A Switch Criterion for Defined A Switch Criterion for Defined Contribution Pension SchemesContribution Pension Schemes

Bas Arts and Elena Vigna

Basic IdeaBasic IdeaInvesting the contributions into equities a certain period and then wait for the “right time” to switch into bonds

Inspired by:• Mean Reversion of Equities• Lifestyle followed by Income Drawdown leads to discontinuity in portfolio composition• The idea of extra saving or reserve required in DC schemes• Considering both the accumulation and the distribution phase

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Assumptions and Target ReturnAssumptions and Target Return

• 2-assets: equities and bonds with lognormal distribution

• Equities, real force of interest t N(, 22), IID

• Bonds, real force of interest t N(, 12), IID

• t and t are uncorrelated

• c, contribution rate (constant)

• Target Return :

(Chisini Average)

)(81)(2

1 22*

μλ σσλμr

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The basic strategyThe basic strategy

The aim is to minimize the probability of failing the target pension

• Find the optimal number of years for investing the contributions into equities: SC• After SC the new contributions are invested into bonds, while the old contributions remain invested into equities (equity fund)• Propose an optimal criterion for switching the equity fund from equities into bonds (SF) using a dynamic approach

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6 important moments in life6 important moments in life

• I : Moment of joining the scheme

• SC : Switch of the contributions

• SF : Switch of the equity fund (maybe)

• R : Age of retirement

• A : Age when annutization is compulsory

• D : Death

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TimelineTimeline

I

SF

SC A DR

Contribution BondsContribution Equities

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Initial SC and SF Initial SC and SF

Looking only at expected returns, we calculate the switch of contributions (SC) as follows:

TargetFund

Equity Fund

Bond Fund

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1

)()(1

0

)()(*)1(

0

))(())(()))((()(R

SCi

iRSCRSC

i

iSCjRrR

j

tti eEceEeEcec

ExampleExample

The following parameter values have been chosen:

µ=4% λ=6% σµ=5% σ λ =15%

With a contribution of c=1 and 40 years to retirement, this results in a Target Return of 5.3125% and in a Target Fund ( ) of 142,50 at retirement and the initial switch of the contributions from equities to bonds (SC) will be 23

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TARIF

SC for different Target ReturnsSC for different Target Returns

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0,00%

1,00%

2,00%

3,00%

4,00%

5,00%

6,00%

7,00%

8,00%

0 14 23 32 40

SC

Ta

rge

t R

etu

rn r

*

Further research: Sensitivity Analysis to take into account risk aversion

Dynamic Switch CriterionDynamic Switch Criterion • From time t=SC on at the beginning of each year we

check whether the projected future value fund of the realized fund at time t together with future contributions is greater than or equal to the Target Fund

• If this is the case then the equity fund will be converted in bonds otherwise it remains invested into equities for at least one more year, while investing the new contributions in bonds

• In formula the SF occurs at the first time the following holds:

)( TARIF

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)( TOTtF

TARI

CBt

CEt FFF

Figure 1:High return on equitiesFigure 1:High return on equities

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0

50

100

150

200

250

1 4 7

10

13

16

19

22

25

28

31

34

37

40

Year

To

tal

Fu

nd

Fund Equities (Return 10%) Fund Bonds (Return 4%)

Figure 2: Lower than expected return Figure 2: Lower than expected return on equitieson equities

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02040

6080

100120

140160

1 4 7

10 13 16 19 22 25 28 31 34 37 40

Year

To

tal F

un

d

Fund Equities (Return 6%) Fund Bonds (Return 4%)

Figure 3: Low return on equitiesFigure 3: Low return on equities

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0

10

20

30

40

50

60

701 4 7

10

13

16

19

22

25

28

31

34

37

40

Year

To

tal

Fu

nd

Fund Equities (Return 2%) Fund Bonds (Returns 4%)

Comparison other strategiesComparison other strategies

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(TARGET FUND = 142,50)

Lifestyle Strategy 40 years 100% equities

Switch strategy (SC=23)

