A Switch Criterion for Defined
Contribution Pension Schemes
Bas Arts and Elena Vigna
Basic Idea
Investing 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 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 : r*  12  (μ  λ)  18  (σ λ2 σ μ2)
(Chisini Average)
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The 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 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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Timeline
SF
I
SC
Contribution Equities
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R
A
D
Contribution Bonds
5
Initial SC and SF
Looking only at expected returns, we calculate
the switch of contributions (SC) as follows:
( R 1)
 c  (e )
r* ( R  j )
j 0
Target
Fund
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SC 1
 (  c  ( E (e ))
i
( SC i )
t
)  ( E (e ))
i 0
Equity
Fund
( R  SC )
R 1
  c  ( E (e t )) ( R i )
i  SC
Bond
Fund
6
Example
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 ( FITAR ) of 142,50 at
retirement and the initial switch of the contributions
from equities to bonds (SC) will be 23
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SC for different Target Returns
8,00%
7,00%
Target Return r*
6,00%
5,00%
4,00%
3,00%
2,00%
1,00%
0,00%
0
14
23
32
40
SC
Further research: Sensitivity Analysis to take into account risk aversion
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Dynamic Switch Criterion
• From time t=SC on at the beginning of each year we
check whether the projected future value fund ( FtTOT )
of the realized fund at time t together with future
contributions is greater than or equal to the Target Fund
( FITAR )
• 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: FCE  FCB  FTAR
t
t
I
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Figure 1:High return on equities
250
Total Fund
200
150
100
50
40
37
34
31
28
25
22
19
16
13
10
7
4
1
0
Year
Fund Equities (Return 10%)
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Fund Bonds (Return 4%)
10
Figure 2: Lower than expected return
on equities
Total Fund
160
140
120
100
80
60
40
37
34
31
28
25
22
19
16
13
10
7
4
1
40
20
0
Year
Fund Equities (Return 6%)
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Fund Bonds (Return 4%)
11
Figure 3: Low return on equities
70
Total Fund
60
50
40
30
20
10
40
37
34
31
28
25
22
19
16
13
10
7
4
1
0
Year
Fund Equities (Return 2%)
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Fund Bonds (Returns 4%)
12
Comparison other strategies
(TARGET FUND = 142,50)
Lifestyle Strategy
40 years 100%
equities
Switch strategy
(SC=23)
201,8
236,4
158,1
136,4
202,6
66,7
48,7
54,3
44,5
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
Mean
Standard Deviation from the
F ITAR
Downside Deviation from the
Mean shortfall from the F ITAR
P ( F RTOT  F RTAR )
P ( F RTOT  F RTAR | SF  41 )
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F ITAR
13
Comments
• 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)
This is because the current Switch Criterion
ignores the fact that bonds have their risk as
well
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Adjustments 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 SC
Year t
Average fund at t
YT SC at t
23
55,4
54,9
24
60,6
58,2
25
65,8
61,7
26
71,8
65,3
27
79,2
69,1
28
85,6
73,1
29
92,9
77,2
30
100,3
81,5
31
108,7
86,0
P ( SF  SC | SC  t )
38,8% 41,6% 44,2% 45% 46,7% 47,7% 50,2% 52,9% 54,7%
P(FRTOT  FRTAR SC  t ) 25,5% 23,7% 21,6% 21,0% 18,6% 18,2% 17,0% 15,9% 14,0%
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
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Comparison other Strategies
Switch Strategy
Lifestyle
strategy
40 years 100%
equities
SC=23
SC=31
Mean
201,8
236,4
158,1
184,8
TAR
Standard Deviation from the FI
136,4
202,6
66,7
104,7
48,7
54,3
44,5
52
42
46,3
35,9
42,7
40,5%
39,4%
43,3%
36,0%
25,5%
14,0%
Downside Deviation from the
FITAR
Mean shortfall from the FITAR
P( FRTOT  FRTAR )
P( FRTOT  FRTAR | SF  41)
Does not apply Does not apply
VaR 95%
69,4
58,3
67,8
60,9
VaR 75%
111,6
110,1
119,6
123,7
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Comments
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
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Reserve 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.
40
39
Simulated future Fund= (YTt   e   c 
i t
New Criterion=
With
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i
i t
40
e
j
)
j  ( i 1)
(FtCE  FtCB )  (1  Re serve t )  FRTAR
SMS t
Re servet 
YTt
19
Percentage of Yearly Target
Linear Regression Reserve
40%
30%
y = -0,0181x + 0,3655
R2 = 0,9879
20%
10%
0%
18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1
Years to Retirement
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Adjusted Switch strategy in
comparison with other strategies
Lifestyle
Strategy
Mean
Standard Deviation from the
Downside Deviation from the
Mean shortfall from the FITAR
P( FRTOT  FRTAR )
FITAR
FITAR
40 years 100%
Switch strategy
equities
(SC=31) with reserve
201,8
236,4
187,6
136,4
202,6
105,1
48,7
54,3
56,4
42,0
46,3
49,0
40,5%
39,4%
32,7%
Does not apply
4,0%
P( FRTOT  FRTAR | SF  41) Does not apply
VaR 95%
69,4
58,3
60,9
VaR 75%
111,6
110,1
117,7
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Comments
• 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
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Distribution Phase
• 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
qt
• We include a bonus factor for pooling (1  )
pt
((f tCE
1
The pensioner annuitizes at age A or before if the fund is
big enough to buy the target pension, i.e. when:
qt
t
t
CB
CB
 min( f t 1  P;0))  e  max( f t 1  P;0)  e )  (1  )  P  ax  t
pt
TOT
Ft
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Ft
TAR
23
Results Distribution Phase
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%
7,5
7,7
P( FRTOT  FRTAR )
P( FRTOT  FRTAR | SF  41)
P(No Drawdown & FRTOT  FRTAR
)
E ( FRTOT | NoDrawdown & FRTOT  FRTAR )
Drawdown/initial people
E ( FRTOT | Drawdown )
P(0  FRTOT
 FRTAR
& Ft TOT  0 ;R<t<(R+10))
10
10
TOT
TAR
E ( FRTOT
10 | FR 10  FR 10 )
P (Ruin )
Average year of ruin after R
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Comments
• 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
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Conclusion
The 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 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
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