of Mathematical FinanceSpringer 1998

[33] J Annaert S V Osselaer and B Verstraete ldquoPerformance eval-uation of portfolio insurance strategies using stochastic domi-nance criteriardquo Journal of Banking and Finance vol 33 no 2 pp272ndash280 2009

[34] P Bertrand and J-L Prigent ldquoOmega performancemeasure andportfolio insurancerdquo Journal of Banking and Finance vol 35 no7 pp 1811ndash1823 2011

[35] Y Yao ldquoAnalysis of the value of portfolio insurance strategyrdquoAdvances in Intelligent and Soft Computing vol 114 pp 291ndash2982012

[36] P H Dybvig ldquoDynamic portfolio strategies or how to throwaway amillion dollars in the stockmarketrdquoTheReview of Finan-cial Studies vol 1 no 1 pp 67ndash88 1988

[37] S Vanduffel A Ahcan L Henrard and M Maj ldquoAn explicitoption-based strategy that outperforms dollar cost averagingrdquoInternational Journal of Theoretical and Applied Finance vol 15no 2 Article ID 1250013 2012

[38] C Bernard M Maj and S Vanduffel ldquoImproving the design offinancial products in a multidimensional black-Scholes mar-ketrdquo North American Actuarial Journal vol 15 no 1 pp 77ndash962011

[39] G M Constantinides ldquoA note on the suboptimality of dollar-cost averaging as an investment policyrdquo Journal of Financial andQuantitative Analysis vol 14 no 2 pp 445ndash450 1979

[40] S Vanduffel J Dhaene andM J Goovaerts ldquoOn the evaluationof savingconsumption plansrdquo Journal of Pension Economics andFinance vol 4 no 1 pp 17ndash30 2005

[41] C Donnelly M Guillen and J P Nielsen ldquoExchanging uncer-tain mortality for a costrdquo Insurance Mathematics and Eco-nomics vol 52 no 1 pp 65ndash76 2013

[42] R C Merton ldquoLifetime portfolio selection under uncertaintythe continuoustime caserdquo The Review of Economics and Statis-tics vol 51 no 3 pp 247ndash257 1969

[43] R C Merton ldquoOptimum consumption and portfolio rules in acontinuous-time modelrdquo Journal of EconomicTheory vol 3 no4 pp 373ndash413 1971

[44] A Geyer and W T Ziemba ldquoThe innovest Austrian pensionfund financial planningmodel InnoALMrdquoOperations Researchvol 56 no 4 pp 797ndash810 2008

[45] R T Baumann and H HMuller ldquoPension funds as institutionsfor intertemporal risk transferrdquo Insurance Mathematics andEconomics vol 42 no 3 pp 1000ndash1012 2008

[46] J Chai W Horneff R Maurer and O S Mitchell ldquoOptimalportfolio choice over the life cycle with flexible work endoge-nous retirement and lifetime payoutsrdquo Review of Finance vol15 no 4 pp 875ndash907 2011

[47] P Boyle J Imai and K S Tan ldquoComputation of optimal port-folios using simulation-based dimension reductionrdquo InsuranceMathematics and Economics vol 43 no 3 pp 327ndash338 2008

[48] S Benartzi and R HThaler ldquoRisk aversion or myopia Choicesin repeated gambles and retirement investmentsrdquoManagementScience vol 45 no 3 pp 364ndash381 1999

[49] A Vanduel J Dhaene andM Goovaerts ldquoOn the evaluation ofldquosavingsconsumptionrdquo plansrdquo Journal of Pension Economics andFinance vol 4 no 1 pp 17ndash30 2005

[50] J W Pratt ldquoRisk aversion in the small and in the largerdquo Econo-metrica vol 32 pp 122ndash136 1964

[51] A K Konicz Performance measurement of empirical and theo-retical pension investment strategies [MS thesis] University ofCopenhagen 2010 httppapersssrncomsol3paperscfmabstract id=1663791

[52] L Shen and R J Elliott ldquoHow to value riskrdquo Expert Systems withApplications vol 39 no 5 pp 6111ndash6115 2012

