# Net contribution, liquidity, and optimal pension management.

IntroductionDuring the recent global financial crisis, one of the most severe crises to afflict the global financial markets, internationally renowned pension service providers (PSPs) showed different reactions to the market turmoil: some PSPs did not alter their asset allocation, whereas others actively adjusted their portfolios to adapt to the adverse market conditions.

When a financial market experiences a slump, investment returns on stocks (risky assets) can fall below returns on bonds (risk-free assets). From a myopic perspective, liquidating risky assets and investing funds in risk-free assets may seem more profitable. Yet, according to our observations, risky asset to total asset ratios of some large-scale PSPs experienced marginal change of only a few percent (see Table D1 in Appendix D). These observations represent the motivation for this research, which focuses on the investment strategies of large-scale PSPs. This research develops theoretical asset allocation schemes to examine the investment strategies of large-scale PSPs.

To lay the theoretical foundations of PSP portfolio management, this article constructs a portfolio balancing strategy that maximizes the expected value of CRRA (constant relative risk aversion) utility function in discrete time where assets consist of one risk-free asset and one risky asset and proportional and fixed costs are paid for risky asset trades. (1)

CRRA criterion may be better suited for normal periods than turbulent periods. However, this research attempts to analyze the asset allocation behaviors of PSPs during both normal and turbulent periods using the widely used CRRA framework.

Within this framework, this research provides a multiperiod optimal portfolio balancing strategy that maximizes the expected utility of the last period while considering periodic (negative or positive) cash flows into the funds at the beginning of each period. For PSPs, these cash flows are usually the net contribution, defined as the difference between the contributions from participating members and the annuity (benefits) payments to members. It is possible to consider net liability instead of net contribution. For example, D'Arcy, Dulebohn, and Oh (1999) study an alternative approach to the optimal management of a pension fund that considered the current pension obligation (such as liabilities for current employees and retirees) and the respective growth rates of pension expenses and the tax base.

Additionally, the impact of market liquidity on the optimal asset portfolio strategies of some large-scale PSPs is examined using the bid-ask spreads as the proxy for the proportional transaction cost, although bid-ask spread could understate the true transaction costs for large investors such as pension funds (Marshall, 2006). (2)

Korn (1998) demonstrates that a fixed transaction cost can be easily incorporated into a continuous time model because this type of cost is incurred whenever an investor rebalances a portfolio. However, incorporating the fixed cost into a discrete time portfolio optimization problem is complex because an investor may or may not rebalance her portfolio at a certain period, in which case the fixed cost is a binary variable. When a binary variable exists, a terminal wealth maximization problem has multiple discontinuous points complicating the application of traditional methods. This article proposes a resolution to this issue by separating the problem into two parts: one part assumes no trading and the other part assumes trading a positive amount of risky assets.

Portfolio optimization is one of the most actively studied subjects in finance. (3) Whereas studies concerning optimal portfolio rebalancing in continuous time abound, the research on portfolio rebalancing in discrete time is relatively scarce despite its practical applications. For example, Zakamouline (2002) proposes a method that generates an approximate solution to the finite horizon portfolio optimization problem for a CRRA investor who pays both fixed and proportional transaction costs (FPTC), whereas this article provides a method that generates an exact solution for the one-period FPTC problem in discrete time.

The FPTC problem presented in this article can be considered as an extension of Boyle and Lin (1997), who consider only proportional transaction costs (PTC) for a CRRA investor whose no-trading region (NTR) can be completely defined by two rebalancing lines. As shown in this research, the NTR of the one-period FPTC problem is an indefinite space that envelops two straight rebalancing lines. A portfolio that falls outside of NTR is rebalanced so that the resulting portfolio falls onto a rebalancing line (Figure 1). In addition to presenting an explicit strategy for the one-period FPTC problem, a heuristic that finds an approximate solution for the multiperiod FPTC problem is introduced that utilizes the one-period optimal strategy. We show that our theoretical analysis of the FPTC problem can be useful to explain the actual asset allocations of PSP as well as their reactions to the financial market crash. (4)

The contributions of this article are as follows. Under a discrete time framework, this article presents an optimal asset allocation strategy for the one-period FPTC problem for the first time and develops a heuristic for the multiperiod FPTC problem that can generate near-optimal solutions. Using theoretical findings and a heuristic for the multiperiod FPTC problem, the asset allocation strategies of a number of largescale PSPs are analyzed and compared considering the annual net contributions and market liquidity. The computational results of this article indicate that the actual investment strategies of some large PSPs are very close to the theoretically optimal investment strategies suggested by this research, which implies that our model can be useful for PSP asset management.

This article is divided into six sections. The "Motivation: PSPs of Interest" section presents the research motivation and basic information concerning the PSPs under consideration. "The Model" section introduces relevant models used to solve the proposed research problem. The optimal trading policy for the one-period FPTC problem is developed in "The Optimal Strategy for the One-Period FPTC Problem" section. The numerical results of our research are presented in the "Numerical Implications" section. In the "Application to Pension Funds" subsection, we apply our asset allocation framework to several world-renowned pension funds and compare our theoretical asset allocation schemes to the actual asset allocations. Finally, in the "Conclusion" section we present a conclusion and several suggestions for future research.

Motivation: PSPs of Interest

A study of the investment strategies of world renowned PSPs during the recent financial crisis revealed that some PSPs actively adjusted their asset allocations whereas others held onto their existing asset allocation schemes. Moreover, the majority of PSPs adjusted their risky asset to net asset ratio by only a few percent. Our study showed that certain factors, such as liquidity and changes in net contribution have a significant impact on PSP asset allocation policies. Five PSPs are chosen for our analysis and are presented in Table 1.

Table 2 shows the PSPs of interest in this research. These PSPs exhibited different behaviors during the stock market crash between 2008 and 2009. For example, CALP and NYSLR decreased their risky asset ratios by 10.12 percent and 13.39 percent between 2007 and 2009, respectively. Since 2009 CALP and NYSLR have steadily increased their risky asset ratios. Although the magnitudes were relatively smaller, AVON, FAPF, and NPS also decreased their risky asset ratios in 2008 by 5.73 percent, 2.67 percent, and 2.82 percent, respectively. Consistent with CALP and NYSLR, AVON, FAPF, and NPS have shown steady increases in risky asset ratios since 2008 with the exception of 9.51 percent drop in FAPF's risky asset ratio between 2010 and 2011.

This research presents a mathematical model that generates a theoretical optimal asset allocation of PSP, which is influenced by various factors such as net contribution, (fixed and variable) trading costs, PSP's risk aversion, risky asset return, and market volatility. Then we attempt to explain PSP's asset allocation behavior using this model.

The results of our computational study, discussed in the "Sensitivity Analysis" subsection, can be summarized as follows.

1. When the stock market's volatility increases, a PSP's NTR is pushed downward. Hence, the optimal policy is to invest less in risky assets when increased stock market volatility is expected to persist in the near future (see Figure 4f later). (5)

2. A PSP's NTR is widened when the (fixed and variable) trading costs are increased, prohibiting it from trading risky assets when the stock market crashes (see Figures 4b and c later).

3. A PSP's NTR moves upward when the annual net contribution increases, allowing the PSP to invest more in risky assets (see Figure 4a later).

When the above-mentioned findings are applied to five PSP cases, we discover that the five PSPs' portfolios are often managed within or near our NTRs and their behavior during the stock market crash between 2008 and 2009 can be well explained by our framework with the exception of NPS. (6) Some discrepancies between our framework and the PSPs' actual allocations may be caused by various factors such as regulations that govern each PSP, taxes, the price impact of trading, pension product types, and culture.

Table 3 shows that world stock market volatility varied from 7.45 percent to 23.64 percent between the years 2002 and 2011. "Scaled" changed from 1 to 4.02. Therefore, we ran tests using different values of [[sigma].sub.s] to reflect the volatility changes. (7)

Table D1 in Appendix D includes the actual asset allocations and net contributions of the PSPs of interest. The columns in these tables include their asset allocations in stocks (risky assets) and bonds (risk-free assets), member contributions, and benefit payments. The currency units for these tables are thousand USD (CALP and NYSLR), thousand GBP (AVON), million SEK (FAPF), and million KRW (NPS). The net asset values in Table D1 represent market values. The tables show a substantial drop in asset values between the years 2008 and 2010 because of the 2008 financial market crash and the ensuing economic crisis. The net asset values of the five PSPs showed common and distinct changes during this period. One common change was the depreciation of stock values. Additionally, the majority of PSPs' bond values also dropped (with the exception of NPS) possibly because of a rush to withdraw funds.

A sudden drop in the net contributions of CALP, NYSLR, and FAPF are observed during the global financial crisis. The net contributions are approximately -2 percent for CALP, -4 percent for NYSLR, -1 percent in 2000 and 1 percent recently for AVON, 2 percent in 2002, negative in 2009 for FAPF, and close to zero for NPS. The net contributions of the majority of PSPs fluctuated when the financial market crashed. However, the net contribution of NPS showed almost no change and AVON has shown an increase in membership contributions since 2006.

The Model

First, we assume that two classes of assets are traded in the financial market: a risky asset (stock) and a risk-free asset (bond). (8) The one-period return of the risk-free asset is denoted by R. We assume two possible outcomes of the one-period return of the risky asset, u and d (0 < d < R < u) with probabilities [[pi].sub.u] and [[pi].sub.d], respectively. u, d, [[pi].sub.u], [[pi].sub.d] are selected to satisfy [[pi].sub.u], [[pi].sub.d] [greater than or equal to] 0, [[pi].sub.u] + [[pi].sub.d] = 1, [[pi].sub.u]u + [[pi].sub.d]d = [r.sub.s], [[pi].sub.u][u.sup.2] + [[pi].sub.d][d.sup.2] = [[sigma].sup.2.sub.s] + [r.sup.2.sub.s], where [r.sub.s] and [[sigma].sub.s] are the one-period expected return and the volatility of the risky asset, respectively. It is also assumed that the financial market has market friction, and therefore, market participants pay fixed and proportional transaction costs when trading risky assets. By ([x.sub.t], [y.sub.t]), we denote a pair of dollar amounts invested in the risk-free and risky assets at time t(t [member of] {0,1,..., Tg). The fixed cost is denoted by K (9) and the proportional transaction cost for sales (purchases) is [k.sub.1] ([k.sub.2]).

The investor considered in this article is a pension fund manager with the following constant relative risk aversion (CRRA)-type utility preference for wealth:

U(w) := [w.sup.1-[gamma]]/[1 - [gamma]], (1)

where [gamma] is the coefficient of relative risk aversion that satisfies [gamma] > 0 and [gamma] [not equal to] 1. We work with ([x.sub.t], [y.sub.t]) that is constrained within the following (solvency) set:

C := {(x, y)\x + (1 - [k.sub.1])y - K [greater than or equal to] 0, x [greater than or equal to] 0, y [greater than or equal to] 0}.

In the definition of C, x + (1 - [k.sub.1])y - K [greater than or equal to] 0 is the non-negativity of the liquidated wealth constraint and the remaining two constraints represent (no) short-sale constraints. In "The Optimal Strategy for the One-Period FPTC Problem" section, we prove that our optimal strategy generates an optimal solution that is in C as long as the initial portfolio ([x.sub.0], [y.sub.0]) [member of] C satisfies [x.sub.0] [greater than or equal to] 2K.

The fund manager considered in our article plans to maximize her expected utility for wealth at the final time T,

E[U([x.sub.T] + [y.sub.T])],

by controlling the investment amount in the risky asset. (10) Here, E[*] is the expectation under the probabilities [[pi].sub.u] and [[pi].sub.d].

We denote a dollar amount of the risky asset for sales (purchases) at time t by [[DELTA].sub.-] ([[DELTA].sub.+]). (11) Then, we obtain the following relationships between asset holdings at time t and (t +1):

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII],

where [z.sub.t] is the price process of the risky asset.

