On improving pension product design · Applications in Finance, Energy, Planning and Logistics, chapter Papers in Finance: Longevity risk management for individual investors. World

Article from:

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


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).



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).


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


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


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.


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.


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.


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


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.


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.


http://www.staff.dtu.dk/agko/Research/~/media/agko/konicz_combined.ashx

Appendix


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.


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 .


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


Documents

On improving pension product design · Applications in Finance, Energy, Planning and Logistics, chapter Papers in Finance: Longevity risk management for individual investors. World