How to Re-Member?
17
How to Re-Member?
Paul J. Campbell
Mathematics and Computing
Beloit College
700 College St.
Beloit, WI 53511-5595
[email protected]
All models are wrong but some are useful.
—George E.P. Box [1979, 202]
Introduction
Like many mathematicians, I belong to a number of professional societies and subscribe to lots of their journals and other magazines. Here I
apply mathematics to the decisions about renewals, continuing the agenda
begun in Campbell [2008] of applying mathematics to mundane daily life.
Renewal of a membership or subscription usually presents choices about
length for the renewal. Although I am favorably disposed toward the
societies—some of which I have belonged to for more than 40 years—and
wish them success and enhancement of their revenues, competing claims
on my finances stimulate me to consider how to optimize (from my point
of view) renewal choices.
Optimization of multi-year subscription pricing from the viewpoint of
the publisher was treated with a simple model by Dudley [1993], who
• surveyed the practices of 39 magazines,
• concluded that “mathematics does not enter into the rate-setting,”
• suggested that failure to apply simple mathematics in such an ordinary
and common setting casts doubt on the value of emphasizing applications in teaching mathematics, and
• concluded that “emphasis on applications is certainly useless and perhaps harmful. . . [so] it is better to present mathematics to students as a
glorious adventure for the mind.”
c
The UMAP Journal 30 (1) (2009) 17–36. !Copyright
2009 by COMAP, Inc. All rights reserved.
Permission to make digital or hard copies of part or all of this work for personal or classroom use
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18
The UMAP Journal 30.1 (2009)
Prof. Dudley and I are engaged elsewhere in a friendly long-running conversation in print about the values of education in mathematics [Campbell
2006; Dudley 2008; Campbell 2009], so I will not contest here his concluding
claim. In fact, I agree wholeheartedly with his assertion that one should
indeed apply mathematics in ordinary settings; and without casting aspersions on individuals who fail to do so (e.g., myself in past years in regard
to membership and subscription renewals!), I will essay an attempt here.
Nievergelt [1996] responded to Dudley’s claim of “the irrelevance of
mathematics to price setting.” Nievergelt
• examined the pricing policies of some mathematical societies for their
journals;
• noted that “in a decade of informal search I had been unable to identify
and document any successful example of optimal pricing” of any good;
• suggested various explanations for failure to apply optimal pricing;
• provided a case study of one mathematical society; and
• asked his own provocative question, “whether business mathematics has
a positive net present value to the business community”—particularly
if undergraduates studying business are unconvinced. He answered by
citing $1.4 billion saved by one airline over three years through use of
linear programming.
The Situation
Society A
Society A offers
• a multi-year regular membership for two to five years at that many times
the cost of a regular one-year membership;
• a retired membership at about 40% of the cost of a regular membership;
and
• a life membership determined by age on January 1:
≥60: 5 times the cost of a regular membership;
≥50: 10 times the cost of a regular membership;
≥40: 15 times the cost of a regular membership.
This society’s renewal form also offers a free “emeritus” membership
“after retirement on account of age or a long-term disability, with membership extending over at least twenty years.” This category, though enshrined in the bylaws of the society for at least the past half century, does
not appear at the society’s membership Web pages. Also, it is not clear
How to Re-Member?
19
on the face of it what “on account of age” is supposed to mean: voluntary retirement at the formerly “customary” age of 65? forced retirement
at a statutory age? or just too feeble to keep working? Fortunately, the
society offered a clarification, which we reveal later.
Society B
Society B offers
• a two-year regular membership at twice the cost of a regular one-year
membership;
• a retired membership at between one-half (for a single journal) and twothirds (for all journals) times the cost of a regular membership; and
• a life membership “based on actuarial calculations,” in 5-year brackets
from age 72 (and up) down to the bracket ages 22–26, at a cost between
about 6 and 13 times a regular membership.
Considerations
What If There Is No Discount for Future Years?
Even if a multi-year membership is priced at no discount from the corresponding successive one-year memberships, it can be more favorable if the
society raises membership costs at a rate higher than the effective annual
rate of interest that the individual could otherwise earn on the money.