The Optimal Strategy for the One-Period FPTC Problem

This section shows the one-period optimal trading strategy for the one-period FPTC problem. (12) Let us define the one-period FPTC sell and buy problems at time 0:

max [f.sub.-] ([[DELTA].sub.-]; [x.sub.0]; [y.sub.0]) (2) s.t. [[DELTA].sub.-] [greater than or equal to] 0, ([x.sub.1], [y.sub.1]) [member of] C (2)

and

max [f.sub.+] ([[DELTA].sub.+]; [x.sub.0]; [y.sub.0]) (3) s.t. [[DELTA].sub.+] [greater than or equal to] 0, ([x.sub.1], [y.sub.1]) [member of] C, (3)

where

[f.sub.-]([[DELTA].sub.-]; x, y) := [[pi].sub.u]U(xR + yu - K x R + ((1 [k.sub.1])R - u)[[DELTA].sub.-]) + [[pi].sub.d]U(xR + yd - K x R + ((1 - [k.sub.1])R - d)[[DELTA].sub.-]) (4)

and

[f.sub.+] ([[DELTA].sub.+]; x, y) := [[pi].sub.u]U(xR + yu - K x R + (-(1 + [k.sub.2])R + u)[[DELTA].sub.+]) + [[pi].sub.d]U(xR + yd - K x R + (-(1 + [k.sub.2])R + d)[[DELTA].sub.+]).

We solved the one-period problem by finding the optimal strategies [[DELTA].sup.*.sub.-] and [[DELTA].sup.*.sub.+] for the sell and buy problems and choosing the strategy that maximizes the expected utility. The optimal investment strategy of our one-period problem is the strategy that yields the maximum,

max{[f.sub.-] ([[DELTA].sup.*.sub.-]; [x.sub.0], [y.sub.0]), [f.sub.+] ([[DELTA].sup.*.sub.+]; [x.sub.0], [y.sub.0]), [f.sub.0](0; [x.sub.0], [y.sub.0])}

where

[f.sub.0](0;[x.sub.0],[y.sub.0]) := [[pi].sub.u]U([x.sub.0]R + [y.sub.0]u) + [[pi].sub.d]U([x.sub.0]R + [y.sub.0]d), (6)

which is the expected utility where no transaction occurs.

This article often works with the sell problem only. We omit the analysis of the buy problem because it is a mirror image of the sell problem's case.

We show, using basic algebra, that (4) and (5) are concave functions.

Proposition 1: [f.sub.-] ([[DELTA].sub.-]; x, y) and [f.sub.+] ([[DELTA].sub.+]; x, y) are concave functions with respect to [[DELTA].sub.-] and [[DELTA].sub.+].

Proof: See Appendix A.

Because [f.sub.-]([[DELTA].sub.-]; x,y)([f.sub.+]([[DELTA].sub.+];x,y)) is a convex function, its maximum is achieved at [[DELTA].sup.*.sub.-] ([[DELTA].sup.*.sub.+]), which is a point that satisfies the first-order optimality condition

[f'.sub.-] ([[DELTA].sup.*.sub.-]; x, y) = 0([f'.sub.+] ([[DELTA].sup.*.sub.+]; x, y) = 0).

The first-order optimality conditions of the sell and buy problems can be written as

[w.sub.1] = xR + yu - KR + ((1 - [k.sub.1])R - u)[[DELTA].sup.*.sub.-]/xR + yd KR + ((1 - [k.sub.1])R d)[[DELTA].sup.*.sub.-] (7)

and

[w.sub.2] = xR + yu - KR + ((1 - [k.sub.1])R - u)[[DELTA].sup.*.sub.+]/xR + yd KR + ((1 - [k.sub.1])R d)[[DELTA].sup.*.sub.+] (8)

where [w.sub.1] and [w.sub.2] are defined as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (9)

This article includes parameters that satisfy (1 - [k.sub.1])R > d and u > (1 + [k.sub.2])R so that [w.sub.1] and [w.sub.2] can be defined for any [gamma]. (13) u > (1 [k.sub.1])R and (1 + [k.sub.2])R > d are always satisfied as long as d < R < u. The nominator and the denominator of (7) represent wealth levels of the "up stock market" and the "down stock market" after trading [[DELTA].sup.*.sub.-] units of a risky asset. Given a set of parameters that satisfy the aforementioned conditions, we can show that [w.sub.1] > 1. (14) Equation (7) indicates that the optimal choice for the FPTC sell problem is the one that maintains the after-trading risky asset to risk-free asset ratio at [w.sub.1]. If we ignore fixed transaction costs, ((1 - [k.sub.1])R - u) (1 - [k.sub.1])R - d)) is the next-period return obtained from the unit sale of the stock in an up (down) stock market. Therefore, we can consider [-w.sup.[gamma].sub.1] to be the relative return for sales of the up event with respect to the down event. The case of Equation (8) is identical to that of (7) with a different constant w2. Additionally, we call [w.sup.[gamma].sub.2] the relative return for purchases of the up event with respect to the down event.

Rearranging (7) and (8), we obtain the closed-form solutions for [[DELTA].sup.*.sub.-] and [[DELTA].sup.*.sub.+]

[[DELTA].sup.*.sub.-] = [-w.sub.1] (xR + yd - KR) + (xR + yu - KR)/[w.sub.1] ((1 - [k.sub.1])R - d) - ((1 - [k.sub.1])R - u) (10)

and

[[DELTA].sup.*.sub.+] = [-w.sub.1] (xR + yd - KR) + (xR + yu - KR)/[w.sub.1] ((1 - [k.sub.2])R - d) - ((1 - [k.sub.2])R - u). (11)

By setting [[DELTA].sup.*.sub.-] = 0 and [[DELTA].sup.*.sub.+] = 0 in (7) and (8), respectively, and rearranging them, we can find the following two lines:

y = R([w.sub.1] - 1)/u - [w.sub.1][d] x -KR([w.sub.1] - 1)/u - [w.sub.1]d and (12)

y = R([w.sub.2] - 1)/u - [w.sub.2][d] x - KR([w.sub.2] - 1)/u - [w.sub.2]d. (13)

(x, y) pairs that are on lines (12) and (13) do not require rebalancing. Suppose that ([x.sub.0], [y.sub.0]) is on (12). Then the corresponding A* is zero. Rebalancing lines (12) and (13) pass through (K, 0). We assume that parameters can be chosen so that u/d > [w.sub.1] and u/d > [w.sub.2] are satisfied. (15)

Proposition 2 summarizes the relationship between the location of the initial portfolio (x, y) and the corresponding values of [[DELTA].sup.*.sub.-] and [[DELTA].sup.*.sub.+].

Proposition 2: The signs of [[DELTA].sup.*.sub.-] and [[DELTA].sup.*.sub.+] are determined by the location of the initial portfolio (x, y) as follows.

[[DELTA].sup.*.sub.-] [greater than or equal to] 0 when (x, y) is on the upper left of (12), [[DELTA].sup.*.sub.-] = 0 otherwise, [+ or -]

and

[[DELTA].sup.*.sub.+] > 0 when (x, y) is on the lower right of (13), [[DELTA].sup.*.sub.+] = 0 otherwise.

Proof: See Appendix A.

We showed that [[DELTA].sup.*.sub.-] and [[DELTA].sup.*.sub.+] are optimal choices for (2) and (3), respectively. Moreover, Proposition 2 suggests that events [[DELTA].sup.*.sub.-] > 0, [[DELTA].sup.*.sub.+] > 0, and [[DELTA].sup.*.sub.-] = [[DELTA].sup.*.sub.+] = 0 cannot happen simultaneously. However, it is still possible that [[DELTA].sup.*.sub.-] ([[DELTA].sup.*.sub.+]) is not an optimal solution even if [[DELTA].sup.*.sub.-] > 0([[DELTA].sup.*.sub.+] > 0) because doing nothing may be preferable due to the existence of fixed trading costs.

Let us define explicit formulae that check the optimality of [[DELTA].sup.*.sub.-] > 0 and [[DELTA].sup.*.sub.+] > 0 as follows:

[[delta].sub.-] ([[DELTA].sub.-]) := [f.sub.-]([[DELTA].sub.-]) - [f.sub.0](0) and [[delta].sub.+] ([[DELTA].sub.+]) := [f.sub.+]([[DELTA].sub.+]) - [f.sub.0] (0).

[[delta].sub.-] ([[DELTA].sub.-])([[delta].sub.+]([[DELTA].sub.+])) has a negative y-intercept because there is a jump at [[DELTA].sub.-] = 0([[DELTA].sub.+] = 0) where the fixed transaction cost disappears. We describe NTR in terms of x and y because an (x, y) pair can completely specify [[DELTA].sup.*.sub.-] and [[DELTA].sup.*.sub.+].

NTR of the one-period FPTC problem can be defined as [[THETA].sub.-] [intersection] [[THETA].sub.+] where [[THETA].sub.+] and [[THETA].sub.+] are defined as

[[THETA].sub.-] := {(x,y) : [delta]([[DELTA].sup.*.sub.-] (x,y)) [less than or equal to] 0}and

[[THETA].sub.+] := {(x,y) : [[delta].sub.+] ([[DELTA].sup.*.sub.+] (x,y)) [less than or equal to] 0}. (14)

The nonemptiness of [[THETA].sub.-](x, y) and [[THETA].sub.+] (x, y) implies that we may not take [[DELTA].sup.*.sub.-] or [[DELTA].sup.*.sub.+] even if they are positive. Equalities are used in (14) because we assume that no transaction occurs even when [[delta].sub.-] ([[DELTA].sup.*.sub.-] (x, y)) = 0 or [[delta].sub.+] ([[DELTA].sup.*.sub.+] (x, y)) = 0. We can check whether a given point (x;y) belongs to NTR using Theorem 1.

Theorem 1: and Q+ can be written in terms of x and y as follows:

[[THETA].sub.-] = {(x, y): ([[pi].sub.u][w.sup.1-[gamma].sub.1] + [[pi].sub.d])u(xR + yd - KR + ((1 - [k.sub.1])R d)[[DELTA].sup.*.sub.-] (x, y)) [less than or equal to] [[pi].sub.u]u (xR + yd) + [[pi].sub.d]u(xR + yu)} (15)

and

[[THETA].sub.+] = {(x, y) : ([[pi].sub.u][w.sup.1-[gamma].sub.2] + [[pi].sub.d])u(xR + yd - KR + (-(1 + [k.sub.2])R d) [[DELTA].sup.*.sub.+] (x, y)) [less than or equal to] [[pi].sub.u]u(xR + yd) + [[pi].sub.d]u(xR + yu)} (16)

Proof: See Appendix A.

Theorem 1 completely describes the NTR of the one-period FPTC problem. In Boyle and Lin (1997), the NTR of the PTC problem is bounded above and below by the two rebalancing lines that pass through the origin. In Figure 1, two lines pass through (K, 0) and its NTR is expanded away from the two rebalancing lines because of the existence of fixed trading costs. We show that the NTR of the one-period FPTC problem is an indefinite (neither concave nor convex) space in Appendix B.

The optimal trading strategy for the one-period FPTC problem is introduced in Theorem 2.

Theorem 2: The following is an optimal strategy for the one-period FPTC problem.

Step 1. Given (x, y), calculate [[DELTA].sup.*.sub.-] and [[DELTA].sup.*.sub.+]

Step 2. If [[DELTA].sup.*.sub.-] > 0 and (x, y) [??] [[THETA].sub.-], liquidate [[DELTA].sup.*.sub.-] of the risky asset.

Step 3. If [[DELTA].sup.*.sub.+] > 0 and (x, y) [??] [[THETA].sub.+], purchase [[DELTA].sup.*.sub.+] of the risky asset.

Step 4. Otherwise, do nothing.

Proof: See Appendix A.

