Embed Size (px)
Citation preview
Dynamic Preferencesfor Popular Investment Strategies
in Pension Funds
Carole Bernard∗ Minsuk Kwak†
October 3, 2013
Abstract
In this paper, we infer preferences that are consistent with somegiven dynamic investment strategies. Two popular dynamic strategiesin the pension funds industry are considered: a constant proportionportfolio insurance (CPPI) strategy and a life-cycle strategy. In bothcases, we are able to infer preferences of the pension fund’s managerfrom her investment strategy, and to exhibit the specific expected util-ity maximization that makes this strategy optimal at any given timehorizon. For example, we show that, in a Black-Scholes market, aCPPI strategy is optimal for a fund manager with HARA utility func-tion, while an investor with a SAHARA utility function (introducedby Chen et al. (2011)) will choose a time decreasing allocation to riskyassets in the same spirit as the life-cycle funds strategy. We also sug-gest how to modify these strategies if the financial market follows amore general diffusion process than in the Black-Scholes market.
Keywords : Forward utility, CPPI strategy, Life-Cycle fund, HARA
utility, SAHARA utility
∗Department of Statistics and Actuarial Science, University of Waterloo, 200 UniversityAvenue West, Waterloo, Ontario, N2L 3G1, Canada, [email protected]†Department of Mathematics and Statistics, McMaster University, 1280 Main Street
West, Hamilton, ON, L8S 4K1, Canada, [email protected]
1
1 Introduction
This paper studies dynamic investment strategies, popular in the industry,especially in pension funds management. Our goal is to infer preferencesfor which a given investment strategy appears optimal. Our approach is incontrast to most of the existing literature on optimal investment in pensionfunds. Typically, research in this area starts by specifying a utility function oran objective to optimize (i.e. investor’s preferences) and a given investmenthorizon (e.g., retirement age) and then derives the optimal investment strat-egy for accumulating money. A lot of previous studies focus on the expectedutility theory. For example, optimal dynamic asset allocation are obtainedby Cairns et al. (2006) in a defined contribution plan for an expected utilitymaximizer (with power utility or log utility) or in a mean-variance setting byViceira (2007). Boyle & Tian (2009) derive “optimal” parameters of an eq-uity indexed annuity (typical retirement product) for a given expected utilitymaximizer. Black & Perold (1992) show that a constant proportion portfolioinsurance (CPPI) strategy can maximize expected utility when a HARA util-ity function is assumed. See also Basak (2002), Bertrand & Prigent (2005)and Prigent (2007). Benninga & Blume (1985) further show that the op-timality of a portfolio insurance strategy heavily depends on the investor’sutility function.
Other studies have been using preference-free approaches to assess opti-mality and compare pension plans and investment strategies. For example,Annaert et al. (2009) and Zagst & Kraus (2011) use stochastic dominanceto compare portfolio insurance strategies. In a framework including a lot ofpractical features (such as periodic charges and contributions as well as bor-rowing constraints), Graf et al. (2012) introduce a new methodology basedon risk-return profiles, i.e. the (forward-looking) probability distribution ofbenefits, to assess the risk-return profiles of existing retirement products andhelp comparing products across the insurance industry. Their study is il-lustrated with CPPI and static option-based strategies, Graf (2012) focusesmore specifically on life-cycle funds.
As Viceira (2007) explained, there are essentially two types of pensionfunds investment strategies: balanced funds (also called constant-mix strate-gies) and life-cycle funds (also called target-date funds). Balanced funds aresimilar to mutual funds with no investment horizon and built on the ideathat a fixed fraction of the funds based on investors’ risk tolerance (and in-dependent of their investment horizon) is allocated to stocks. This is referredas “risk-based investing” and on average (in the 15 largest balanced fundson the market at the end of 2006), 66% was invested in the stock marketand 34% in fixed income instruments (see Pang & Warshawsky (2008)). On
2
the contrary, life-cycle funds are “age-based” with the idea that investorsshould allocate a larger share of their long-term savings to stocks when theyare young and decrease this allocation over time. The asset allocation istypically adjusted to become more conservative as the retirement age of theinvestor approaches. Life-cycle funds and constant-mix portfolio strategies(which are a special case of CPPI strategies) are often set as default choice ofasset allocation in numerous defined contribution schemes or related old ageprovision products. Incorporating labor, Gomes et al. (2008) show that life-cycle funds are superior and that equities are the preferred asset for younghouseholds, with the optimal share of equities generally declining prior toretirement.
In this paper, we propose to study both types of strategies and the linkbetween preferences and investment strategies. We focus on two populardynamic strategies in the pension funds industry, a dynamic portfolio insur-ance strategy (CPPI with the constant-mix or balanced strategy obtained asa special case) and a strategy with decreasing allocation to risky assets (inthe same spirit of a life-cycle fund). The advantage of the CPPI strategyabove the balanced strategy is to propose a guarantee. The importance andimplications of such minimum guarantee are discussed in Pezier & Scheller(2011). We discuss how these latter strategies can be optimal for an ex-pected utility investor and what changes in utility throughout the lifetime ofthe investor are consistent with them.
