EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 3, 2019, 1315-1336 ISSN 1307-5543 – www.ejpam.com Published by New York Business Global Strategic Asset Allocation for Life Insurers with Stochastic Liability† C. Atkinson1, S. Mokkhavesa2,∗, P. Ingpochai3 1 Mathematics Department, Imperial College, London, 180 Queen’s Gate, London SW7 2AZ, U.K. 2 Muang Thai Life Assurance PCL, 250 Rachadaphisek Rd., Huaykwang, Bangkok 10320, Thailand 3 Aigen, 250 Rachadaphisek Rd., Huaykwang, Bangkok 10320, Thailand Abstract. The problem of strategic asset allocation and product mix choice of a life insurance company is considered where account is taken of the stochastic risk associated with both assets and liabilities. Using the methods of stochastic dynamic programming we derive equations for optimal weights of both risky and riskless assets under continuous time. The resulting equations can be solved exactly for some parameter values and utility functions. When this is not possible a general perturbation expansion method is set up for which explicit solutions are derived for the first terms but the method can be generalized to any order in the expansion parameter. 2010 Mathematics Subject Classifications: 35Q93, 49J20, 49L20, 91B16, 91B30, 91G80 Key Words and Phrases: Optimal control problems involving partial differential equations, Stochastic Dynamic Programming, Utility Theory, Insurance, stochastic control, PDE 1. Introduction The financial management of an insurance company involves the analysis of the assets and liabilities on a unified basis. The premiums paid by the policyholder should be pru- dently invested so that the company can honor the contractual obligation that comes with the policy with a high level of confidence. In its most basic form, asset-liability manage- ment (ALM) dictates the investment choice on the asset side and the choice of products marketed on the liability side. Moreover, ALM deals with the planning of financial re- sources with uncertainty about the economic, capital markets and actuarial conditions. The two controls available to the management of the insurance company are thus the †The authors would like to thank Mrs Vanee Lamsam for her kind support. ∗Corresponding author. DOI: https://doi.org/10.29020/nybg.ejpam.v12i3.3478 Email addresses: c.atkinson@imperial.ac.uk (C. Atkinson), drsutee@gmail.com (S. Mokkhavesa), hall ingpochai@hotmail.com (P. Ingpochai) http://www.ejpam.com 1315 c© 2019 EJPAM All rights reserved. C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1316 allocation of the premiums received from the policyholders to the various invested assets as well as the decision to market different insurance products†. Insurers currently use cashflow projection of liabilities based on Monte Carlo simu- lation as a starting point for the strategic asset allocation exercise. Several papers e.g. Gerstner, Griebel and Holtz (2009) used deterministic numerical simulation to model Asset-Liability management (ALM). These numerical techniques are often cumbersome and lack the transparency offered by analytical method which we apply in this paper. Here, we show that if the utility of the company is of the constant relative risk aversion (CRRA) form then the solution can be found exactly for the case where the correlation ρ between the risky asset and the risky insurance liability is zero. For the non-zero corre- lation case, we find that we can solve for the solution provided that the interest rates for savings (ra) and loans (rl) are the same. To solve for the scenario where the interest rates on savings and loans differ only slightly, we devise a perturbation method and find a first and second order solution though the method can be generalized to any order. This work is based on Merton’s continuous time optimal portfolio selection and con- sumption rule, Merton (1990). Here an investor wishes to maximise the expected utility of consumption over his life time. For the case of the utility functions of the type constant relative risk aversion (CRRA), one can find the optimal allocation and consumption rule which will maximise the lifetime expected utility of the investor. 