Proposition 3 shows that the optimal solution of the one-period FPTC is in the feasible region when its initial portfolio satisfies the simple condition [x.sub.0] [greater than or equal to] 2K. This condition can be easily satisfied because we assume that K is a small portion of the net asset.

Proposition 3: When an initial portfolio ([x.sub.0], [y.sub.0]) [member of] C satisfies [x.sub.0] [greater than or equal to] 2K, the corresponding optimal portfolios of "down" and "up" stock markets are also in C.

Proof: See Appendix A.

NUMERICAL IMPLICATIONS

This section examines the numerical properties of the multiperiod FPTC problem. We develop a heuristic to solve the multiperiod FPTC problem and it is displayed in Appendix C. We set the periodic percentage change in the net contribution to be [phi] for the analysis.

Benchmark Parameters

The following benchmark parameters are used for testing: return on bond R = 1.0371, return on risky assets [r.sub.s] = 1.0771, volatility of risky asset [[sigma].sub.s] = 0.2, risk aversion parameter [gamma] = 2, proportional cost rate [k.sub.1] = [k.sub.2] = 0.005, and fixed cost rate K is 0.1 percent of total wealth. R = 1.0371 is taken from Bodie, Marcus, and Kane (2005) and [r.sub.s] is set to 1.0771 so that the risk premium becomes 4 percent. In reality, calculating an exact fixed cost of risky asset trading for a PSP is challenging. We use K = (0.1 percent) x ([x.sub.t] + [y.sub.t]) as a rough estimate of fixed cost because the operating (or managerial) costs of the PSPs of interest are approximately 0.1 percent of net asset values (estimated from the annual report of each PSP). Using these parameters, we compute [[pi].sub.d] = 0.5, [[pi].sub.u] = 0.5, d = 0.8771, and u = 1.2771. These parameters are used throughout this article unless otherwise stated.

Three values, -5 percent, 0 percent, and +5 percent, are used for f. At the beginning of each period, [x.sub.t] is increased (or decreased when [phi] < 0) by ([x.sub.t] + [y.sub.t]) x [phi]. For instance, [x.sub.t] is decreased by 0.05 x ([x.sub.t] + [y.sub.t]) at the beginning of each period when [phi] = -5 percent. (16) Considering net contributions with respect to optimal asset allocation is worthwhile for the following reasons. First, most PSPs possess accurate long-term forecasts of f, which are influenced by economic and demographic factors. Second, the "Sensitivity Analysis" subsection shows that [phi] has a significant impact on optimal asset allocation. To test the effect of applying different values of [phi] on the optimal asset allocation of PSPs, we implement the part that makes corresponding periodic adjustments in [x.sub.t] in the heuristic algorithm in Appendix C.

NTR

Figure 2 shows NTR and the rebalancing lines of the benchmark three-period FPTC problem. Hereafter, our test results are based on three-period FPTC problems. When a given (x, y) pair lies between the two dashed lines, inside NTR, no rebalancing is required. However, when a given (x, y) pair is outside NTR, it is moved (rebalanced) to the nearest solid (rebalancing) line. The NTR is bounded above and below by two curves, not straight lines. The interior formed by these lines is an indefinite space.

Figure 3 shows that the size of the NTR reduces as the number of periods increases because it is riskier to invest for more periods. Hence, the portfolio is managed within a narrower NTR when T is larger.

Sensitivity Analysis

This subsection addresses the effect of a change in the six parameters ([phi], K, k, [gamma], [r.sub.s], and [[sigma].sub.s]) on the NTR of the threeperiod FPTC benchmark problem. We solve the three-period FPTC problem using various combinations of (x, y) [member of] [0,4000] x [0,4000] by changing each parameter while holding the others fixed. The results of the tests are summarized in Table 4.

The changes in NTR in response to parameter changes are as follows. As the net contribution rate [phi] increases, NTR is lifted. When [phi] is a positive number, we expect that a PSP will receive a lump sum of cash at the beginning of each year. If a PSP has more wealth on y, the x-to-y ratio would be optimally balanced by the receipt of more cash in x. However, when [phi] is a negative number, we expect that a PSP will lose a portion of x at the beginning of each year, requiring a greater allocation of its wealth in x. Therefore, Figure 4a illustrates that a positive [phi] would lift the NTR upward allowing a PSP to invest more in risky assets. (The "Sensitivity Analysis" subsection shows that PSPs with a significant [phi] achieve a more precise optimal asset allocation by taking the net contribution into account.)

Figure 4b shows that NTR widens as fixed cost K increases. This phenomenon can be explained by a simple example. Suppose that K is 0.1 percent of the total value of assets and the maximum expected profit from trading is 0.5 percent of the total value of assets. In this case, trading risky assets is preferable. However, if K is increased to 1 percent, it is not optimal to trade anymore. Thus when K is increased NTR would be widened. Figure 4b shows that changes in K substantially affect the size of NTR. When K is about 1 percent of the total asset, NTR covers a large portion of the first quadrant.

Increasing proportional cost k widens NTR for the same reason as the case of K (see Figure 4c). However, the impact of changes in k is less than the impact of changes in K partly because k is paid proportionally to the risky asset only.

When NTR is enlarged by increased k and K, the trading region of a PSP shrinks. A shrunken trading region allows a PSP less room to maneuver. A PSP with a portfolio in the sell (buy) region would experience difficulty liquidating (purchasing) the risky asset when trading costs are high. The "Sensitivity Analysis" subsection verifies the effect of increased trading costs on the theoretical optimal asset allocations of PSPs.

We use three values for the risk aversion parameter [gamma]: 1.9,2.0, and2.1. Figure 4d shows that smaller g results in reduced investor sensitivity to risk and allows an investor to invest more in a risky asset, causing the NTR to rise.

Figure 4e presents larger values of [r.sub.s] that provide investors with an incentive to purchase risky assets more. Hence, increasing [r.sub.s] causes the NTR to rise.

Figure 4f depicts the NTR changes for different values of [[sigma].sub.s]. Similar to the case of g, smaller values of [[sigma].sub.s] lift NTR because less volatility in a risky asset implies that the risky asset is a more attractive investment.

Application to Pension Funds

This section compares the actual asset allocation schemes of five world-renowned pension funds to our benchmark asset allocations. Tables 5 and 6 summarize the impact of stock market volatility, liquidity (or proportional trading cost), and net contribution, respectively, on theoretical optimal asset allocations. Tables D2D6 in Appendix D include the test results for the five PSPs of interest.

Table 5, Panel A shows that the actual risky asset ratios of CALP and AVON are higher than the benchmark allocations where [[sigma].sub.s] = 20 percent. NYSLR and FAPF's asset allocations are close to our benchmark where [[sigma].sub.s] = 20 percent. NPS's risky asset ratio is significantly lower than our benchmark.

Volatility and the bid-ask spread increases when the stock market crashes. According to the test presented in Table 5 Panel A a myopic investor might try to liquidate up to 10 percent of risky assets when market volatility increases from 15 percent to 20 percent. However, an investor may also recognize that doing so is not optimal because of increased trading cost (see Table 5, Panels A and B).

The result of the test in Table 5 indicates that as trading costs increase threefold and fivefold, the optimal risky asset ratio changes by between 3 percent and 4 percent. Changes in optimal risky asset ratios have different signs depending on where the initial portfolio lies. If an initial portfolio (x, y) lies on the upper left side of NTR, an increase in trading cost would increase the risky asset ratio as in the cases of NYSLR and NPS. However, an increased trading cost would decrease the risky asset ratio when (x, y) lies on the lower right of NTR as in the cases of CALP, AVON, and FAPF.

When the stock market crashes, PSPs with a high risky asset ratio might try to liquidate as much as they could. However, increased trading costs would decrease the amount that PSPs can liquidate. A PSP with a low risky asset ratio would also experience difficulty acquiring risky assets because of increased trading costs (see Table 5, Panels A and B).

Table 6 shows that the net contribution, f, has a significant impact on optimal asset allocations. Because large-scale PSPs manage billions of dollars, even a change of a few percent in their asset allocations can be significant. For example, Bikker et al. (2012) discover that participants' age distribution has a significant impact on the asset allocation of Dutch pension funds. The far right column of Table 6 contains the average value of different combinations of parameters and may not represent an actual phenomenon. However, it is sufficient to show that considering net contributions in calculating optimal asset allocation has a significant impact on optimal asset allocations. Table 6 shows that CALP, NYSLR, and AVON have relatively large values of [phi] that lead to 0.55 percent to 4.41 percent differences in the risky asset ratio (AVONS's averaged [phi] appears small because it turned positive from negative in 2006). In particular, AVON shows that it needs to acquire 4.41 percent more risky assets when [phi] is considered.

Using the results in Tables D2-D6 in Appendix D, we compare our results to PSPs' actual asset allocations during the financial market crash. With respect to CALP, our optimal allocations (without applying net contributions) were 66.02 percent (2008), 60.86 percent (2009), and 63.28 percent (2010) where [[sigma].sub.s] = 20 percent and k = 2.5 percent, and the actual asset allocations were 67.12 percent (2008), 60.86 percent (2009), and 63.28 percent (2010). According to this test, their actual allocations during the financial crisis were close to our results.

Similar to CALP, the asset allocations of NYSLR and FAPF matched our optimal asset allocations where [[sigma].sub.s] = 20 percent and k = 2.5 percent without applying the net contribution. This test shows that CAPS would ideally have lowered the risky asset ratio to 58 percent. However, an increase in trading cost would have prevented CALP from liquidating as much, forcing them to maintain a higher risky asset ratio.

The actual risky asset ratios of NYSLR between 2008 and 2011 were 72.38 percent (2008), 61.53 percent (2009), 71.71 percent (2010), and 73.35 percent (2011). These risky asset ratios fall within NTR of our test results where [[sigma].sub.s] = 20 percent and k = 2.5 percent. This test shows that NYSLR's risky asset ratio is on the lower right of NTR. Without an increase in trading cost, more risky assets might have been acquired. However, an increase in trading cost prevented NYSLR from doing so.

Although the asset allocations of CALP and NYSLR fall within NTR of our tests, public pension funds have been criticized for increasing risky asset ratios for the last two decades to maintain a high discount rate and lower liabilities (Andonov, Bauer, and Cremers, 2013).

FAPF's asset allocations also fall within NTR of our tests. FAPF's actual risky asset ratios between 2008 and 2011 were 57.17 percent (2008), 62.58 percent (2009), 64.00 percent (2010), and 54.49 percent (2011); these risky asset ratios fall within NTR of our test results where [[sigma].sub.s] = 20 percent and k = 2.5 percent.

These results show that the reactions of CALP, NYSLR, and FAPF were almost identical to the theoretical optimal choices during the financial market crash, whereas the asset allocations of AVON and NPS were somewhat different from our test results.

CONCLUSION

It has been observed that a number of large PSPs maintained stable asset allocations during the financial turmoil of 2008 and 2009, while others actively adjusted their portfolios. This article was motivated by this observation and examines the optimal investment strategies of pension service providers. To model large PSP asset allocation behavior, this article constructed an optimal asset allocation scheme for the multiperiod FPTC problem and compared the theoretical optimal portfolios to the actual asset allocations of five large pension service providers.

Bid-ask spread and stock market volatility are indicators of a stock market condition. In the event of a stock market crash, a myopic optimal choice for an investor in the sell (buy) region is to liquidate (purchase) as much stock as possible. However, an increase in a bidask spread, which often occurs during a stock market crash, increases trading costs. This pattern prevents an investor in the sell (buy) region from liquidating (purchasing) additional stock. This liquidity problem is considered in the optimal portfolio strategies.

Moreover, we show that net contribution has a significant impact on the optimal asset allocation. Unlike other unpredictable factors, most pension service providers have accurate estimates for future expected changes in net contribution. Therefore, we strongly encourage a pension service provider to consider future expected changes in net contribution to determine optimal asset allocations.

