Contribution and solvency risk in a defined benefit pension scheme
Steven Haberman
∗, Zoltan Butt, Chryssoula Megaloudi
Department of Actuarial Science and Statistics, City University, Northampton Square, London EC1V 0HB, UK
Received April 2000; received in revised form June 2000; accepted June 2000
Abstract
This paper presents a stochastic investment model for a defined benefit pension scheme, in the presence of IID real rates of return. The spread method of adjustment to the normal cost is used to deal with surpluses or deficiencies. Two types of risk are identified, the “contribution rate risk” and the “solvency risk” which are concerned with the stability of the contributions and the security of the pension fund, respectively. A performance criterion is introduced to deal with the simultaneous minimisation of these two risks, using the fraction of the unfunded liability paid off (k) or the spread period (M) as the control variable. A full numerical investigation of the optimal values ofkandMis provided. The results lead to practical conclusions about the optimal funding strategy and, hence, about the optimal choice of the contribution rate subject to the constraints needed for the convergence of the performance criterion. © 2000 Elsevier Science B.V. All rights reserved.
JEL classification: C61
Keywords: Contribution risk; Solvency risk; Defined benefit pension scheme; Stochastic investment returns
1. Introduction
1.1. Risk in a defined benefit pension scheme
As defined by Lee (1986), occupational pension schemes are arrangements by means of which employers or groups of employers provide pensions and related benefits to their employees. We are interested in defined benefit pension schemes where the benefits promised are the defined quantity and the contributions are the dependent variable. The determination of these contributions takes place through the valuation process, which is performed by the actuary at regular intervals.
The method by which the scheme is valued and the contribution rate determined is called the actuarial funding method. In this paper, we shall consider the class of individual funding methods (which involve an actuarial liability and a normal cost). It is inevitable that the actual experience of the pension scheme will deviate from the assumptions (of demographic and economic variables) on which the previous valuation was based so that a surplus or a deficiency will emerge. In the light of the particular situation revealed by the valuation, appropriate action will be taken by way of an adjustment to the contribution rate so as to remove the shortfall or to utilise the surplus. For individual funding methods, the most common ways of dealing with this adjustment are the spread method and the amortisation of losses method. We will consider the spread method under which the unfunded liability is spread into the future
∗Corresponding author. Tel.:+44-0171-477-8470; fax:+44-0171-477-8572.
E-mail address: [email protected] (S. Haberman).
over a certain period. The choice of this period, i.e. called the spread period, will be related to the required balance between the different types of risk facing the pension scheme.
We investigate two types of risk. The first one is the “contribution rate risk”. According to Lee (1986), the sponsor of the scheme will look for a contribution plan which will not be disturbed by significant changes so that the contribution rate will remain reasonably stable in the future. The second type of risk is the “solvency risk”. As Lee (1986) explains, the trustees and the employees will be concerned that the accumulated assets represent reasonable security for the growing pension rights of the members, independently of the sponsoring employer, at any time or when the scheme is wound up. We note also that an excessive build-up of assets relative to liabilities (i.e. overfunding) can also have serious economic consequences, e.g. for the employer (Thornton and Wilson, 1992; Exley et al., 1997). In this paper, we will use a mathematical model to represent the financial structure of a defined benefit pension scheme with stochastically varying investment returns. As noted by many commentators (e.g. Thornton and Wilson, 1992) one of the principal sources of surplus or deficiency for pension schemes has been the rate of investment return on the scheme assets. We will consider methods for controlling the above types of risk by using the spread period as our control variable. As noted by Owadally and Haberman (1999), the spread period plays an important role in “the stochastic dynamics of the fund over time and careful consideration should be given to its choice”.
1.2. Formulation of the problem
The approach described is based on the earlier investigations of Haberman (1997a,b). The optimal contribution rate will be determined by minimising a quadratic performance criterion, that includes both the contribution rate risk and the solvency risk. The problem is described below using a discrete time formulation (as in Haberman and Sung (1994)).
We wish to find the contribution ratesC(s), C(s+1), . . . , C(T −1)over the finite time period (s,T) which minimise the quadratic performance criterionssJT defined below. We use the following notation:
C(t )=contribution rate for the time period(t, t+1).
F (t )=fund level at timet, measured in terms of the market value of the assets.
CT(t )=contribution target for the period(t, t+1).
FT(t )=fund target for the period(t, t+1).
w=(1+j )−1for somej >0,wis a discount factor used to weight the different contributions to the criterionJ
over time.
θ =a weighting factor to reflect the relative importance of the solvency risk against the contribution rate risk. Then we definesJT as
The first term provides a measure of the contribution rate risk and the second term the solvency risk. The quadratic nature of (1) means that high values ofC(t )(relative to the target) are considered to be as undesirable as low values, and similarly forF (t ). This characteristic is justified on the grounds that we seek to penalise instabilities inC(t )as well as either underfunding or overfunding of the liabilities (as noted in Section 1.1).
The expectation operator is necessary because we are interested in the stochastic case so as to recognise the random nature of investment returns. (In a continuous time formulation, the mathematical approach would be based on an integral version of (1).)
Thus, the risk of the pension fund is defined as a “time-weighted” sum of the weighted average of the future variances of the fund level and contribution rate.θis determined according to which of the variability of the fund or the contribution is more important for the employer. This balance will influence the choice of the funding strategy, since some methods (e.g. prospective benefit methods) aim more at stabilising the contribution rate, whereas others (e.g. accrued benefit methods) have as their main purpose the funding of the actuarial liability.w =(1+j )−1is used to discount the variances. A low value ofw(high value ofj) indicates that more emphasis is placed on the shorter-term position of the pension fund rather than the longer term, thereby providing a mechanism for weighting in time. We will comment in places on the special case,j =i, so thatw=v=(1+i)−1, whereiis the valuation rate of interest.
1.3. The mathematical problem
We consider the behaviour ofC(t )by using a stochastic investment model of a defined benefit pension scheme. Its main features are a stationary population and independent and identically distributed rates of return. As noted earlier, we shall work in discrete time(t =0,1,2, . . . ).
When an actuarial valuation takes place, the actuary estimatesC(t )andF (t )based only on the active and retired members of the scheme at timetunder the following assumptions:
• The valuation interest rate is fixed and isi(but see assumption (4) below).
• The contribution income and benefit outgo cash flows occur at the start of each scheme year.
• Valuations are carried out at annual intervals.
The following recurrence relations for the pension fund’s assets and the actuarial liability hold:
F (t+1)=(1+i(t+1))(F (t )+C(t )−B(t )), (3)
AL(t+1)=(1+i)(AL(t )+NC(t )−B(t )) (4)
fort =0,1,2, . . ., based on the following notation:
i(t +1)=rate of investment return earned during the period(t, t+1), defined in a manner consistent with the definition ofF (t )(but see assumption (4) below).
AL(t+1)=actuarial liability at the end of the period(t, t+1)with respect to the active and retired members.
B(t )=overall benefit outgo for the period(t, t+1).
NC(t )=normal cost for the period(t, t+1).
We make the following further simplifying assumptions:
1. The experience is in accordance with all the features of the actuarial basis, except for investment returns. 2. The population is stationary from the start. We could alternatively assume that the population is growing at a
fixed, deterministic rate.
3. There is no promotional salary scale. Salaries increase at a deterministic rate of inflation. This inflation component is used to reduce the assumed rate of investment return to give a real rate of investment return. We also assume that benefits in payment increase at the same rate as salaries and then consider all variables to be in real terms (i.e. AL(t ), B(t ),NC(t ), F (t ), C(t )).
4. Following the previous assumption,i(t+1)is the real rate of investment return earned during the period(t, t+1)
andE i(t )=i, whereiis the real valuation rate of interest. This means that funds are assumed in the valuation to be invested in future to earn the mean rate of investment return. We also defineσ2 =Var(i(t ))and assume
σ2<∞.