[53] M Guillen J P Nielsen A M Perez-Marın and K S PetersenldquoPerformance measurement of pension strategies a case studyof Danish life-cycle productsrdquo Scandinavian Actuarial Journalno 1 pp 49ndash68 2013

[54] T Bjork Arbitrage Theory in Continuous Time Oxford Univer-sity Press 2004

Research ArticleLong-Run Savings and Investment Strategy Optimization

Russell Gerrard1 Montserrat Guilleacuten2

Jens Perch Nielsen1 and Ana M Peacuterez-Mariacuten2

1 Cass Business School City University London 106 Bunhill Row London EC1Y 8TZ UK2Department of Econometrics Riskcenter-IREA University of Barcelona Avenue Diagonal 690 08034 Barcelona Spain

Correspondence should be addressed to Montserrat Guillen mguillenubedu

Received 10 November 2013 Accepted 8 January 2014 Published 23 February 2014

Academic Editors J M Calvin C-T Chang and M Pal

Copyright copy 2014 Russell Gerrard et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We focus on automatic strategies to optimize life cycle savings and investment Classical optimal savings theory establishes thatgiven the level of risk aversion a saver would keep the same relative amount invested in risky assets at any given time We showthat when optimizing lifecycle investment performance and risk assessment have to take into account the investorrsquos risk aversionand the maximum amount the investor could lose simultaneously When risk aversion and maximum possible loss are consideredjointly an optimal savings strategy is obtained which follows from constant rather than relative absolute risk aversionThis result isfundamental to prove that if risk aversion and themaximumpossible loss are both high then holding a constant amount invested inthe risky asset is optimal for a standard lifetime savingpension process and outperforms some other simple strategies Performancecomparisons are based on downside risk-adjusted equivalence that is used in our illustration

1 Introduction

We study long term investment strategies characterised by afirst period of savings followed by a period of consumptionafter retirement This is the most common form of optimiza-tion problem that is faced by most citizens who seek a way tobuild up a pension payment stream that provides a balancebetween risk and reward Most investors request downsideprotection and desire upside potential

There are two competing magnitudes related to risk inclassical optimality theory when studying lifetime savingsand retirement investmentsOne is themaximumamount thesaver is allowing to lose during his life cycle The other is hisrisk aversion a parameter that is linked to the power of apower utility function However risk aversion andmaximumpossible loss have not been studied together Most articlesfocus on one and consider the other parameter as fixedWhenlooking at the distribution of terminal wealth our resultsshow that those two parameters should not be consideredseparately when optimizing long-term expected returnswhile controlling for downside risk

In this paper we use the risk adjusted ad hoc performancemeasure recently developed in [1 2] to evaluate the conse-quences of choosing a given combination of maximum loss

and risk aversion We study optimality of a constant absoluterisk aversion investment strategy that differs slightly from butis reminiscent of standard Constant Proportion PortfolioInsurance (CPPI) which was introduced by [3] (see also[4ndash6]) It is unsurprising that combining a high maximumpossible long-term loss with a high risk aversion that is alower power in the power utility function will be superior inthe long run to the risk-adjusted equivalent with a lower long-termmaximumpossible loss and a lower risk aversion that ishigher power in the utility function However the first com-bination is a good deal better than the second when studyingthe distribution of terminal wealth leading us to furtherinvestigations in this direction We take this point of view toits logical extreme and let the maximum loss go to infinity atthe same time as the power in the utility function goes to zerowhich means that we also raise risk aversion to infinity Ourlimit behaviour is carried out in such a way that the limit ofpower utility functions is also a utility function which hap-pens to be the exponential utility and which gives a fixed con-stant absolute risk aversion It turns out that both the optimalstrategy in the constant relative risk aversion and the perfor-mancemeasures converge smoothly to the equivalent entitiesderived directly from the exponential utility

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 510531 13 pageshttpdxdoiorg1011552014510531






















119906120574 (119888) = 120574minus1119888120574 (1)



119877 (119888) = minus119888

11990610158401015840

120574(119888)

1199061015840120574(119888)

= 1 minus 120574 (2)


119860 (119888) = minus

11990610158401015840

120574(119888)

1199061015840120574(119888)

=1 minus 120574

119888 (3)



119906120574119870 (119888) = 120574minus1(119888 + 119870)

120574 (4)