With respect to NTR changes in response to parameter variance, we found that NTR shrinks as the number of periods decreases. Additionally, increasing the fixed and proportional trading costs widens NTR because increased trading costs would decrease investor willingness to trade. Moreover, the change in the size of NTR is more sensitive to the fixed cost than the proportional cost. Decreasing volatility in the stock market lifts NTR, allowing an investor to allocate additional wealth to a risky asset.

Because the tests in this article are conducted without reflecting each PSP's detailed environment, our theoretical optimal allocation is different from a PSP's actual allocation. However, the actual asset allocations of some pension service providers can be well explained by the test outcomes of this article. For example, the actual asset allocations of CALP, NYSLR, and FAPF are well explained by the test outcomes although AVON seems to invest about 10 percent more in risky assets than the optimal asset allocation schemes modeled. Overall it is shown that the actual portfolio management strategies of some pension service providers are close to the theoretical optimal allocations although there is room for improvement. For instance, the asset allocation of NPS on risk-free assets seems too high compared to our optimal levels; however, NPS has steadily increased its risky asset ratio.

This article showed that asset allocation behaviors of some large-scale PSPs can be well explained using our relatively simple model. However, it is likely that one can come up with a more sophisticated normative model with a greater explanatory power by considering more options. It would be an interesting future research to look for a better normative model by studying the suitability of various utility functions and considering various factors (such as taxes, price impacts of trading risky assets, longevity risks (17), and regulations (18)) that affect PSP asset allocation decisions.

Incorporating the solvency (19) regulation into our model and studying how it affects an insurance company's investment decisions would be an another interesting research topic. (See Filipovic, Kremslehner, and Muermann, forthcoming, for a related study.)

APPENDIX A: PROOFS

Proof of Proposition 1: Because [f".sub.-] ([[DELTA].sub.-]; x, y) [less than or equal to] 0 and [f".sub.+] ([[DELTA].sub.+]; x, y) [less than or equal to] 0, [f.sub.-] and [f.sub.+] are concave functions.

Proof of Proposition 2: We solve [[DELTA].sup.*.sub.-]'s case only. [[DELTA].sup.*.sub.+]'s case is a mirror image. Suppose that ([x.sub.1], [y.sub.1]) is on the upper left of (12). Then, ([x.sub.1], [y.sub.1]) can be written as (x, y + [alpha]) where [alpha] [greater than or equal to] 0 and (x, y) satisfies (12). From (7) we have

[w.sub.1](xR + yd - KR + ((1 - [k.sub.1])R - d) [[DELTA].sup.*.sub.-]) + [alpha][w.sub.1]d = xR + yu - KR + ((1 - [k.sub.1])R - u) [[DELTA].sup.*.sub.-]) + [alpha]u.

Because (x, y) satisfies (12), we have

[w.sub.1] (xR + yd - KR) = xR + yd - KR.

Subtracting these two equations, we obtain

[w.sub.1] ((1 - [k.sub.1])R - d) [[DELTA].sup.*.sub.-] + [alpha][w.sub.1]d = ((1 - [k.sub.1])R - u) [[DELTA].sup.*.sub.-] + [alpha]u.

Rearranging the equation above, we obtain

[[DELTA].sup.*.sub.-] = [alpha](u - [w.sub.1]d)/[w.sub.1]((1 - [k.sub.1])R - d) - ((1 - [k.sub.1.])R - u) = [alpha](u [w.sub.1]d)/([w.sub.1] - 1)(1 - [k.sub.1])R + (u - [w.sub.1]d) [greater than or equal to] 0.

Therefore, [[DELTA].sup.*.sub.-] [greater than or equal to] 0 when (x, y) is on the upper left of (13). The remainder of the proposition can be proven using similar arguments. ?

Proof of Theorem 1: We discuss the FPTC sell problem's case only. The optimal policy for the one-period FPTC problem is to take [[DELTA].sup.*.sub.-] > 0 if selling [[DELTA].sup.*.sub.-] is preferable to not trading at all. Therefore, we specify the NTR by stating [[THETA].sub.-] in terms of (x, y). Because [[DELTA].sup.*.sub.-] satisfies (7), we write [f.sub.-] ([[DELTA].sup.*.sub.-]; x, y) as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (A1)

Because [[DELTA].sup.*.sub.-] is a function of (x, y), (15) specifies [[THETA].sub.-] in terms of (x, y) only. The case of [[THETA].sub.+] is omitted because it is a mirror image of the case of [[THETA].sub.-].

Proof of Theorem 2: Based on Proposition 2, only three mutually exclusive cases are possible:

Case 1: [[DELTA].sup.*.sub.-] > 0,

Case 2: [[DELTA].sup.*.sub.+] > 0,

Case 3: [[DELTA].sup.*.sub.-] = [[DELTA].sup.*.sub.+] = 0.

When [[DELTA].sup.*.sub.-] > 0([[DELTA].sup.*.sub.+] > 0) this is an optimal solution for the FPTC sell (buy) problem because [f.sub.-] ([f.sub.+]) is a concave function. We accept [[DELTA].sup.*.sub.-] or [[DELTA].sup.*.sub.+] as an optimal solution only if it is better than [f.sub.0]([DELTA] = 0). Liquidating (purchasing) [[DELTA].sup.*.sub.-] ([[DELTA].sup.*.sub.+]) of a risky asset is better than not trading only if (x, y) [??] [[THETA].sub.-]((x, y) [??] [[THETA].sub.+]) from Theorem 3. Therefore, taking the steps above is an optimal strategy.

Proof of Proposition 3: This proof is in three parts: selling [[DELTA].sup.*.sub.-] units of risky assets is optimal, purchasing [[DELTA].sup.*.sub.+] units of risky assets is optimal, and not trading is optimal. First, let us consider the case where selling A* units of its risky assets is the optimal choice for the one-period FPTC problem with an initial point ([x.sub.0], [y.sub.0]). For this problem, the current portfolio is rebalanced to (x' = [x.sub.0] + (1 - [k.sub.1]) [[DELTA].sup.*.sub.-] - K, y = [y.sub.0] - [[DELTA].sup.*.sub.-]), where (x', y') is on (12). When [x.sub.0] [greater than or equal to] 2K, we have x' [greater than or equal to] K. Moreover, we have y = [y.sub.0] - [[DELTA].sup.*.sub.-] [greater than or equal to] 0 because x' [greater than or equal to] K and (x', y') is on line (12). (Because (12) passes through (K, 0) and has a positive slope, y' [greater than or equal to] 0 when x [greater than or equal to] K.)

Next, we can prove that the portfolio of the "down" stock market (Rx' + (1 [k.sub.1])d. y' - K) is in C when [x.sub.0] [greater than or equal to] 2K because Rx' + (1 [k.sub.1])d x y' - K = R([x.sub.0] + (1 - [k.sub.1]) [[DELTA].sup.*.sub.-] - K) + (1 - [k.sub.1]) ([y.sub.0] - [[DELTA].sup.*.sub.])d - K [greater than or equal to] 0. (20) We do not show the "up" stock market case because the "up" stock market portfolio is in C when the "down" stock market portfolio is in C.

The case of [[DELTA].sup.*.sub.+] [greater than or equal to] 0 is slightly different. According to our settings, the slope of the line (13) can be either positive or negative. When the sign is negative, the corresponding part of NTR is formed in the fourth quadrant and an initial point ([x.sub.0], [y.sub.0]) would be rebalanced onto this line only when ([x.sub.0], [y.sub.0]) [??] C. Therefore, we only consider the nontrivial case where the slope of (13) is positive. We consider the one-period FPTC problem with an initial point ([x.sub.0], [y.sub.0]). The optimal choice is to purchase [[DELTA].sup.*.sub.+] units of risky assets. Then, the current portfolio is rebalanced to (x' = [x.sub.0] - (1 + [k.sub.2]) [[DELTA].sup.*.sub.+] - K, y' = [y.sub.0] + [[DELTA].sup.*.sub.+]), where (x', y') is on (13). Because [[DELTA].sup.*.sub.+] [greater than or equal to] 0 and [y.sub.0] [greater than or equal to] 0, we have y' [greater than or equal to] 0. From y' [greater than or equal to] 0, we have x' [greater than or equal to] K because (x', y') is on line (13) that passes through (K, 0) and has a positive slope. The portfolio of the "down" stock market Rx' + (1 [k.sub.1])d x y' - K is non-negative because Rx' + (1 - [k.sub.1])d x y' - K = R([x.sub.0] (1 + [k.sub.2]) [[DELTA].sup.*.sub.+] K) + (1 - [k.sub.1]) ([y.sub.0] + [[DELTA].sup.*.sub.+])d - K [greater than or equal to] 0. (21)

When not trading is the optimal choice, the portfolio value in the case of the "down" stock market [Rx.sub.0] + d x [y.sub.0] (1 - [k.sub.1]) is non-negative when [x.sub.0] [greater than or equal to] 2K.

APPENDIX B: CONVEXITY ANALYSIS OF THE ONE-PERIOD FPTC PROBLEM

An intersection of two convex sets is again, a convex set. Therefore the NTR of the oneperiod FPTC problem would be a convex set if both [[THETA].sub.-] and [[THETA].sub.+] are convex sets. From (14) and basic algebra, we can prove that [[THETA].sub.-]([[THETA].sub.+]) is a convex set if the Hessian of [[delta].sub.-]([[delta].sub.+]) is positive semidefinite. Unfortunately, our study indicated that the Hessians of and d+ are usually indefinite.

Let us define q(x, y):= [(ax + by + c).sup.[gamma]-1]/([gamma] - 1). Then the Hessian matrix is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (B1)

Note that (B1) is a rank-one negative definite matrix. Using (18), we can write the Hessian matrix of [[delta].sub.-]([[DELTA].sup.*.sub.-] (x, y)) as below. (The buy problem's case is omitted because it is a mirror image of the sell problem's case.)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (B2)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

In(B2), A, A', B, and B' are rank-one positive semidefinite matrices. Because A' and B' have positive signs, [[nabla].sup.2][[delta].sup.-] becomes positive semidefinite when [[pi].sub.u]([(xR + yu) [[DELTA].sup.*.sub.-]).sup.-[gamma]1] and [[pi].sub.d]([(xR + yd) [[DELTA].sup.*.sub.-]).sup.-[gamma]-1] are significantly larger than [[pi].sub.u] (xR + yu - KR + ([(1 - [k.sub.1]) R - u)[[DELTA].sup.*.sub.-]).sup.-[gamma]-1] and [[pi].sub.d](xR + yd - KR + ([(1 - [k.sub.1])R - d) [[DELTA].sup.*.sub.-]).sup.-[gamma]-1], which would indicate that [THETA]- is a convex space of (x, y). However, according to our tests, [[nabla].sup.2][[delta].sub.-] and [[nabla].sup.2][[delta].sub.+] usually have one positive and one negative eigenvalue. Hence, [[nabla].sup.2][[delta].sub.-] and [[nabla].sup.2][[delta].sup.+] are usually indefinite matrices. Therefore the NTR of the one-period FPTC problem is usually an indefinite space.

APPENDIX C: A HEURISTIC FOR THE MULTIPERIOD FPTC PROBLEM

[Tree, FV] = RMPO(x, y, [k.sub.1], [k.sub.2], K, d, u, [[pi].sub.d], [[pi].sub.u], [r.sub.b], [phi], [epsilon], p, m).

[L01] Set x = x +(x + y) x [phi].

[L02] If p < T.

[L03] Initialize [FV.sub.u] = -[infinity], [FV.sub.d] = -1, [Tree.sub.u] = empty, and [Tree.sub.d] = empty.

[L04] Loop [delta] = -y : [epsilon] : x.

[L05] If [delta] < 0, set x' = x - K + [delta][k.sub.1] - d and y' = y + [delta].

[L06] else if [delta] > 0, set x' = x - K - [delta][k.sub.2] - [delta] and y' = y + [delta].

[L07] else set x' = x and y' = y.

[L08] If (x, y) [member of] C.