5. The earned real rates of returni(t )are independent identically distributed random variables with Prob(i(t ) >
−1)=1.
6. The initial value of the fund at time zero is known, i.e. Prob[F (0)=F0]=1 for someF0.
period(s, s+1)):
NC(t )=NC; AL(t )=AL;
B(t )=B.
Then, combining these results with (4), we obtain
AL=(1+i)(AL+NC−B). (5)
1.4. Individual funding methods
For an individual funding method, the unfunded liability denotes the difference between the plan’s actuarial liability and its assets.
UL(t )=AL(t )−F (t ), (6)
where UL(t )is the unfunded liability at timet.
These methods involve an actuarial liability and a normal cost which is then adjusted to deal with the unfunded liability. There are a number of choices for the ADJ(t )term. We will consider the spread method, under which
C(t )=NC(t )+ADJ(t ), (7)
ADJ(t )=kUL(t ), (8)
where ADJ(t )is the adjustment to the contribution rate at timet,k=1/a¨¬
M is calculated at the valuation rate of
interest,Mis the “spread period”.
So the unfunded liability is spread overMyears andkcan be thought of as a penalty rate of interest that is being charged on the unfunded liability. The choice ofM, as we will see later on, is of great importance and influences the funding strategy.
The above definition of ADJ(t )implies that the spread period is always the same whether there is a surplus or a deficit. According to Winklevoss (1993), this may not always be the case in practice with a shorter spread period being used to eliminate deficiencies than for surpluses. (This asymmetric approach has recently been investigated by Haberman and Smith (1997) using simulation.)
Finally, from (6)–(8) and the previous assumptions
C(t )=NC+k(AL−F (t )). (9)
1.5. Moments ofC(t )andF (t )
Dufresne (1988) has shown that, given our mathematical formulation,
EF(t )=qtF0+AL(1−qt)=qtF0+
r(1−qt)
(1−q) , t ≥0, (10)
whereq =(1+i)(1−k),r=(1+i)(k−d)AL. So from (9),
EC(t )=NC+k(AL−EF(t )), (11)
and limt→∞EF(t )=AL,limt→∞EC(t )=NC, provided 0< q <1 (i.e.d < k <1 orM >1).
Dufresne also proves that
VarF (t )=b
t X
j=1
whereb=σ2v2anda=q2(1+b)=(1−k)2(1+i)2(1+σ2v2), and that choice of the spread period (see Sections 1.6 and 2.2).
1.6. The general form
q <1 andw <1 so that these conditions are satisfied.
Our aim is to find the value(s) ofk(or equivalently the spread period, ask=1/a¨¬
M) which minimises the above
equation. Then, we can find the optimalC(t )via Eq. (9).
We note thatq =(1+i)(1−k)⇒k=1−qvanda=q2(1+σ2v2)and sinceq →kis a 1:1 mapping with domain
g(q)is a ratio of two polynomials of degree 5. It is a complex function ofq and may have a number of turning points, both minima and maxima. The solution of (18) reduces to find the roots of a polynomial of degree 8. It is clear from the form ofg(q)that its graph has three vertical asymptotes, i.e.g(q) → ±∞, whenq → qi for
i=1,2,3, where
q1=w−1, q2=w−1/2, q3=(w(1+σ2v2))−1/2.
Given thatw < 1 we have thatq1 > q2 > q3. In order to ensure that the limiting variances ofF (t )andC(t )
are finite (as given by (14)), we will restrict our attention to the range of values (0, qmax), whereqmaxis such that
a=1. Thus
qmax=(1+σ2v2)−1/2
and we only considerq < qmax. We note also thatqmax< q3, so that the above mentioned asymptotes fall outside
2. Risk as a time-weighted mechanism
2.1. Introduction
We wish to solve Eq. (18) and find the values ofqat whichJ∞is minimised. For all the calculations, we assume
for convenience (and without loss of generality) that AL=1. As noted above, solutions are restricted to values such that
q < qmax<1, qmax=(1+b)−1/2, b=σ2v2.
We verify that the chosen values ofqare the minimum points by detailed numerical calculations, and demonstrate some of the results by the relevant graphs ofJ∞plotted againstq. The results indicate that, in some cases, there is a single global minimum and, in others, there is a global minimum and a local minimum. It is also possible that, in some cases, the minimum value ofJ∞confined to the range(0, qmax)occurs at one of the endpoints.
In any particular case, calculation of the minimising value(s) ofq allows us to find the corresponding values of
kandMfrom
q =(1+i)(1−k), M= −log(1−d/(1−qv))
log(1+i) .
The tables in Section 2.3 onwards provide the optimal values ofkandMas a function ofi,σ,j,zandθ (to the nearest integer) and the values ofkandM which are marked with † correspond to the minimum value ofJ
occurring atqmax. Detailed investigations have been carried out for a range of values ofzandθbut detailed results
are reported here only for the casesz= −AL,−21AL and 0 and selected values ofθ.
2.2. The maximum feasible values of the spread period
As noted earlier, the requirementa <1 (for convergence) places a restriction on the choice ofq. So the optimal values ofqmust be restricted to values such thatq < qmax. Table 1 provides rounded values of the maximum spread
periodM1which correspond toqmaxfor different combinations ofσ andiand indicates the extent to whichM1
decreases asσ andieach increase. Similarly, there is a minimum value ofk,k1corresponding toqmaxandM1. 2.3. Initial funding level of 0%
Tables 2–6 provide the optimal values ofk and the spread period Mfor F0 = 0 (i.e.z = −1) and selected
combinations ofσ, i, j andθ, corresponding to the global and local minima ofg(q)as appropriate. The top panel of each table presents the global minima and the bottom panel the local minima.
Tables 2–6 (and other results not tabulated here for reasons of space) indicate that:
1. There is a value ofkorM(k∗andM∗, respectively) which leads to a global minimum inJ∞(and henceg(q)).
2. For certain combinations ofθandσ (see below), there is a local minimum forJ∞.
Table 1
Maximum spread period,M1, such thata <1 i σ
0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35
0.01 535 318 223 112 66 42 30 22 17
0.03 218 145 111 68 46 33 25 19 15
Table 2
0 0.0099/535.0† 0.0103/318.1† 0.9900/1.01 0.9901/1.01 0.9902/1.01 0.9904/1.01 0.9906/1.01 0.9908/1.01 0.9911/1.01 0.5 0.0099/535.0† 0.0103/318.1† 0.6103/1.64 0.6124/1.64 0.6159/1.65 0.6207/1.62 0.6266/1.61 0.6337/1.58 0.6417/1.56 0.75 0.0099/535.0† 0.0103/318.1† 0.111/222.8† 0.4293/2.35 0.4339/2.32 0.4402/2.29 0.4482/2.24 0.4577/2.20 0.4686/2.15 0.85 0.0099/535.0† 0.0103/318.1† 0.111/222.8† 0.3361/3.01 0.3412/2.96 0.3484/2.90 0.3574/2.82 0.3682/2.47 0.3806/2.65 0.95 0.0099/535.0† 0.0103/318.1† 0.111/222.8† 0.1995/5.12 0.2056/4.96 0.2140/4.76 0.2248/4.53 0.2379/4.27 0.2530/4.01 1 0.0099/535.0† 0.0103/318.1† 0.111/222.8† 0.0147/112.2† 0.0227/57.7 0.0394/29.1 0.0596/18.3 0.0830/12.8 0.1092/9.55 Local minimum ofg(q)
0 0.9900/1.01 0.9900/1.01 0.0111/222.8† 0.0147/112.2† N/A N/A N/A N/A N/A 0.5 0.6096/1.65 0.6098/1.65 0.0111/222.8† 0.0147/112.2† N/A N/A N/A N/A N/A 0.75 0.4256/2.37 0.4259/2.36 0.4265/2.36 0.0147/112.2† N/A N/A N/A N/A N/A 0.85 0.3320/3.04 0.3323/3.04 0.3330/3.03 0.0147/112.2† N/A N/A N/A N/A N/A 0.95 0.1946/5.25 0.1950/5.24 0.1958/5.21 0.0147/112.2† N/A N/A N/A N/A N/A
1 N/A N/A N/A N/A N/A N/A N/A N/A N/A
3. Whenσ is small,M∗is very large and equal to its maximum permitted value, so thata < 1,M1. Similarlyk∗
is close to its minimum permitted valuek1, so thata <1.