2. Governing Equations 2.1. Evolution of the Liability L Let L be the liabilities of the company. L is made up of the risky insurance liability uL and riskless liability (1− u)L. The risky insurance liability comes from the stochastic nature of the benefits outgo (claims payment, surrender, and maturity benefits). The riskless liability is defined to be liabilities that are deterministic in value such as debt issued by the company. Suppose that we assume that the liability of a life insurance company evolves according to the stochastic differential equation dLt = (cuLt −muLt + (1− u)Ltrl)dt− qLtudZ (t) (1) where Lt is the liability at time t, c (orm) represents the exponential growth (or decay) rate in L, qdZ is the stochastic component which represents the random nature of insurance, dZ is an infinitesimal Wiener increment. The stochastic term −qLtdZ (t) can increase or decrease the liability of the insurance company. The variable rl represents the interest rate at which the riskless liability grows; equivalently this is the same as the interest rate charged for a loan. The control variable u allows the management to vary the proportion of the policyholder’s liability between the deterministic (riskless) liability which grows at the rate rl and the stochastic (risky) liability with some known probability distribution. †Life insurance products have vastly different risk profiles. The decision to market a particular product should be based on the ability to find suitable assets to invest in. C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1317 Figure 1: Illustration of the insurer’s balance sheet and the associated control variables w and u. In reality, this can be done through risk transfer techniques such as reinsurance or hedging with swaps. Intuitively, suppose that there is a death claim then the company’s liability decreases (i.e. once the company has paid out the claim of this particular policy, the contract is terminated). Also, the liability can increase if the actuary decides to strengthen the reserve. 2.2. Evolution of the total Asset W Let us assume that the company’s assets are comprised of the risky asset, S1 (= wW ) and the risk-free asset S0 (= (1− w)W ) with dS1 = αS1dt+ σS1dX (2) dS0 = raS0dt (3) then we can write dW = [w (α− ra) + ra]Wdt+ wWσdX (4) where dX is an infinitesimal Wiener increment. The total asset of the company W is by definition W ≡ L+ E (5) where E stands for equity and represents the portion of assets owned by the shareholders once all the liabilities have been paid out. The illustration of the insurer’s balance sheet is shown in Figure (1). Then dE = dW − dL (6) C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1318 = {[w (α− ra) + ra]W − cuLt − ((1− u) rl −mu)Lt} dt + qLtudZ + wWσdX (7) The problem for choosing the strategic asset allocation and insurance product mix strategy is formulated as follows; maxE {∫ T 0 U [W (t) , L (t)] dt+B [W (T ) , L (T ) , T ] } subject to the budget equation (4), and W (t) > 0;W (0) = W0 > 0. Let us assume U (W,L) to be a strictly concave utility function and B [W (t) , T ] is a specified bequest valuation function. More details on the bequest valuation function can be found in Merton (1969). Here E is short for E (0), the conditional expectation operator given W0 is known. The differential equations governing the optimal solution of this stochastic control problem for the portfolio selection (w is the weight of risky investment) and liability mix (u is the weight of the risky insurance liability) can be derived by the methods of stochastic