Dynamic Preferences for Popular Investment Strategies in Pension Funds Carole Bernard and Minsuk Kwak Paris, June 2013 Bernard Carole (University of Waterloo) June 2013 1 / 24
Outline Motivation & Contributions 1 Dynamic preferences: “Forward utility” 2 Dynamic Preferences for CPPI 3 Dynamic Preferences for Life-cycle Funds 4 Conclusions 5 Bernard Carole (University of Waterloo) June 2013 2 / 24
Motivation Utility function The way we measure satisfaction from consumption or wealth Increasing function : economic agent prefers a higher level of consumption or wealth to lower one. Concave function : marginal utility is decreasing Classical optimal portfolio choice problem Choose a utility function ⇒ Find the optimal investment strategy Opposite way Given an investment strategy ⇒ Infer the utility for it to be optimal? Bernard Carole (University of Waterloo) June 2013 3 / 24
Contributions Infer the utility for a dynamic strategy: ◮ no specific horizon ◮ the type of strategy is associated to a class of utility. ◮ the parameters of the strategy are related to the risk aversion level. Work specifically on 2 examples CPPI strategies and Life Cycle Funds A standard CPPI strategy is optimal in a Black-Scholes model for HARA utility but it needs to have a dynamically updated multiple to be optimal for a HARA utility in a more general market. Some type of life-cycle funds can be optimal for the SAHARA utility (optimality of a decreasing proportion in risky asset over time). However, a constant decrease over time may not be optimal. Bernard Carole (University of Waterloo) June 2013 4 / 24
Strategy ⇒ Utility : Literature Review Similar perspective, but different approach Dybvig and Rogers (1997) : “Recovery of Preferences from Observed Wealth in a Single Realization” Cuoco and Zapatero (2000) : “On the Recoverability of Preferences and Beliefs” Cox, Hobson, and Obloj. (2012) : “Utility Theory Front to Back - Inferring Utility from Agents’ Choices” Bernard, Chen, Vanduffel (2013): “All Investors are Risk Averse Expected Utility Maximizers” Forward investment performance or Forward utility Musiela and Zariphopoulou (2009, 2010, 2011) Berrier, Rogers, and Tehranchi. (2010) Bernard Carole (University of Waterloo) June 2013 5 / 24
Outline Forward Utility Define “Forward Utility” 1 Illustrate Key Idea to find the forward utility 2 CPPI Strategy Introduce CPPI strategy 1 Find the corresponding “Forward Utility” (which is a HARA utility at 2 fixed time) corresponds to CPPI strategy Life-Cycle Funds Introduce Life-Cycle Funds 1 Introduce SAHARA utility 2 Find the corresponding “Forward Utility” (which is a SAHARA 3 utility at fixed time) and corresponding investment strategy which is a kind of Life-Cycle Funds Bernard Carole (University of Waterloo) June 2013 6 / 24
Financial Market & Portfolio Value Process One-dimensional market with two assets: a risky asset S t and a risk-free bond B t dS t = S t ( µ t dt + σ t dW t ) , S 0 > 0 , dB t = r t B t dt , B 0 = 1 , r t , µ t and σ t may be stochastic but are adapted to the filtration F t Market price of risk (or instantaneous Sharpe ratio) λ t � µ t − r t σ t eraire. Then, X π Risk-free bond B t is used as num´ t : present value(value at time 0) of the portfolio at time t , with strategy π X π t = π 0 t + π t ◮ π 0 t amount invested in the risk-free asset B t ◮ π t amount invested in the risky asset S t . Since B t is used as num´ eraire, d π 0 dX π t = 0 , t = d π t = π t [( µ t − r t ) dt + σ t dW t ] = σ t π t ( λ t dt + dW t ) . Bernard Carole (University of Waterloo) June 2013 7 / 24
Definition of Forward Utility Definition 2.1 (Forward utility) An F t -adapted process U t ( x ) is a “Forward utility” if : x → U t ( x ) is strictly concave and increasing 1 for each π ∈ A (i.e. for each attainable X π s ), and t ≥ s, 2 E [ U t ( X π t ) |F s ] ≤ U s ( X π s ) , there exists π ∗ ∈ A , for which for all t ≥ s, 3 E [ U t ( X π ∗ ) |F s ] = U s ( X π ∗ s ) , t for t ≥ 0 and x ∈ D where D is an interval of R Bernard Carole (University of Waterloo) June 2013 8 / 24
Explanation for the Definition of Forward Utility For a fixed t , x → U t ( x ) is a concave, increasing function. For some T > 0, let us define v ( x , t ) as E [ U T ( X π T ) |F t , X π v ( x , t ) � sup t = x ] (1) π ∈A where U t ( x ) is a forward utility defined in the previous page. Let π ∈ A and π ∗ is the optimum. Then, by dynamic programming principle, ( v ( X π s , s )) s : Supermartingale for each π ( v ( X π ∗ s , s )) s : Martingale for π ∗ Under some conditions, we can prove that v ( x , t ) = U t ( x ) , 0 ≤ t ≤ T . ⇒ This is why the forward utility is defined as in the previous page! Bernard Carole (University of Waterloo) June 2013 9 / 24