[L09] [[Tree.sub.u], [FV.sub.u]] = RMPO(x'[r.sub.b], y'u, [k.sub.1], [k.sub.2], K, d, u, [[pi].sub.u], [[pi].sub.d], [phi], [epsilon], p + 1).

[L10] [[Tree.sub.d], [FV.sub.d]] = RMPO(x'[r.sub.b], y'd, [k.sub.1], [k.sub.2], K, d, u, [[pi].sub.u], [[pi].sub.d], [phi], [epsilon], p + 1).

[L11] If FV < [[pi].sub.u][FV.sub.u] + [[pi].sub.d][FV.sub.d], set FV = [[pi].sub.u][FV.sub.u] + [[pi].sub.d][FV.sub.d].

[L12] Tree.[delta] = [delta], Tree. [Tree.sub.u] = [Tree.sub.u], and Tree. [Tree.sub.d] = [Tree.sub.d].

[L13] End of the loop that begins at [L04].

[L14] Else if p = T.

[L15] Solve one-period optimal problem using the method in "The Optimal Strategy for the One-Period FPTC Problem" section and return.

[L16] One-period optimal d, FV, an empty Treeu, andan empty Treed.

In this algorithm, input arguments [k.sub.1], [k.sub.2], K, d, u, [[pi].sub.d], [[pi].sub.u], and [r.sub.b] (or R) are constant values; [phi] is the annual changes in total assets; and [epsilon] > 0 is an offset. In the algorithm, "Loop [delta] = -y : [epsilon] : x" on [L04] repeats [L05] - [L12] with different values of [delta], which goes from -y to x with a step size 2.

This algorithm can be explained as follows. Suppose that [DELTA] [member of] [[[DELTA].sub.min], [[DELTA].sub.max]] at a certain period, where [[DELTA].sub.min] and [[DELTA].sub.max] are the minimum and maximum possible values of A, respectively ([[DELTA].sub.min] can be a negative number when we decide to liquidate risky assets). Then we try [[DELTA].sub.min], [[DELTA].sub.min] + [epsilon], [[DELTA].sub.min] + 2 x [epsilon], ... , [[DELTA].sub.max] as possible values of [DELTA] to test the optimal choice for the expected final wealth. This is done at each period with the exception of the last period, where we utilize the one-period optimal trading policy. We can obtain a near-optimal solution when a small 2 is used. The algorithm in this subsection can solve the multi-period FPTC problem approximately where the number of periods T is relatively large. Since the running time of our heuristic increases exponentially as the number of periods T increases, it may be difficult to solve a multiperiod FPTC with a large T with a relatively small step size 2 using this algorithm.

Utilizing the one-period optimal trading strategy in [L15] - [L16] in this algorithm reduces the running time significantly because the strategy provides the oneperiod optimal solutions at [2.sup.T-1] nodes at time T - 1. In other words, trying n possible values at these nodes would require [2.sup.T-1] x n trials.

We did not include the part that checks insolvency because we have not encountered such cases while running tests. If necessary, we can force the algorithm to avoid insolvency by adding a part that sets [FV.sub.u] and [FV.sub.d] to a huge negative number when insolvency occurs.