Precisely, our contributions are as follows. We are able to directly relatethe type of strategy chosen by the manager with a class of utility functionsfor which it is optimal, as well as the choice of its parameters with themanager’s risk aversion level and market conditions. We show that CPPIstrategies are optimal portfolio choice for an investor with a HARA utilityfunction at all horizon whereas funds with a decreasing allocation to riskyassets are optimal for investors with the SAHARA utility function introducedby Chen et al. (2011). The result on CPPI extends Black & Perold (1992)in the sense that the market price of risk may not be constant and thecorresponding utility function to optimize in the expected utility setting isnow state-dependent and involve the financial market. In the specific case ofthe Black-Scholes setting, we show how CPPI, constant-mix strategies andsome type of life-cycle funds are optimal for some explicit standard expectedutility maximizers at any given time horizon. When the instantaneous Sharperatio of the underlying risky asset is stochastic (that is, when the financialmarket is not Black-Scholes model), we explain how to modify these strategiesto account for changes in instantaneous Sharpe ratio and preserve optimality.
The techniques used throughout this paper are based on the recent workof Musiela & Zariphopoulou (2009, 2010, 2011). In the standard expected
3
utility setting, an optimal investment problem consists of a given fixed invest-ment horizon and a utility function. The optimization of expected utility ofterminal wealth among all admissible strategies is then performed to obtainthe optimal portfolio selection. The approach of Musiela & Zariphopoulou(2009, 2010, 2011) is quite different: they introduce the so-called “forwardperformance criterion” which allows to define dynamic preferences based on adynamic investment strategy. Our work is done in a continuous-time setting.In practice dynamic strategies are implemented in discrete-time. However thecontinuous-time setting is a good approximation of the discrete-time setting.Moreover Pezier & Scheller (2013) show that CPPI strategies still dominatesstatic option based strategies when implemented discretely under real-worldconditions (e.g., in the presence of jumps).
The rest of the paper is organized as follows. The problem and notationare given in Section 2. In particular we recall the definition of a “forwardutility”. Examples of dynamic investment strategies are then studied in fulldetails: the CPPI and constant-mix strategies in Section 3, the life-cyclefunds in Section 4. Section 5 concludes.
2 Setting
2.1 Financial Market
We consider a one-dimensional market1 with two assets: a risky asset anda risk-free bond with the following dynamics under the physical probabilitymeasure P
dSt = St(µtdt+ σtdWt), S0 > 0,
dBt = rtBtdt, B0 = 1, (1)
where we assume that rt, µt and σt may be stochastic but are adapted tothe filtration generated by the Brownian motion Wt that drives the price ofthe risky asset. Similarly as Musiela & Zariphopoulou (2009, 2010, 2011), weassume that the market price of risk (or instantaneous Sharpe ratio) definedby
λt ,µt − rtσt
1The extension to a multidimensional market is straightforward and would only com-plicate the notation. As our objective is to better understand the concept of forwardperformance criterion and illustrate it on some examples, we will only consider the casewhen there is one risk-free asset and one risky asset in the financial market. A multidi-mensional market is considered for example in Musiela & Zariphopoulou (2009).
4
is bounded by a deterministic constant c > 0 (for all t > 0, |λt| ≤ c).We can assume that all amounts are expressed in terms of present values,
equivalently Bt is used as numeraire. A dynamic investment strategy consistsof an amount π0
t at time t invested in the risk-free asset and an amount πtinvested in the risky asset. Let Xπ
t denote the present value of the strategyat time t when the strategy π is implemented. In particular,
Xπt = π0
t + πt.
Since Bt is used as numeraire, it follows that
dXπt = dπt = σtπt(λtdt+ dWt). (2)
We define the set A of admissible strategies as
A ,
{π :
Self-financing and adapted strategy
∀t > 0,E(∫ t
0|σsπs|2ds
)<∞
}.
Throughout this paper, Ft denotes the information available at time t andEt(·) , E(·|Ft) denotes conditional expectation on the information at t.
2.2 Preferences
Musiela & Zariphopoulou (2009, 2010, 2011) propose to model preferences ofan investor by a forward performance criterion, that we will also call “forwardutility” similarly as Berrier et al. (2010). Let us recall its definition given forexample in Musiela & Zariphopoulou (2010) (Definition 1 page 329)
Definition 2.1 (Forward investment performance or Forward utility). AnFt-adapted process Ut(x) is a forward investment performance if for t ≥ 0and x ∈ D where D is an interval of R:
1. x→ Ut(x) is strictly concave and increasing
2. for each π ∈ A (i.e. for each attainable Xπs ), and t ≥ s,
Es(Ut(Xπt )) ≤ Us(X
πs ),
3. there exists π∗ ∈ A, for which for all t ≥ s,
Es(Ut(Xπ∗
t )) = Us(Xπ∗
s ).