4. Asσ is increased, there is a dramatic change in the optimal values withM∗decreasing andk∗increasing. For example, we note that wheni=1%, j =1% andθ=0,M∗=318.1 andk∗=0.0103 whenσ =0.03, but for
σ ≥0.05, M∗=1.01 andk∗=0.9900.
For the permitted range of values ofMork,J∞has a turning point but the choice of the location of the global minimum depends on the relationship between the value ofJ∞at the turning point and at the boundary valuesM1
ork1.
For smallσ, the global minimum occurs atM1ork1(i.e. to makeMas large as possible, orkas small as possible).
Asσ is increased further the shape ofJ∞changes, so that there is only one minimum point.
We have noted in Section 2.2, that whenσincreases,M1decreases andk1increases. Further analysis shows that
there is a critical value ofσ,σ¯ for whichJ∞has a minimum at the two pointsM2≪M3(andk2≫k3), wherek3
0 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.9696/1.03 0.9700/1.03 0.9705/1.03 0.9711/1.03 0.9718/1.03 0.9726/1.03 0.5 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.5904/1.69 0.5956/1.67 0.6020/1.66 0.6097/1.64 0.6183/1.61 0.75 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.4034/2.46 0.4105/2.42 0.4194/2.37 0.4299/2.31 0.4418/2.25 0.85 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.3074/3.22 0.3157/3.14 0.3260/3.04 0.3382/2.93 0.3520/2.82 0.95 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.1756/5.58 0.1895/5.18 0.2054/4.79 0.2231/4.41 1 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.0281/32.8† 0.0396/23.5 0.0638/14.9 0.0902/10.6 Local minimum ofg(q)
0 0.9693/1.03 0.9693/1.03 0.9694/1.03 0.0145/67.8† 0.0202/45.8† N/A N/A N/A N/A 0.5 0.5835/1.71 0.5837/1.71 0.5842/1.71 0.5866/1.70 0.0202/45.8† N/A N/A N/A N/A 0.75 0.3940/2.52 0.3943/2.52 0.3950/2.52 0.3982/2.49 0.0202/45.8† N/A N/A N/A N/A 0.85 0.2964/3.34 0.2968/3.33 0.2976/3.32 0.3013/3.28 0.0202/45.8† N/A N/A N/A N/A 0.95 0.1469/6.64 0.1476/6.61 0.1489/6.55 0.1548/6.31 0.1639/5.97 N/A N/A N/A N/A
Table 4
0 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.9488/1.05 0.9496/1.05 0.9507/1.05 0.9519/1.05 0.9533/1.05 0.5 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.5656/1.76 0.5730/1.74 0.5817/1.71 0.5914/1.69 0.75 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.3749/2.65 0.3853/2.58 0.3975/2.50 0.4110/2.42 0.85 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.2759/3.58 0.2887/3.42 0.3033/3.26 0.3193/3.10 0.95 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.0281/32.8† 0.1429/6.82 0.1662/5.89 0.1885/5.20 1 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.0281/32.8† 0.0378/24.5† 0.0494/19.0† 0.0660/14.4 Local minimum ofg(q)
0 0.9476/1.06 0.9477/1.06 0.9477/1.05 0.9481/1.05 0.0202/45.8† 0.0281/32.8† N/A N/A N/A 0.5 0.5518/1.81 0.5521/1.80 0.5527/1.80 0.5553/1.79 0.5597/1.78 0.0281/32.8† N/A N/A N/A 0.75 0.3551/2.79 0.3555/2.79 0.3564/2.78 0.3602/2.75 0.3665/2.71 0.0281/32.8† N/A N/A N/A 0.85 0.2504/3.94 0.2510/3.93 0.2521/3.91 0.2572/3.72 0.2653/32.8 0.0281/32.8† N/A N/A N/A
0.95 N/A N/A N/A N/A N/A 0.1106/8.75 N/A N/A N/A
so thatM3is the maximum feasible value for the spread period corresponding toσ¯ andk3is the corresponding
minimum feasible value fork.
ForM2< M < M3(ork3< k < k2),J∞is higher than at either of the end points of this interval. So when the
choice ofσ makesM1 < M3(i.e.σ >σ¯),J∞is only minimised atM2. WhenMis allowed to exceedM3,J∞
decreases. Hence, when the choice ofσ makesM1> M3(σ <σ¯), the optimal choice becomesM1.
Approximate values ofσ¯ for different values ofj, θandiare shown in Table 7. We note the dependence ofσ¯ on these parameters.
0 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.9696/1.03 0.9700/1.03 0.9704/1.03 0.9710/1.03 0.9717/1.03 0.9725/1.03 0.5 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.5932/16.7 0.5983/1.66 0.6046/1.64 0.6120/1.62 0.6204/1.60 0.75 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.4058/2.42 0.4128/2.38 0.4215/2.33 0.4319/2.27 0.4436/2.22 0.85 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.3092/3.14 0.3175/3.06 0.3278/2.97 0.3399/2.81 0.3536/2.76 0.95 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.1760/5.35 0.1904/4.97 0.2067/4.60 0.2247/4.25 1 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.0274/27.5† 0.0434/18.6 0.0702/12.3 0.0977/9.17 Local minimum ofg(q)
0 0.9693/1.03 0.9693/1.03 0.9694/1.03 0.0144/51.1† 0.0199/36.6† N/A N/A N/A N/A 0.5 0.5866/1.69 0.5868/1.69 0.5873/1.69 0.5896/1.68 0.0199/36.6† N/A N/A N/A N/A 0.75 0.3965/2.47 0.3969/2.47 0.3976/2.46 0.4007/2.44 0.0199/36.6† N/A N/A N/A N/A 0.85 0.2982/3.25 0.2986/3.24 0.2994/3.24 0.3031/3.20 0.0199/36.6† N/A N/A N/A N/A 0.95 0.1456/6.38 0.1463/6.35 0.1477/6.29 0.1540/6.05 0.1637/5.72 N/A N/A N/A N/A
Table 6
Optimal values ofk∗andM∗: whenF
0=0,i=5%,j=5% θ σ
0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Global minimum ofg(q)
0 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.9487/1.05 0.9496/1.05 0.9506/1.05 0.9518/1.05 0.9532/1.05 0.5 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.5680/1.74 0.5752/1.72 0.5836/1.70 0.5931/1.67 0.75 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.3762/2.60 0.3865/2.53 0.3985/2.46 0.4120/2.38 0.85 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.0274/27.5† 0.2888/3.35 0.3035/3.19 0.3196/3.04 0.95 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.0274/27.5† 0.1384/6.78 0.1640/5.71 0.1875/5.04 1 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.0274/27.5† 0.0368/21.4† 0.0480/17.1† 0.0693/12.4 Local minimum ofg(q)
0 0.9476/1.05 0.9476/1.05 0.9477/1.05 0.9481/1.05 0.0199/36.6† 0.0274/27.5† 0.0368/21.4† N/A N/A 0.5 0.5546/1.78 0.5549/1.78 0.5554/1.78 0.5580/1.77 0.5622/1.76 0.0274/25.5† 0.0368/21.4† N/A N/A 0.75 0.3566/2.74 0.3570/2.74 0.3579/2.13 0.3617/2.70 0.3679/2.65 0.0274/25.5† 0.0368/21.4† N/A N/A 0.85 0.2500/3.84 0.2505/3.83 0.2517/3.82 0.2569/3.74 0.2651/3.63 0.2759/3.50 0.0368/21.4† N/A N/A
0.95 N/A N/A N/A N/A N/A N/A 0.0368/21.4† N/A N/A
1 N/A N/A N/A N/A N/A N/A N/A N/A N/A
Tables 2–6 also indicate that forσ <σ¯, this optimal choice ofMorkdoes not depend onθas neitherM1nor
k1depends onθ. On the other hand, forσ >σ¯, an increase inθhas a dramatic effect onM∗andk∗. For example,
wheni=1%,j =1% andσ =0.05, the optimal spread periodM∗=1.64 (withk∗ =0.6103) forθ=0.5, but
M∗=222.8 (andk∗=0.0111) forθ=0.85.