Mean 201,8 236,4 158,1

Standard Deviation from the 136,4 202,6 66,7

Downside Deviation from the 48,7 54,3 44,5

Mean shortfall from the 42 46,3 35,9

40,5 39,40% 43,30%

Does not apply Does not apply 25,50%

VaR 95% 69,4 58,3 67,8

VaR 75% 111,6 110,1 117,8

TARIF

)( TARR

TOTR FFP

)41|( SFFFP TARR

TOTR

TARIF

TARIF

CommentsComments

• The mean of the Switch Strategy is much lower than the mean of the other strategies and at the same time the probability of failing the target fund is higher

• The higher standard deviation of the other strategies can for a great part be explained by the surplus of the final fund on the Target Fund ( the other risk measures are comparable)

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This is because the current Switch Criterion ignores the fact that bonds have their risk as well

Adjustments to Basic StrategyAdjustments to Basic Strategy

• Investing the contributions some extra years in equities affects SC (& SF)

• Including a reserve when calculating SF

affects SF

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Flexible SCFlexible SCY e a r t 2 3 2 4 2 5 2 6 2 7 2 8 2 9 3 0 3 1 A v e ra g e fu n d a t t 5 5 ,4 6 0 ,6 6 5 ,8 7 1 ,8 7 9 ,2 8 5 ,6 9 2 ,9 1 0 0 ,3 1 0 8 ,7

SCYT a t t 5 4 ,9 5 8 ,2 6 1 ,7 6 5 ,3 6 9 ,1 7 3 ,1 7 7 ,2 8 1 ,5 8 6 ,0

)|( tSCSCSFP 3 8 ,8 % 4 1 ,6 % 4 4 ,2 % 4 5 % 4 6 ,7 % 4 7 ,7 % 5 0 ,2 % 5 2 ,9 % 5 4 ,7 %

)( tSCFFP TAR

R

TOT

R 2 5 ,5 % 2 3 ,7 % 2 1 ,6 % 2 1 ,0 % 1 8 ,6 % 1 8 ,2 % 1 7 ,0 % 1 5 ,9 % 1 4 ,0 %

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Comments: the difference between the average fund at t and the target fund increases with time because the fund remains invested longer in equities; the probability that SC and SF coincide increases; the probability of failing the target remarkably decreases when SC increases

Comparison other StrategiesComparison other Strategies

Switch StrategyLifestyle strategy

40 years 100% equities

SC=23 SC=31

Mean 201,8 236,4 158,1 184,8

Standard Deviation from the 136,4 202,6 66,7 104,7

Downside Deviation from the 48,7 54,3 44,5 52

Mean shortfall from the 42 46,3 35,9 42,7

40,5% 39,4% 43,3% 36,0%

Does not apply Does not apply 25,5% 14,0%

VaR 95% 69,4 58,3 67,8 60,9

VaR 75% 111,6 110,1 119,6 123,7

TARIF

)( TARR

TOTR FFP

)41|( SFFFP TARR

TOTR

TARIF

TARIF

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CommentsComments

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Comparing the SC=31 with SC=23 strategy:• the mean is higher while the probability of failing

the Target Fund is lower• the standard deviation, the downside deviation and

the mean shortfall are slightly higher (but considering the 36% lowest values - in case of failure - of SC=23 these risk measures are very similar)

• in the worst cases (VaR95%) the final fund is lower for SC=31 while the VaR75% is higher

Reserve at time tReserve at time t

Estimating the mean shortfall of the final fund for each year t (SMS), given that the yearly target (YT) is exactly satisfied at time t.

