ARCH 2014.1 Proceedings July 31-August 3, 2013 On improving pension - PDF document

Article from: ARCH 2014.1 Proceedings July 31-August 3, 2013

On improving pension product design Agnieszka K. Konicz 1 and John M. Mulvey 2 1 Technical University of Denmark, DTU Management Engineering, Management Science 2 Princeton University, Department of Operations Research and Financial Engineering, Bendheim Center for Finance ARC, Philadelphia August 2, 2013

On improving pension product design (jg Focus on DC pension plans: ◮ Quickly expanding, ◮ Easier and cheaper to administer, ◮ More transparent and flexible so they can capture individuals’ needs. However, ◮ If too much flexibility (e.g. U.S.), the participants do not know how to manage their saving and investment decisions. ◮ If too little flexibility (e.g. Denmark), the product is generic and does not capture the individuals’ needs. Agnieszka K. Konicz - Technical University of Denmark 1/9

On improving pension product design (jg Asset allocation, payout profile and level of death benefit capture the individual’s personal and economical characteristics: ◮ current wealth, expected lifetime salary progression, mandatory and voluntary pension contributions, expected state retirement pension, risk preferences, choice of assets, health condition and bequest motive. Combine two optimization approaches: ◮ Multistage stochastic programming (MSP) ◮ Stochastic optimal control (dynamic programming, DP). Agnieszka K. Konicz - Technical University of Denmark 2/9

On improving pension product design (jg Asset allocation, payout profile and level of death benefit capture the individual’s personal and economical characteristics: ◮ current wealth, expected lifetime salary progression, mandatory and voluntary pension contributions, expected state retirement pension, risk preferences, choice of assets, health condition and bequest motive. Combine two optimization approaches: ◮ Multistage stochastic programming (MSP) ◮ Stochastic optimal control (dynamic programming, DP). Agnieszka K. Konicz - Technical University of Denmark 2/9

Optimization approaches (jg stochastic optimal control (DP) - explicit solutions ✪ explicit solution may not exist ✦ ideal framework - produce an optimal policy that is easy to ✪ difficult to solve when dealing with understand and implement details stochastic programming (MSP) - optimization software ✦ general purpose decision model ✪ difficult to understand the solution with an objective function that can ✪ problem size grows quickly as a take a wide variety of forms function of number of periods and ✦ can address realistic considerations, scenarios such as transaction costs ✪ challenge to select a representative ✦ can deal with details set of scenarios for the model Agnieszka K. Konicz - Technical University of Denmark 3/9

Combined MSP and DP approach (jg n 0 , x 0 = 550 Benefits 34.4 Purchases Sales Allocation Cash Bonds 300.6 0.58 Dom. Stocks 177.3 0.34 Int. Stocks 37.7 0.08 n 1 Benefits 31.6 Purchases Sales Allocation Returns Cash 0.030 Bonds 98.8 0.49 -0.039 Dom. Stocks 8.3 0.44 -0.093 Int. Stocks 4.4 0.07 -0.169 Agnieszka K. Konicz - Technical University of Denmark 4/9

Objective (jg Maximize the expected utility of total retirement benefits and bequest given uncertain lifetime, T − 1 Parameters: � � s , B tot � � max s p x u · prob n T R retirement time, s , n end of decision horizon T n ∈N s s =max( t 0 , T R ) and beginning of DP, T − 1 t p x probability of surviving to age x + t � � � s , I tot � given alive at age x , + s p x q x + s Ku · prob n s , n q x mort. rate for an x -year old, s = t 0 n ∈N s probability of being in node n , prob n K weight on bequest motive. � � � � X → + T p x V T , · prob n i , T , n Variables: B tot total benefits at time t , node n , n ∈N T i t , n I tot bequest at time t , node n , t , n X → amount allocated to asset i , i , t , n period t , node n . Richard, S. F. (1975), Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics , 2(2):187–203. Agnieszka K. Konicz - Technical University of Denmark 5/9

Conclusions I (jg Equally fair payout profiles given CRRA utility: γ w 1 − γ u ( t , B t ) = 1 B γ t , w t = e − 1 / (1 − γ ) ρ t t * Optimal benefits, B t 50 * Optimal benefits, B t 40 36 Traditional product B * t : γ =−3, ρ =0.04 1000 EUR 30 34 B * t : γ =−3, ρ =−0.02 32 B * t : γ =−1, ρ =0.04 20 Traditional product, 25 years 1000 EUR B * t : γ =−1, ρ =−0.02 30 B * t : γ =−3, ρ =r, µ t =10 ν t 10 B * t : γ =−3, ρ =0.04, µ t =10 ν t 28 B * t : γ =−3, ρ =−0.02, µ t =10 ν t 26 0 65 70 75 80 85 90 age 24 22 Subjective mortality rate µ t = 10 ν t : 65 70 75 80 85 90 age 30% chances to survive until age 75, Sensitivity to risk aversion 1 − γ < 1% chance to survive until age 85. and impatience (time preference) factor ρ . � � T � s X t � � 1 γ a ∗ e − ¯ r +¯ µ τ d τ ds , B ∗ ¯ r = 1 − γ ρ − 1 − γ r ¯ = t = t y + t a ∗ ¯ , 1 γ µ τ = ¯ 1 − γ µ τ − 1 − γ ν τ t y + t ���� ���� subj . obj . Savings upon retirement X TR = 550 , 000 EUR, b state = 0, risk-free investment, no insurance. TR Agnieszka K. Konicz - Technical University of Denmark 6/9