The results thus indicate that, forσ large enough, there is a minimising value ofM∗ork∗. Ifσ is small, then there are two minima and the global minimum corresponds to the choiceM1andk1, wherek1is close tod. We can
interpret this feature as follows: ifF0=0 andkis chosen to be equal tod, then from (10),r=0 and EF(t )=0 for all t
and from (12),
VarF (t )=0 for all t.
Thus, we would be operating a pay-as-you-go system with
EC(t )=NC+dAL=B.
Table 7
Critical values ofσ¯: whenF0=0
i j θ
0 0.25 0.5 0.75 0.85 0.95 1
0.01 0.01 0.04 0.043 0.046 0.051 0.055 0.064 0.14
0.03 0.01 0.045 0.046 0.047 0.053 0.056 0.066 0.13
0.03 0.03 0.097 0.105 0.11 0.125 0.135 0.16 0.245
0.03 0.05 0.14 0.155 0.167 0.186 0.205 0.23 0.33
0.05 0.03 0.098 0.105 0.115 0.12 0.135 0.16 0.24
0.05 0.05 0.147 0.157 0.17 0.19 0.205 0.24 0.33
So, the case whereσ is small leads to a funding policy with a very small fund and the sponsor paying the interest on the unfunded liability in addition to NC. So the analysis implies the existence of two possible funding positions, with the combination of parameters determining which is optimal.
A funding policy based on a high value forM∗(low value ofk∗) would aim at keeping the contributions stable by ignoring positive or negative fluctuations in investment returns and hence inF (t ). This would work in the short term (i.e. small values oft) but, in due course, the level ofF (t )would need to be taken into account and the fluctuations in fund size will require compensating changes inC(t )to control the system. On the other hand, a funding policy based on a value ofM∗close to 1 (andk∗close to 1) implies dealing immediately with any surplus or deficiencies as they emerge. Thus, the resultingC(t )will be volatile in the short term but in the long term the relative stability ofF (t )will compensate and lead to lower values of VarC(t )(Dufresne, 1988; Cairns and Parker, 1997; Owadally and Haberman, 1999). The balance between these two extremes will depend on the value ofσ, j (representing the relative weight to be given to the short term versus the long term in the overall measureJ∞) andθ (representing
the relative weight to be given to fund stability and contribution stability).
We next consider Eq. (17) as a function ofθ, 0 < θ < 1. We recall that θ controls the balance between the solvency risk and the contribution rate risk. The risk (as represented byJ∞) is a decreasing function ofθ:
∂J∞
∂θ α(k
2−1) <0,
but this decrease in risk is significant only for large values ofq (i.e. large values ofMor small values ofk). So, whenθincreases, the risk decreases but this downward shift in risk is not smooth.J∞decreases markedly whenM
is large (kis small) and remains approximately constant whenMis small (kis large), thereby, making the optimal spread periodM∗larger (andk∗smaller).
Tables 3–6 indicate that forσ <σ¯, when a higher discounting factor,w, is used (a lowerj) the optimal choices
M∗andk∗remain the same (equal toM1andk1, respectively). This is explained as neitherM1nork1depends on
j. On the other hand, forσ >σ¯, we observe that the optimal choiceM∗becomes smaller (andk∗larger) whenj
is decreased. For example, whenσ =0.15,θ =0.75 andi =3%,M∗ =45.8 andk∗ =0.0202 whenj =5%, butM∗ =2.46 andk∗ =0.4034 whenj =3%. Fig. 1 shows the graph ofJ∞for the corresponding case when θ=0.25.
We observe that whenj rises, the risk as represented byJ∞ decreases, with less weight being given to the
elements of formula (15) corresponding to higher values oft. This downward shift in risk is much more significant for large values ofM, making the optimal spread period longer. Hence, whenj =3% the riskJ∞is minimised for
Fig. 2. Graph ofJ∞(j )whenσ=0.15,θ=0.25 andi=3%.
M ∼=1. Whenj =5% (more emphasis is placed on the shorter-term state of the pension fund), the risk remains approximately the same forM∼=1 but decreases considerably forM∼=46 and the optimal spread period becomes
M∗∼=46 (=M1).
We can explain these results in a different way. We consider Eq. (17) as a function ofjand choose for convenience the particular parameter valuesi=3%,σ =0.15 andθ=0.25. Fig. 2 shows the graph of this function for different values ofM, selected carefully in the light of the earlier results.
Fig. 2 demonstrates what we have already claimed. The risk as represented byJ∞is a decreasing function ofj
and this decrease in risk becomes more significant as the values ofMbecome larger.
From Tables 2–6, it can also be seen that an increase in the assumed rate of returnicauses a significant decrease inM∗whenσ <σ¯ and a slight decrease inM∗whenσ >σ¯. We recall that whenσ <σ¯, the optimal choice is
M1which depends oni, and which changes in the way shown by Table 1. Whenσ >σ¯, the optimal choice for
Mis smaller (and forkis longer) and remains the same or decreases slightly (or increases fork) wheniincreases. For example, whenj =5%,θ =1 andσ =0.35,M∗ =14.4 andk∗=0.0660 fori =3% butM∗ =12.4 and
k∗=0.0693 fori=5%. Fig. 3 illustrates these two cases.
Fig. 3 shows thatJ∞remains approximately the same for low values ofMand slightly increases for high values ofMwheniis increased.
We consider Eq. (17) as a function ofiand choose for convenience the particular parameter values,σ =0.35,
θ=1 andj =5%. Fig. 4 shows this function for different values ofM.
Fig. 4 demonstrates the sensitivity ofJ∞to changes ini, indicating that the optimal choice is influenced only
slightly when the assumed rate of return changes. It also indicates that for the case when the rate of interest used for discounting(j )is equal to the valuation rate of interesti(see Tables 2, 3 and 6), the changes inJ∞are principally
attributable to changes in the discounting rate of interest.
2.4. Initial funding level of 25%
The detailed results for this case ofF0 =0.25 (z= −0.75) are not presented here. However, these show that,
for a higher initial funding level, the optimal choice forM, spread period is much lower (and much higher fork) for many combinations ofσ, i, j andθ. For example, wheni =1%,j =1%, σ =0.01 andθ =0 the optimal choice isM∗ =535.0 andk∗ =0.0099 whenF0 =0, andM∗ =1.01 andk∗ =0.9925 whenF0=0.25. The
Fig. 3. Graph ofJ∞whenσ =0.35,θ=1 andj=5%.
the other parameters are such that we do not have the case ofM1. For example, for the combination ofj =10%,
i = 5%,σ = 0.01, the higher initial funding level of 25% is not sufficient to change the optimal choice which remainsM∗ =143.6 andk∗ =0.0099. Hence, for this case, the effect of the high value ofj is more significant than the magnitude of the initial funding level.
When the optimal spread period is shorter thanM1(σ > σ¯), an increase inj leads to a higher optimal choice.
For example, wheni=5%,θ =0.25 andσ =0.01,M∗=1.01 forj =5% butM∗=143.6 forj =10%. Fig. 5 showsJ∞plotted againstM.