APPENDIX D: DATA AND TEST RESULT

Table D1 Assets and Net Contributions (NC) of Five PSPs Panel A: Assets and Net Contributions of CALP Year Stock Bond Stock (1) Ratio Contribution 2002 80,220,734 40,064,811 66.69% 2,955,706 2003 86,135,240 37,904,304 69.44% 3,812,969 2004 102,505,858 42,991,871 70.45% 6,527,792 2005 114,838,218 54,340,778 67.88% 8,950,901 2006 129,887,184 52,026,254 71.40% 9,175,908 2007 149,704,501 61,218,906 70.98% 9,705,083 2008 122,375,605 59,934,932 67.12% 10,754,877 2009 80,229,982 51,597,650 60.86% 10,794,731 2010 91,941,405 53,359,863 63.28% 10,333,916 2011 116,731,425 53,066,388 68.75% 11,065,486 Panel B: Assets and Net Contributions of NYSLR Year Stock Bond Stock (1) Ratio Contribution 2003 51,357,030 32,019,681 61.60% 980,853 2004 74,876,438 29,691,227 71.61% 1,585,474 2005 80,917,186 29,310,816 73.41% 3,314,919 2006 88,550,861 28,888,995 75.40% 3,117,876 2007 100,164,486 33,536,212 74.92% 3,100,572 2008 95,853,118 36,571,473 72.38% 3,030,236 2009 58,434,690 36,541,603 61.53% 2,885,457 2010 85,473,716 33,726,066 71.71% 2,719,494 2011 94,860,086 31,037,855 75.35% 4,578,479 Panel C: Assets and Net Contributions of AVON Year Stock Bond Stock (1) Ratio Contribution 2002 1,132,219 296,399 79.25% 56,382 2003 801,006 302,153 72.61% 63,347 2004 1,083,154 354,532 75.34% 71,492 2005 1,341,347 370,825 78.34% 75,361 2006 1,565,453 427,810 78.54% 93,403 2007 1,683,838 446,958 79.02% 105,149 2008 1,360,519 495,827 73.29% 112,646 2009 1,067,809 355,936 75.00% 125,349 2010 1,463,791 487,930 75.00% 134,681 2011 1,602,545 534,182 75.00% 139,519 Panel D: Assets and Net Contributions of FAPF Year Stock Bond Stock (1) Ratio Contribution 2002 66,580 48,611 57.80% 40,186 2003 79,082 55,498 58.76% 41,481 2004 90,659 62,358 59.25% 42,904 2005 110,338 74,643 59.65% 44,883 2006 118,487 81,420 59.27% 45,906 2007 126,222 84,704 59.84% 47,603 2008 89,576 67,120 57.17% 50,783 2009 117,526 70,262 62.58% 50,678 2010 126,470 71,135 64.00% 51,267 2011 104,531 87,303 54.49% 53,895 Panel E: Assets and Net Contributions of NPS Year Stock Bond Stock (1) Ratio Contribution 2004 12,702,300 120,596,100 9.53% 17,143 2005 20,394,900 141,482,400 12.6% 18,544 2006 21,986,300 164,432,400 11.79% 20,152 2007 38,422,600 174,834,000 18.02% 21,670 2008 34,263,500 191,124,000 15.2% 22,986 2009 49,720,000 215,085,300 18.78% 23,858 2010 74,973,200 229,166,000 24.65% 25,285 2011 82,061,600 238,071,000 25.63% 27,430 Panel A: Assets and Net Contributions of CALP Year (2) NC: (1) - (2) NC Ratio Benefits 2002 6,534,405 -3,578,699 -2.97% 2003 7,105,939 -3,292,970 -2.65% 2004 7,790,611 -1,262,819 -0.86% 2005 8,549,355 401,546 0.23% 2006 9,407,002 -231,094 -0.12% 2007 10,252,129 -547,046 -0.25% 2008 11,066,832 -311,955 -0.17% 2009 12,018,619 -1,223,888 -0.92% 2010 13,154,844 -2,820,928 -1.94% 2011 14,600,037 -3,534,551 -2.08% Panel B: Assets and Net Contributions of NYSLR Year (2) NC: (1) - (2) NC Ratio Benefits 2003 5,029,766 -4,048,913 -4.86% 2004 5,423,277 -3,837,803 -3.67% 2005 5,690,865 -2,375,946 -2.16% 2006 6,072,868 -2,954,992 -2.52% 2007 6,431,731 -3,331,159 -2.49% 2008 6,883,034 -3,852,798 -2.91% 2009 7,265,499 -4,380,042 -4.61% 2010 7,718,872 -4,999,378 -4.19% 2011 8,520,223 -3,941,744 -3.13% Panel C: Assets and Net Contributions of AVON Year (2) NC: (1) - (2) NC Ratio Benefits 2002 68,004 -11,622 -0.81% 2003 70,482 -7,135 -0.65% 2004 75,354 -3,862 -0.27% 2005 79,196 -3,835 -0.22% 2006 81,324 12,079 0.61% 2007 94,038 11,111 0.52% 2008 100,908 11,738 0.63% 2009 111,161 14,188 1.00% 2010 121,232 13,449 0.69% 2011 121,745 17,774 0.83% Panel D: Assets and Net Contributions of FAPF Year (2) NC: (1)-(2) NC Ratio Benefits 2002 37,939 2,247 1.95% 2003 39,057 2,424 1.80% 2004 40,696 2,208 1.44% 2005 42,268 2,615 1.41% 2006 44,033 1,873 0.94% 2007 46,405 1,198 0.57% 2008 49,796 987 0.63% 2009 54,348 -3,670 -1.95% 2010 55,050 -3,783 -1.91% 2011 54,919 -1,024 -0.53% Panel E: Assets and Net Contributions of NPS Year (2) NC: (1)-(2) NC Ratio Benefits 2004 2,914 14,229 0.0107% 2005 3,584 14,960 0.0092% 2006 4,360 15,792 0.0085% 2007 5,182 16,488 0.0077% 2008 6,180 16,806 0.0075% 2009 7,471 16,387 0.0062% 2010 8,636 16,649 0.0055% 2011 9,819 17,611 0.0055% TABLE D2 CALP Three-Period Optimal FPTC Allocation Test Results [phi] Not Applied PSP Information World Stock Market Year St. NA Spread Scaled Volatility Ratio 2002 66.69% NA 0.27 4.02 19.30% 2003 69.44% NA 0.18 2.66 12.26% 2004 70.45% NA 0.14 2.03 8.19% 2005 67.88% NA 0.15 2.15 8.19% 2006 71.40% NA 0.15 2.13 7.45% 2007 70.98% NA 0.16 2.36 9.33% 2008 67.12% NA 0.23 3.36 23.63% 2009 60.86% NA 0.09 1.36 23.27% 2010 63.28% NA 0.08 1.13 20.60% 2011 68.75% NA 0.07 1.00 17.97% Average 67.69% NA 0.15 2.22 15.02% [phi] Applied PSP Information World Stock Market Year St. NA Spread Scaled Volatility Ratio 2002 66.69% -2.98% 0.27 4.02 19.30% 2003 69.44% -2.65% 0.18 2.66 12.26% 2004 70.45% -0.87% 0.14 2.03 8.19% 2005 67.88% 0.24% 0.15 2.15 8.19% 2006 71.40% -0.13% 0.15 2.13 7.45% 2007 70.98% -0.26% 0.16 2.36 9.33% 2008 67.12% -0.17% 0.23 3.36 23.63% 2009 60.86% -0.93% 0.09 1.36 23.27% 2010 63.28% -1.94% 0.08 1.13 20.60% 2011 68.75% -2.08% 0.07 1.00 17.97% Average 67.69% -1.18% 0.15 2.22 15.02% FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% Ratio 2002 66.69% 92.12 81.17 73.12 85.08 67.00 66.69 2003 69.44% 92.10 81.14 73.06 85.07 69.44 69.44 2004 70.45% 92.10 81.13 73.05 85.06 70.45 70.45 2005 67.88% 92.11 81.16 73.09 85.07 67.88 67.88 2006 71.40% 92.09 81.12 73.03 85.06 71.40 71.40 2007 70.98% 92.10 81.12 73.04 85.06 70.98 70.98 2008 67.12% 92.11 81.17 73.11 85.08 67.12 67.12 2009 60.86% 92.14 81.25 72.20 85.10 67.06 60.86 2010 63.28% 92.13 81.22 72.16 85.09 67.04 63.28 2011 68.75% 92.11 81.15 73.08 85.07 68.75 68.75 Average 67.69% 92.11 81.16 72.89 85.07 68.71 67.69 FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% Ratio 2002 66.69% 89.09 78.11 71.04 83.06 68.74 68.74 2003 69.44% 89.08 79.09 71.33 83.05 71.33 71.33 2004 70.45% 91.09 80.11 72.02 84.05 71.07 71.07 2005 67.88% 92.11 81.16 73.10 85.07 67.72 67.72 2006 71.40% 92.09 81.12 73.03 85.06 71.49 71.49 2007 70.98% 92.10 81.12 72.02 85.06 71.16 71.16 2008 67.12% 92.11 81.17 72.09 85.08 67.24 67.24 2009 60.86% 91.13 80.22 72.19 84.09 67.06 61.43 2010 63.28% 90.11 79.17 71.12 83.08 66.01 64.53 2011 68.75% 90.09 79.10 71.01 83.05 70.21 70.21 Average 67.69% 90.90 80.04 71.90 84.07 69.20 68.49 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k Year St. 0.5% 1.5% 2.5% Ratio 2002 66.69% 58.03 62.04 66.69 2003 69.44% 58.03 62.07 66.06 2004 70.45% 58.04 62.08 66.07 2005 67.88% 58.03 62.05 66.03 2006 71.40% 58.04 62.09 66.09 2007 70.98% 58.04 62.08 66.08 2008 67.12% 58.03 62.05 66.02 2009 60.86% 58.01 60.86 60.86 2010 63.28% 58.02 62.01 63.28 2011 68.75% 58.03 62.06 66.05 Average 67.69% 58.03 61.94 65.32 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k Year St. 0.5% 1.5% 2.5% Ratio 2002 66.69% 56.04 60.08 64.08 2003 69.44% 56.04 60.10 64.12 2004 70.45% 57.04 61.09 66.08 2005 67.88% 58.03 62.05 67.01 2006 71.40% 58.04 62.09 66.09 2007 70.98% 58.04 61.09 66.09 2008 67.12% 58.03 62.05 66.02 2009 60.86% 57.01 61.00 61.43 2010 63.28% 57.02 61.03 64.53 2011 68.75% 57.04 60.09 65.08 Average 67.69% 57.23 61.07 65.05 Note: This table compares PSP actual asset allocations to the optimal asset allocations of three- period FPTC problems with different values of f, stock market volatility, and ([k.sub.1], [k.sub.2]). For this test, we used the actual (x, y) and [psi] of a PSP at each year, while unspecified parameters are identical to the benchmark parameters. The upper half of the table includes test results that do not reflect [phi], while the bottom half does. At each row, we solved nine different FPTC problems with nine possible combinations of [[psi].sub.s] [member of] {10%, 15%, 20%} x (k) G {0.5%, 1.5%, 2.5%}. Table D3 NYRLS Three-Period Optimal FPTC Allocation Test Results [phi] Not Applied PSP Information World Stock Market Year St. Ratio NA Spread Scaled Volatility 2003 61.60% NA 0.18 2.66 12.26% 2004 71.61% NA 0.14 2.03 8.19% 2005 73.41% NA 0.15 2.15 8.19% 2006 75.40% NA 0.15 2.13 7.45% 2007 74.92% NA 0.16 2.36 9.33% 2008 72.38% NA 0.23 3.36 23.63% 2009 61.53% NA 0.09 1.36 23.27% 2010 71.71% NA 0.08 1.13 20.60% 2011 75.35% NA 0.07 1.00 17.97% Average 70.88% NA 0.14 2.02 14.54% [phi] Applied PSP Information World Stock Market Year St. Ratio NA Spread Scaled Volatility 2003 61.60% --4.86% 0.18 2.66 12.26% 2004 71.61% --3.67% 0.14 2.03 8.19% 2005 73.41% --2.35% 0.15 2.15 8.19% 2006 75.40% --2.73% 0.15 2.13 7.45% 2007 74.92% --2.69% 0.16 2.36 9.33% 2008 72.38% --5.58% 0.23 3.36 23.63% 2009 61.53% --5.19% 0.09 1.36 23.27% 2010 71.71% --4.70% 0.08 1.13 20.60% 2011 75.35% --3.55% 0.07 1.00 17.97% Average 70.88% --3.92% 0.14 2.02 14.54% FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. Ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2003 61.60% 95.16 88.35 82.42 91.13 79.21 71.17 2004 71.61% 95.11 88.22 83.24 91.09 79.09 71.61 2005 73.41% 95.10 88.19 83.20 91.08 79.07 73.41 2006 75.40% 95.09 88.17 83.16 91.07 79.04 75.40 2007 74.92% 95.10 88.17 83.17 91.07 79.05 74.92 2008 72.38% 95.11 88.21 83.22 91.08 79.08 72.38 2009 61.53% 95.16 88.35 82.42 91.13 79.21 71.17 2010 71.71% 95.11 88.22 83.24 91.09 79.09 71.71 2011 75.35% 95.09 88.17 83.16 91.07 79.04 75.35 Average 70.88% 95.11 88.23 83.03 91.09 79.10 73.01 FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. Ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2003 61.60% 91.12 83.23 78.26 87.10 75.12 67.04 2004 71.61% 92.08 85.14 80.11 88.06 77.03 74.33 2005 73.41% 93.08 86.14 81.12 90.07 78.03 75.03 2006 75.40% 93.07 86.11 81.07 89.05 77.35 77.35 2007 74.92% 93.08 86.12 81.08 89.05 78.01 76.83 2008 72.38% 92.08 86.15 80.11 89.06 77.03 74.55 2009 61.53% 91.12 84.25 78.26 87.10 76.13 68.06 2010 71.71% 91.07 84.12 79.08 88.06 76.01 74.85 2011 75.35% 92.07 85.09 80.04 89.05 77.78 77.78 Average 70.88% 92.09 85.15 79.90 88.51 76.94 73.98 Volatility 20% PSP Information k Year St. Ratio 0.5% 1.5% 2.5% 2003 61.60% 76.05 70.09 61.60 2004 71.61% 76.02 71.61 71.61 2005 73.41% 76.01 73.41 73.41 2006 75.40% 76.00 75.40 75.40 2007 74.92% 76.00 74.92 74.92 2008 72.38% 76.01 72.38 72.38 2009 61.53% 76.06 70.09 61.53 2010 71.71% 76.02 71.71 71.71 2011 75.35% 76.00 75.35 75.35 Average 70.88% 76.02 72.77 70.88 Volatility 20% PSP Information k Year St. Ratio 0.5% 1.5% 2.5% 2003 61.60% 73.03 67.02 64.74 2004 71.61% 74.33 74.33 74.33 2005 73.41% 75.03 75.03 75.03 2006 75.40% 77.35 77.35 77.35 2007 74.92% 76.83 76.83 76.83 2008 72.38% 74.55 74.55 74.55 2009 61.53% 73.03 67.03 64.50 2010 71.71% 74.85 74.85 74.85 2011 75.35% 77.78 77.78 77.78 Average 70.88% 75.20 73.86 73.33 Notes: See Notes to Table D2. Table D4 NYRLS Three-Period Optimal FPTC Allocation Test Results [phi] Not Applied PSP Information World Stock Market Year St. Ratio NA Spread Scaled Volatility 2002 79.25% NA 0.27 4.02 19.30% 2003 72.61% NA 0.18 2.66 12.26% 2004 75.34% NA 0.14 2.03 8.19% 2005 78.34% NA 0.15 2.15 8.19% 2006 78.54% NA 0.15 2.13 7.45% 2007 79.02% NA 0.16 2.36 9.33% 2008 73.29% NA 0.23 3.36 23.63% 2009 75.00% NA 0.09 1.36 23.27% 2010 75.00% NA 0.08 1.13 20.60% 2011 75.00% NA 0.07 1.00 17.97% Average 76.14% NA 0.15 2.22 15.02% [phi] Applied PSP Information World Stock Market Year St. ratio NA Spread Scaled Volatility 2002 79.25% --0.81% 0.27 4.02 19.30% 2003 72.61% --0.65% 0.18 2.66 12.26% 2004 75.34% --0.27% 0.14 2.03 8.19% 2005 78.34% --0.22% 0.15 2.15 8.19% 2006 78.54% 0.61% 0.15 2.13 7.45% 2007 79.02% 0.52% 0.16 2.36 9.33% 2008 73.29% 0.63% 0.23 3.36 23.63% 2009 75.00% 1.00% 0.09 1.36 23.27% 2010 75.00% 0.69% 0.08 1.13 20.60% 2011 75.00% 0.83% 0.07 1.00 17.97% Average 76.14% 0.23% 0.15 2.22 15.02% FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. Ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2002 79.25% 92.06 81.02 79.25 85.02 79.25 79.25 2003 72.61% 92.09 81.10 72.61 85.05 72.61 72.61 2004 75.34% 92.08 81.07 75.34 85.04 75.34 75.34 2005 78.34% 92.06 81.03 78.34 85.03 78.34 78.34 2006 78.54% 92.06 81.03 78.54 85.03 78.54 78.54 2007 79.02% 92.06 81.02 79.02 85.03 79.02 79.02 2008 73.29% 92.09 81.09 73.29 85.05 73.29 73.29 2009 75.00% 92.08 81.07 75.00 85.04 75.00 75.00 2010 75.00% 92.08 81.07 75.00 85.04 75.00 75.00 2011 75.00% 92.08 81.07 75.00 85.04 75.00 75.00 Average 76.14% 92.07 81.06 76.14 85.04 76.14 76.14 FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2002 79.25% 91.05 80.00 79.90 84.02 79.90 79.90 2003 72.61% 91.08 80.08 73.09 85.05 73.09 73.09 2004 75.34% 92.08 81.07 75.54 85.04 75.54 75.54 2005 78.34% 92.06 81.03 78.51 85.03 78.51 78.51 2006 78.54% 92.06 81.04 78.06 85.03 78.06 78.06 2007 79.02% 92.06 81.03 78.62 85.03 78.62 78.62 2008 73.29% 92.09 81.10 73.00 85.05 72.83 72.83 2009 75.00% 93.09 82.10 74.26 86.05 74.26 74.26 2010 75.00% 93.09 81.08 74.49 86.05 74.49 74.49 2011 75.00% 93.09 81.08 74.38 86.05 74.38 74.38 Average 76.14% 92.18 80.96 75.99 85.24 75.97 75.97 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k 2.5% Year St. Ratio 0.5% 1.5% 2002 79.25% 58.06 62.16 66.22 2003 72.61% 58.04 62.10 66.11 2004 75.34% 58.05 62.12 66.15 2005 78.34% 58.06 62.15 66.20 2006 78.54% 58.06 62.15 66.21 2007 79.02% 58.06 62.16 66.22 2008 73.29% 58.04 62.11 66.12 2009 75.00% 58.05 62.12 66.15 2010 75.00% 58.05 62.12 66.15 2011 75.00% 58.05 62.12 66.15 Average 76.14% 58.05 62.13 66.17 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k Year St. ratio 0.5% 1.5% 2.5% 2002 79.25% 57.07 61.17 66.23 2003 72.61% 57.05 61.11 66.12 2004 75.34% 57.05 61.13 66.16 2005 78.34% 57.06 61.16 66.21 2006 78.54% 58.06 62.15 67.19 2007 79.02% 58.06 62.15 67.20 2008 73.29% 58.04 62.10 67.10 2009 75.00% 58.05 62.11 67.12 2010 75.00% 58.05 62.12 67.13 2011 75.00% 58.05 62.12 67.12 Average 76.14% 57.65 61.73 66.76 Note: See Notes to Table D2. Table D5 NYRLS Three-Period Optimal FPTC Allocation Test Results [phi] Not Applied PSP Information World Stock Market Year St. Ratio NA Spread Scaled Volatility 2002 57.80% NA 0.27 4.02 19.30% 2003 58.76% NA 0.18 2.66 12.26% 2004 59.25% NA 0.14 2.03 8.19% 2005 59.65% NA 0.15 2.15 8.19% 2006 59.27% NA 0.15 2.13 7.45% 2007 59.84% NA 0.16 2.36 9.33% 2008 57.17% NA 0.23 3.36 23.63% 2009 62.58% NA 0.09 1.36 23.27% 2010 64.00% NA 0.08 1.13 20.60% 2011 54.49% NA 0.07 1.00 17.97% Average 59.28% NA 0.15 2.22 15.02% [phi] Applied PSP Information World Stock Market Year St. Ratio NA Spread Scaled Volatility 2002 57.80% 1.95% 0.27 4.02 19.30% 2003 58.76% 1.80% 0.18 2.66 12.26% 2004 59.25% 1.44% 0.14 2.03 8.19% 2005 59.65% 1.41% 0.15 2.15 8.19% 2006 59.27% 0.94% 0.15 2.13 7.45% 2007 59.84% 0.57% 0.16 2.36 9.33% 2008 57.17% 0.63% 0.23 3.36 23.63% 2009 62.58% -1.95% 0.09 1.36 23.27% 2010 64.00% -1.91% 0.08 1.13 20.60% 2011 54.49% -0.53% 0.07 1.00 17.97% Average 59.28% 0.43% 0.15 2.22 15.02% FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. Ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2002 57.80% 92.16 81.28 72.26 85.12 67.09 57.80 2003 58.76% 92.15 81.27 72.24 85.11 67.08 58.76 2004 59.25% 92.15 81.27 72.23 85.11 67.08 59.25 2005 59.65% 92.15 81.26 72.22 85.11 67.07 59.65 2006 59.27% 92.15 81.26 72.23 85.11 67.08 59.27 2007 59.84% 92.15 81.26 72.22 85.11 67.07 59.84 2008 57.17% 92.16 81.29 72.27 85.12 67.10 57.17 2009 62.58% 92.14 81.22 72.17 85.10 67.04 62.58 2010 64.00% 92.13 81.21 72.14 85.09 67.03 64.00 2011 54.49% 92.17 81.32 72.32 85.13 67.13 57.04 Average 59.28% 92.15 81.26 72.23 85.11 67.08 59.54 FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. Ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2002 57.80% 93.17 82.31 74.32 86.13 68.12 57.00 2003 58.76% 93.16 82.30 73.28 86.12 68.10 57.72 2004 59.25% 93.16 82.29 73.27 86.12 68.10 58.41 2005 59.65% 93.16 82.29 73.26 86.12 68.09 58.82 2006 59.27% 93.16 81.27 73.26 86.12 68.09 58.72 2007 59.84% 92.15 81.26 73.25 85.11 68.09 59.50 2008 57.17% 92.16 81.30 73.30 85.12 68.11 56.81 2009 62.58% 90.12 79.18 71.13 83.08 66.02 63.83 2010 64.00% 90.11 79.16 71.10 83.07 66.01 65.25 2011 54.49% 91.17 80.30 72.31 85.13 67.12 56.02 Average 59.28% 92.15 81.17 72.85 85.21 67.59 59.21 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k Year St. Ratio 0.5% 1.5% 2.5% 2002 57.80% 57.80 57.80 57.80 2003 58.76% 58.00 58.76 58.76 2004 59.25% 58.00 59.25 59.25 2005 59.65% 58.00 59.65 59.65 2006 59.27% 58.00 59.27 59.27 2007 59.84% 58.01 59.84 59.84 2008 57.17% 57.17 57.17 57.17 2009 62.58% 58.01 62.01 62.58 2010 64.00% 58.02 62.02 64.00 2011 54.49% 54.49 54.49 54.49 Average 59.28% 57.55 59.03 59.28 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k Year St. Ratio 0.5% 1.5% 2.5% 2002 57.80% 56.69 56.69 56.69 2003 58.76% 57.72 57.72 57.72 2004 59.25% 58.41 58.41 58.41 2005 59.65% 58.00 58.82 58.82 2006 59.27% 58.00 58.72 58.72 2007 59.84% 58.00 59.50 59.50 2008 57.17% 56.81 56.81 56.81 2009 62.58% 57.02 61.03 63.83 2010 64.00% 57.02 61.04 65.00 2011 54.49% 54.78 54.78 54.78 Average 59.28% 57.25 58.35 59.03 Notes: See Notes to Table D2. Table D6 NYRLS Three-Period Optimal FPTC Allocation Test Results [phi] Not Applied PSP Information World Stock Market Year St. Ratio NA Spread Scaled Volatility 2004 9.53% NA 0.14 2.03 8.19% 2005 12.60% NA 0.15 2.15 8.19% 2006 11.79% NA 0.15 2.13 7.45% 2007 18.02% NA 0.16 2.36 9.33% 2008 15.20% NA 0.23 3.36 23.63% 2009 18.78% NA 0.09 1.36 23.27% 2010 24.65% NA 0.08 1.13 20.60% 2011 25.63% NA 0.07 1.00 17.97% Average 17.03% NA 0.13 1.94 14.83% [phi] Applied PSP Information World Stock Market Year St. Ratio NA Spread Scaled Volatility 2004 9.53% 0.01% 0.14 2.03 8.19% 2005 12.60% 0.01% 0.15 2.15 8.19% 2006 11.79% 0.01% 0.15 2.13 7.45% 2007 18.02% 0.01% 0.16 2.36 9.33% 2008 15.20% 0.01% 0.23 3.36 23.63% 2009 18.78% 0.01% 0.09 1.36 23.27% 2010 24.65% 0.01% 0.08 1.13 20.60% 2011 25.63% 0.01% 0.07 1.00 17.97% Average 17.03% 0.01% 0.13 1.94 14.83% FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k k Year St. Ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2004 9.53% 92.38 80.85 72.11 85.32 67.58 56.66 2005 12.60% 92.37 80.82 73.09 85.31 67.55 56.61 2006 11.79% 92.37 80.83 73.10 85.31 67.56 56.63 2007 18.02% 92.34 80.75 72.98 85.29 67.50 56.54 2008 15.20% 92.35 80.79 73.04 85.30 67.52 56.58 2009 18.78% 92.34 80.74 72.97 85.28 67.49 56.53 2010 24.65% 92.31 80.67 72.86 85.26 67.43 56.44 2011 25.63% 92.31 80.66 72.84 85.25 67.42 56.43 Average 17.03% 92.35 80.76 72.87 85.29 67.51 56.55 FPTC Optimal Risky Asset Ratio Volatility 10% Volatility 15% PSP Information k ]k Year St. Ratio 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% 2004 9.53% 92.38 80.85 73.14 85.32 67.58 56.66 2005 12.60% 92.37 80.82 73.09 85.31 67.55 56.61 2006 11.79% 92.37 80.83 73.10 85.31 67.56 56.63 2007 18.02% 92.34 80.75 72.99 85.29 67.50 56.54 2008 15.20% 92.35 80.79 73.04 85.30 67.52 56.58 2009 18.78% 92.34 80.74 72.97 85.28 67.49 56.53 2010 24.65% 92.31 80.67 72.86 85.26 67.43 56.44 2011 25.63% 92.31 80.66 72.84 85.25 67.42 56.43 Average 17.03% 92.35 80.76 73.00 85.29 67.51 56.55 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k Year St. Ratio 0.5% 1.5% 2.5% 2004 9.53% 49.10 45.24 41.33 2005 12.60% 49.09 45.22 41.29 2006 11.79% 49.09 45.23 41.30 2007 18.02% 49.08 45.18 41.24 2008 15.20% 49.08 45.20 41.27 2009 18.78% 49.07 45.18 41.23 2010 24.65% 49.06 46.15 41.17 2011 25.63% 49.06 46.14 41.16 Average 17.03% 49.08 45.44 41.25 FPTC Optimal Risky Asset Ratio Volatility 20% PSP Information k Year St. Ratio 0.5% 1.5% 2.5% 2004 9.53% 49.10 45.24 41.33 2005 12.60% 49.09 45.22 41.29 2006 11.79% 49.09 45.23 41.30 2007 18.02% 49.08 45.18 41.24 2008 15.20% 49.08 45.20 41.27 2009 18.78% 49.07 46.19 41.23 2010 24.65% 49.06 46.15 41.17 2011 25.63% 49.06 46.14 41.16 Average 17.03% 49.08 45.57 41.25 Notes: See Notes to Table D2.