5
Although this setting is not equivalent to the well-known expected utilitymaximization (see for example Musiela & Zariphopoulou (2010)), there aresome connections and the definition of a forward utility is quite natural giventhe following observations. Assume that the investor has a finite tradinghorizon T and maximizes over all admissible strategies
v(x, t) , supπ∈A
Et (V (XπT )|Xπ
t = x)
where V is a concave increasing utility function. Under some general condi-tions, the value function v satisfies the dynamic programming principle, thatis for 0 ≤ s ≤ t ≤ T ,
v(x, s) = supπ∈A
Es (v(Xπt , t)|Xπ
s = x) .
Let π ∈ A and π∗ be the optimal strategy in A. Then, (v(Xπs , s))s and(
v(Xπ∗s , s)
)s
are respectively a supermartingale and a martingale with Ft.Musiela & Zariphopoulou (2009, 2010, 2011) develop several examples of
correspondence between a forward utility and a dynamic investment strategy.They find sufficient conditions for a forward utility to exist and explain theoptimality of a dynamic strategy. This forward utility is always formulatedas
Ut(x) = u(x,At)
where At ,∫ t0λ2sds, t ≥ 0 may be stochastic.2 We show how their work can
be applied to understand CPPI strategies and life-cycle funds.
2.3 Dynamic Investment Strategies
The next sections present two relevant examples for the industry. Our firstexample (Section 3) is the CPPI strategy. As a special case we will look atthe constant-mix strategy. Typically, these strategies do not have a givenhorizon. We exhibit the corresponding forward utility, and show that atany time t, it belongs to the family of HARA utilities. We also prove thatthis forward utility can be interpreted as the utility function that makes theCPPI strategy optimal at any given horizon. Our second example (Section 4)concerns life-cycle funds and more generally strategies that involve reducingthe investment in risky asset over some time period. We will discuss how thisstrategy can be optimal for an expected utility investor at any horizon and
2When At is stochastic then the forward utility is stochastic and thus not necessar-ily law-invariant anymore. This departs from the traditional setting for expected utilitymaximization.
6
what changes in utility throughout the lifetime of the investor are consistentwith such a strategy. In particular, we will show that the SAHARA utility asintroduced in Chen et al. (2011) can justify the optimality of some life-cyclefunds.
3 CPPI and Constant-mix Strategies
The constant proportion portfolio insurance (CPPI) introduced by Black &Perold (1992) is popular in the insurance industry to manage pension fundsand variable annuities (see for example Bernard & Le Courtois (2012)). CPPIis indeed a good way to hedge long-term guarantees when the maturity dateis not known in advance or when regulators require the guarantee to be metat all times. Let us first recall what a CPPI strategy is.
3.1 CPPI and Constant-mix Strategies
The CPPI strategy consists of rebalancing a portfolio between the risky andrisk-free assets in order to maintain the value of this portfolio always superiorto a predefined floor level Gt ≥ 0 at time t ≥ 0, whilst keeping the possibilityof large returns. We assume that the floor behaves dynamically like a savingsaccount compounded at the risk-free rate rt:
dGt = Gt rt dt, G0 = G.
Then, one defines the cushion to be the difference between the portfoliovalue at time t, Vt, and the floor level, Gt = GBt at time t:
∀t ∈ [0, T ], Ct = Vt −Gt.
As we work with present values, it can be rewritten in terms of discountedvalue of the portfolio, Xt, and we get
∀t ∈ [0, T ],CtBt
= Xt −G.
The investment rule is to maintain an exposure to the risky asset proportionalto the cushion. In other words, the fund manager defines and chooses attime 0 a number m > 0 called the multiple, and rebalances dynamically itsportfolio such as
∀t ∈ [0, T ], πt = mCtBt
= m(Xt −G).
7
When G = 0, note that it is a constant-mix strategy. The amount of risk-freeasset is therefore at all times
∀t ∈ [0, T ], π0t = Xt − πt,
and one can write
Xt = G+CtBt
= πt + π0t .
The first equality is a virtual decomposition of the discounted portfolio valueinto the guarantee G and the discounted cushion, whilst the second equalitygives the actual decomposition into risky and risk-free assets as discussed inSection 2.
3.2 Forward utility corresponding to the CPPI strat-egy
To ensure that the CPPI strategy is optimal for an expected utility maximizerat any time horizon in the general market described in Section 2.1, we con-sider a slightly generalized CPPI strategy with random multiple mt = λt/λ0
σt/σ0m
adapted at any time t to the information available. In the case of a Black-Scholes model, it corresponds to a standard CPPI strategy with fixed multiplem (because both λt and σt are constant).
Proposition 3.1 (General Case). The dynamic CPPI investment strategyconsisting of
π∗t =λt/λ0σt/σ0
m(X∗t −G) (3)
invested in the risky asset (i.e. a CPPI strategy with an adapted multipleλt/λ0σt/σ0
m) gives the following portfolio value at time t
X∗t , Xπ∗
t = (x−G) exp
((γ − 1
2γ2)At + γMt
)+G, X∗0 = x > G (4)
where γ = σ0m/λ0, At ,∫ t0λ2sds and Mt ,
∫ t0λsdWs. This portfolio value
process corresponds to the optimum for the forward utility Ut(x) = u(x,At)where u(x, s) is given for x ∈ (G,∞) and s ≥ 0 by
u(x, s) =
γγ−1(x−G)
γ−1γ e−
γ−12s, γ ∈ (0, 1) ∪ (1,∞),
ln (x−G)− s2, γ = 1.
