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ARCH 2014.1 Proceedings
July 31-August 3, 2013
On improving pension product design
Agnieszka K. Konicz1 and John M. Mulvey2
1 Technical University of Denmark, DTU Management Engineering, Management Science2 Princeton University, Department of Operations Research and Financial Engineering,
Bendheim Center for Finance
ARC, PhiladelphiaAugust 2, 2013
On improving pension product design(jg
Focus on DC pension plans:I Quickly expanding,
I Easier and cheaper to administer,
I More transparent and flexible so they can capture individuals’ needs.
However,I If too much flexibility (e.g. U.S.), the participants do not know how to
manage their saving and investment decisions.
I If too little flexibility (e.g. Denmark), the product is generic and doesnot 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 theindividual’s personal and economical characteristics:
I current wealth, expected lifetime salary progression, mandatory andvoluntary pension contributions, expected state retirement pension, riskpreferences, choice of assets, health condition and bequest motive.
Combine two optimization approaches:I Multistage stochastic programming (MSP)
I 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 theindividual’s personal and economical characteristics:
I current wealth, expected lifetime salary progression, mandatory andvoluntary pension contributions, expected state retirement pension, riskpreferences, choice of assets, health condition and bequest motive.
Combine two optimization approaches:I Multistage stochastic programming (MSP)
I Stochastic optimal control (dynamic programming, DP).
Agnieszka K. Konicz - Technical University of Denmark 2/9
Optimization approaches(jg
stochastic optimal control (DP) - explicit solutions
! ideal framework - produce anoptimal policy that is easy tounderstand and implement
% explicit solution may not exist
% difficult to solve when dealing withdetails
stochastic programming (MSP) - optimization software
! general purpose decision modelwith an objective function that cantake a wide variety of forms
! can address realistic considerations,such as transaction costs
! can deal with details
% difficult to understand the solution
% problem size grows quickly as afunction of number of periods andscenarios
% challenge to select a representativeset of scenarios for the model
Agnieszka K. Konicz - Technical University of Denmark 3/9
Combined MSP and DP approach(jg
n0, x0 = 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
n1 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 uncertainlifetime,
maxT−1∑
s=max(t0,TR )
∑n∈Ns
spxu(s,B tot
s,n
)· probn
+T−1∑s=t0
∑n∈Ns
spx qx+sKu(s, I tot
s,n
)· probn
+ Tpx
∑n∈NT
V
(T ,∑
i
X→i,T ,n
)· probn
Parameters:
TR retirement time,
T end of decision horizon
and beginning of DP,
t px probability of surviving to age x + t
given alive at age x ,
qx mort. rate for an x-year old,
probn probability of being in node n,
K weight on bequest motive.
Variables:
Btott,n total benefits at time t, node n,
I tott,n bequest at time t, node n,
X→i,t,n amount allocated to asset i ,
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:
u(t,Bt) = 1γw
1−γt Bγt , wt = e−1/(1−γ)ρt
65 70 75 80 85 9022
24
26
28
30
32
34
36
age
1000
EU
R
Optimal benefits, Bt*
Traditional product
B*t: γ=−3, ρ=0.04
B*t: γ=−3, ρ=−0.02
B*t: γ=−1, ρ=0.04
B*t: γ=−1, ρ=−0.02
Sensitivity to risk aversion 1− γand impatience (time preference) factor ρ.
65 70 75 80 85 900
10
20
30
40
50
age
1000
EU
R
Optimal benefits, Bt*
Traditional product, 25 years
B*t: γ=−3, ρ=r, µ
t=10ν
t
B*t: γ=−3, ρ=0.04, µ
t=10ν
t
B*t: γ=−3, ρ=−0.02, µ
t=10ν
t
Subjective mortality rate µt = 10νt :30% chances to survive until age 75,<1% chance to survive until age 85.
a∗y+t =
∫ T
t
e−∫ s
t
(r+µτ
)dτds, B∗t =
Xt
a∗y+t
,r = 1
1−γ ρ−γ
1−γ r
µτ = 11−γ µτ︸︷︷︸
subj.
− γ1−γ ντ︸︷︷︸
obj.
Savings upon retirement XTR= 550, 000 EUR, bstate
TR= 0, risk-free investment, no insurance.