Taxes1
Under certain circumstances, an individual can avoid paying U.S. federal, state, and local income tax on expenses related to employment by
deducting those expenses from taxable income. The individual must “itemize deductions” and can deduct professional expenses from federal taxes
only under the category of “miscellaneous deductions” in excess of 2% of
adjusted gross income.
Moreover, per the Tax Reform Act of 1986 [Kott 1987], the deductions for
a multi-year membership or subscription should be allocated and deducted
pro rata over the period of the membership. How deductions should be
apportioned for a lifetime membership paid all at once is not addressed in
IRS publications and may even not have been resolved definitively by a
Private Letter Ruling or in tax court.
Such a deduction is not available to an individual after the individual is
no longer employed in the profession (e.g., fired or retired).
1 Nothing
in this article should be construed as offering tax or legal advice.
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The UMAP Journal 30.1 (2009)
If the individual is self-employed (e.g., as a consultant), such professional expenses can be offset directly against income rather than be deducted under miscellaneous deductions, with their floor of 2% of adjusted
gross income.
We assume that the member before retirement
• has enough deductible expenses to justify itemizing deductions instead
of taking the fixed-amount standard deduction;
• is entitled to deduct the membership expense; and
• renews at the end of a year, so that the year’s tax savings are virtually
immediate;
but after retirement receives no tax benefit from the membership.
Maybe You Can Get It for Free
If an individual serves at an institution with an institutional membership
in a society, the institutional membership may include the right to nominate
a certain number of individuals for complimentary (cost-free) individual
memberships.
No Refunds
An individual who dies or otherwise becomes unable to benefit from a
membership loses the remaining value of membership costs already paid
(there are no refunds).
Dudley’s Models for a Two-Year Renewal
As a starting point, we recapitulate Dudley’s models for a two-year
subscription, translated here from a subscription to a membership.
Notation
c = cost (advertising, etc.) of obtaining one new one-year membership,
s = price of a one-year membership,
t = price of a two-year membership,
p = probability that a one-year member will renew for the second year,
q = probability of member “disenchantment” (and hence disinclination to
renew) before the end of the first year,
How to Re-Member?
21
i = annual rate of interest at which the society raises the cost of membership,
and
j = annual rate of interest that the member could earn instead on the money
if not spent on membership.
Dudley’s Model for the Society
Dudley equates the present value of a two-year membership paid today
with the present value of two consecutive one-year memberships (one paid
today, one paid a year from now), arriving at
µ
∑
∏∂
p
s−c
tS = s 1 +
,
1+i
s
where tS is revenue-neutral pricing for the society in terms of net present
value.
Nievergelt suggests in his case study of one mathematical society that
the failure to use tS and offer a discount on a two-year membership may
reflect that “the society’s management might prefer a higher profit to a
larger readership.”
Dudley’s Model for the Member
For the member, we have
µ
∂
1−q
tM = s 1 +
,
1+i
where tM is revenue-neutral pricing for the member in terms of net present
value. Dudley suggests that 1 − q > p because an individual buying a twoyear membership is less likely to become disenchanted in the first year than
one buying a one-year membership.
Specific Situation
Consider a typical member of a society who
• intends to remain a member for the rest of the individual’s life (i.e., we
disregard the possibility of disenchantment);
• plans to retire z years from now;
• itemizes deductions for membership on tax returns each year until retirement (but not after), for a net tax benefit of proportion t of the cost of
membership (i.e., the individual’s marginal tax rate is t);
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The UMAP Journal 30.1 (2009)
• at retirement will qualify for the retirement membership rate;
• is x years old;
• anticipates an average remaining lifetime, according to a standard life
table for the individual’s sex (though mathematicians may on average
live longer than such tables prescribe); and
• confronts the choice of a multi-year membership for a period of n years
at cost now of nP0 vs. a sequence of annual memberships at cost P0 ,
P0 (1 + i), . . . , P0 (1 + i)n−1 .
For convenience, we collect the notation:
x = age of member,
z = number of years to retirement,
t = member’s marginal tax rate, and
P0 = cost of a one-year membership now.