Simulated future Fund=

New Criterion=

With

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39 40

)1(

40

)(ti ijti

tji eceYT

TARRt

CBt

CEt F)serveRe1()FF(

t

tt YT

SMSserve Re

Linear Regression ReserveLinear Regression Reserve

y = -0,0181x + 0,3655

R2 = 0,9879

0%

10%

20%

30%

40%

18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

Years to Retirement

Per

cen

tag

e o

f Y

earl

y T

arg

et

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Adjusted Switch strategy in Adjusted Switch strategy in comparison with other strategiescomparison with other strategies

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Lifestyle Strategy

40 years 100% equities

Switch strategy (SC=31) with reserve

Mean 201,8 236,4 187,6

Standard Deviation from the 136,4 202,6 105,1

Downside Deviation from the 48,7 54,3 56,4

Mean shortfall from the 42,0 46,3 49,0

40,5% 39,4% 32,7%

Does not apply Does not apply 4,0%

VaR 95% 69,4 58,3 60,9

VaR 75% 111,6 110,1 117,7

TARIF

)( TARR

TOTR FFP

)41|( SFFFP TARR

TOTR

TARIF

TARIF

CommentsComments

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• The mean in comparison with the other strategies remains lower but the probability of failing the Target Fund is lower as well

• The VaR95% is lower than the Var95% of the lifestyle strategy, while the VaR75% is higher than in both the other strategies

• The probability of failing the Target Fund, given that the SF occurred, is only 4%. This is important for the Income Drawdown option in the distribution phase

Distribution PhaseDistribution Phase

txt

tCB1t

CB1t

CE1t aP)

p

q1()e)0;Pfmax(e))0;Pfmin(f(( tt

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TOTtF

TARtF

• The Criterion changes: Income Drawdown (only if the switch of the equity fund SF didn’t occur)

• For the pension P we take the pension that would have been obtained with the Target Fund

• While the fund in bonds >= 0 the pension will be deducted from this fund else it will be deducted from the equity fund

• We include a bonus factor for pooling

The pensioner annuitizes at age A or before if the fund is big enough to buy the target pension, i.e. when:

)1(t

t

p

q

Results Distribution PhaseResults Distribution Phase

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40 years 100% equities

Switch SC=31 with buffer

39,4% 32,7%

Does not apply 4,0%

Does not apply 2,8%

Does not apply 133,9

39,4% 29,9%

96,2 87,4

P(Switch between R and A) 16% 8,7%

Average year switch after R 3,6 3

6,3% 4,9%

18,1 19,9

17,1% 16,3%

Average year of ruin after R 7,5 7,7

)( TARR

TOTR FFP

)41|( SFFFP TARR

TOTR

)|( DrawdownFE TOTR

)|( 101010TARR

TOTR

TOTR FFFE

)(RuinP

NoDrawdownFETOTR|( & )TARR

TOT

RFF

P(No Drawdown & TARR

TOTR FF )

0&0( 1010 TOT

t

TAR

R

TOT

R FFFP ;R<t<(R+10))

Drawdown/initial people

CommentsComments

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• Total probability of failing the Target Fund for the Switch Strategy now becomes 16,3%+4,9%+2,8%=24% and 23,4% for the 100% equity strategy

• The 100% equity strategy has a higher probability of reaching the desired pension than the other strategies if we take into consideration the income drawdown option

• The average of the fund in the cases where the Target is not reached is higher for the Switch Strategy and the SF occurs on average earlier

• Income drawdown for the lifestyle strategy has not been done, because the fund is fully invested in bonds at retirement

ConclusionConclusionThe adjusted Switch Strategy seems to be suitable for DC schemes

for the following reasons:• it allows for a first partial switch of the fund from equities into

bonds (in order to limit the risk), but considers also actual returns from the financial market through a dynamic criterion for the second and definitive switch

• numerical results are good in comparison with other investment strategies for DC schemes

• it considers both the accumulation and the distribution phase so that discontinuity in portfolio composition when applying income drawdonw (like lifestyle) is avoided

Furthermore, investing fully in equities seems to be less risky than usually considered

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Further ResearchFurther Research

• Finding a more appropriate estimate for the reserve

• Introduce deferred annuities as a third investment possibility

• Taking into account the current yield on bonds at any time t, instead of considering a constant expected return

AFIR 2003 27