Fig. 5 illustrates what we have already claimed. The risk as represented byJ∞is a decreasing function ofj and
is more sensitive to changes inj for the higher values ofM(see also Fig. 2), and the correspondingly lower values ofk. Therefore, when we are more interested in the shorter-term position of the pension fund (j =10%), the risk decreases to a greater extent forM∼=144 than forM∼=1 and the optimal spread period becomesM∗=143.6.
For an initial funding level of 25% and for theσ > σ¯ cases, the detailed results show that an increase in the assumed rate of return leads to the same results as for a zero initial funding level. The optimal choice forMdoes
Fig. 5. Graph ofJ∞whenσ=0.01,θ=0.25 andi=5%.
not change or decreases slightly. Therefore, when the valuation rate of interestiis equal to the rate of interest used in discounting,j, changes inMandkare principally in response to changes inj.
As for the caseF0=0, the results show that the increase inθcauses a significant change in the optimal choices
forMandk. The reason for this change has already been discussed. The risk as represented byJ∞is a decreasing function ofθand the level of this decrease is considerable only for the case of highM(or lowk). Therefore, when
θincreases, the risk decreases for high values ofM(low values ofk) and the optimal choice becomes larger forM
(and lower fork). For example, wheni=1% andσ =0.01,M∗=1.01 forθ=0, butM∗=535.0 forθ=1. Table 8 presents the critical values ofσ¯(where they exist) for certain combinations of parameters. The dependence ofσ¯ on these parameters is clear.
2.5. Initial funding level of 50%
Tables 9–13 provide the optimal values ofkandM forF0 =0.50 (z= −0.50) and selected combinations of
θ, σ, j andi, corresponding to the global and local minimum ofg(q)as appropriate.
The results presented by Tables 9–13 show less dramatic variation than the corresponding results in Section 2.3: this arises because of the higher initial funding level. For low values ofθ, we observe that the optimal choice ofM
is not affected by changes inσ, iorj. For example, whenθ =0.5,M∗∼=2 andklies in the range (0.581, 0.646) for each value ofi, jandσ investigated.
Table 8
Critical values ofσ¯: whenF0=0.25
i j θ
0 0.25 0.5 0.75 0.85 0.95 1
0.01 0.01 – – – – – 0.125
0.03 0.01 – – – – – – 0.11
0.03 0.03 – – – 0.02 0.055 0.095 0.215
0.03 0.05 – – 0.055 0.095 0.12 0.17 0.32
0.05 0.03 – – – 0.025 0.06 0.096 0.195
0.05 0.05 – – 0.06 0.99 0.125 0.175 0.29
Table 9
0 0.9950/1.01 0.9950/1.01 0.9950/1.01 0.9951/1.01 0.9951/1.00 0.9952/1.00 0.9953/1.00 0.9954/1.00 0.9956/1.00 0.5 0.6146/1.63 0.6148/1.63 0.6152/1.63 0.6173/1.63 0.6207/1.62 0.6254/1.60 0.6312/1.59 0.6381/1.57 0.6460/1.55 0.75 0.4309/2.34 0.4312/2.33 0.4318/2.33 0.4345/2.32 0.4390/2.29 0.4452/2.26 0.4531/2.22 0.4624/2.18 0.4731/2.13 0.85 0.3376/2.99 0.3380/2.99 0.3386/2.98 0.3416/2.96 0.3466/2.91 0.3536/2.85 0.3624/2.78 0.3729/2.70 0.3850/2.62 0.95 0.2012/5.08 0.2014/5.07 0.2021/5.05 0.2056/4.96 0.2114/4.82 0.2196/4.64 0.2300/4.42 0.2427/4.19 0.2575/3.94 1 0.0099/535.0† 0.0103/318.1† 0.0111/222.8† 0.0147/112.2† 0.0266/46.8 0.0428/26.4 0.0628/17.2 0.0861/12.3 0.1121/9.29 Local minimum ofg(q)
Whenθincreases (except for the cases ofθ=0.95 andθ=1), the optimal choice forMincreases slightly and forkdecreases. Given the initial funding level of 50%, changes inj and/orσ do not cause any dramatic change in
M∗ork∗. Similarly the range of the optimal values is reduced relative to the earlier cases ofF0=0 orF0=0.25.
For higher values ofθ(θ=0.95 orθ=1) and whenσ >σ¯, an increase injleads to a higher optimal choice for
M(and a lower choice fork). For example, wheni=5%, θ=0.95 andσ =0.1,M∗=5.15 andk∗=0.1833 for
j =3%, M∗=6.03 andk∗ =0.1547 forj =5% butM∗ =51.1 andk∗=0.0144 (not tabulated) forj =10%. Whenσ <σ¯, a change in value ofj does not cause any changes inM∗(=M1for these cases) unlike the assumed
rate of return which affects the optimal choice (as noted earlier). So, wheniincreases, the maximum feasible spread period decreases, as can be seen from Table 1. Forσ >σ¯,M∗does not change markedly in response to changes in
i. Hence, in the case ofi=j (see Tables 9, 10 and 13), the optimal choice is principally affected byj. Table 14 shows the values ofσ¯ (where they exist) for combinations ofi, jandθ(=0.95 and 1.0).
0 0.9848/1.02 0.9848/1.02 0.9848/1.02 0.9850/1.02 0.9851/1.02 0.9854/1.01 0.9857/1.01 0.9860/1.01 0.9864/1.01 0.5 0.5996/1.66 0.5999/1.66 0.6003/1.66 0.6025/1.65 0.6060/1.65 0.6108/1.63 0.6169/1.62 0.6240/1.60 0.6321/1.58 0.75 0.4123/2.41 0.4126/2.41 0.4132/2.40 0.4161/2.39 0.4209/2.36 0.4274/2.33 0.4355/2.38 0.4453/2.23 0.4564/2.18 0.85 0.3169/3.12 0.3172/3.12 0.3179/3.11 0.3212/3.08 0.3265/3.03 0.3339/2.97 0.3432/2.89 0.3543/2.80 0.3670/2.70 0.95 0.1763/5.56 0.1767/5.54 0.1776/5.52 0.1816/5.40 0.1883/5.21 0.1974/4.97 0.2089/4.71 0.2227/4.42 0.2384/4.13 1 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.0314/29.4 0.0513/18.3 0.0743/12.8 0.1000/9.64 Local minimum ofg(q)
0 0.0099/218.5† N/A N/A N/A N/A N/A N/A N/A N/A
0.5 0.0099/218.5† 0.0103/144.6† N/A N/A N/A N/A N/A N/A N/A
0.75 0.0099/218.5† 0.0103/144.6† N/A N/A N/A N/A N/A N/A N/A
0.85 0.0099/218.5† 0.0103/144.6† N/A N/A N/A N/A N/A N/A N/A
0.95 0.0099/218.5† 0.0103/144.6† N/A N/A N/A N/A N/A N/A N/A
Table 11
0 0.9743/1.03 0.9743/1.03 0.9743/1.03 0.9745/1.03 0.9748/1.03 0.9753/1.03 0.9758/1.02 0.9764/1.02 0.9770/1.02 0.5 0.5810/1.72 0.5812/1.71 0.5817/1.71 0.5840/1.71 0.5878/1.70 0.5930/1.68 0.5994/1.66 0.6071/1.64 0.6157/1.62 0.75 0.3903/2.54 0.3906/2.54 0.3913/2.54 0.3944/2.52 0.3995/2.49 0.4064/2.44 0.4152/2.39 0.4255/2.34 0.4373/2.27 0.85 0.2930/3.37 0.2934/3.37 0.2942/3.36 0.2977/3.22 0.3036/3.26 0.3116/3.18 0.3216/3.08 0.3334/2.97 0.3469/2.86 0.95 0.1490/6.55 0.1495/6.53 0.1505/6.48 0.1553/6.29 0.1529/6.00 0.1732/5.65 0.1858/5.28 0.2005/4.90 0.2172/4.53 1 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.0281/32.8† 0.0378/24.5† 0.0541/17.4 0.0791/12.1 Local minimum ofg(q)
0 0.0099/218.5† 0.0103/144.6† N/A N/A N/A N/A N/A N/A N/A
0.5 0.0099/218.5† 0.0103/144.6† N/A N/A N/A N/A N/A N/A N/A
0.75 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† N/A N/A N/A N/A N/A N/A
0.85 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† N/A N/A N/A N/A N/A N/A
0.95 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† N/A N/A N/A N/A N/A N/A
1 N/A N/A N/A N/A N/A N/A N/A N/A N/A
2.6. Initial funding level of 75%
As in Section 2.4, the detailed results for this case ofF0=0.75 (z= −0.25) are not presented here.