What should such an individual do in regard to membership renewals
for Society A and Society B?
Assumptions
The following stay constant for the duration of the membership term:
• the interest rates i and j ,
• the individual’s marginal tax rate t,
• the portion of the life table relevant to the individual.
Multi-Year vs. Sequential One-Years
A member who is sure to live through the term of membership without retiring should elect a multi-year membership at the corresponding
multiple of the one-year membership as long as j < i, that is, as long the
member’s interest rate is less than the annual rate of increase in membership cost. Of course, a member cannot predict future membership costs,
only observe immediate past ones (and should take into account that the
costs are rounded up or down to the nearest dollar).
But a member cannot count on living through the term of the membership. We assume that a deceased member will not renew membership
further and that the tax benefit of a multi-year membership will also terminate with death. We take these factors into account by dealing in terms of
expected value. Actuaries use the notation
How to Re-Member?
23
px = the probability of a person alive at age x surviving to age x + 1, and
= the probability of a person alive at age x surviving to age x + k ,
where
k px
k px
=
k−1
Y
px+m .
m=0
The values of px are commonly recorded in life tables. For convenience,
we use values derived from the table of the U.S. Social Security Administration [2004] (which gives not px but instead qx = 1 − px ). (Members of
academic societies are generally likely to live longer than the general population. Richardson [2008, 12, Table 9] illustrates this fact by comparing
life expectancy for ages 62–66 according to the 2007 Social Security table vs.
the 2008 mortality table of the Teachers Insurance and Annuity Association
(TIAA), a pension fund that serves mainly employees in higher education.
We use the Social Security table because TIAA does not publish the full
table that it uses for its annuitants.)
In addition, we let
τ = the tax benefit as a proportion of the cost of an annual membership.
Using those notations, for the multi-year membership to be the better
deal, its expected cost must be less than the expected cost of successive
one-year memberships, with adjustment for the individual’s interest rate j
and for
r = the annual rate of inflation.
Thus, an investment D at rate j will grow to D(1 + j) after one year but in
terms of purchasing power will have value D(1 + j)/(1 + r). Correspondingly, a payment P a year from now has value today P (1 + r)/(1 + j).
We set out the two situations in Figures 1 and 2, using the extremely
useful representation in terms of a timeline, as employed systematically in
Daniel and Vaaler [2007; 2009]. In the timelines, payments by the member
are entered as negative values and tax benefits to the member by positive
values. “VALUE” refers to present value at time 0.
Converting to net present value, we write for each situation what is
called in financial mathematics a time-0 equation of value [Daniel and Vaaler
2007, 79; 2009, 76]. For the n-year membership, we have expected present
value (also called actuarial present value) of the cost (as a negative number)
Cn = −nP0 + τ P0 + τ P0
n−1
X
k=1
k px
µ
1+r
1+j
∂k
.
VALUE:
TIME:
PAYMENT:
VALUE:
TIME:
···
···
−(1 − τ )P0 · · ·
0
−(1−τ )P0
1+r
1+j
∂1
···
···
···
τ P0 (k px )
k
µ
1+r
1+j
τ P0
≥
1+r
1+j
¥k
···
···
···
···
···
···
n−1
−(1−τ )P0 (1+i)n−1
≥
1+r
1+j
n−1
µ
∂n−1
1+r
τ P0 (n−1 px )
1+j
τ P0
−(1 − τ )P0 (n−1 px )(1 + i)n−1
∂k
Figure 2. Timeline for n successive one-year memberships.
−(1 − τ )P0 (k px )(1 + i)k
k
−(1−τ )P0 (1+i)k
Figure 1. Timeline for an n-year membership.
1
µ
τ P0
−nP0 + τ P0 τ P0 (1 px )
0
PAYMENT: −nP0 + τ P0
¥n−1
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The UMAP Journal 30.1 (2009)
How to Re-Member?
25
The present value (also negative) for the n successive one-year memberships is
µ
∂k
n−1
X
1+r
k
Cn×1 = −(1 − τ )P0 − (1 − τ )P0
(k px )(1 + i)
.