119877 (119888) = minus119888

11990610158401015840

120574119870(119888)

1199061015840

120574119870(119888)= (1 minus 120574)

119888

119888 + 119870 (5)


119860 (119888) = minus

11990610158401015840

120574119870(119888)

1199061015840

120574119870(119888)= (1 minus 120574)

1

119888 + 119870 (6)







119906120574119870 (119888) =1

120574119870120574minus1(119888 + 119870)

120574 (7)


119906120574119870 (119888) =Λ (1 minus 120574)

120574(1 +

119888

Λ(1 minus 120574))

120574

(8)


119906 (119888) = minusΛ119890minus119888Λ

(9)











(10)


119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905) (11)





max120587

E [1

120574(119883 (119879) + 119870)

120574] (12)


120587 (119905)119883 (119905) = 119860 (119870 + 119883 (119905) + 119892 (119905)) (13)










119883 (119905) + 119892 (119905) + 119870 = 119885 (119905) (14)


119885 (119905) = 119885 (0) exp (120572119860 minus 1212059021198602) 119905 + 119860120590119882(119905) (15)







119889119883 (119905) =1205722Λ

1205902119889119905 +

120572Λ

120590119889119882(119905) (16)


119883(119905) =1205722Λ

1205902119905 +120572Λ

120590119882(119905) (17)







ES120579 = E [119883 (119879) | 119883 (119879) lt V120579] (18)


P [119883 (119879) gt V120579] = 120579 (19)



119879minus1

sum

119905=0

Δ119862 (119905) (1 + 119903int)119879minus119905minus 119883119898 (119879) = 0 (20)







V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(21)



ES120579 = minus119870 +1





119870119887 =ES119887120579


(23)

We will call 119870119887ES119887




119883119898 (119879) = minus119870 + 119870 exp (120572119860 minus 1212059021198602)119879 (24)


119883119898 (119879) =1205722Λ

1205902119879 (25)



1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0 (26)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (27)






119903 asymp1


8119883119898 (119879)

119888119879) (28)



119883 (119879) = 119886

1198792minus1

sum

119904=0

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

minus 119886

119879minus1

sum

119904=1198792

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

(29)


E [119883 (119879)] = 119886

1198792minus1

sum

119904=0

119890120572120587119887(119879minus119904)

minus 119886

119879minus1

sum

119904=1198792

119890120572120587119887(119879minus119904)

= 119886119890120572120587119887(1198901205721205871198871198792minus 1)

2

119890120572120587119887

minus 1

(30)





5 Illustration




120572

(1 minus 120574) 1205902(119883 (119905) + 119892 (119905) + 119870) (31)













120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887





0

5

10

15

20

25

30

35

40

0 05 1 15 2

()



0

10

20

30

40

50

60

70

80

0 05 1 15 2

()


A














Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References













































































119906120574 (119888) = 120574minus1119888120574 (1)



119877 (119888) = minus119888

11990610158401015840

120574(119888)

1199061015840120574(119888)

= 1 minus 120574 (2)


119860 (119888) = minus

11990610158401015840

120574(119888)

1199061015840120574(119888)

=1 minus 120574

119888 (3)



119906120574119870 (119888) = 120574minus1(119888 + 119870)

120574 (4)


119877 (119888) = minus119888

11990610158401015840

120574119870(119888)

1199061015840

120574119870(119888)= (1 minus 120574)

119888

119888 + 119870 (5)


119860 (119888) = minus

11990610158401015840

120574119870(119888)

1199061015840

120574119870(119888)= (1 minus 120574)

1

119888 + 119870 (6)







119906120574119870 (119888) =1

120574119870120574minus1(119888 + 119870)

120574 (7)


119906120574119870 (119888) =Λ (1 minus 120574)

120574(1 +

119888

Λ(1 minus 120574))

120574

(8)


119906 (119888) = minusΛ119890minus119888Λ

(9)











(10)


119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905) (11)





max120587

E [1

120574(119883 (119879) + 119870)

120574] (12)


120587 (119905)119883 (119905) = 119860 (119870 + 119883 (119905) + 119892 (119905)) (13)










119883 (119905) + 119892 (119905) + 119870 = 119885 (119905) (14)


