Dynamic formation of investment strategies for DC pension plan participants: two new approaches
Vadim Prudnikov
USATU, Ufa, Russia
Radon Workshop on Financial and Actuarial Mathematics for Young Researchers
Linz, Austria, May 30 - 31, 2007
The investment in a pension fund
• The investment – is very important field of any pension fund
– for DB plan participants (the actuarial rate)
– for DC plan participants (the pension value, attractiveness for new clients)
• The only relevant characteristic for DC plan participants is the length of the investment horizon
Overview of the problem
• Funds propose to their participants a various number of portfolios
• Two opposite opinions occur regarding the question whether or not investment risk depends on the length of investment horizon
Procedure 1
• Purpose: reproduce the distribution function of the daily logreturns of assets
• Process of the daily logreturns is defined by the following model:
(1)
)1,0(*)2(
)1,0(*)1(
)2(*)2(
)1(*)1(*
*
2
2
NS
NSE
EF
EFSGKS
EXAX
Procedure 2
• Purpose: reproduce the autocorrelation function of the process of the daily logreturns
• According to the signal processing theory:
– to get the power spectrum of the signal
– to go to the gain-frequency characteristic of the signal
– to estimate the parameters of the filter given the gain-frequency characteristic, the order of the filter and the method of estimation
– to pass ‘white’ noise through this filter
First approach, optimization problemt
t
• Consider a participant who has intervals up to retirement, each interval has days long
• Criterion: (2)
• Constrains:(3)
(4)
(5)
- is the minimally allowed proportion of participants’ portfolio invested in each asset
- is the structure of “equal-weighted” portfolio
max)(* txm
0***)()(**)( xVxtAtxVtx TT
ItxD )(
1)(* txI T
TdddD ,...,,
d
TI 1,...,1,1
TMMMx /1,...,/1,/1
t
Estimating of ,• 3 methods for efficiencies calculation, using :
– by direct statistical approach(6)
– by the simulation of the financial market trajectories (using procedure 1 or procedure 2) (7)
- is the value of the quote of asset “i” if the market will go by trajectory “l”
• (8)
(9)
m V
L
llii R
Nm
1,*
1
L
ljljiliji mRmR
Lv
1,, **
1
1,
1,
,,
Ki
lili Q
QMR
1,
,,
li
dtlili Q
QR
kiQ ,
liQM ,
Function A(t), features• - is the number of intervals in the investment horizon of the most
young participants • 2 opinions:
- investment risk of a participant is independent of the length of his investment horizon:
(10)- investment risk of preretired participants must be fewer than it is for young participants
- is a droningly increased function, and
(11)(12)
ttA 1)(
perefT
tA
11A 1perefTA
Function A(t), forms
11
)1*2(*)(
peref
peref
T
TttA
1)1(
)1(
*2)( 2
2t
TtA
peref
1)*21**2()1(
*2)( 2
2 perefperefperef
TtTtT
tA
11
*2
)*2(*sin*)(
peref
peref
T
TttA
otherwise
TttA
peref
,1
2/,1)(
otherwise
TtT
Tt
tA perefperef
peref
,1
2/*23/,1
3/,1
)(
• Linear (13)
• concave (14)
• convex (15)
• S-typed (16)
• 2-stepwised (17)
• 3-stepwised (18)
Second approach
• let to be a minimally allowed step in the change of the portfolio value invested in any particular asset
• to realize the second approach we need:
– to form the countable set of the investment strategies
– to simulate the matrix which contain trajectories of financial market probably motion, for days each
• let S(j,z) be the participants’ portfolio value at the moment of retirement while he applies investment strategy x(j) and financial market moves over trajectory z
• (19)
(20)
PP Ztt *
)(),( tsszjS
tztiPP
ztiPPijxss
ss M
i
,...,1,),1)1(*,(
),*,(*),(*)1(
0,1
)(
1
Second approach, optimization• we treat all the strategies as ‘participants’ of Z independently provided
heats• we transform the matrix of S(j,z) to the matrix of places that the
‘participants’ have achieved, Places(j,z) • we consider as optimal the investment strategy that has the minimal sum of