Conclusions II (jg More aggressive investment strategy and higher benefits given state retirement pension b state T R b state b state = 0 = 5 TR TR Expected asset allocation \ Age 65-90 65 70 75 80 85 90 Cash 20% 4% 5% 6% 7% 7% 7% Bonds 44 53 52 52 51 51 51 Dom. Stocks 25 30 30 29 29 29 29 Int. Stocks 11 13 13 13 13 13 13 Expected benefits \ Age 65 70 75 80 85 90 t , b state Benefits B ∗ = 0 32,7 34,8 36,9 39,1 41,5 44,1 TR t , b state Benefits B ∗ = 5 34,4 36,5 38,7 41,1 43,6 46,3 TR * Optimal benefits based on historical data, B * Expected optimal benefits, B t t 65 65 Traditional product Traditional product B * t : γ =−3, ρ =0.0189 * , γ =−3, ρ =0.0189 B t 55 B * t : γ =−3, ρ =0.04 55 * , γ =−3, ρ =0.04 B t B * t : γ =−3, ρ =−0.02 * , γ =−3, ρ =−0.02 B t 1000 EUR 45 1000 EUR 45 35 35 25 25 15 15 65 70 75 80 85 90 2000 2002 2004 2006 2008 2010 2012 age year Left plot: expected optimal benefits. Right plot: optimal benefits based on historical returns: 3-m U.S. T-Bills, Barclays Agg. Bond, S&P500, MSCI EAFE. Agnieszka K. Konicz - Technical University of Denmark 7/9

Conclusions III (jg Possible to adjust the investment strategy such that B tot ∗ ≥ b min t t i X → i , t , n ≥ x min Possible to adjust the investment strategy such that � t (a) immediate annuity, age 0 = 65, x 0 = 550, b state = 5 TR Optimal benefits, B tot* Optimal benefits, B tot* 85 85 75 75 65 65 1000 EUR 1000 EUR 55 55 45 45 35 35 25 25 15 15 65 66 69 72 T=75 65 66 69 72 T=75 age age (b) deferred annuity, age 0 = 45, x 0 = 130, l 0 = 50, p fixed = 15%, p vol = 10% (right plot only), b state = 5, ins fixed = 150 TR 0 140 140 120 120 Number of scenarios 100 Number of scenarios 100 80 80 60 60 40 40 20 20 0 0 200 400 600 800 1000 1200 1400 1600 200 400 600 800 1000 1200 1400 1600 Savings at retirement Savings at retirement Agnieszka K. Konicz - Technical University of Denmark 8/9

Conclusions IV (jg Possible to include individual’s preferences on portfolio composition, � � X i , t , n ≥ d i X i , t , n , X i , t , n ≤ u i X i , t , n i i e.g. d bonds = 50% and u bonds = 70%. Though any additional constraints lead to a suboptimal solution (= ⇒ lower of more volatile benefits). Optimal investment vs. optimal fixed-mix portfolio: Optimal asset allocation Cash Cash 1 1 Bonds Bonds Dom. Stocks Dom. Stocks Int. Stocks Int. Stocks 0.8 0.8 0.6 0.6 0.4 0.4 0.2 0.2 0 0 45 50 55 60 T=65 70 75 45 50 55 60 T=65 70 75 age age Deferred life annuity. 20% lower expected benefits given the same risk level. Left: optimal investment, E [ B tot ∗ ] = 46 , 200 EUR. Right: fixed-mix portfolio, E [ B tot ∗ ] = 37 , 700 EUR. t t Agnieszka K. Konicz - Technical University of Denmark 9/9

Selected references (jg Høyland, K., Kaut, M., and Wallace, S. W. (2003). A Heuristic for Moment-Matching Scenario Generation. Computational Optimization and Applications , 24(2-3):169–185. Kim, W. C., Mulvey, J. M., Simsek, K. D., and Kim, M. J. (2012). Stochastic Programming. Applications in Finance, Energy, Planning and Logistics , chapter Papers in Finance: Longevity risk management for individual investors. World Scientific Series in Finance: Volume 4. Konicz, A. K., Pisinger, D., Rasmussen, K. M., and Steffensen, M. (2013). A combined stochastic programming and optimal control approach to personal finance and pensions. http://www.staff.dtu.dk/agko/Research/~/media/agko/konicz_combined.ashx . Milevsky, M. A. and Huang, H. (2011). Spending retirement on planet Vulcan: The impact of longevity risk aversion on optimal withdrawal rates. Financial Analysts Journal , 67(2):45–58. Mulvey, J. M., Simsek, K. D., Zhang, Z., Fabozzi, F. J., and Pauling, W. R. (2008). Assisting defined-benefit pension plans. Operations research , 56(5):1066–1078. Richard, S. F. (1975). Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics , 2(2):187–203. Agnieszka K. Konicz - Technical University of Denmark 10/9