(5)
8
Proof. See Appendix A.
We observe that the forward utility u(·, s) belongs to the HARA familyfunction of utilities at all time s.
Proposition 3.2. Reciprocally, given any time T > 0 and initial wealth x ∈(G,∞), consider the following portfolio optimization problem to maximizethe utility of wealth at time T
maxπ∈A
E (u(XπT , AT )|Xπ
0 = x) , (6)
where AT =∫ T0λ2sds and where u(·, ·) is given by (5) and defined over
(G,∞) × [0,∞). Then the optimal allocation is a dynamic CPPI strategy
π∗t = λt/λ0σt/σ0
m(X∗t −G).
Proof. See Appendix A.
Proposition 3.2 shows that a dynamic CPPI is the optimal solution to autility maximization. However, (6) is not a standard expected utility maxi-mization as the objective is not law invariant given that the utility functiondepends on AT , which can be stochastic. The optimal strategy (3), dynamicCPPI, in stochastic environment implies that we have to rebalance the in-vestment strategy depending on λt and σt. It is obvious that we have toadjust the investment strategy dynamically if the market is stochastic andthe investment opportunity changes dynamically. We obtain the optimalityof the standard CPPI strategy in the Black-Scholes model as a corollary ofPropositions 3.1 and 3.2. In the setting of Black Scholes, we find optimalityin the standard expected utility setting.
Corollary 3.1 (Black-Scholes Case). Assume that the risky asset follows aBlack-Scholes model (i.e. µ, r and σ are constant and λ , (µ−r)/σ). Defineγ = σm/λ. Then, we have the following results.
• With the CPPI strategy π∗t = m(X∗t − G), the portfolio value is givenby
X∗t = (x−G) exp
((σλm− 1
2σ2m2
)t+ σmWt
)+G, X∗0 = x > G
and the corresponding forward utility is Ut(x) = u(x, λ2t) with u(·, ·) isgiven by (5).
9
• Given any time T > 0 and initial wealth x ∈ (G,∞), consider the fol-lowing portfolio optimization problem to maximize the utility of wealthat time T
maxπ∈A
E(u(Xπ
T , λ2T )|Xπ
0 = x),
with u(·, ·) given by (5). Then the optimal allocation is CPPI strategyπ∗t = m(X∗t −G) where the multiple is m = λγ
σ.
It is known that in the Black-Scholes setting, the portfolio value of aCPPI strategy X is a shifted lognormal process (see Prigent (2007)). Notethat m is an indicator of the riskiness of the strategy. It is clear that anincreased risk aversion decreases γ and thus m. An increase in the volatilityσ decreases m.
4 Life-Cycle Funds
In recent years, most pension plans have defined contributions (DC) but nodefined benefits (DB). An employee with a DC pension plan has to make herown savings and investment decisions to fund her income after retirement.But, as Viceira (2007) pointed out, a large number of DC plan participantsusually adopt the investment strategy suggested by the DC plan sponsor asa default option and rarely rebalance their portfolio by themselves. A typical(and popular) asset allocation strategy for managing equity risk during theaccumulation phase of a defined contribution (DC) pension plan is the “life-cycle funds”. At the beginning of the plan, the contributions are investedentirely (or almost entirely) in equities. Then on a predetermined date priorto retirement (e.g., ten years), the assets are switched gradually into bondsat a rate equal to the inverse of the length of the switchover period (e.g.,10% per year). By the date of retirement, all (or almost all) the assets areheld in bonds, which are then sold to purchase a life annuity to provide thepension payments. Cairns et al. (2006) extend this strategy to “stochasticlifestyling” and compare it against various static and deterministic life-cyclefunds.
In this section, we present the Symmetric Asymptotic Hyperbolic Abso-lute Risk Aversion (SAHARA) class of utility functions introduced by Chenet al. (2011) and give the corresponding forward utility and optimal strategy.It is shown that this optimal strategy displays the age-based investing featureof life-cycle funds which means that the optimal investment in risky asset isa decreasing function of time.
10
4.1 SAHARA utility functions and related forward util-ity
A SAHARA utility function is given by U(x), x ∈ R, whose absolute riskaversion γA(x) = −Uxx(x)/Ux(x) satisfies
γA(x) =1√
a2(x− d)2 + b2,
with a > 0, b > 0 and d ∈ R. We can assume that d = 0 without loss ofgenerality. Then U(x) is given as follows (up to a linear transformation)
• If a = 1,
U(x) =1
2ln(x+√x2 + b2
)+
1
2b2x(√
x2 + b2 − x). (7)
• If a 6= 1,
U(x) =a(a+ 1)
(ax2 + x
√a2x2 + b2
)+ b2
(a2 − 1)(ax+
√a2x2 + b2
)1+ 1a
. (8)
The formula for the utility function can be obtained from Proposition 2.2 inChen et al. (2011). Details are given in B.