119885 (119905) = 119885 (0) exp (120572119860 minus 1212059021198602) 119905 + 119860120590119882(119905) (15)







119889119883 (119905) =1205722Λ

1205902119889119905 +

120572Λ

120590119889119882(119905) (16)


119883(119905) =1205722Λ

1205902119905 +120572Λ

120590119882(119905) (17)







ES120579 = E [119883 (119879) | 119883 (119879) lt V120579] (18)


P [119883 (119879) gt V120579] = 120579 (19)



119879minus1

sum

119905=0

Δ119862 (119905) (1 + 119903int)119879minus119905minus 119883119898 (119879) = 0 (20)







V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(21)



ES120579 = minus119870 +1





119870119887 =ES119887120579


(23)

We will call 119870119887ES119887




119883119898 (119879) = minus119870 + 119870 exp (120572119860 minus 1212059021198602)119879 (24)


119883119898 (119879) =1205722Λ

1205902119879 (25)



1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0 (26)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (27)






119903 asymp1


8119883119898 (119879)

119888119879) (28)



119883 (119879) = 119886

1198792minus1

sum

119904=0

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

minus 119886

119879minus1

sum

119904=1198792

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

(29)


E [119883 (119879)] = 119886

1198792minus1

sum

119904=0

119890120572120587119887(119879minus119904)

minus 119886

119879minus1

sum

119904=1198792

119890120572120587119887(119879minus119904)

= 119886119890120572120587119887(1198901205721205871198871198792minus 1)

2

119890120572120587119887

minus 1

(30)





5 Illustration




120572

(1 minus 120574) 1205902(119883 (119905) + 119892 (119905) + 119870) (31)













120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887





0

5

10

15

20

25

30

35

40

0 05 1 15 2

()



0

10

20

30

40

50

60

70

80

0 05 1 15 2

()


A














Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References



























































119906120574119870 (119888) =1

120574119870120574minus1(119888 + 119870)

120574 (7)


119906120574119870 (119888) =Λ (1 minus 120574)

120574(1 +

119888

Λ(1 minus 120574))

120574

(8)


119906 (119888) = minusΛ119890minus119888Λ

(9)











(10)


119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905) (11)





max120587

E [1

120574(119883 (119879) + 119870)

120574] (12)


120587 (119905)119883 (119905) = 119860 (119870 + 119883 (119905) + 119892 (119905)) (13)










119883 (119905) + 119892 (119905) + 119870 = 119885 (119905) (14)


119885 (119905) = 119885 (0) exp (120572119860 minus 1212059021198602) 119905 + 119860120590119882(119905) (15)







119889119883 (119905) =1205722Λ

1205902119889119905 +

120572Λ

120590119889119882(119905) (16)


119883(119905) =1205722Λ

1205902119905 +120572Λ

120590119882(119905) (17)







ES120579 = E [119883 (119879) | 119883 (119879) lt V120579] (18)


P [119883 (119879) gt V120579] = 120579 (19)



119879minus1

sum

119905=0

Δ119862 (119905) (1 + 119903int)119879minus119905minus 119883119898 (119879) = 0 (20)







V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(21)



ES120579 = minus119870 +1





119870119887 =ES119887120579


(23)

We will call 119870119887ES119887




119883119898 (119879) = minus119870 + 119870 exp (120572119860 minus 1212059021198602)119879 (24)


119883119898 (119879) =1205722Λ

1205902119879 (25)



1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0 (26)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (27)






119903 asymp1


8119883119898 (119879)

119888119879) (28)



119883 (119879) = 119886

1198792minus1

sum

119904=0

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

minus 119886

119879minus1

sum

119904=1198792

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

(29)


E [119883 (119879)] = 119886

1198792minus1

sum

119904=0

119890120572120587119887(119879minus119904)

minus 119886

119879minus1

sum

119904=1198792

119890120572120587119887(119879minus119904)

= 119886119890120572120587119887(1198901205721205871198871198792minus 1)

2

119890120572120587119887

minus 1

(30)





5 Illustration




120572

(1 minus 120574) 1205902(119883 (119905) + 119892 (119905) + 119870) (31)













120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887





0

5

10

15

20

25

30

35

40

0 05 1 15 2

()