DOI: 10.1111/jori.12072

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Changhui Choi is at the Department of Financial Strategy, Korea Insurance Research Institute. Bong-Gyu Jang is at the Department of Industrial and Management Engineering, POSTECH. Changki Kim is at Korea University Business School, Korea University. Sang-youn Roh is at the Performance Evaluation Team, National Pension Research Institute. The authors can be contacted via e-mail: cchoi@kiri.or.kr, bonggyujang@postech.ac.kr, changki@korea.ac.kr, and riskhunter@nps.or.kr, respectively. We thank journal editor Keith J. Crocker and the anonymous referee for the careful reading of our manuscript and the valuable comments. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2014S1A3A2036037) and by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science, and Technology (NRF-2012R1A1A2038735, NRF2013R1A2A2A03068890).

(1) According to Martellini and Milhau (2008), a series of papers (Leland, 1980; Benninga and Blume, 1985; Franke and Subrahmanyam, 1998) examined the investor preferences and market characteristics that would support the idea of including derivatives features in insurance and investment portfolios. The authors mostly find that holding such derivatives payoffs can be justified under usually severe forms of market incompleteness and/or the presence of background risk. This finding is supported by five PSPs that carry a small amount of financial derivatives (the value is usually less than 0.1 percent of the total assets). As discussed in Leland (1980), many authors consider maximizing the terminal wealth and minimizing the probability of insolvency as appropriate goals for DC (defined-contribution) pension plans and DB (defined-benefit) pension plans. However, a PSP that manages a DB pension plan can still pursue long-term wealth maximization if there is little chance of insolvency, which is the case for many large public PSPs. van Binsbergen and Brandt (2007) and Martellini and Milhau (2008) refer to cases where maximizing the expected terminal funding ratio (asset/liability) (or terminal net asset (asset-liability)) is used as an objective for DB pension plans.

(2) According to Sarr and Lybek (2002), liquidity measures include the following: transaction cost measures, volume-based measures, price-based measures, market-impact measures, and other econometric techniques. Plerou, Gopikrishnan, and Stanley (2005) analyze market liquidity using bid-ask spread. Wei and Zheng (2010) measure the liquidity of individual equity options using the bid-ask spread, which can be expressed as the absolute value or the ratio of a gap between the bid price and the ask price. For example, Marshall and Young (2003) calculate the bid-ask spread using every Wednesday's closing bid and ask prices. According to Marshall (2006), an order-based measure such as the bid-ask spread is effective for measuring the liquidity of small investors; however, it is not effective for measuring the liquidity of larger investors. Marshall insists that weighted order value can compensate for the weak points of traditional liquidity proxies by incorporating the bid-ask spread and the market depth; weighted order value is one of the liquidity proxies used by Aitken and Comerton (2003).

(3) Merton (1971) obtains a closed-form solution for a two-asset portfolio optimization problem in the absence of transaction costs. Following this paper, Cvitanic and Karatzas (1992), Xu and Shreve (1992), and He and Pearson (1993) analyze portfolio optimization with strategic constraints. Davis and Norman (1990), Magill and Constantinides (1976), Taksar, Klass, and Assaf (1988), and Jang et al. (2007) examine portfolio optimization problems with proportional stock trading cost. Constantinides (1979) shows that the no-trading region (NTR) is a convex cone for an investor with the power utility and a proportional transaction cost. An approximate solution to this problem is provided in Constantinides (1986), and Gennotte and Jung (1994) numerically identify the approximate boundary values of NTR for portfolio optimization with a finite terminal date. Moreover Eastham and Hastings (1988) extend the problem by considering both fixed and proportional transaction costs in their impulse control approach, whereas Grossman and Vila (1992) study optimal dynamic trading with leverage constraints. The approach employed by Eastham and Hastings (1988) is improved further by Korn (1998), who proposes using an optimal stopping criteria method. Shreve and Soner (1994) apply the theory of viscosity solution to Hamilton-Jacobi-Bellman (HJB) equations. Dumas and Luciano (1991) provide an exact solution to the portfolio choice problem for a CRRA investor who pays a proportional cost in an infinite horizon. More recently Zakamouline (2002) proposes a solution to finite-horizon portfolio optimization in continuous time with both fixed and proportional costs using a Markov approximation. Finally 0ksendal and Sulem (2002) consider the portfolio choice problem for a CRRA investor who pays both proportional and fixed transaction costs in an infinite horizon as a combination of a stochastic control problem and an impulse control problem and provided an optimal solution procedure for this problem.