An important feature of SAHARA utility functions is that they give closedform convex dual U(y) , supx∈R (U(x)− xy) and the inverse marginal utilityI(y) , (Ux)
−1(y). The next proposition exhibits a dynamic strategy π∗t andcorresponding portfolio value X∗t with decreasing allocation to the risky assetover time and the corresponding forward utility at t for which it is an optimalstrategy.
Proposition 4.1 (General Case). Consider an investment strategy
π∗t =λtσt
√a2(X∗t )2 + b2e−a2At , (9)
then π∗t gives the following portfolio value at time t
X∗t =b
ae−
12a2At sinh
(ln
(a
bx+
√a2
b2x2 + 1
)+ aAt + aMt
), x ∈ R, (10)
where a > 0, b > 0, At =∫ t0λ2sds, and Mt =
∫ t0λsdWs. Then the correspond-
ing forward utility Ut(x) = u(x,At) where u(x, s) is defined over R× [0,∞)and given by
11
• If a = 1,
u(x, s) =1
2ln(x+
√x2 + b2e−s) +
es
2b2x(√x2 + b2e−s − x)− s
4. (11)
• If a 6= 1,
u(x, s) = e12(1−a)sa(a+ 1)(ax2 + x
√a2x2 + b2e−a2s) + b2e−a
2s
(a2 − 1)(ax+√a2x2 + b2e−a2s)1+
1a
. (12)
Proof. See Appendix A.
We observe that at any time t, the forward utilities obtained in (11) and(12) are SAHARA utilities with varying parameters. Proposition 4.2 makesthis fact clearer and is needed to discuss the risk aversion of the forwardutility naturally.
Proposition 4.2. Given any time T > 0, consider the following portfoliooptimization problem to maximize the utility of wealth at time T
maxπ∈A
E (u(XπT , AT )|Xπ
0 = x) ,
where AT =∫ T0λ2sds and where u(·, ·) is defined over R × [0,∞) and given
by (11) and (12) for a > 0 and b > 0. Then the optimal allocation is givenby (9).
Proof. See Appendix A.
Let us consider a Black-Scholes market model (with constant µ, r, andσ), then we have the following corollary of Propositions 4.1 and 4.2.
Corollary 4.1 (Black-Scholes Case). Assume that µ, r, and σ are constant.Then, we have the following results.
• The following investment strategy
π∗t =λ
σ
√a2(X∗t )2 + b2e−a2λ2t, (13)
in the risky asset gives the portfolio value at time t below
X∗t =b
ae−
12a2λ2t sinh
(ln
(a
bx+
√a2
b2x2 + 1
)+ aλ2t+ aλWt
), x ∈ R,
where a > 0 and b > 0. Moreover, we have a corresponding forwardutility Ut(x) = u(x, λ2t) where u(x, s) is given by (11) and (12).
12
• Reciprocally, given any time T > 0, consider the following portfoliooptimization problem to maximize the utility of wealth at time T
maxπ∈A
E(u(Xπ
T , λ2T )|Xπ
0 = x),
where u(·, ·) is defined over R× [0,∞) and given by (11) and (12) fora > 0 and b > 0. Then the optimal allocation is given by (13).
4.2 Link to life-cycle funds
Let us define the local (absolute) risk aversion function
γ(x, s) , −uxx(x, s)/ux(x, s) (14)
from the forward utility u(·, ·), where u(·, ·) is given by (11) and (12). Then,we have
γ(x, s) =1√
a2x2 + b2e−a2s. (15)
It is easily seen that the local risk aversion function (15) is an increasingfunction of s. If there is an economic agent with a SAHARA utility function,her optimal investment strategy becomes more conservative as time goesbecause her local risk aversion is an increasing function of time s. As aconsequence, the optimal allocation to the risky asset (13) is a decreasingfunction of time. So the investment strategy (13) can be seen as a life-cyclefund for an agent whose preference is given by SAHARA utility function.
However, note that for the HARA utility as in (5) the local risk aversion(14) is equal to γ(x, s) = 1
ax(with G = 0). Asymptotically the local risk
aversion of the SAHARA utility converges to the local risk aversion of theHARA utility (for a = γ in (5)). In terms of strategy, it can be seen asthe optimal allocation of a SAHARA utility maximizer becomes closer to aCPPI strategy (with G = 0) but not really to the 100% bond strategy whichis typically seen in a life-cycle fund. We leave for future research to determinethe exact utility which would explain for instance a time declining parameterm in the CPPI to an ultimate value of 0 at the horizon.
The optimal strategy (9) in the general case shares similar features, butwe have to rebalance the investment taking into account λt and σt becausethe market is stochastic. This is consistent with Viceira (2007) who suggestedthat the market conditions should be involved in determining the asset allo-cation path of life-cycle funds. The standard life-cycle funds, consisting ofa linear decrease of the percentage invested in risky asset does not appearoptimal. The way to decrease the allocation over time, depends on changesin market conditions and risk aversion.