1
+
j
k=1
The amount of savings of the multi-year membership over the successive
one-year memberships, in units of the cost of a one-year membership, is
Sn,n×1 ≡
Cn − Cn×1
.
P0
We distinguish with the subscript M or F the respective quantities for
males and females of px and Sn,n×1 . Since empirically for all ages we have
qM,x > qF,x ,
or
pM,x < pF,x ,
then for all ages x and terms n, we also have
SM,n,n×1 < SF,n,n×1 .
Examples
Example: Let x = 60, n = 2, r = 3%, i = 3% (so that the society is
raising its prices at just the rate of inflation), and j = 4% (meaning
that the member would otherwise invest the money at 4%).
For a male, we have
pM,60 = 1 − qM,60 = 1 − .011858 = .988142,
pM,61 = 1 − qM,61 = 1 − .012966 = .987034,
and for τ = .3155 ( a marginal tax bracket of 25% federal plus 6.55%
state), we find that
SM,n,n×1 = −0.0012 < 0;
for a female, we have
pF,60 = 1 − qF,60 = 1 − .007445 = .992555,
pF,61 = 1 − qF,61 = 1 − .008187 = .991813,
SF,2,2×1 = 0.0032 > 0.
So, what to do depends on whether the member is male or female,
though it is a very close call in both cases—the respective savings per
$100 of membership dues are −$0.12 and $0.32. For these interest
rates, marginal tax rate, and this age, the same opposing conclusions
for males vs. females hold for all n > 1.
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The UMAP Journal 30.1 (2009)
Example: As in the previous example but with j = 2%. Now matters
are clear-cut for both sexes:
SM,2,2×1 = 0.0183,
SF,2,2×1 = 0.0229.
In fact, a multi-year membership, for any number of years, is better
for either sex.
Example: As in the previous example but with i = 2%, j = 2%, and
r = 1%. Matters are again clear-cut for both sexes:
SM,2,2×1 = −0.0082,
SF,2,2×1 = −0.0037.
The probability of dying (1.2%, 0.7%) during the first year of the term
outweighs the “benefit” of inflation, and the multi-year membership
is marginally worse.
Example: As in the previous examples but with all interest rates at
3%. For either sex, the multi-year membership is better over a term of
up to 20 years, because the rate of inflation outweighs the probability
of dying during the term. Mortality finally overcomes inflation after
20 years for men and 30 for women.
Observations
• If the member projects that the cost of an annual membership will increase at a rate greater than the rate of interest that the member can
obtain elsewhere (i.e., i > j ), it would seem better to buy a multi-year
membership. However, the decision is not clear-cut; the member’s rate of
interest j still can affect the decision, as can the member’s life expectancy
and marginal tax rate.
• If it is favorable for a member to buy a 2-year membership, then it is
increasingly favorable (in terms of net present value) to buy an even
longer multi-year membership, up to the point when mortality becomes
the dominant consideration.
• Because the probabilities px are greater at lower ages, if it is favorable
to buy a multi-year membership at age 60, it is favorable to do so at all
younger ages, too. I picked the age of 60 for the examples because a fiveyear membership would take the member to the traditional retirement
age of 65. (Not coincidentally, I am 60 as I write this and try to decide
about renewal options for Society A and Society B!)
Life Membership vs. Sequential One-Years
We compare a life membership with sequential annual memberships
(but don’t consider the alternatives of successive multi-year memberships).
How to Re-Member?
27
For a life membership, we assume that the member deducts from taxable
income each year the cost of a one-year membership until the entire cost of
the life membership has been deducted. (Although we have no basis in tax
publications or rulings for this assumption, it appears in good agreement
with the tax code provision that the tax deductions should allocated and
deducted pro rata over the period of the membership.) Let
L(x) = the cost of a life membership at age x,
L(x)
= the cost of a life membership at age x as a multiple of the cost
P0
of an annual membership at the time,
λ=
ρ = the cost of a retired membership as a proportion of the cost of an annual
membership at the time, and
eˆx = life expectancy of a person x years old.