With an initial funding level of 75%, the results are similar to those for 50% funding. Except for the case of
θ=1, an increase inj and/or inθdoes not cause any dramatic change inM∗for all values ofσ. The effect of the valuation rate of interestionM∗is also of minor significance (except for the case ofθ=1).
Whenθ =1, the results do indicate some dramatic changes inM∗. For example, whenσ =0.01,M∗=43.8 andk∗=0.0211 fori=3% andj =1% andM∗=535.0 andk∗=0.0099 fori=1% andj =1%. The higher initial funding level, means that the effect of the valuation rate of interest is more significant than for lower initial funding levels. This is also illustrated in Fig. 6 which presents the variation ofJ∞viewed as a function ofi.
Table 12
Optimal values ofk∗andM∗: whenF0=12AL,i=5%, j=3% θ σ
0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Global minimum ofg(q)
0 0.9848/1.02 0.9848/1.02 0.9848/1.02 0.9849/1.02 0.9851/1.01 0.9854/1.01 0.9856/1.01 0.9860/1.01 0.9864/1.01 0.5 0.6029/1.64 0.6031/1.64 0.6036/1.64 0.6056/1.64 0.6091/1.63 0.6137/1.62 0.6196/1.60 0.6265/1.58 0.6344/1.56 0.75 0.4153/2.36 0.4156/2.36 0.4162/2.36 0.4191/2.34 0.4237/2.32 0.4301/2.28 0.4381/2.24 0.4477/2.20 0.4586/2.15 0.85 0.3194/3.04 0.3198/3.04 0.3205/3.03 0.3237/300 0.3290/2.96 0.3364/2.89 0.3456/2.82 0.3566/2.74 0.3692/2.65 0.95 0.1778/5.30 0.1782/5.28 0.1791/5.26 0.1833/5.15 0.1902/4.97 0.1996/4.75 0.2113/4.50 0.2252/4.24 0.2412/3.98 1 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0208/35.1 0.0401/19.9 0.0614/13.8 0.0848/10.4 0.1102/8.22 Local minimum ofg(q)
0 0.0099/143.6† 0.0103/98.7† N/A N/A N/A N/A N/A N/A N/A
0.5 0.0099/143.6† 0.0103/98.7† N/A N/A N/A N/A N/A N/A N/A
0.75 0.0099/143.6† 0.0103/98.7† N/A N/A N/A N/A N/A N/A N/A
0.85 0.0099/143.6† 0.0103/98.7† N/A N/A N/A N/A N/A N/A N/A
0.95 0.0099/143.6† 0.0103/98.7† N/A N/A N/A N/A N/A N/A N/A
Table 13
0 0.9743/1.03 0.9743/1.03 0.9743/1.03 0.9745/1.03 0.9748/1.03 0.9752/1.03 0.9757/1.02 0.9763/1.02 0.9769/1.02 0.5 0.5841/1.70 0.5843/1.69 0.5848/1.69 0.5870/1.69 0.5907/1.68 0.5957/1.66 0.6020/1.65 0.6094/1.63 0.6178/1.60 0.75 0.3927/2.49 0.3931/2.49 0.3937/2.49 0.3968/2.47 0.4018/2.44 0.4087/2.40 0.4173/2.35 0.4275/2.30 0.4390/2.24 0.85 0.2947/3.29 0.2951/3.28 0.2958/3.27 0.2994/3.24 0.3052/3.18 0.3132/3.10 0.3232/3.01 0.3349/2.91 0.3483/2.80 0.95 0.1480/6.28 0.1486/6.26 0.1497/6.22 0.1547/6.03 0.1627/5.75 0.1734/5.42 0.1865/5.06 0.2016/4.71 0.2187/4.36 1 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† 0.0144/51.1† 0.0199/36.6† 0.0274/27.5† 0.0391/20.3 0.0620/13.7 0.0874/10.1 Local minimum ofg(q)
0 0.0099/143.6† 0.0103/98.7† N/A N/A N/A N/A N/A N/A N/A
0.5 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† N/A N/A N/A N/A N/A N/A
0.75 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† N/A N/A N/A N/A N/A N/A
0.85 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† N/A N/A N/A N/A N/A N/A
0.95 0.0099/143.6† 0.0103/98.7† 0.0110/78.1† N/A N/A N/A N/A N/A N/A
1 N/A N/A N/A N/A N/A N/A N/A N/A N/A
Fig. 6 demonstrates that the risk as represented byJ∞ is an increasing function ofi for high values ofM.
Therefore, an increase ini leads to an upwards shift in the risk for the long spread periods, making the optimal choice ofMshorter.
Whenθincreases, the results show that the optimal choice increases. Whenθ=1, for low values ofσ, the risk is minimised whenM∗ = M1. For higher values ofσ, the optimal choice forM decreases and forkincreases.
Table 15 indicates the critical values ofσ¯ whenθ=1.
It is clear that the influence of the high initial funding level is more significant than any of the other parameters, making the optimal choice shorter for most cases, when compared with theF0=0 andF0=0.50 cases discussed
Fig. 6. Graph ofJ∞(i)whenσ=0.01,θ=1 andj=1%.
Table 15
Critical values ofσ¯: whenθ=1 andF0=0.75
i j σ¯
0.01 0.01 0.095
0.03 0.03 0.13
0.03 0.05 0.24
0.05 0.05 0.17
0.05 0.1 0.37
2.7. Initial funding level of 100%
Tables 16–20 provide the optimal values ofkandM∗whenF0=1 and for selected combinations ofθ, σ, j and
i. We note in this case, that there is only a global minimum forg(q).