the places over all the heats: (21)
• 2 opinions– the investment risk of a participant is independent of the length of his
investment horizon, then: optimization is provided at the start time only and the strategies that were once found applies continuously up to retirement
– the investment risk does depend on the participants’ investment horizon, then: optimization is provided in the beginning of every interval
Z
z
zjPlacesj1
* ),(minarg
The mechanism of efficiency checking (1)• group all participants by the length of their investment horizon up to
retirement, accurate within one year
– T is the number of groups we obtained
– the youngest participants has T years or intervals up to retirement
• variant – any approach with specific parameters:
– for the first approach:• the form of A(t)
• the value of
• the method of calculating the efficiencies
– for the second approach:• procedure used to simulate matrix PP
• the option whether the optimization is accomplished periodically or at the start time only
tTTperef
250
*
The mechanism of efficiency checking (2)• input data
– matrix P, containing V trajectories of actual market motion during the period of days. Should be obtained by cutting from actual market trajectories
– parameters of the variants to be checked
• checking of the variants is provided for every group of participants:
– applying the variants of forming strategies over the trajectories of the matrix P
– providing the analysis of the empirical distribution of the value of the portfolio that could be accumulated up to retirement while using one or another variant
KTt peref *
The mechanism of efficiency checking (3)• analysis is provided by means of the stochastic dominance criteria of the
first, the second and the third order for every pair of variants separately
– we consider the variant X as better than variant Y if the following inequality holds:
(22)
(23)
(24)
)(int)(int YspoXspo
T
tt
ttXpXspo1
),()(int
orderunderYiantoveratesdoXiantif
orderunderYiantoveratesdoXiantif
orderunderYiantoveratesdoXiantif
ttgroupforYiantbyateddoisXiantif
ttXp
1varminvar,3
2varminvar,2
3varminvar,1
varminvar,0
),(
Data used for checking
• US financial market indexes (Dow Jones Industrials, index of 13-week US Treasury notes)
• period from 18.09.1986 to 30.04.2007 (5 199 daily quotes for every index)• other parameters were chosen as , , ,so • each trajectory of P must therefore have 40*125+100 = 5 100 daily quotes• the cutting: first trajectory starts 18.09.1986, and every next starts one day
after the preceding (then, V is equal to 5 199 – 5 100 + 1 = 100).
20T 125t 40perefT100K
Approach 1. Experiment 1
Variant 1:
A(t) – linear
Variant 2:
A(t) – concave
Variant 3:
A(t) – convex
Variant 4:
A(t) – S-typed
Variant 5:
A(t) – 2-stepwised
Variant 6:
A(t) – 3-stepwised
0 28 23 24 19 21 115
19 0 22 21 17 17 96
30 29 0 47 6 12 124
16 29 3 0 3 5 56
24 27 34 44 0 11 140
18 32 28 40 6 0 124
Approach 1. Experiment 2
Variant 1:
Variant 2:
Variant 3:
0 4 43 47
43 0 45 88
8 8 0 16
05,0
1,0
025,0
Approach 1. Experiment 3
Variant 1:
Calculation the efficiencies using direct statistical approach
Variant 2:
Calculation by simulation the trajectories using procedure 1
Variant 3:
Calculation by simulation the trajectories using procedure 2
0 36 33 69
0 0 0 0
0 34 0 34
Approach 1. Experiment 4
Variant 1:
A(t) – 2-stepwised
Calculation the efficiencies using direct statistical approach
Variant 2:
A(t) = 1 for all t
1,0
0 49 49
2 0 2
Approach 2. Experiment 5
Variant 1:
Matrix PP is simulated using procedure 1
Variant 2:
Matrix PP is simulated using procedure 2
0 19 19
2 0 2
Approach 2. Experiment 6
Variant 1:
Optimization is provided periodically
Variant 2:
Optimization is provided at the start time only
0 0 0
22 0 22
Experiment 7. Comparison of the approaches
Variant 1:
Approach 1
Variant 2:
Approach 2
Variant 3:
Approach of ‘equal-weighted’ portfolio
0 32 8 40
3 0 0 3
14 23 0 37