dynamic programming see e.g. Merton (1990). Define the value function J (W (t) , L (t) , t) by J = maxEt {∫ T t U (W (t) , L (t)) dt+B (W (T ) , L (T ) , T ) } = maxEt { ∫ t+δt t U (W (t) , L (t)) dt +J (W (t+ δt) , L (t+ δt) , t+ δt) } (8) where Et is the conditional expectation operator given W (t) is known. If one now expands the term in J on the right hand side of (8) by Taylor’s series using equations (1) and (4), apply Ito’s Lemma, and then take the limit as δt tends to zero one arrives at the Hamilton- Jacobi equation for the value function given by 0 = φ [w∗, u∗;W,L, t] ≡ max {u,w} { U (W,L) + ∂J ∂t + [w (α− ra) + ra]W ∂J ∂W + 1 2 σ2w2W 2 ∂ 2J ∂W 2 + {cu−mu+ (1− u) rl}L ∂J ∂L +qLWuwσρ ∂2J ∂L∂W + 1 2 q2L2u2 ∂2J ∂L2 } (9) The optimality condition may be rewritten as φ [w∗, u∗;W,L, t] = 0, (10) for fixed W and L at anytime t. The optimal strategy of (9) is given by w∗and u∗. We differentiate (10) with respect to w and u and set to zero to find the interior maximum. C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1319 This gives (α− ra)W ∂J ∂W + σ2wW 2 ∂ 2J ∂W 2 + qLWuσρ ∂2J ∂L∂W = 0 (11) and {c−m− rl}L ∂J ∂L + qLWwσρ ∂2J ∂L∂W + uq2L2 ∂ 2J ∂L2 = 0 (12) Note that w ∂φ ∂w + u∂φ∂u gives 1 2 σ2w2W 2 ∂ 2J ∂W 2 + 1 2 u2q2L2 ∂ 2J ∂L2 + qLWwuσρ ∂2J ∂L∂W = 1 2 { −{c−m− rl}uL ∂J ∂L − (α− ra)wW ∂J ∂W } (13) Substituting this into φ = 0 gives 0 = max {u,w} { U (W,L) + ∂J ∂t + [ 1 2 w (α− ra) + ra ] W ∂J ∂W + { 1 2 cu− 1 2 mu+ ( 1− 1 2 u ) rl } L ∂J ∂L } (14) subject to the terminal condition J [W (T ) , L (T ) , T ] = B [W (T ) , L (T ) , T ] and the solu- tion being feasible. 3. Utility U (W ) = W γ γ For this particular case, the utility function is a function of W only. This suggests that the management is most interested in maximising the firm’s total asset. This is not as nonsensical as it may first seem as financial institutions are constantly seeking to reduce their expense ratio; increasing the size of the asset under management is one way of achieving this. Suppose that U (W ) = W γ γ and assume that the value function has the form J = W γF (L, t), where F (L, t) is an arbitrary function independent of W , then the optimality conditions for w and u in equations (11) and (12) become (α− ra)F + σ2w(γ − 1)F + qLuσρ ∂F ∂L = 0 (15) and (c−m− rl + qwσργ) ∂F ∂L + uq2L ∂2F ∂L2 = 0. (16) The value function as given in (9) becomes 0 = φ [w∗, u∗;W,L, t] ≡ max {u(s),w(s)} { 1 γ + ∂F ∂t C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1320 + ( w (α− ra) + ra + 1 2 σ2w2(γ − 1) ) γF + (cu−mu+ (1− u) rl + quwσργ)L ∂F ∂L + 1 2 q2u2L2∂ 2F ∂L2 } . (17) 3.1. Zero Correlation Case In this particular scenario, we assume that the correlation between the risky asset and the risky liability is zero. Note that if ρ = 0 then from (15) and (16) we obtain w∗ = (α− ra) σ2(1− γ) (18) and u∗ = ( − (c−m− rl) ∂F∂L q2L∂ 2F ∂L2 ) (19) Then from (17) 0 = 1 γ + ∂F ∂t + ( ra + 1 (α− ra)2 2σ2 (1− γ) ) γF + ( rl − (c−m− rl)2 ∂F∂L 2q2L∂ 2F ∂L2 ) L ∂F ∂L (20) If we choose F (L, t) = A (t)Lλ +B (t) , (21) then ∂F ∂L = A (t)λLλ−1, ∂2F ∂L2 = A (t)λ (λ− 1)Lλ−2 , ∂F∂t = Ȧ (t)Lλ + Ḃ (t) . Substituting this into (20) gives 0 = 1 γ + Ȧ (t)Lλ + Ḃ (t) + ( ra + 1 2 (α− ra)2 σ2(1− γ) ) γ ( A (t)Lλ +B (t) ) + ( rl − (c−m− rl)2 2q2 (λ− 1) ) AλLλ . (22) This implies that 0 = 1 γ + Ḃ (t) + ( ra + 1 2 (α− ra)2 σ2(1− γ) ) γB (t) (23) and 0 = Ȧ (t) + ( ra + 1 2 (α− ra)2 σ2(1− γ) ) γA (t) C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1321 + ( rl − (c−m− rl)2 2q2 (γ − 1) ) λA (t) . (24) Therefore, A (t) = A0 exp [ − ( raγ + 1 2 (α− ra)2 γ σ2(1− γ) + rlλ− (c−m− rl)2 λ 2q2 (λ− 1) ) t ] . (25) We now solve for A (t) and B (t) with the terminal condition F (L, T ) = A (T )L2 +B (T ) , with A(t) given in (25) then u∗ = c−m− rl q2 (1− λ) . (26) If this final condition is F (L, T ) = 0 then from (21) we obtain B (T ) = 0 and A (T ) = 0 and then B (t) = 1 γ2 ( ra + 1 2 (α− ra)2 σ2 (1− γ) )( exp [ − { ra + 1 2 (α− ra)2 σ2 (1− γ) } γ (t− T ) ] − 1 ) A (t) ≡ 0. Hence the solution is independent of u. This is aligned with our intuition that if the utility depends only on W and that the risky insurance liability is not related to the risky asset then we would expect to find that the strategic allocation of the risky asset, w, is the same as Merton’s solution and the optimal product mix u can take on any value. 