When the initial unfunded liability is zero, a different value of the interest rate(j )used for discounting does not lead to a markedly different optimal choice forkorM— except for the case ofθ =1. From Tables 16–20, it can be seen that, for low values ofθ, the results forkandMdepend little onσ,iorj. For example, whenθ=0.5,M∗
lies in the range (1.52, 1.63) andk∗in the range (0.6134, 0.6511) for the values ofσ, iandj investigated. Whenθ increases, there is a slight increase in the optimal choice for M, as for the other cases discussed in Sections 2.3–2.6. When we are only concerned with stabilising the contribution rate (θ =1), the optimal choice
Table 16
Optimal values ofk∗andM∗: whenF
0=AL,i=1%, j=1% θ σ
0.01 0.03 0.05 0.1 0.15 0.2 0.25 0.3 0.35
Global minimum ofg(q)
0 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1 1/1
Table 17
0.5 0.6188/1.61 0.6190/1.61 0.6194/1.61 0.6214/1.60 0.6246/1.60 0.6290/1.59 0.6345/1.57 0.6411/1.56 0.6486/1.54 0.75 0.4340/2.29 0.4343/2.29 0.4348/2.29 0.4374/2.27 0.4417/2.25 0.4476/2.22 0.4551/2.19 0.4640/2.14 0.4742/2.10 0.85 0.3400/2.91 0.3404/2.91 0.3410/2.91 0.3439/2.88 0.3486/2.84 0.3553/2.79 0.3636/2.73 0.3737/2.66 0.3854/2.58 0.95 0.2025/4.85 0.2029/4.84 0.2036/4.83 0.2069/4.75 0.2125/4.63 0.2202/4.47 0.2302/4.28 0.2424/4.07 0.2567/3.84 1 0.0099/218.5† 0.0107/121.6 0.0122/33.7 0.0191/48.7 0.0305/30.2 0.0459/20.4 0.0650/14.6 0.0873/11.0 0.1124/8.61
Table 18
0.5 0.6134/1.63 0.6136/1.62 0.6140/1.62 0.6160/1.62 0.6192/1.61 0.6237/1.60 0.6293/1.58 0.6359/1.57 0.6434/1.55 0.75 0.4269/2.33 0.4272/2.33 0.4277/2.32 0.4304/2.31 0.4346/2.29 0.4405/2.26 0.4480/2.22 0.4569/2.18 0.4671/2.13 0.85 0.3322/2.98 0.3325/2.98 0.3331/2.97 0.3360/2.95 0.3407/2.91 0.3473/2.85 0.3557/2.79 0.3657/2.71 0.3773/2.63 0.95 0.1936/5.07 0.1939/5.06 0.1946/5.04 0.1979/4.96 0.2033/4.83 0.2109/4.66 0.2207/4.46 0.2327/4.23 0.2468/4.00 1 0.0099/218.5† 0.0103/144.6† 0.0111/110.9† 0.0145/67.8† 0.0202/45.8† 0.0281/32.8† 0.0468/20.0 0.0696/13.7 0.0952/10.1
Table 19
0.5 0.6222/1.59 0.6224/1.59 0.6228/1.59 0.6247/1.59 0.6278/1.58 0.6321/1.57 0.6375/1.56 0.6438/1.54 0.6511/1.52 0.75 0.4375/2.25 0.4377/2.24 0.4383/2.24 0.4408/2.23 0.4450/2.21 0.4508/2.18 0.4581/2.15 0.4669/2.11 0.4769/2.07 0.85 0.3434/2.84 0.3437/2.83 0.3443/2.83 0.3472/2.81 0.3519/2.77 0.3584/2.72 0.3668/2.66 0.3767/2.59 0.3882/2.52 0.95 0.2059/4.62 0.2062/4.61 0.2070/4.59 0.2103/4.52 0.2160/4.41 0.2239/4.27 0.2340/4.09 0.2463/3.90 0.2606/3.69 1 0.0288/26.3 0.0296/25.8 0.0310/24.8 0.0375/21.1 0.0482/17.0 0.0628/13.5 0.0809/10.8 0.1021/8.81 0.1259/7.29
Table 20
forM, as previously, is as large as possible. Therefore, forσ < σ¯,M∗ =M1which decreases whenirises but
remains the same whenj changes. Forσ >σ M¯ ∗is shorter. Given the initial funding level of 100%, the optimal spread period is more sensitive to changes in the valuation rate of interest. For example, whenσ =0.01,j =3% andθ =1,M∗ =26.3 andk∗=0.0288 fori=5%, butM∗=218.5 andk∗=0.0099 fori=3%. Therefore, when the valuation rate of interestiis equal to the rate of interest used in the discounting termj (see Tables 16, 17 and 20), changes in the optimal choice are attributable to changes ini.
For the specific cases tabulated, whenθ=1,i=3% andj =5%,σ¯ =0.21.
3. Further comments
3.1. Minimising the solvency risk
Ifθ =0, we are minimising the solvency risk. The degree of security will depend on the speed with which the shortfall is removed by means of special contributions. In this case, the best course of action would be to pay the full amount of the shortfall as it arises without spreading any payments into the future (i.e.M∗=1,k∗=1). But this may not always be attractive, or even possible, from the sponsoring employer’s point of view.
However, Tables 2–6 show that when the initial assets are much less than the initial liability (e.g.F0 =0), the
optimal spread period is much longer, especially, for low values ofσ. In particular, forσ <σ¯, the optimal choice is the maximum feasible spread periodM1which decreases as the mean returniincreases. Whenσ >σ¯, the optimal
spread period is approximately equal to 1 (andk∗∼=1), for all values investigated for the parametersi, jandz. The results of Sections 2.5 and 2.7 show that asF0is increased, the optimal choices for minimising the solvency
risk areM∗∼=1 andk∗∼=1.
3.2. Minimising the contribution rate risk
Ifθ =1, we are minimising the contribution rate risk. We are concerned with stabilising the contribution rate by spreading the unfunded liability over as long a period as possible. Stable contributions enable the employer to plan more effectively future cash flows. Therefore, in order to make the call on the employer’s resources more stable, the actuary should choose the period for the extinguishing of the unfunded liability to be as long as possible (and making the corresponding value ofkas small as possible), otherwise the range of variation of the contribution rates is increased.
The length of the spread period decreases asσ increases. Forσ <σ¯, the optimal choice isM1. Forσ >σ¯,M∗
becomes shorter (andk∗ larger) according to the particular combinations ofi, j andz. When the initial funding level (represented byz) decreases, the contribution rate required rises. If our objective is one of minimising the contribution rate risk, then the optimal spread period increases,M∗, and the optimal choice ofkdecreases.
For theσ <σ¯ cases, the optimal choiceM∗=M1does not change whatever the value of the interest rate used
in the discounting process (M1does not depend onj). Forσ > σ¯, a higher value ofj leads to a longer optimal
choice forM∗ (and a lower value fork∗). An increase in j means that greater emphasis is being placed on the shorter-term state of the pension fund. For the case of an initial funding deficit (low value ofF0), this means that
a higher adjustment to the contribution rate is required. If we are concerned with minimising the contribution rate risk, a higher value ofM∗should be chosen (or lower value ofk∗) so as to reduce the variation in the contribution rate. The higher is the initial funding deficit, the greater is the impact ofjon the optimal choice.
The results are also sensitive to changes in the interest rate(i). The optimal choiceM∗decreases (andk∗increases) wheniincreases for each value ofσ. Forσ <σ¯, the changes inM∗arising from changes inicorrespond to Table 1. For theσ >σ¯ cases, the extent to which the results are affected by changes in the investment assumption depends on the initial level of assets. If the pension fund had no assets initially (F0=0), the impact onM∗andk∗would be
on the assets and consequently to a lower contribution and a smallerM∗. Hence, increasingihas a larger impact on the optimal choice,M∗, when the initial funding level as represented byzis high.
Dufresne (1988) considers the trade-off between the limiting variances of the contribution rate risk and of the fund level, recognising that improved security may imply regularly adjusted contribution rates and, conversely, stable contribution rates may be achieved by a greater variation in the fund level. He minimises the ultimate level of these variances and finds a region forM(1, M∗), where
He calls this region an optimal one, in the sense that, forM > M∗, both limiting variances increase with increasing
Mand, forM≤M∗, the limiting variance of the contribution rate risk decreases and the limiting variance of the fund level increases with increasingM. (We note that this line of research has been taken further by Cairns and Parker (1997) and Owadally and Haberman (1999).)
Therefore, we may consider as a measure of the contribution rate risk the variance of C(t )in the limit, i.e.
VarC(∞). We recall Eq. (14): VarC(∞) = k2(bAL2/(1−a)). Dufresne (1988) shows thatk∗ (as defined by
Hence, we must find the values ofkfor whichJ∞or
β(k)= k
2[z2q2(1−w)(1−wq)+AL2(1−wq2)(1−wq)+2zALq(1−wq2)(1−w)]
[1−w(1−k)2y](1−wq2)(1−wq) (20)
is minimised.
We consider the case ofz= −AL for convenience. Fig. 7 shows that the spread periodMwhich minimisesJ∞
is longer than the spread period which Dufresne defines as the optimal choice for minimising VarC.
In our formulation of the problem, the criterion of minimising the contribution rate risk is defined as a time-weighted sum of the VarC(t ). Hence, the discounting factor is the weight applied to the variance which means that forj >0 (w <1), more emphasis is placed on the shorter-term variances. On the other hand, minimising VarC(∞)means that we consider only the ultimate situation (t → ∞).