(4) A number of previous studies argue that asset allocation of a PSP is influenced by politics. For example, Pennacchi and Rastad (2001) state that pension plans tend to take more risk when they have greater representation on the board of trustees by pension plan participants. Dobra and Lubich (2013) discuss governance influence on investment decisions and risk profiles of public pension systems.

(5) This test result is consistent with the findings of Ang and Bekaert (2002).

(6) Regulations that govern NPS have strict limitations on risky asset investments.

(7) For computational testing, this article assumes that the proportional trading cost changes in proportion to the bid-ask spread.

(8) In actual investment operations, there are no true risk-free assets. However, in Table D1 in Appendix D, we considered all bonds to be risk-free assets and all stocks to be risky assets. It is still possible to lose money from these bonds when their market value drops. However, we considered all the bonds as risk-free assets because most PSPs usually invest only in bonds with high credit ratings and low default probability. Asset types other than stocks and bonds are not considered in this article.

(9) We assume that a fixed portion (e.g., 0.1 percent) of the total asset value ([x.sub.0] + [y.sub.0]) is paid as a fixed cost at the beginning of each period when a PSP rebalances its portfolio.

(10) Maximizing the expected utility of the last period is used widely in the literature (e.g., Mossin, 1968; Boyle and Lin, 1997).

(11) We do not always mark [[DELTA].sub.-] and [[DELTA].sub.+] with the time variable t for convenience.

(12) Net contribution is not considered in this section because we assume that assets are adjusted by net contribution at the beginning of each period. In other words, one can apply the technique in this section to both one-period FPTC problems: one with net contribution and the other without. When net contribution is not considered, one can use the technique in this section. Otherwise, one can adjust the initial portfolio according to net contribution and apply the same technique. See the "Numerical Implications" section for details.

(13) These conditions can be easily satisfied because d < R < u and ([k.sub.1], [k.sub.2]) are small numbers.

(14) Using basic algebra, it can be shown that this condition is equivalent to [r.sub.s] > (1 - [k.sub.1])R.

(15) Reasonable parameters, such as long-term estimates of the world stock market, usually satisfy these conditions. These conditions are more easily satisfied for a large g because w1 and w2 are decreasing functions with respect to [gamma].

(16) It is implemented on line [L01] of the heuristic for the multiperiod FPTC problem in Apendix C.

(17) See Fong, Piggott, and Sherris (forthcoming), who assess longevity risks (longevity selection, mortality improvements, and unexpected systematic shocks) faced by public sector, DB plan providers.

(18) ALM (funding level) requirements, taxes on foreign investment profits, and restrictions on the composition of portfolio are well-known regulations on PSP asset allocation decisions.

(19) RBC (risk-based capital) in the United States.

(20) Note that R([x.sub.0] + (1 - [k.sub.1]) [[DELTA].sup.*.sub.-] - K) [greater than or equal to] K and [y.sub.0] [[DELTA].sup.*.sub.-] [greater than or equal to] 0.

(21) x' = [x.sub.0] - (1 + [k.sub.2]) [[DELTA].sup.*.sub.+] - K [greater than or equal to] K from y' [greater than or equal to] 0 and (1 - [k.sub.1])([y.sub.0] + [[DELTA].sup.*.sub.+])d [greater than or equal to] 0.

Caption: FIGURE 1 NTR and Rebalancing Lines of the One-Period FPTC Problem

Caption: FIGURE 2 Benchmark Three-Period Optimal Trading Policy With [phi]=0

Caption: FIGURE 3 NTRs of Different Values of Terminal Time T

Caption: FIGURE 4 Changes in NTR for Different Values of Parameters

Table 1 PSPs of Interest Name Abbrev. Country Net Asset Valuation (Bill. Date USD) CalPERS CALP United 241,761 June 2011 States New York State and NYSLR United 149,548 March 2011 Local Retirement States System Avon Pension Fund AVON United 4,098 March 2011 Kingdom Forsta AP-Fonden FAPF Sweden 33,883 December 2011 National Pension NPS South 312,934 December 2011 Service Korea Notes: This table contains analyzed PSP information including name, abbreviated name, base country of operation, estimated net asset value in billion USD, and valuation date. Table 2 Risky Asset (Stock) Ratio Changes of Five PSPs Between 2007 and 2011 Risky Asset Ratio Change PSP a. 2007 b-a b. 2008 c-b c. 2009 d-c CALP 70.98 -3.86 67.12 -6.26 60.86 2.42 NYSLR 74.92 -2.54 72.38 -10.85 61.53 10.18 AVON 79.02 -5.73 73.29 1.71 75 0 FAPF 59.84 -2.67 57.17 5.41 62.58 1.42 NPS 18.02 -2.82 15.2 3.58 18.78 5.87 Risky Asset Ratio Change PSP d. 2010 e-d e. 2011 CALP 63.28 5.47 68.75 NYSLR 71.71 3.64 75.35 AVON 75 0 75 FAPF 64 -9.51 54.49 NPS 24.65 0.98 25.63 Notes: Columns 2, 4, 6, 8, and 10 are actual stock to total wealth (stock and bond) ratios of five PSPs. Columns 3, 5, 7, and 9 are the changes in the stock ratios between years. Table 3 World Stock Market Indices Year Spread Scaled [[sigma].sub.s] 2002 0.27 4.02 19.30% 2003 0.18 2.66 12.26% 2004 0.14 2.03 8.19% 2005 0.15 2.15 8.19% 2006 0.15 2.13 7.45% 2007 0.16 2.36 9.33% 2008 0.23 3.36 23.63% 2009 0.09 1.36 23.27% 2010 0.08 1.13 20.60% 2011 0.07 1.00 17.97% Notes: This table includes world stock market indices such as bid-ask spread (Spread), scaled spread (Scaled) and stock market volatility ([[sigma].sub.s]). We calculated "Scaled" by dividing "Spread" by the smallest spread between 2002 and 2011, which is 0.07 (Spread of 2011). Table 4 NTR Change in Response to the Parameters Parameter Par. Par. Values Figure NTR Change Notation # Net contrib. [phi] -5%, 0%, +5% 4a Pushed up Fixed cost K 0.1%, 0.5%, 1% 4b Widened Prop. cost k 0.1%, 0.5%, 1% 4c Widened Risk aversion [gamma] 1.9, 2.0, 2.1 4d Pushed down Risky asset [r.sub.s] 1.0771, 1.0810, 4e Pushed up return 1.0732 Volatility [S.sub.s] 1.9, 2.0, 2.1 4f Pushed down Notes: "Parameter" tells us which parameter's sensitivity was tested. "Par. Notation" shows the notation for each parameter. "Par. Values" are the values tried for each parameter. "Figure #" points to a figure that contains sensitivity test for each parameter. Finally, "NTR Change" represents NTR changes as a parameter increases. Table 5 Average Optimal Risky Asset Ratios for Different Values of [[sigma].sub.s] and k Panel A: Theoretical Optimal Asset Allocation for Different Values of [[sigma].sub.s] PSP Information Volatility ([[sigma].sub.s]) Name Stock a. 10% b--a b. 15% c--b c. 20% Ratio CALP 67.69% 82.06 --8.23 73.82 --12.06 61.76 NYSLR 72.19% 88.79 --7.72 81.07 --7.84 73.22 AVON 76.14% 83.09 --3.99 79.11 --16.99 62.12 FAPF 59.28% 81.88 --11.31 70.57 --11.96 58.62 NPS 17.03% 81.99 --12.21 69.78 --24.53 45.26 Panel B: Theoretical Optimal Asset Allocation for Different Values of k PSP Information k Name Stock a. 0.5% b - a b. 1.5% c - b c. 2.5% Ratio CALP 67.69% 58.03 3.91 61.94 3.38 65.32 NYSLR 72.19% 76.02 -3.25 72.77 -1.89 70.88 AVON 76.14% 58.05 4.08 62.13 4.04 66.17 FAPF 59.28% 57.55 1.48 59.03 0.25 59.28 NPS 17.03% 49.08 -3.64 45.44 -4.19 41.25 Notes: Panel A contains the average risky asset ratios of each PSP for different volatilities (or [[sigma].sub.s]), where [phi] is not applied. Columns 3, 5, and 7 are the average values of optimal asset allocations in the upper part of Tables D2-D6 in Appendix D for the volatilities 10%, 15%, and 20%, respectively. For example, 82.06 is the annual average of the risky asset ratios of CALP where [[sigma].sub.s] = 10% and k [member of] {0.5%, 1.5%, 2.5%}. This test shows that the theoretical optimal asset allocation decreases by roughly 10% when the market volatility is increased by 5%. Panel B shows how optimal risky asset ratios change as the trading cost increases. Columns 3, 5, and 7 are averages of optimal risky asset ratios where trading costs are 0.5%, 1.5%, and 2.5%, respectively, where volatility ([[sigma].sub.s]) is fixed at 20%. Table 6 Summary of the Impact of Net Contribution Volatility 10% [phi] [k.sub.1],[k.sub.2] Name Average Applied 0.5% 1.5% 2.5% CALP -1.1769% No 92.11 81.16 72.89 Yes 90.90 80.04 71.90 NYSLR -3.9237% No 95.11 88.23 83.03 Yes 92.09 85.15 79.90 AVON 0.2324% No 92.07 81.06 76.14 Yes 91.57 80.84 73.45 FAPF 0.4341% No 92.15 81.26 72.23 Yes 92.15 81.17 72.85 NPS 0.0076% No 92.35 80.76 72.87 Yes 92.35 80.76 73.00 Volatility 15% Volatility 20% Average [k.sub.1],[k.sub.2] k Name 0.5% 1.5% 2.5% 0.5% 1.5% 2.5% CALP 85.07 68.71 67.69 58.03 61.94 65.32 72.55 84.07 69.20 68.49 57.23 61.07 65.05 71.99 NYSLR 91.09 79.10 73.01 76.02 72.77 70.88 81.03 88.51 76.94 73.98 75.20 73.86 73.33 79.88 AVON 85.04 76.14 76.14 58.05 62.13 66.17 74.77 84.74 69.46 64.03 55.26 56.54 57.36 70.36 FAPF 85.11 67.08 59.54 57.55 59.03 59.28 70.36 85.21 67.59 59.21 57.25 58.35 59.03 70.31 NPS 85.29 67.51 56.55 49.08 45.44 41.25 65.68 85.29 67.51 56.55 49.08 45.57 41.25 65.71 Name Difference CALP 0.55% NYSLR 1.30% AVON 4.41% FAPF 0.05% NPS -0.03% Notes: This table summarizes how the net contribution affects the optimal asset allocation of five PSPs. According to the test results in this table, considering net contribution generates -0.03% -4.41% changes in the optimal asset allocations of five PSPs. Two rows for each PSP are the annual average risky asset ratios from Tables Dl-D-6 in Appendix D where the first and second rows contain the averages of the upper ([phi] not applied) and lower ([phi] applied) parts of Tables Dl-D-6. The last column contains the average values of risky asset ratio differences of two cases: ([phi] applied and ([phi] not applied.

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Author: | Choi, Changhui; Jang, Bong-Gyu; Kim, Changki; Roh, Sang-youn |
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Publication: | Journal of Risk and Insurance |

Article Type: | Report |

Geographic Code: | 1USA |

Date: | Dec 1, 2016 |

Words: | 17698 |

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