4. Utility U (W,L) = (W−L)γ γ The power utility function assumed in this paper belongs to the class of utility functions known as constant relative risk aversion (CRRA). Utility of this class are sensible because we observe that companies strive to link the size of the risk that they take to their capacity for risk absorption. 4.1. Non-zero correlation and ra = rl By setting the utility of the form (W−L)γ γ , we are assuming that the management is interested in maximising the utility based on shareholder’s equity. It can be argued that a utility function that depends on the shareholder’s equity is more sensible than a utility which depends solely on the size of the company as shown in the earlier example. Suppose that the utility function has the form U (E) = Eγ γ = (W−L)γ γ and we seek the solution to the value function of the form J (W,L, t) = A (t) (W−L)γ γ Thus from (11) and (12) we have σ2wW (γ − 1)− quLσρ (γ − 1) = − (α− ra) (W − L) (27) C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1322 and −qwWσρ (γ − 1) + q2uL (γ − 1) = {c−m− rl} (W − L) (28) which can be rewritten as( σ2 (γ − 1) −qσρ (γ − 1) −qσρ (γ − 1) q2 (γ − 1) )( wW uL ) = ( − (α− ra) (W − L) {c−m− rl} (W − L) ) . (29) The equation (29) can then be solved to give wW = Q1 (W − L) (30) uL = Q2 (W − L) (31) where Q1 = − (α− ra) q2 + qσρ (c−m− rl) σ2q2 (γ − 1) (1− ρ2) (32) and Q2 = −qσρ (α− ra) + σ2 (c−m− rl) σ2q2 (γ − 1) (1− ρ2) . (33) Note that the weights w and u are time independent depending on the parameters α, ra, rl, q, γ, σ, ρ, c, m. and also on the ratio E W . Also these weights involve both rα and rl and they have only been assumed equal to solve the full Hamilton-Jacobi equation to this order. When we go to the next order, order ε, we must keep ra and rl in the above equations. With the value function of the form J (W,L, t) = A (t) (W−L)γ γ and its respective derivatives, we can substitute (30) and (31) into (9), which gives 0 = { (W − L)γ γ +A′ (W − L)γ γ + (α− ra)Q1A (W − L)γ + 1 2 σ2Q2 1 (γ − 1)A (W − L)γ −{c−m− rl}Q2A (W − L)γ + raWA (W − L)γ−1 − rlLA (W − L)γ−1 −qσρQ1Q2 (γ − 1)A (W − L)γ + 1 2 q2Q2 2 (γ − 1)A (W − L)γ } (34) Dividing (34) by (W − L)γ gives 0 = { 1 γ + A′ γ + ( (α− ra)Q1 + 1 2 σ2Q2 1 (γ − 1)− {c−m− rl}Q2 −qσρQ1Q2 (γ − 1) + 1 2 q2Q2 2 (γ − 1) ) A + (raW − rlL)A (W − L) } . (35) C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1323 4.1.1. Case: ra ≡ rl The equation above has a special solution when ra ≡ rl. In this particular case, we can cancel out W−L and solve for A (t) . From a practical perspective, equating ra to rl implies that the risk-free interest rate on the asset side is assumed to be the same as the borrowing rate on the liability side. This situation is likely to be found in developed markets where policyholders exhibit a high level of financial literacy and hence will demand that the compensation for opportunity cost be on par with the observed riskfree rate. Hence, for this case, we can solve for A (t). To solve for A (t), consider 0 = 1 +A′ + ( (α− ra)Q1 + 1 2 σ2Q2 1 (γ − 1)− {c−m− rl}Q2 −qσρQ1Q2 (γ − 1) + 1 2 q2Q2 2 (γ − 1) + ra ) γA. (36) We can rewrite (34) by letting κ = ( (α− ra)Q1 + 1 2σ 2Q2 1 (γ − 1)− {c−m− rl}Q2 −qσρQ1Q2 (γ − 1) + 1 2q 2Q2 2 (γ − 1) + ra ) γ then 0 = dA dt + κA+ 1 and A (t) = −1 κ + e−κtC1. 