We note also, that in the special case wherez=0, so thatF0=AL, (20) simplifies to
β(k)= k
2AL2
1−w(1−k)2y,
which, in a similar vein to Dufresne’s result (19), has a minimum atk=1−1/wy.
3.3. Minimising the risk: the general case (0< θ <1)
Complete security or complete stability is not always an over-riding principle (0< θ <1). In this more general case, it is necessary to consider how quickly a particular contribution arrangement would meet the underlying pension scheme liability, in order to build up the security of the benefits.
We note that typically VarF (t )is much larger that VarC(t )(forM >1 ork <1). Thus, the discussion of Section 1.5 and Eq. (11) demonstrate that VarC(t )=k2VarF (t ). This means that the most interesting values ofθ lie in the range (0.9, 1). The results in Sections 2.3–2.7 bear out this feature, showing a relative insensitivity to the value ofθwhenθis below 0.9.
An alternative formulation based on scaling VarF (t )(e.g. by AL2) and VarC(t )(e.g. by NC2) leads to a similar problem in that the latter is then much larger than the former for low values ofM.
Whenσ¯ exists, we observe that, forσ <σ¯, the optimal choice isM1, so that we want to spread the unfunded
liability for as long as possible. Forσ >σ¯,M∗is much shorter and depends on the particular combinations of
i, j, zandθ.
From the tables in Section 2 it can be seen that, whenθincreases, the optimal spread period increases (and the optimal choice ofkdecreases) because we are more concerned with the criterion of stability. The sensitivity of the risk toθdepends on the particular values of the other parameters. Tables 2–6 indicate that, for low values ofσ, the optimal choice (=M1) is independent ofθ(F0=0). Table 10 shows thatM∗increases andk∗decreases whenθ
increases for each value ofθ(F0=0.50). Table 17 and unpublished tables for the casesF0=0.25 andF0=0.75
show similar features.
When the initial funding level(F0)rises, the risk as represented byJ∞is minimised for shorter spread periods.
We consider Eq. (17) as a function ofz,−AL ≤z ≤ 0.J∞is an increasing function ofzand is more sensitive to changes inzfor high values ofq (i.e. high values ofMand low values ofk). For low values ofq, the risk is approximately constant. So, when the initial funding level rises, the riskJ∞remains approximately constant for low values ofM, but it increases to a substantial extent for high values ofM. With the objective of minimising the riskJ∞, the optimal spread period becomes shorter. The extent to which the optimal choice is decreased, when the
initial funding level rises, depends on the particular combination of the other parameters. From a comparison of the results (see, e.g. Tables 4 and 11), it can be observed that, for low values ofσ, a lower initial funding level makes the optimal spread period jump from a low value to become equal to the maximum feasible spread periodM1. For
higher values ofσ, the effect of the initial funding level is of lesser significance.
The higher is the initial level of assets, the greater is the impact of the assumed rate of return (i) on the optimal choice. Because of the interest earned on the plan’s high initial funding level, the criterion of security is satisfied when a small spread period is chosen, without leading to variations in contribution rate. Hence, the values of the optimal period range from 1.5 to 5.1 (andk∗ranges from 0.1936 to 0.6459) whenz=0, as Tables 16–20 indicate for the range of parameters investigated.
We note that, in the special case wherez=0, the form ofg(q)simplifies to
g(q)= (θ (1−qv)
2+1−θ )AL2
a ratio of two polynomials of degree 2 inq. The solution of(d/dq)g(q)=0 then becomes the simpler matter of solving a quadratic equation inq.
With the objective of placing more emphasis on the shorter-term state of the pension fund (a higher value ofj), the results become more dependent onj, the lower is the initial funding level and forσ >σ¯ (forσ <σ¯,M∗=M1
which is independent ofj). When the short-term state of the pension fund is to be emphasised and a large initial unfunded liability exists, minimisation of the riskJ∞can be achieved by small changes to the contribution rate.
This means that longer spread periods (and smaller values ofk) should be chosen, as illustrated by Tables 4 and 6.
4. Conclusions
From the results in Section 2, it is clearly seen that, the lower is the initial funding level (F0), the greater is the
range of the optimal choices. Hence, when the funding strategy involves a low initial funding level, a high choice forM(or a low choice fork) is necessary (under particular choices of the other parameters) in order to remove the initial funding deficit while limiting the variation in the contribution rate.
The choice of the parameterθis of great importance as it reflects which of the variability of the fund or of the contribution is required to be more stable from the employer’s point of view. The results are presented for different values ofθand allow a comparison of the optimal choices for valuation methods with different principal objectives (e.g. more emphasis on stable contribution rates or on funding of the actuarial liability).
The use of a discounting factorw 6= v clarifies the effect of the assumed rate of return (i) and of the rate of interest for discounting variances (j) on the range of the optimal spread periods. The results support the conclusion that, the greater is the departure from a 100% initial funding level, the less is the effect of the assumed rate of return. Given a high initial funding level, an alteration of the assumed rate of return does have an important effect on the range of the optimal spread periods. In particular, the interest earned on the pension plan’s assets leads to a small range of optimal choices.
The rate of return used in the discounting process (j) indicates which of the short-term or the long-term state of the pension fund is to be more emphasised. The conclusion is that the lower is the initial funding level, the greater is the impact ofj on the range of the optimal choices. We also demonstrate that in the long term, the risk as represented byJ∞is independent of the initial funding level and the range of the optimal choices forMandkis much diminished.
Finally, it is seen that, the range of the optimal spread periods is large for particular combinations of the parameters. For these cases and for low values ofσ, the optimal choice is to makeMas large as possible (andkas small as possible), effectively leading to a pay-as-you-go solution, as discussed in Section 2.3. The critical values ofσ, which make the optimal spread period jump from a low value to the maximum feasible spread periodM1, are shown in
Section 2.
It is worth noting that, in UK, the values of the spread period used from most pension schemes in practice may be different from what the tables in Section 2 indicate. As far as the solvency risk is concerned, the minimum funding requirement (MFR) rules, as given in Guidance Note (1998), place a restriction on the choice ofM.
In particular, if the funding level (which is defined as the ratio between the values of assets and liabilities under a prescribed set of assumptions) is less than 90%, the deficit has to be eliminated within 1 year. If the funding level is more than 90%, the unfunded liability has to be removed within 5 years. WhenF0=0 andσ <σ¯, Tables 2–6
show that the spread period which minimises the riskJ∞is longer than would be allowed by the MFR rules. For all the other cases tabulated, the results presented in Section 2 are consistent with MFR rules.
Section 2. Thus, Fig. 1 would implyM∗∼=1 and Fig. 5 would implyM∗∼=25 for the combinations of parameters depicted for the cases whereσ < σ¯. The results of this paper suggest that, forσ < σ¯, if we are interested in minimising the riskJ∞, values ofMwhich are much smaller than this indicative range 20–25 (traditionally chosen
by practitioners) would be optimal. These conclusions are in line with those of Dufresne (1988) and Owadally and Haberman (1999), in relation to gains and losses arising from investment return fluctuations, and those of Kryvicky (1981) in relation to gains and losses arising from benefits amendments.
As noted earlier, the relative magnitudes of the variances ofF (t )andC(t )mean that only a narrow range ofθis of practical interest. An alternative formulation would be to return to the performance criterion and allow a choice of non-negative weights without the restriction of their having to sum to 1, e.g.
sJT = T−1 X
t=s
wt[θ0VarC(t )+θ1VarF (t )].
At a more general level, the problem as formulated requires an initial choice ofM(ork) to be made and to be permanently maintained thereafter, irrespective of the information available. Although many practitioners use a fixed value ofM, an alternative, less restrictive and possibly superior approach would be to relatekto the information as it develops over time: this is the subject of current investigation.
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