Musiela and Zariphopoulou (2009, 2010, 2011) Musiela and 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 the optimality of a dynamic strategy. This forward utility is formulated as U t ( x ) = u ( x , A t ) (2) � t 0 λ 2 where A t � s ds , t ≥ 0. ⇒ We show how their work can be applied to understand CPPI strategies and life-cycle funds. Bernard Carole (University of Waterloo) June 2013 10 / 24
Key Idea to find forward utilities For each strategy π ∈ A , assume that U t ( X π t ) = u ( X π t , A t ) . By applying Itˆ o’s formula, we have dU t ( X π t ) = u x ( X π t , A t ) σ t π t dW t (3) � � t , A t ) α t + 1 + λ 2 u t ( X π t , A t ) + u x ( X π 2 u xx ( X π t , A t ) α 2 dt , t t where α t � σ t π t /λ t . Goal For each strategy π ∈ A , non-positive drift of U t ( X π t ) t , A t ) α t + 1 u t ( X π t , A t ) + u x ( X π 2 u xx ( X π t , A t ) α 2 t ≤ 0 For optimal strategy π ∗ , zero drift of U t ( X π ∗ ) t , A t ) α t + 1 u t ( X π ∗ , A t ) + u x ( X π ∗ 2 u xx ( X π ∗ , A t ) α 2 t = 0 t t t Bernard Carole (University of Waterloo) June 2013 11 / 24
CPPI Strategy (1) Constant Proportion Portfolio Insurance Introduced by Black and Perold (1992) Key feature : at any time... Value of portfolio ≥ Predefined floor level Good way to hedge long-term guarantees when ◮ the maturity date is not known in advance ◮ regulators require the guarantee to be met at all times Popular in the insurance industry to manage pension funds and variable annuities Bernard Carole (University of Waterloo) June 2013 12 / 24
CPPI Strategy (2) G t > 0: predefined floor level. Assume that dG t = G t r t dt , G 0 = G . ⇒ G t = GB t . V t : portfolio value at time t C t = V t − G t : cushion Define X t = V t / B t , the present value of V t , then C t = X t − G . B t Maintain an exposure to the risky asset S t proportional to the cushion. ( m : multiple) π t = m C t = m ( X t − G ) (4) B t The amount of risk-free asset is therefore at all times π 0 t = X t − π t . Bernard Carole (University of Waterloo) June 2013 13 / 24
Adapted Random Multiple To ensure that the CPPI strategy is optimal for an expected utility maximizer at any time horizon in the general market (stochastic parameters), we consider a slightly generalized CPPI strategy with random multiple m t = λ t /λ 0 m , π t = m t ( X t − G ) (5) σ t /σ 0 At any time t , m t is adapted to F t , the information available. In the case of a Black-Scholes model (constant parameters), π t = m t ( X t − G ) corresponds to a standard CPPI strategy with fixed multiple m π t = m ( X t − G ) because both λ t and σ t are constant. Bernard Carole (University of Waterloo) June 2013 14 / 24
Proposition 2.1 (General Case) The dynamic CPPI investment strategy consisting of t = λ t /λ 0 π ∗ m ( X ∗ t − G ) (6) σ t /σ 0 invested in the risky asset (i.e. a CPPI strategy with an adapted multiple λ t /λ 0 σ t /σ 0 m) corresponds to the optimum for the forward utility U t ( x ) = u ( x , A t ) where u ( x , s ) is given for x ∈ ( G , ∞ ) and s ≥ 0 by γ − 1 γ e − γ − 1 γ 2 s , γ − 1 ( x − G ) γ ∈ ( 0 , 1 ) ∪ ( 1 , ∞ ) , u ( x , s ) = (7) ln ( x − G ) − s 2 , γ = 1 . � t 0 λ 2 where γ = σ 0 m /λ 0 and A t � s ds. ⇒ The forward utility u ( · , s ) belongs to the HARA utility class at all s . Bernard Carole (University of Waterloo) June 2013 15 / 24
Proposition 2.2 Reciprocally, given any time T, consider the following portfolio optimization problem to maximize the utility of wealth at time T max π ∈A E [ u ( X T , A T )] , � T 0 λ 2 where A T = s ds and u ( · , · ) is given by (7) and defined over ( G , ∞ ) × [ 0 , ∞ ) . Then the optimal allocation is a dynamic CPPI strategy t = λ t /λ 0 π ∗ m ( X ∗ t − G ) . σ t /σ 0 This proposition holds for any given time T with u ( X T , A T ) . ⇒ Forward utility: Dynamically consistent utility functions! We have to rebalance the investment strategy depending on λ t and σ t in stochastic environment. (Dynamically changing investment opportunity) Bernard Carole (University of Waterloo) June 2013 16 / 24
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