4.1.2. Case: ra 6= rl It is clear from equation (35) that the choice of J (W,L, T ) = A (T ) (W−L)γ γ is a solution only in the case when ra = rl because only then do W and L disappear in equation (35) leaving an ordinary differential equation for A (t) . However, if the difference between ra and rl is assumed small it is possible to solve equations (11), (12), and (13) as a power series in ε where ε is the difference between ra and rl. In less developed markets, the policyholders may not demand that the return on their policies to match the observed risk free asset unlike the previous case. We assume here that |ε| � ra and describe the method for determining the order ε correction we generalise the method to get higher order terms in the appendix. We write J = J0 + εJ1 where J0 is the value function determined in the last section. Note that when ε = 0, we have the previous case and the value function J . Now we write wW = w, C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1324 uL = u and expand as w = w0 + εw1 + o ( ε2 ) u = u0 + εu1 + o ( ε2 ) where w0 and u0 are the solutions of equations (30) and (31) where J = J0. With the above notation we can expand equations (11) and (12) to get for O (1) A0 ( w1 u1 ) = ( − (α− ra) J1W − σ2w0J1WW − qσρu0J1LW −cJ1L − q2u0J1LL − qσρw0J1LW + J0L ) where c = c−m− ra and A0 = ( σ2 ∂ 2J0 ∂W 2 qσρ ∂2J0 ∂L∂W qσρ ∂2J0 ∂L∂W q2 ∂ 2J0 ∂L2 ) . For the Hamilton-Jacobi equation (9) when U (E) = Eγ γ it is best to use equation (14) i.e. 0 = U (E) + ∂J ∂t + 1 2 (α− ra)w ∂J ∂W + 1 2 cu ∂J ∂L + ra ( W ∂J ∂W + L ∂J ∂L ) + εL ∂J ∂L (37) when rl = ra + ε. Then for zero order we have 0 = U (E) + ∂J0 ∂t + 1 2 (α− ra)w0 ∂J0 ∂W + 1 2 cu0 ∂J0 ∂L + ra ( W ∂J0 ∂W + L ∂J0 ∂L ) (38) which we can write as U (E) + L0 (J0) = 0. For order ε, from (37) we also get 0 = L0 (J1) + L ∂J0 ∂L + 1 2 (α− ra)w1 ∂J0 ∂W + 1 2 cu1 ∂J0 ∂L (39) We have solved the zero order problem for U (E) = Eγ γ = (W−L)γ γ by choosing J0 = A (t) (W−L)γ γ and solving for A (t) . The extra terms in the o (ε) equation depending on J0 gives L0 (J1) + (W − L)γ−1A (t) [ −L+ 1 2 (α− ra)w1 − 1 2 cu1 ] = 0. (40) We now look for J1 in the form J1 = [B (t)L+B1 (t)W ] (W − L)γ−1 (41) C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1325 so that L0 (J1) = [ Ḃ (t)L+ Ḃ1W ] (W − L)γ−1 + (W − L)γ−1 {( 1 2 (α− ra)w0B1 (t) + 1 2 cu0B (t) + ra (B1W +BL) )} + [BL+B1W ] {( 1 2 (α− ra)w0 − 1 2 cu0 ) (γ − 1) } Eγ−2 + raE γ−1 (γ − 1) [BL+B1W ] but note that from the zero order equation (38) we have 0 = ( 1 2 (α− ra)w0 − 1 2 cu0 ) A (t)Eγ−1 + raA (t)Eγ + Ȧ (t)Eγ γ + Eγ γ (42) and since from equations (30) and (31), w0 and u0 are proportional to E = W − L, this equation reduces to 0 = {[ 1 2 (α− ra)Q1 − 1 2 cQ2 ] + ra } A (t) + Ȧ (t) γ + 1 γ (43) from which we can determine A (t). This expression agrees with that of equation (36) when ra = rl though they look superficially different. Furthermore, using w0 = Q1 (W − L) u0 = Q2 (W − L) in Equation (40) gives to order ε. 0 = ( Ḃ (t)L+ Ḃ1W ) + 1 2 (α− ra)Q1B1 (W − L) + 1 2 cBQ2 (W − L) + ra (B1W +BL) + (γ − 1) (BL+B1W ) { 1 2 (α− ra)Q1 − 1 2 cQ2 } + ra (γ − 1) (BL+B1W ) +A (t) [ −L+ 1 2 (α− ra)w1 − 1 2 cu1 ] (44) where c = c −m − rl. If we now equate to zero coefficients of L and W, we get coupled differential equations for B (t) and B1 (t) in terms of A (t) . To proceed further we need w1 C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1326 and u1 which we have from the o (ε) equations of (11) and (12). With the above choice of J0 and J1 these become A (t) ( σ2 −qσρ −qσρ q2 )( w1 u1 ) = ( Ξ1 Ξ2 ) (45) where Ξ1 = − (α− ra) [ BL+ B1γW (γ − 1) − B1L (γ − 1) ] − σ2Q1 [B1 {γW − 2L}+ (γ − 2)BL] − qσρQ2 [B {W − (γ − 1)L} − (γ − 1)B1W +B1L] (46) and Ξ2 = −c [ BW (γ − 1) − BLγ (γ − 1) −B1W ] − q2Q2 [−2BW + γBL+ (γ − 2)B1W ] − qσρQ1 [BW − (γ − 1)BL+B1L− (γ − 1)B1W ] −A (t) W − L γ − 1 . (47) From this equation we can see that A (t)w1 and A (t)u1 are expressions linear in W and L so when substituted in (44), we get two linear coupled differential equations in B (t) and B1 (t) . Further as can be seen in (44) there is one term which is −A (t)L which stands alone hence both B (t) and B1 (t) will depend on A (t) . Finally since the condition at t = T has been satisfied by A (t) we will have B (T ) = 0 and B1 (T ) = 0. The functional form of A (t) is derived in the Appendix. 