To arrive at the actuarial present value of a life membership, we diminish
its cost by adding the present value of the immediate and future annual tax
deduction each year of the cost of an annual membership at the time of
purchase of the life membership, with potentially a lesser deduction in the
last year, when less than a full annual cost may remain to be deducted. We
also must weight the tax benefit by the probability of survival to that age
(we disregard any potential tax benefit to the individual’s estate).
A ≡ −L(x) + τ P0 + t
"bλc−1
X
k=1
P0 (k px )
µ
1+r
1+j
∂k
+ (bλc px ) [L(x) − P0 bλc]
µ
1+r
1+j
∂bλc #
.
For the life membership to be advantageous compared to a succession
of one-year memberships, its cost must be less than the present value of the
cost of one-year memberships until retirement (with their tax deductions)
plus one-year retired memberships after retirement (with no tax deductions).
Although potentially we should sum the benefits to infinity, we have no
life table data for survival beyond age 120 (the limit of the Social Security
life table is 119, for survival to 120), and at this writing the oldest living
person in the world has just died at 115. Hence we must cut off our horizon
at 120. Since it is not be realistic to count on living that long—though in
2000 mathematician Dirk Struik died at 106—we note that the results below
are affected negligibly by reducing the time horizon to age 100, and are not
affected qualitatively for a time horizon equal to the life expectancy eˆx of a
person of the age at which the life membership is purchased.
We also assume that the member retires before age 120(!).
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The UMAP Journal 30.1 (2009)
B ≡ −P0 (1 − τ ) − (1 − τ )
z−1
X
k
P0 (k px )(1 + i)
k=1
−r
119
X
µ
1+r
1+j
∂k
k
P0 (k px )(1 + i)
k=z
µ
1+r
1+j
∂k
.
Following the custom in mathematical modeling of working with dimensionless quantities, we divide by P0 and express the difference in terms
of λ, the cost of a life membership as a multiple of the cost of a one-year
membership at the time. Consequently, the savings of the life membership
over successive sequential memberships, in units of annual membership
cost, is
A−B
Slife,sequential ≡
=
P0
"bλc−1
µ
µ
∂k
∂bλc #
X
1+r
1+r
−λ+τ
(k px )
+ (λ − bλc)bλc px
1+j
1+j
k=0
µ
µ
∂k
∂k
z−1
119
X
X
1+r
1+r
k
k
(k px )(1 + i)
+r
(k px )(1 + i)
.
+ (1 − τ )
1
+
j
1
+
j
k=0
k=z
For λ an integer, this expression simplifies to
µ
∂k
λ−1
X
1+r
Slife,sequential = −λ + τ
(k px )
1+j
k=0
µ
µ
∂k
∂k
z−1
119
X
X
1
+
r
1
+
r
+ (1 − τ )
(k px )(1 + i)k
+r
(k px )(1 + i)k
.
1
+
j
1
+
j
k=0
k=z
Examples
Example: Society A Suppose that the member is age 40, so x = 40; at
that age, the proportional cost of a life membership is λ40 = 15. Then
for retirement at age 65, we have z = 25, and the proportional cost of
a retired membership is ρ = .3780.
Let i = j = r = 3% and suppose again that t = .3155.
Then
SM,life,sequential = 27.8,
SF,life,sequential = 33.3.
The lifetime membership is clearly a bargain. For the cost of 15 − τ ≈
14.7 annual memberships paid now, the member saves a net present
How to Re-Member?
29
value of about 28 (if male) or 33 (if female) annual memberships. In
other words, for a payment of about 15 annual memberships now,
the member receives a net present value of about 43 (if male) or 48 (if
female) annual memberships.
Example: Society A As in the previous example but with j = 11%
(approximately the average annual return on the New York Stock Exchange 1993–2002). Then
SM,life,sequential = −1.6,
SF,life,sequential = −1.2.
The lifetime membership is almost a competitive investment.
Example: Society A As in the previous examples, but the member
instead is age 60, so x = 60 and the proportional cost of a life membership is λ60 = 5. For retirement at 65, we have z = 5 and we still
have ρ = .3780.
Let i = j = r = 3% and suppose again that t = .3155. Then
SM,life,sequential = 9.1,
SF,life,sequential = 11.5.