0

10

20

30

40

50

60

70

80

0 05 1 15 2

()


A














Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References




























































119883 (119905) + 119892 (119905) + 119870 = 119885 (119905) (14)


119885 (119905) = 119885 (0) exp (120572119860 minus 1212059021198602) 119905 + 119860120590119882(119905) (15)







119889119883 (119905) =1205722Λ

1205902119889119905 +

120572Λ

120590119889119882(119905) (16)


119883(119905) =1205722Λ

1205902119905 +120572Λ

120590119882(119905) (17)







ES120579 = E [119883 (119879) | 119883 (119879) lt V120579] (18)


P [119883 (119879) gt V120579] = 120579 (19)



119879minus1

sum

119905=0

Δ119862 (119905) (1 + 119903int)119879minus119905minus 119883119898 (119879) = 0 (20)







V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(21)



ES120579 = minus119870 +1





119870119887 =ES119887120579


(23)

We will call 119870119887ES119887




119883119898 (119879) = minus119870 + 119870 exp (120572119860 minus 1212059021198602)119879 (24)


119883119898 (119879) =1205722Λ

1205902119879 (25)



1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0 (26)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (27)






119903 asymp1


8119883119898 (119879)

119888119879) (28)



119883 (119879) = 119886

1198792minus1

sum

119904=0

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

minus 119886

119879minus1

sum

119904=1198792

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

(29)


E [119883 (119879)] = 119886

1198792minus1

sum

119904=0

119890120572120587119887(119879minus119904)

minus 119886

119879minus1

sum

119904=1198792

119890120572120587119887(119879minus119904)

= 119886119890120572120587119887(1198901205721205871198871198792minus 1)

2

119890120572120587119887

minus 1

(30)





5 Illustration




120572

(1 minus 120574) 1205902(119883 (119905) + 119892 (119905) + 119870) (31)













120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887





0

5

10

15

20

25

30

35

40

0 05 1 15 2

()



0

10

20

30

40

50

60

70

80

0 05 1 15 2

()


A














Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References


























































V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(21)



ES120579 = minus119870 +1





119870119887 =ES119887120579


(23)

We will call 119870119887ES119887




119883119898 (119879) = minus119870 + 119870 exp (120572119860 minus 1212059021198602)119879 (24)


119883119898 (119879) =1205722Λ

1205902119879 (25)



1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0 (26)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (27)






119903 asymp1


8119883119898 (119879)

119888119879) (28)



119883 (119879) = 119886

1198792minus1

sum

119904=0

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

minus 119886

119879minus1

sum

119904=1198792

119890(120572120587119887minus(12)120590

21205871198872)(119879minus119904)+120590120587

119887(119882(119879)minus119882(119904))

(29)


E [119883 (119879)] = 119886

1198792minus1

sum

119904=0

119890120572120587119887(119879minus119904)

minus 119886

119879minus1

sum

119904=1198792

119890120572120587119887(119879minus119904)

= 119886119890120572120587119887(1198901205721205871198871198792minus 1)

2

119890120572120587119887

minus 1

(30)





5 Illustration




120572

(1 minus 120574) 1205902(119883 (119905) + 119892 (119905) + 119870) (31)













120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887





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Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References

























































5 Illustration




120572

(1 minus 120574) 1205902(119883 (119905) + 119892 (119905) + 119870) (31)













120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887





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Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References


























































120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 481 201 287 086 lowast

90 minus17485 569 216 312 097 lowast

100 minus20536 668 227 337 110 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887



80 minus14767 226 201 285 084 lowast

90 minus17485 268 216 310 095 lowast

100 minus20536 315 227 335 108 lowast



120587119887 ES (95) 119870 119903

119887

int 119903119901

int 119903119901

int minus 119903119887





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Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References

























































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()


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Appendices




J (119905 119909 120587)def= E [119906 (119883 (119879)) | 119883 (119905) = 119909




119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References


























































119889119883 (119905) = 120572120587 (119905)119883 (119905) 119889119905 + 120590120587 (119905)119883 (119905) 119889119882 (119905) + 119889119862 (119905)

(A2)