5. Conclusion & Final remarks This paper investigates the optimal portfolio selection for insurers with stochastic liabilities. The model considers characteristics of the insurers balance sheet and the de- pendence on control variables w of the wealth investment (for which wW is the risky part see Figure 1) and u of the liability of which uL is the amount of risky liability. These controls are found for a variety of utility functions where portfolio selection is found via a maximum expected utility over a life time. For the simplest case (Section 3) where the utility ( U = W γ γ ) depends only on wealth we find (Equation 18) that the optimal allocation to the risky asset is w∗ = (α− ra) σ2(1− γ) and u can take any value. C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1327 In section 4, we consider a utility function of the form U (W,L) = (W − L)γ γ . For this case if ra = rl (i.e. asset interest rate is equal to liability interest rate) and for any correlation ρ, we have the optimal weights w and u given as w0 and u0 below which are exact in this case since ε = rl − ra = 0. For the situation when ra 6= rl but |ε| � ra, we use an expansion such as that described below (see also the Appendix). We have solved for the optimal weight w and u w = wW = w0 + εw1 + ε2w2 +O ( ε3 ) (48) u = uL = u0 + εu1 + ε2u2 +O ( ε3 ) (49) This can be solved to give to O (1) as w0 = w0W = Q1 (W − L) (50) u0 = u0L = Q2 (W − L) (51) where Q1 = [ − (α− ra) q2 + qσρ {c−m− rl} ] σ2q2 (γ − 1) (1− ρ2) (52) and Q2 = [ −qσρ (α− ra) + σ2 {c−m− rl} ] σ2q2 (γ − 1) (1− ρ2) (53) w0 and u0 have no explicit dependence of time. The O (ε) approximation can be deduced from the equation A (t) ( σ2 −qσρ −qσρ q2 )( w1 u1 ) = ( Ξ1 Ξ2 ) (54) with Ξ1 = − (α− ra) [ BL+ B1γW (γ − 1) − B1L (γ − 1) ] − σ2Q1 [B1 {γW − 2L}+ (γ − 2)BL] − qσρQ2 [B {W − (γ − 1)L} − (γ − 1)B1W +B1L] (55) and Ξ2 = −c [ BW (γ − 1) − BLγ (γ − 1) −B1W ] − q2Q2 [−2BW + γBL+ (γ − 2)B1W ] − qσρQ1 [BW − (γ − 1)BL+B1L− (γ − 1)B1W ] C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1328 −A (t) W − L γ − 1 , (56) where A (t) follows from Equation (43) and B and B1 can be deduced from a pair of first order coupled differential equation in time. From (54) we can write A (t)w1 = q2Ξ1 σ2q2 (1− ρ2) + qσρΞ2 σ2q2 (1− ρ2) ≡ t1L+ t2W (57) A (t)u1 = qσρΞ1 σ2q2 (1− ρ2) + σ2Ξ2 σ2q2 (1− ρ2) ≡ t3L+ t4W (58) Equating coefficients of L and W in equation (44) gives 0 = · B − 1 2 (α− ra)Q1B1 +B { raγ − 1 2 cQ2γ + 1 2 (γ − 1) (α− ra)Q1 } + (α− ra) 2 t1 − 1 2 ct3 −A (t) (59) and 0 = · B1 − 1 2 cQ2B +B1 { raγ + 1 2 (α− ra) γQ1 − 1 2 cQ2 (γ − 1) } + (α− ra) 2 t2 − 1 2 ct4 (60) 5.1. Example: Results for ra = rl, ρ 6= 0 For this case, the ratios wW E , uLE , (E = W − L) are time independent for the example we consider ra = rl, we plot Q1 and Q2 for the parameters α = 0.1, ra = 0.03, q = 0.25, σ = 0.3, c = 1.5, m = 1.3, rl = 0.025, γ = 0.5 but with ρ and c varying, see Figure 2 and Figure 3. C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1329 Figure 2: Plot of Q1 when ra = rl. Figure 3: Plot of Q2 when ra = rl. As is evident from the plot of Q1, we must exclude regions where Q1 < 0 since it is inadmissible given our short selling constraint of the risky asset. If we examine Q1 when c = 0, we see the convexity structure and if ρ = 0, then Q1 = 1.56, which implies the amount of risky asset is 1.56 times that of the shareholders equity W −L, refer to Figure 4. For ease of explanation, we plot Q2 (optimal allocation into risky liability) for the case where c = 0, we