Under these circumstances, the member should elect the lifetime membership.
Example: Society B As in the immediately previous example but with
j = 11%. For Society B, we have λ60 = 9.8456 and ρ = .6448. Then
SM,life,sequential = 0.2,
SF,life,sequential = 1.0.
The relatively higher rate for retired membership makes a life membership competitive, even with such a high alternative rate of return.
Paradox: Maybe You Should Retire Earlier
Let all the interest rates be 3%. It may seem paradoxical that for Society B
at x = 60 and z = 5 that we have
SM,life,sequential = 12.4,
but for z = 0 we have
SM,life,sequential = 12.9.
Avoiding 5 years more at full membership dues in favor of 5 years at the
the lesser retired dues would seem to give greater savings in the first case
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The UMAP Journal 30.1 (2009)
than in the second. But in the second case, the tax deductions are spread
over more years farther into the future, allowing the inflation rate to give
them greater leverage.
However, for Society A, the results are reversed: At x = 60 we have for
z=5
SM,life,sequential = 9.1,
but for z = 0 we have
SM,life,sequential = 8.2.
Why the difference? The retired membership in Society A is a much lower
proportion (.38) of annual membership than in Society B (.64), so the moving
of tax deductions into the retired years is overbalanced by the lower savings
on retired memberships.
Conclusions
What the mathematics will recommend depends on the member’s
• assessment of the rate at which the annual membership cost will increase,
• assessment of the rate of interest that the member could otherwise earn
on funds,
• current and future tax situation (including state of residence),
• age,
• assessment of likely longevity,
• retirement plans,
• (as we will see in connection with Society A) how long the member has
belonged to the society, and
• the value the member places in minimizing long-term cost to the member
vs. possibly subsidizing to some extent a perhaps beloved society.
Whether to elect a multi-year membership in either society is sensitive
to the first three considerations, but the potential benefit may be more in
convenience than savings.
Whether to elect a life membership in either society is further sensitive
to the remaining considerations, but in most circumstances it seems to be
financially beneficial to the member to do so.
How to Re-Member?
31
Questions about Life Memberships
Why Don’t More Members Go for Such Good Deals?
Good question. In some instances, a life membership seems to be, in
common parlance, “a heckuva deal.” But
• most people don’t regard dues to professional societies as in any sense
investments;
• a life membership requires a large initial cash outlay;
• members may think that the society prices life memberships actuarially
in a revenue-neutral manner, so that it should make little difference which
membership option a member elects;
• long-time members may be happy to continue annual “support” of their
society;
• members may suspect that what is a good deal for them in a life membership may not be good for the society; and
• members may sense that for them the rate j of interest that they could
otherwise earn may turn out to be less than the rate i of increase in annual
membership cost (perhaps even j = 0).
How Can Societies Afford to Offer Such Good Deals?
Given that the two societies considered here are mathematics societies,
life membership prices may be based in part on actuarial considerations
(though rather coarsely, with the 5- and 10-year age brackets).
How then could it be so beneficial to a member to elect a life membership
without it being detrimental to the society? The answers are
• the member is being subsidized by the government through the tax deduction, a fact likely not taken into account in the society’s pricing;
• the society’s calculations are likely based on a different rate of interest
from what the member uses;
• the society may use revenue from publications to subsidize memberships; and
• if the society miscalculates, life memberships may indeed be detrimental
to its welfare.
Pertinent to the member’s decision are the rate of taxation and the rate
of increase in the cost of one-year memberships (which may roughly parallel inflation). What is important to the society instead is the interest rate
that it can earn from putting funds from life memberships into long-term
investments, a rate much higher (we hope) than inflation.
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The UMAP Journal 30.1 (2009)
A simple accounting for life memberships would have the society credit
the life membership and debit a corresponding annuity-due [Daniel and
Vaaler 2007, 128ff; 2009, 123ff] in the amount of an annual membership for
the remaining years until the member is 65.
Consider a society whose marginal cost of servicing a retired life membership is negligible and which has only a small proportion of life members.