13
5 Conclusion and Future Research Directions
The study of two popular dynamic investment strategies in the pension fundsindustry shows that the HARA and SAHARA utilities may play a key role inexplaining fund manager’s decisions or in modeling optimal decision making.Future research directions include proving the existence and giving an explicitconstruction of the forward utility for any type of investment strategies. Inparticular, this would help in explaining how caps and floors (that are oftenembedded in pension products) may appear optimal.
Acknowledgement
C. Bernard acknowledges support from the Natural Sciences and EngineeringResearch Council of Canada, from the Society of Actuaries Centers of Ac-tuarial Excellence Research Grant and from the Humboldt foundation. M.Kwak acknowledges that this work was mostly carried out at the Universityof Waterloo as a postdoctoral fellow.
14
A Proofs of Propositions
Proof of Proposition 3.1. First of all, let us show that the dynamic CPPIinvestment strategy (3) gives the portfolio value (4). Let X∗t = Xπ∗
t be theportfolio value at time t with dynamic CPPI investment strategy (3) andinitial portfolio value X∗0 = x > G, then X∗t satisfies
dX∗t = (X∗t −G)(γλ2tdt+ λtdWt), X∗0 = x > G,
where γ = σ0m/λ0. Let us define Xt , X∗t − G, then Xt is a geometricBrownian motion which satisfies
dXt = Xt(γλ2tdt+ γλtdWt), X0 = x−G > 0.
Hence, Xt is given by
Xt = X0 exp
((γ − 1
2γ2)∫ t
0
λ2sds+ γ
∫ t
0
λsdWs
), X0 = x−G > 0,
and equivalently, X∗t is given by (4).Now we show that Ut(x) = u(x,At), where u(x, s) is given by (5) satisfies
the conditions to be a forward utility. It is easily seen that u(x, s) satisfiesux(x, s) > 0 and uxx(x, s) < 0 for x ∈ (G,∞) and s ≥ 0. Since Ut(x) isconstructed by Ut(x) = u(x,At), we conclude that x → Ut(x) is strictlyconcave and increasing. For each strategy π ∈ A, by applying Ito’s formulato Ut(X
πt ) = u(Xπ
t , At), we have
dUt(Xπt ) =ux(X
πt , At)σtπtdWt
+ λ2t
[ut(X
πt , At) + ux(X
πt , At)αt +
1
2uxx(X
πt , At)α
2t
]dt,
where αt , σtπt/λt. Note that u(x, s) in (5) satisfies
us(x, s) =u2x(x, s)
2uxx(x, s),
for x ∈ (G,∞) and s ≥ 0. Since uxx(x, t) < 0 and u(x, t) satisfies ut(x, t) =u2x(x, t)/2uxx(x, t), we have
ut(Xπt , At) + ux(X
πt , At)αt +
1
2uxx(X
πt , At)α
2t
=1
2uxx(X
πt , At)
(αt +
ux(Xπt , At)
uxx(Xπt , At)
)2
+ ut(Xπt , At)−
u2x(Xπt , At)
2uxx(Xπt , At)
≤ ut(Xπt , At)−
u2x(Xπt , At)
2uxx(Xπt , At)
= 0,
15
and this means that the drift of dUt(Xπt ) is non-positive. Hence we conclude
that
Es (Ut(Xπt )) ≤ Us(X
πs ), (16)
for 0 ≤ s ≤ t. It remains to show that (16) holds as equality with thedynamic CPPI strategy (3). For γ ∈ (0, 1)∪ (1,∞), Ut(X
∗t ) = u(X∗t , At) can
be written as
Ut(X∗t ) =
γ
γ − 1(x−G)
γ−1γ exp
(−1
2
∫ t
0
(γ − 1)2λ2sds+
∫ t
0
(γ − 1)λsdWs
).
By applying Ito’s formula to Ut(X∗t ), we have
dUt(X∗t ) = γ(x−G)
γ−1γ exp
(−1
2
∫ t
0
(γ − 1)2λ2sds+
∫ t
0
(γ − 1)λsdWs
)λtdWt,
and this means that the drift of dUt(X∗t ) is zero. For γ = 1, Ut(X
∗t ) can be
written as
Ut(X∗t ) = ln(x−G) +
∫ t
0
λsdWs.
Thus, for any γ > 0, it is concluded that
Es (Ut(X∗t )) = Us(X
∗s ),
for 0 ≤ s ≤ t and this completes the proof.
Proof of Proposition 3.2. In the proof of Proposition 3.1, it is proved thatUt(x) = u(x,At) with u(x, s) defined in (5) is a forward utility. Therefore,by definition of forward utility, we have
E(UT (XπT )|Xπ
0 = x) ≤ U0(Xπ0 |Xπ
0 = x) = U0(x) (17)
for any admissible strategy π ∈ A. However, we also proved in the proof ofProposition 3.1 that the dynamic CPPI strategy π∗ in (3) satisfies
Es(Ut(X
π∗
t ))
= Us(Xπ∗
s ),
for 0 ≤ s ≤ t. By choosing s = 0 and t = T , we have
E(UT (Xπ∗
T )|Xπ∗
0 = x) = U0(Xπ∗
0 |Xπ∗
0 = x) = U0(x). (18)
From (17) and (18), we obtain
E(u(XπT , AT )|Xπ
0 = x) = E(UT (XπT )|Xπ
0 = x)
≤ E(UT (Xπ∗
T )|Xπ∗
0 = x) = E(u(Xπ∗
T , AT )|Xπ∗
0 = x).