see that the Q2 = 0 when ρ = 0, refer to Figure 5. This is sensible because the movements of the risky assets are not correlated to that of the risky liabilities hence the policy should suggest we minimise the unnecessary volatility this stochastic liability may cause. As the correlation increases, Q2 increases monotonically to help match the movements of the assets. C. Atkinson, S. Mokkhavesa, P. Ingpochai / Eur. J. Pure Appl. Math, 12 (3) (2019), 1315-1336 1330 -0.6 -0.4 -0.2 0.2 0.4 0.6 ρ 2.0 2.5 3.0 Q1 Figure 4: Plot of Q1 when ra = rl and c = 0. -0.6 -0.4 -0.2 0.2 0.4 0.6 ρ -2 -1 1 2 Q2 Figure 5: Plot of Q2 when ra = rl and c = 0. 5.2. Example: Results for ra 6= rl but |ε| � ra O (1) solution as above but now ra 6= rl w0W (W − L) = Q1 u0L (W − L) = Q2 To correct for order ε the behavior will depend on B (t) and B1 (t) and will potentially require us to correct for the optimal ratios as t evolves. The general result for w1 and u1 can be deduced from equations (54), (55) and (56). To this order we deduce wW = Q1 (W − L) + ε [P1 (W − L) + P2L] +O ( ε2 ) REFERENCES 1331 and uL = Q2 (W − L) + ε [P3 (W − L) + P4W ] +O ( ε2 ) where P1 and P3 are constants depending on the various parameters of the problem. Note that higher order terms could be attained if necessary, see the appendix. Also a full numerical solution could be obtained by solving Equation (9) or Equation (14) as a backward diffusion equation together with the constraints Equations (11) and (12). References [1] Atkinson, C. and Al-Ali, B. (1997) On investment consumption model with transac- tion costs: an asymptotic analysis. Applied Mathematical Finance 4, 109-133. [2] Atkinson, C. and Ingpochai, P. (2012) The Effect of Correlation and Transaction Costs on the Pricing of Basket Options. Applied Mathematical Finance 19, 131-179. [3] Atkinson, C. and Ingpochai, P. (2010) Optimisation of N-Risky Asset Portfolios with Stochastic Variance and Transaction Costs. Journal of Quantitative Finance 10(5), 503-514. [4] Atkinson, C. and Ingpochai, P. (2006) The influence of correlation on multi-asset port- folio optimization with transaction costs Journal of Computational Finance 10(2), 53-96. [5] Atkinson, C. and Mokkhavesa, S. (2001) Towards the determination of utility prefer- ence from optimal portfolio selections. Applied Mathematical Finance 8, 1-26. [6] Atkinson, C., Pliska, S.R. and Wilmott, P. (1997) Portfolio management with trans- action costs. Proceedings of the Royal Society, London A 453, 551-562. [7] Atkinson, C. and Wilmott, P. (1995) Portfolio Management with transaction costs: an asymptotic analysis. Mathematical Finance 5, 357-367. [8] Gerstner, T., Griebel, M. and Holtz, M. (2009) Efficient deterministic numerical sim- ulation of stochastic asset-liability management models in life insurance, Insurance: Mathematics and Economics 44(3), 434-446. [9] Merton, R.C. (1969) Lifetime Portfolio Selection Under Uncertainty: The Continuous- Time Case, The Review of Economics and Statistics 51, 247-257. [10] Merton, R.C. (1971) Optimal consumption and portfolio rules in a continuous time model. J. Economic Theory 3, 373-413. [11] Merton, R.C. (1992) Continuous Time Finance. Blackwell, Cambridge, MA. [12] Mokkhavesa, S. and Atkinson, C. (2002) Perturbation solution of optimal portfolio theory with transaction costs for any utility function. IMA Journal of Management Mathematics 13, 131-151. REFERENCES 1332 [13] Morton, A. and Pliska, S. (1995) Optimal portfolio management with fixed transac- tion costs. Mathematical Finance 5, 337-356. APPENDIX 1333 Appendix We assume here that |ε| � ra and show how the method for determining the order ε correction can be extended to obtain higher order terms. J (W,L, t) = J0 + εJ1 + ε2J2 +O ( ε3 ) wW = w0 + εw1 + ε2w2 +O ( ε3 ) uL = u0 + εu1 + ε2u2 +O ( ε3 ) J (W,L, t) = A (t) (W − L)γ γ + ε (BL+B1W ) (W − L)γ−1 + ε2 ( C1W 2 + C2LW + C3L 2 ) (W − L)γ−2 where B (t), B1 (t) are to be determined in terms of A (t) . To this order, equations (11) and (12) (the conditions for the maximum) give 0 = (α− r) [ J0W + εJ1W + ε2J2W ] + σ2 [ J0WW + εJ1WW + ε2J2WW ] [ w0 + εw1 + ε2w2 ] + qσρ [ J0LW + εJ1LW + ε2J2LW ] [ u0 + εu1 + ε2u2 ] (A.1) and 0 = c [ J0L + εJ1L + ε2J2L ] + q2 [ J0LL + εJ1LL + ε2J2LL ] [ u0 + εu1 + ε2u2 ] + qσρ [ J0LW + εJ1LW + ε2J2LW ] [ w0 + εw1 + ε2w2 ] . (A.2) So we have for O (1) 0 = (α− ra) J0W + σ2w0J0WW + qσρu0J0LW and 0 = cJ0L + q2u0J0LL + qσρw0J0LW . For O (ε) 0 = (α− ra) J1W + σ2w0J1WW + qσρu0J1LW + σ2w1J0WW + qσρu1J0LW and 0 = cJ1L + q2u0J1LL + qσρw0J1LW + q2u1J0LL + qσρw1J0LW . APPENDIX 1334 For O ( ε2 ) 0 = w2σ 2J0WW + u2qσρJ0LW (α− ra) J2W + σ2w0J2WW + qσρu0J2LW + σ2w1J1WW + qσρu1J1LW and 0 = w2qσρJ0LW + u2σ 2J0LL cJ2L + qσρw0J2LW + q2u0J2LL + qσρw1J1LW + q2u1J1LL. Our equation (9) can be written as (or alternatively one can use the version in (14)), 0 = U [E] + ∂J ∂t + cu ∂J ∂L + (α− ra)w ∂J ∂W + ( raW ∂J ∂W + rlL ∂J ∂L ) + 1 2 σ2w2 ∂ 2J ∂W 2 + qσρuw ∂2J ∂L∂W + 1 2 q2u2 ∂2J ∂L2 , (A.3) where u = uL, w = wW. For O (i) 0 = 1 2 σ2w2 0JiWW + qσρu0w0JiLW + 1 2 q2u20JiLL + cu0JiL + (α− ra)w0JiW + Jit + other terms (involving Ji−1, etc.). So write L (Ji) = ( 1 2 σ2w2 0 ∂2 ∂W 2 + qσρu0w0 ∂2 ∂L∂W + 1 2 q2u0 ∂2 ∂L2 +cu0 ∂ ∂L + (α− ra)w0 ∂ ∂W + ∂ ∂t ) Ji and note if rl = ra + ε then raW ∂J ∂W + rlL ∂J ∂L = ra ( W ∂J ∂W + L ∂J ∂L ) + εL ∂J ∂L . We have U (E) = Eγ γ = (W−L)γ γ with J0 = A (t) (W−L)γ γ W ∂J0 ∂W + L ∂J0 ∂L = A (t) (W − L)γ APPENDIX 1335 and raW ∂J ∂W + rlL ∂J ∂W = raL1 (J) + εL ∂J ∂L , where L1 (J) ≡ ( W ∂ ∂W + L ∂ ∂L ) J so from our equation we have O (1) U (E) + L (J0) + raL1 (J0) = 0 and O (ε) 0 = L (J1) + raL1 (J1) + L ∂J0 ∂L + cu1J0L + (α− ra)w1J0W + 1 2 σ2 (2w1w0) J0WW + qσρ (u0w1 + u1w0) J0LW + 1 2 q2 (2u1u0) J0LL Call this Lij (J0) , therefore, 0 = L10 (J0) O ( ε2 ) 0 = L (J2) + raL1 (J2) + L ∂J1 ∂L + raL20 (J0) + L10 (J1) , where L20 (J0) = cu2J0L + (α− ra)w2J0W + σ2 ( w2w0 + w2 1 2 ) J0WW + qσρ (u0w2 + u2w0 + u1w1) J0LW + q2 ( u2u0 + u21 2 ) J0LL and L10 (J0) = cu1J0L + (α− ra)w1J0W + σ2w1w0J0WW + qσρ (u0w1 + u1w0) J0LW + q2u1u0J0LL. With the choice J0 = A (t) (W − L)γ γ , APPENDIX 1336 the O (1) equation reduces to the equation we had previously, which has the form Ȧ (t) γ + 1 γ +DA = 0, where D is constant and equals κ γ . Thus A (t) = − 1 γD + 1 γD e(−κ(t−T )) with A (T ) = 0. From the O (1) equations for w0 and u0 we find that w0 and u0 are both proportional to (W − L). When we come to the O (ε) equations, we write J1 = (BL+B1W ) (W − L)γ−1 and note that L10 (J0) will have terms like (W − L)γ−1 multiplying a linear combina- tion of terms involving u1 and w1. The term L∂J0∂L is again of the form (W − L)γ−1 (L). Then L1 (J1) has this form also on account of( W ∂ ∂W + L ∂ ∂L ) (W − L)γ−1 = (γ − 1) (W − L)γ−1 , L (J1) will also have this form. So if u1 and w1 turn out to be proportional to a linear combination of L and W we get the coupled ODE’1 for B and B1 by equating to zero coefficients of L and W . This behaviour of u1 and w1 follows from the O (ε) equations of (A.1) and (A.2). For the O (ε) equations, note that L10 (J1) will be of the form (W − L)γ−2. Similarly L∂J1∂L has the same form. L20 (J0) has the form u2 () (W − L)γ−1 + w2 () (W − L)γ−1 and the determined terms w2 1 2 J0WW etc are of the quadratic× (W − L)γ−2. If we assume J2 = ( C1W 2 + C2LW + C3L 2 ) (W − L)γ−2 , then L (J2) = quadratic× (W − L)γ−2 + ( C ′1W 2 + C ′2LW + C ′3L 2 ) (W − L)γ−2 Similarly, rL1 (J2) is quadratic× (W − L)γ−2. Finally the O ( ε2 ) equation gives (W − L)γ−2 ( · · · · · · · · · · · · )( w2 u2 ) = (quadratic) (W − L)γ−3 . This gives w2 and u2 as quadratic (W−L) . Thus the term in L20 (J0) has the form (quadratic)(W − L)γ−2 . Finally equating coefficients of W 2, LW and L2 gives 3 ODE for C1, C2 and C3.