Agnieszka K. Konicz - Technical University of Denmark 6/9
Conclusions II(jg
More aggressive investment strategy and higher benefits given stateretirement pension bstate
TR
bstateTR
= 0 bstateTR
= 5
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
Benefits B∗t , bstateTR
= 0 32,7 34,8 36,9 39,1 41,5 44,1
Benefits B∗t , bstateTR
= 5 34,4 36,5 38,7 41,1 43,6 46,3
65 70 75 80 85 9015
25
35
45
55
65
age
1000
EU
R
Expected optimal benefits, Bt*
Traditional product
B*t: γ=−3, ρ=0.0189
B*t: γ=−3, ρ=0.04
B*t: γ=−3, ρ=−0.02
2000 2002 2004 2006 2008 2010 201215
25
35
45
55
65
Optimal benefits based on historical data, B*t
year
1000
EU
R
Traditional product
Bt*, γ=−3, ρ=0.0189
Bt*, γ=−3, ρ=0.04
Bt*, γ=−3, ρ=−0.02
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∗t ≥ bmin
t
Possible to adjust the investment strategy such that∑
i X→i,t,n ≥ xmin
t
(a) immediate annuity, age0 = 65, x0 = 550, bstateTR
= 5
65 66 69 72 T=7515
25
35
45
55
65
75
85
age
Optimal benefits, Btot*
1000
EU
R
65 66 69 72 T=7515
25
35
45
55
65
75
85
age
Optimal benefits, Btot*
1000
EU
R
(b) deferred annuity, age0 = 45, x0 = 130, l0 = 50, pfixed = 15%, pvol = 10% (right plot only), bstateTR
= 5, insfixed0 = 150
200 400 600 800 1000 1200 1400 16000
20
40
60
80
100
120
140
Num
ber
of s
cena
rios
Savings at retirement200 400 600 800 1000 1200 1400 16000
20
40
60
80
100
120
140N
umbe
r of
sce
nario
s
Savings at retirement
Agnieszka K. Konicz - Technical University of Denmark 8/9
Conclusions IV(jg
Possible to include individual’s preferences on portfolio composition,
Xi,t,n ≥ di
∑i
Xi,t,n, Xi,t,n ≤ ui
∑i
Xi,t,n
e.g. dbonds = 50% and ubonds = 70%.
Though any additional constraints lead to a suboptimal solution(=⇒ lower of more volatile benefits).
Optimal investment vs. optimal fixed-mix portfolio:
45 50 55 60 T=65 70 750
0.2
0.4
0.6
0.8
1
age
CashBondsDom. StocksInt. Stocks
45 50 55 60 T=65 70 750
0.2
0.4
0.6
0.8
1
age
Optimal asset allocation
CashBondsDom. StocksInt. Stocks
Deferred life annuity. 20% lower expected benefits given the same risk level.Left: optimal investment, E [Btot∗
t ] = 46, 200 EUR. Right: fixed-mix portfolio, E [Btot∗t ] = 37, 700 EUR.
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, chapterPapers 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 andpensions.
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 optimalwithdrawal 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 ina continuous time model.
Journal of Financial Economics, 2(2):187–203.
Agnieszka K. Konicz - Technical University of Denmark 10/9
Appendix
Agnieszka K. Konicz - Technical University of Denmark 11/9
Constraints I(jg
Budget equation while the person is alive, t ∈ {t0, . . . ,T − 1}, n ∈ Nt :
Bt,n1{t≥TR} + νt Itott,n +
∑i
X buyi,t,n = P tot
t,n 1{t<TR} +∑
i
X selli,t,n + νt
∑i
X→i,t,n
Value of the savings at the beginning of period t:before rebalancing in asset i , t ∈ {t0, . . . ,T}, n ∈ Nt , i ∈ A,
X→i,t,n = xi,01{t=t0} + (1 + ri,t,n)Xi,t−,n−1{t>t0},
after rebalancing in asset i , t ∈ {t0, . . . ,T − 1}, n ∈ Nt , i ∈ A,
Xi,t,n = X→i,t,n + X buyi,t,n − X sell
i,t,n,
Purchases and sales, t ∈ {t0, . . . ,T − 1}, n ∈ Nt , i ∈ A,
X buyi,t,n ≥ 0, X sell
i,t,n ≥ 0.
Agnieszka K. Konicz - Technical University of Denmark 12/9
Constraints II(jg
Premiums, t ∈ {t0, . . . ,T − 1}, n ∈ Nt ,
P tott,n = Pt,n + pfixed lt ,
Pt,n ≤ pvol lt ,
Benefits, t ∈ {t0, . . . ,T − 1}, n ∈ Nt ,
B tott,n = Bt,n + bstate
t ,
B tott,n ≥ bmin
t ,
Insurance, t ∈ {t0, . . . ,T − 1}, n ∈ Nt ,
I tott,n = It,n + insfixed
t ,
It,n ≥ insmin∑
i
X→i,t,n,
Portfolio composition, t ∈ {t0, . . . ,T − 1}, n ∈ Nt , i ∈ A,
Xi,t,n ≤ ui
∑i
Xi,t,n, Xi,t,n ≥ di
∑i
Xi,t,n,
Minimum savings, t ∈ {t1, . . . ,T}, n ∈ Nt ,∑i
X→i,t,n ≥ xmint .
Agnieszka K. Konicz - Technical University of Denmark 13/9
End effect(jg
DP - very specific and simplified model: power utility, risk-free asset, risky assetsfollowing GBM, Gompertz-Makeham mortality rate model, deterministic laborincome and state retirement pension, no constraints on portfolio composition andno constraints on the size of savings or benefits.
Utility:
u(t,Bt) = 1γw 1−γ
t Bγt , wt = e−1/(1−γ)ρt
Optimal value function (end effect):
V (t, x) = 1γf 1−γt
(x + gt
)γOptimal controls:
benefits: B∗t = wtft
(Xt + gt)− bstatet
sum insured: I tot∗t =
(K µtνt
)1/(1−γ)wtft
(Xt + gt)
proportion in risky assets: π∗t = α−rσ2(1−γ)
Xt +gtXt
gt - present value of future cashflows (labor income, retirement state pension, insurance price)
ft - optimal life annuity
Agnieszka K. Konicz - Technical University of Denmark 14/9