Since retired life members cost nothing, it doesn’t matter how long they live,
so we can also disregard survival. Setting to zero the net present value of the
income stream of the purchase of the life membership and the corresponding annuity-due to age 65 makes the life membership revenue-neutral; life
members who die before age 65 enhance revenue to the society.
Applying this naive and unrealistic model to the rates of Society A results
in approximate effective interest rates for the annuity of 5% at age 40, 7% at
age 50, and 8% at age 60. These are interest rates over and above inflation
(pace Fisher’s effect [Nievergelt 1988]) but in line with traditional rates of
growth for values of securities.2
On the other hand, offering life memberships represents a risk to a society that depends heavily on income from memberships (as opposed to sales
of books and journals). Future expenses and future numbers of members
are difficult to predict.
What I Did
Assumptions
• I am 60 and anticipate retiring at age 70. (My wife is 5 years younger. To
maintain health insurance for both of us until we reach Medicare at age
65, either she or I must work until she is 65. I cannot be sure that she will
live or be working until I am 65 (to cover me), much less to when she is
65 (to cover her)—which is when I will be 70.)
• I optimistically anticipate r = 3%, together with i = 3% for both societies, and pessimistically (or is it optimistically?) expect j = 3% for my
own funds—that is, both the increased cost of annual membership and
my own otherwise investment of such funds will proceed at the same
rate as 3% inflation.
Multi-Year Membership
For either society, for a two-year membership I face only a very small advantage. So a multi-year membership would be attractive only if I expected
a big hike in rates next year.
2 I would be interested in learning the details of any society’s pricing of life memberships. Prof.
Nievergelt would no doubt be elated to find a society whose pricing is in any sense optimal.
How to Re-Member?
33
Life Membership
Society A
An inquiry to Society A revealed that because I have been a member for
more than 20 years, I am entitled to a cost-free emeritus membership upon
retirement! The respondent replied “Many members don’t understand that
they have to have 20 consecutive years of membership . . . our membership
department has decided that it is best that this category is offered rather
than chosen.” The society does not advertise this emeritus membership,
but members who feel that they qualify can feel free to inquire directly.
The net present value to me of a life membership until retirement is 7.1
annual memberships more than the 4.7 annual memberships that I would
pay now (5 memberships minus immediate tax benefit). Of course, this
advantage might decrease or disappear if I were to retire sooner (but see
above under Paradox).
So based on my plans, a life membership would be attractive—except
that for 2009 I have been awarded an annual membership as a benefit of
my institution’s institutional membership and as a result of decisions by
my colleagues.3
A year from now, the net present value to me of a life membership will
be somewhat less; the added value will drop to 6.4 annual memberships.
Saving 1 annual membership in the meanwhile, however, I can wait to
decide whether to take advantage of that opportunity—hoping that the
life membership will still be available without great change in the pricing
structure.
Society B
An inquiry to Society B produced information about current practice.
After concluding in 2002 that life memberships were way underpriced, the
Society repriced them:
• using a merged life table (65% male);
• setting a real rate of return (above inflation) of about 4.3%; and
• increasing rates since 2002 by the same percentage increase as regular
dues.
As before, let
P0 = price of a regular one-year membership, and
L(x) = price of a life membership.
3 My
department awards as many of its allocated annual memberships as possible to students
aiming toward graduate school in mathematics; for any remaining memberships, the department
has decided that faculty should take turns. I personally feel that in accepting such a benefit it is
incumbent on me to contribute to the College the amount of the benefit.
34
The UMAP Journal 30.1 (2009)
Let also
c = average cost to the society of a regular one-year membership, and
bx = present value of a stream of payments of (P0 − c) (excess of dues over
expenses) of a sequence of one-year memberships over the life of the
member.
Society B decided to set
L(x) = cmx + bx ,
where mx was calculated to provide $1 per year over the life of a member
starting at age x. For example, in 2002, m62 = $15.33. The multiplier m62
was applied to the average cost to the Society for a member, and e62 was
determined to be about $50.
Payments for life memberships go into a fund. At the start of a life
membership, L(x) is paid into the fund and bx is transferred from the fund
to dues income; then each year during the life of the member, the current
price of a one-year membership is transferred from the fund to dues income.