119881 (119905 119909) = sup120587

J (119905 119909 120587) (A3)


sup120587

[119881119905 + 120572120587119909119881119909 +1

2120587212059021199092119881119909119909] = 0 (A4)


119881 (119905minus 119909) = 119881 (119905 119909 + 119889119862 (119905)) (A5)


= minus120572

1205902119909sdot119881119909

119881119909119909

(A6)


119881119905 minus1205722

21205902

1198812

119909

119881119909119909

= 0 (A7)


120574minus1(119909 + 119870)


119881 (119905 119909) =1

120574119901(119905)1minus120574(119909 + 119902 (119905))

120574 (A8)


119881 (119879 119909) = 119906 (119909) =1

120574(119909 + 119870)

120574 (A9)

We find that

119881 (119905 119909) =1

120574(119909 + 119870 + 119892 (119905))

120574 exp(120572120574

21205902 (1 minus 120574)(119879 minus 119905))

(A10)


(119905 119909) =120572

1205902 (1 minus 120574)

119909 + 119870 + 119892 (119905)

119909 (A11)








Then

119881119905 = minus1198691015840(119905) exp(minus 119909

Λ)

119881119909 =119869 (119905)

Λexp(minus 119909

Λ)

119881119909119909 = minus119869 (119905)

Λ2exp(minus 119909

Λ)

(B2)


[1198691015840(119905) minus

1205722

21205902119869 (119905)] exp(minus 119909

Λ) = 0 (B3)





1205722

21205902(119879 minus 119905))

119909 (119905 119909) =120572

1205902Λ

(B4)





(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References



























































(1 minus 120579)

= P [119883 (119879) lt V120579]

= P [119885 (119879) lt V120579 + 119870]

= P [119882 (119879)

radic119879lt

1

119860120590radic119879


211986021205902)119879]]

(C1)


V120579 = 119870exp ((120572119860 minus1

211986021205902)119879 + Φ


(C2)



E [119883 (119879) | 119883 (119879) lt V120579]

= E [minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times 119890119860120590radic119879119884

| 119883 (119879) lt V120579]

= minus119870 + 119870 exp((120572119860 minus 1211986021205902)119879)

times E [119890119860120590radic119879119884

| 119884 lt Φminus1(1 minus 120579)]

(C3)




ES120579 = minus119870 +1

1 minus 120579119870 exp((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910

times1


1199102

2)119889119910

= minus119870 +1

1 minus 120579119870 exp ((120572119860 minus 1

211986021205902)119879)

times int


minusinfin

119890119860120590radic119879119910 1


1199102

2)

= minus119870 +1

1 minus 120579119870 exp (120572119860119879)

times int


minusinfin

1


2(119910 minus 119860120590radic119879)

2

)

= minus119870 +1


(C4)



1 + 119903

119903((1 + 119903)

1198792minus 1)2

(D1)


1

1 + 119903minus1

119903+ 119879

(1 + 119903)1198792minus1

(1 + 119903)1198792minus 1

(D2)


1 + (119903119879 minus 1) (1 + 119903)1198792

119903 (1 + 119903) ((1 + 119903)1198792minus 1)

(D3)





1198792minus1

sum

119904=0

119886(1 + 119903)119879minus119904

minus

119879minus1

sum

119904=1198792

119886(1 + 119903)119879minus119904minus 119883 (119879) = 0

(E1)

Summing

119886(1 + 119903)

119879minus (1 + 119903)

1198792

1 minus (1 + 119903)minus1

minus 119886(1 + 119903)

1198792minus 1

1 minus (1 + 119903)minus1= 119883 (119879) (E2)


119886((1 + 119903)1198792minus 1)2

= (1 minus (1 + 119903)minus1)119883 (119879) (E3)



((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References


























































((1 + 119903)1198792minus 1)2

= (1 +119879

2119903 +

1

8119879 (119879 minus 2) 119903

2+ 119874 (119903

3) minus 1)

2

=11987921199032

4(1 +

1

2(119879 minus 2) 119903 + 119874 (119903

2))

(E4)


4119883119898 (119879)

1198881198792= 119903 (1 +

1

2119879119903 + 119874 (119903

2)) (E5)



119903 asymp1


8119883119898 (119879)

119888119879) (E6)



Acknowledgments


References
























































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