Taking the supremum over all admissible strategies in A directly shows thatπ∗ is the optimal strategy.
16
Proof of Proposition 4.1. Instead of starting from the strategy π∗ as we didin the proof of Proposition 3.1, let us apply Ito’s formula to X∗t in (10) first.Then it can be verified that X∗t satisfies
dX∗t =be−12a2At cosh
(ln
(a
bx+
√a2
b2x2 + 1
)+ aAt + aMt
)(λ2tdt+ λtdWt)
=√a2(X∗t )2 + b2e−a2At(λ2tdt+ λtdWt) = σtπ
∗t (λtdt+ dWt),
where π∗t is given by (9). This implies that π∗t in (9) gives the portfolio valueX∗t in (10), that is X∗t = Xπ∗
t .It remains to show that Ut(x) = u(x,At), where u(x, s) is given by (11)
and (12) satisfies the conditions to be a forward utility. For each a > 0, letus define a function y(x, s; a) by
y(x, s; a) =a
be
12a2sx+
√a2
b2ea2sx2 + 1,
for x ∈ R and s ≥ 0. Then it can be verified that u(x, s) given by (11), fora = 1, can be written as
u(x, s) =1
2ln y(x, s; 1)− 1
4y(x, s; 1)2− 1
2s+
1
2ln b+
1
4,
and u(x, s) given by (12), for a 6= 1, can be written as
u(x, s) =b1−
1a
ae−
12(a2−1)s
(a
2(a− 1)y(x, s; a)
a−1a − a
2(a+ 1)y(x, s; a)−
a+1a
).
We observe that, for all a > 0, ux(x, s), uxx(x, s), and us(x, s) can be writtenas
ux(x, s) =b−1a e
12ty(x, s; a)−
1a , (19)
uxx(x, s) =− 2b−1−1a e(
12a2+ 1
2)t y(x, s; a)1−
1a
y(x, s; a)2 + 1, (20)
ut(x, s) =− 1
4b1−
1a e(−
12a2+ 1
2)ty(x, s; a)−1−
1a (y(x, s; a)2 + 1). (21)
Since y(x, s; a) is always positive and b > 0, it is easily verified that ux(x, s) >0 and uxx(x, s) < 0 for all a > 0 from (19) and (20). So we conclude thatx → Ut(x) is strictly concave and increasing for all a > 0. From (19), (20),and (21), we observe that u(x, s) satisfies us(x, s) = u2x(x, s)/2uxx(x, s) for
17
all a > 0, x ∈ R, and s ≥ 0. In the proof of Proposition 3.1, we have alreadyseen that this implies that
Es (Ut(Xπt )) ≤ Us(X
πs ),
for any π ∈ A. Now it is enough to show that, for 0 ≤ s ≤ t,
Es (Ut(X∗t )) = Us(X
∗s ), (22)
where X∗t is given by (10). For all a > 0, it can be seen that
a
be
12a2AtX∗t = sinh
(ln
(a
bx+
√a2
b2x2 + 1
)+ aAt + aMt
)and√
a2
b2ea2At(X∗t )2 + 1 = cosh
(ln
(a
bx+
√a2
b2x2 + 1
)+ aAt + aMt
).
Thus we have
y(X∗t , At; a) =a
be
12a2AtX∗t +
√a2
b2ea2At(X∗t )2 + 1
=
(a
bx+
√a2
b2x2 + 1
)eaAt+aMt . (23)
Using (23), we observe that Ut(X∗t ) = u(X∗t , At) can be written as, for a = 1,
Ut(X∗t ) =
1
2Mt + k1e
−2At−2Mt + k2, (24)
and for a 6= 1,
Ut(X∗t ) = k3e
− 12(a−1)2At+(a−1)Mt − k4e−
12(a+1)2At−(a+1)Mt , (25)
where k1, k2, k3, and k4 are given by
k1 =1
4
(1
bx+
√1
b2x2 + 1
)−2,
k2 =1
2ln
(1
bx+
√1
b2x2 + 1
)+
1
2ln b+
1
4,
k3 =b1−
1a
2(a− 1)
(a
bx+
√a2
b2x2 + 1
)a−1a
,
k4 =b1−
1a
2(a+ 1)
(a
bx+
√a2
b2x2 + 1
)−a+1a
.
18
Note that k1, k2, k3, and k4 are constant for given a > 0, b > 0, and x ∈ R.By applying Ito’s formula to (24) and (25), we have, for a = 1,
dUt(X∗t ) =
(1
2− 2k1e
−2At−2Mt
)λtdWt, (26)
and for a 6= 1,
dUt(X∗t ) =
(k3(a− 1)e−
12(a−1)2At+(a−1)Mt + k4(a+ 1)e−
12(a+1)2At−(a+1)Mt
)λtdWt.