If average costs increase faster than regular annual dues (as seems to
have been the case since 2002 for Society B, according to my correspondent),
life memberships can become underpriced according to this model.
Of course, one can question the financial model. (These days, we are
led to question all financial models!) For example, to what extent should
a model be based on marginal cost rather than on average cost?
I find for my present circumstances and retirement plans that
SM,life,sequential = 12.7.
Considering that the immediate investment is (9.8456 − 0.3155) ≈ 9.53,
taking out a life membership would cost about 10 annual memberships
but give me a net present value of about 23 annual memberships—-in effect, doubling my money! So maybe life memberships in Society B are
underpriced—from my point of view, whether or not they are from the
point of view of the society.
In two more years, I would be 62, and for Society B the current value of
λ62 is 7.5676. I would have x = 62, z = 8, and eˆ62 = 18.85. Assuming that
that the cost of a life membership would increase over those two years by
two years of inflation, a factor of 1.032 , I find
SM,life,sequential = 12.0,
with most of the benefit taking place after retirement.
So, as far as life membership goes, I expect to be able to get about the
same deal two years from now. I’m tempted by the opportunity to almost
double my money; but why commit now, amid the uncertainties of current
economic conditions?
I renewed with a one-year membership.
How to Re-Member?
35
References
Box, George E.P. 1979. Robustness in the strategy of scientific model building. In Robustness in Statistics, edited by Robert L. Launer and Graham
N. Wilkinson, 201–236. New York: Academic Press.
Campbell, Paul J. 2006. Calculus is crap. The UMAP Journal 27 (4): 415–430.
. 2008. Always buy two (or more)?. The UMAP Journal 29 (4):
445–456.
. 2009. All mathematics is crap as far as humanists are concerned;
or, a Scottish verdict for calculus and mathematics in general. The UMAP
Journal. To appear.
Daniel, James W., and Leslie Jane Federer Vaaler. 2007. Mathematical Interest
Theory. Upper Saddle River, NJ: Pearson Prentice Hall. 2009. 2nd ed.
Washington, DC: Mathematical Association of America.
Dudley, Underwood. 1993. Two-year magazine subscription rates. American Mathematical Monthly 100 (1): 34–37.
. 2008. Calculus isn’t crap. The UMAP Journal 29 (1): 1–4.
Klott, Gary. 1987. Deduction seekers find benefits and barricades. New York
Times (19 February 1987) http://query.nytimes.com/gst/fullpage.
html?res=9B0DE5D81630F93AA25751C0A961948260&sec=&spon=&
pagewanted=all .
Nievergelt, Yves. 1988. Fisher’s effect: Real growth is not interest less
inflation. Mathematics Teacher 81 (7) (October 1988): 546–547.
. 1996. Is optimal pricing a myth from business calculus? Is
business calculus an oxymoron? American Mathematical Monthly 103 (2):
143–148.
Richardson, David P. 2008. Social Security: Factors affecting the decision
of “when to begin” benefits. TIAA-CREF Institute Trends and Issues (December 2008).
http://www.tiaa-crefinstitute.org/articles/tr120108a.html .
U.S. Social Security Administration, Office of the Chief Actuary. 2008. Period life table, 2004. http://www.ssa.gov/OACT/STATS/table4c6.
html . Last reviewed or modified March 27, 2008. Accessed 1 January
2009.
Acknowledgment
I thank the staff at societies who provided information, and I am grateful
for numerous helpful suggestions from the referee.
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The UMAP Journal 30.1 (2009)
About the Author
Paul Campbell is professor of mathematics
and computer science at Beloit College. His interests include environmental modeling, probability and statistics, computer science, combinatorial games, and the history of mathematics.
In 2004–05, he was in a statistics group at the
University of Augsburg, Germany. For several
years, he enjoyed doing a seminar on the mathematics behind the television series Numb3rs. Responding to student demand, he has begun teaching actuarial science and encouraging actuarial
careers. Doing so helped inspire this article and
facilitate its analysis. He has edited this Journal
since 1984.
As Albert Einstein for
Fasching (Mardi Gras), 2005