(27)
In (26) and (27), we observe that the drift of dUt(X∗t ) is zero for all a > 0.
Thus, for any a > 0, it is concluded that (22) holds for 0 ≤ s ≤ t and thiscompletes the proof.
Proof of Proposition 4.2. The proof follows from a similar argument to thatin Proposition 3.2.
B Derivation of SAHARA utility function in
Section 4
In Chen et al. (2011), absolute risk aversion is given by
A(x) =α√
β2 + x2.
So α = 1/a and β = b/a for a > 0 and b > 0 in our model.If α = 1, U(x) is given by
U(x) =1
2ln(x+
√β2 + x2) +
1
2β−2x(
√β2 + x2 − x)
in Proposition 2.2 of Chen et al. (2011). Since β = c, it is straightforwardthat U(x) = U(x) where U(x) is given by (7).
If α 6= 1, U(x) is given by
U(x) = − 1
α2 − 1
(x+
√β2 + x2
)−α (x+ α
√β2 + x2
)in Proposition 2.2 of Chen et al. (2011). Using α = 1/a and β = b/a, U(x)
19
can be rewritten as follows,
U(x) =a2
a2 − 1
(x+
√b2
a2+ x2
)− 1a(x+
1
a
√b2
a2+ x2
)
=a
1a
a2 − 1
a(a+ 1)(ax2 + x
√a2x2 + b2
)+ b2(
ax+√a2x2 + b2
) 1a+1
= a1aU(x),
where U(x) is given in (8).
20
References
Annaert, J., Osselaer, S., & Verstraate, B. (2009). Performance evaluation ofportfolio insurance strategies using stochastic dominance criteria. Journalof Banking and Finance, 33 , 272–280.
Basak, S. (2002). A comparative study of portfolio insurance. Journal ofEconomic Dynamics and Control , 26 , 1217–1241.
Benninga, S., & Blume, M. (1985). On the optimality of portfolio insurance.Journal of Finance, 40 , 1341–1352.
Bernard, C., & Le Courtois, O. (2012). Asset risk management of partici-pating contracts. Asia-Pacific Journal of Risk and Insurance, 6 .
Berrier, F., Rogers, L., & Tehranchi, M. (2010). A characterization of forwardutility functions. Preprint , .
Bertrand, P., & Prigent, J.-L. (2005). Portfolio insurance strategies: OBPIversus CPPI. Finance, 26 , 5–32.
Black, F., & Perold, A. (1992). Theory of constant proportion portfolioinsurance. Journal of Economic Dynamics and Control , 16 , 403–426.
Boyle, P., & Tian, W. (2009). Optimal design of equity-linked products witha probabilistic constraint. Scandinavian Actuarial Journal , 4 , 253–280.
Cairns, A., Blake, D., & Dowd, K. (2006). Stochastic lifestyling: Optimaldynamic asset allocation for defined contribution pension plans. Journalof Economic Dynamics and Control , 30 , 843–877.
Chen, A., Pelsser, A., & Vellekoop, M. (2011). Modeling non-monotone riskaversion using sahara utility functions. Journal of Economic Theory , 146 ,2075–2092.
Gomes, F., Kotlikoff, L., & Viceira, L. M. (2008). Optimal life-cycle investingwith flexible labor supply: A welfare analysis of life-cycle funds. AmericanEconomic Review: Papers & Proceedings , 98 , 297–303.
Graf, S. (2012). A note on life-cycle funds. Working paper , .
Graf, S., Kling, A., & Russ, J. (2012). Financial planning and risk-returnprofiles. European Actuarial Journal , 2 , 77–104.
Musiela, M., & Zariphopoulou, T. (2009). Portfolio choice under dynamicsinvestment performance criteria. Quantitative Finance, 9 , 161–170.
21
Musiela, M., & Zariphopoulou, T. (2010). Portfolio choice under space-timemonotone performance criteria. SIAM Journal on Financial Mathematics ,1 , 326–365.
Musiela, M., & Zariphopoulou, T. (2011). Initial investment choice optimalfuture allocation under time-monotone performance criteria. InternationalJournal of Theoretical and Applied Finance, 14 , 61–81.
Pang, G., & Warshawsky, M. (2008). Default investment options in definedcontribution plans: A quantitative comparison. Pensions: An Interna-tional Journal , 13 , 221–226.
Pezier, J., & Scheller, J. (2011). Optimal investment strategies and perfor-mance sharing rules for pension schemes with minimum guarantee. Journalof Pension Economics and Finance, 10 , 119–145.
Pezier, J., & Scheller, J. (2013). Best portfolio insurance for long-term in-vestment strategies in realistic conditions. Insurance: Mathematics andEconomics , 52 , 263–274.
Prigent, J.-L. (2007). Portfolio Optimization and Performance Analysis .Chapman and Hall.
Viceira, L. M. (2007). Life-cycle funds. Working paper , .
Zagst, R., & Kraus, J. (2011). Stochastic dominance of portfolio insurancestrategies obpi versus cppi. Annals of Operations Research, 185 , 75–103.
22
