Simple maths of a fairer USS deal


In yesterday’s post I showed a graph, followed by some comments to suggest that future USS proposals with a flatter (or even increasing) “percent lost” curve would be fairer (and, as I argued earlier in my Robin Hood post, more affordable at the same time).

It’s now clear to me that my suggestion seemed a bit cryptic to many (maybe most!) who read it yesterday.  So here I will try to show more specifically how to achieve a flat curve.  (This is not because I think flat is optimal.  It’s mainly because it’s easy to explain.  As already mentioned, it might not be a bad idea if the curve was actually to increase a bit as salary levels increase; that would allow those with higher salaries to feel happy that they are doing their bit towards the sustainable future of USS.)

Flattening the curve

The graph below is the same as yesterday’s but with a flat (blue, dashed) line drawn at the level of 4% lost across all salary levels.


I drew the line at 4% here just as an example, to illustrate the calculation.  The actual level needed — i.e, the “affordable” level for universities —  would need to be determined by negotiation; but the maths is essentially the same, whatever the level (within reason).

Let’s suppose we want to adjust the USS contribution and benefits parameters to achieve just such a flat “percent lost” curve, at the 4% level.  How is that done?

I will assume here the same adjustable parameters that UUK and UCU appear to have in mind, namely:

  • employee contribution rate E (as percentage of salary — currently 8; was 8.7 in the 12 March proposal; was 8 in the January proposal)
  • threshold salary T, over which defined benefit (DB) pension entitlement ceases (which is currently £55.55k; was £42k in the 12 March proposal; and was £0 in the January proposal)
  • accrual rate A, in the DB pension.  Expressed here in percentage points (currently 100/75; was 100/85 in the 12 March proposal; and not relevant to the January proposal).
  • employer contribution rate (%) to the defined contribution (DC) part of USS pension.  Let’s allow different rates C_1 and C_2 for, respectively, salaries between T and £55.55k, and salaries over £55.55k. (Currently C_1 is irrelevant, and C_2 is 13 (max); these were both set at 12 in the 12th March proposal; and were both 13.25 in the January proposal.)

I will assume also, as all the recent proposals do, that the 1% USS match possibility is lost to all members.

Then, to get to 4% lost across the board, we need simply to solve the following linear equations.  (To see where these came from, please see this earlier post.)

For salary up to T:

 (E - 8) + 19(100/75 - A) + 1] = 4.

For salary between T and £55.55k:

  -8 + 19(100/75) - C_1 + 1 = 4.

For salary over £55.55k:

 13 - C_2 = 4.

Solving those last two equations is simple, and results in

 C_1 = 14.33, \qquad C_2 = 9.

The first equation above clearly allows more freedom: it’s just one equation, with two unknowns, so there are many solutions available.  Three example solutions, still based the illustrative 4% loss level across all salary levels, are:

 E=8, \qquad A = 1.175 = 100/85.1

 E = 8.7, \qquad A = 1.21 = 100/82.6

 E = 11, \qquad A = 100/75.

At the end here I’ll give code in R to do the above calculation quite generally, i.e., for any desired percentage loss level.  First let me just make a few remarks relating to all this.


Choice of threshold

Note that the value of T does not enter into the above calculation.  Clearly there will be (negotiable) interplay between T and the required percentage loss, though, for a given level of affordability.

Choice of C_2

Much depends on the value of C_2.

The calculation above gives the value of C_2 needed for a flat “percent lost” curve, at any given level for the percent lost (which was 4% in the example above).

To achieve an increasing “percent lost” curve, we could simply reduce the value of C_2 further than the answer given by the above calculation.  Alternatively, as suggested in my earlier Robin Hood post, USS could apply a lower value of C_2 only for salaries above some higher threshold — i.e., in much the same spirit as progressive taxation of income.

Just as with income tax, it would be important not to set C_2 too small, otherwise the highest-paid members would quite likely want to leave USS.  There is clearly a delicate balance to be struck, at the top end of the salary spectrum.

But it is clear that if the higher-paid were to sacrifice at least as much as everyone else, in proportion to their salary, then that would allow the overall level of “percent lost” to be appreciably reduced, which would benefit the vast majority of USS members.

Determination of the overall “percent lost”

Everything written here constitutes a methodology to help with finding a good solution.  As mentioned at the top here, the actual solution — and in particular, the actual level of USS member pain (if any) deemed to be necessary to keep USS afloat — will be a matter for negotiation.  The maths here can help inform that negotiation, though.

Code for solving the above equations

## Function to compute the USS parameters needed for a
## flat "percent lost" curve
## Function arguments are:
## loss: in percentage points, the constant loss desired
## E: employee contribution, in percentage points
## A: the DB accrual rate
## Exactly one of E and A must be specified (ie, not NULL).
## Example calls:
## flatcurve(4.0, A = 100/75)
## flatcurve(2.0, E = 10.5)
## flatcurve(1.0, A = 100/75)  # status quo, just 1% "match" lost

flatcurve <- function(loss, E = NULL, A = NULL){

    if (is.null(E) && is.null(A)) {
        stop("E and A can't both be NULL")}
    if (!is.null(E) && !is.null(A)) {
        stop("one of {E, A} must be NULL")}

    c1 <- 19 * (100/75) - (7 + loss)
    c2 <- 13 - loss

    if (is.null(E)) {
        E <- 7 + loss - (19 * (100/75 - A))

    if (is.null(A)) {
        A <- (E - 7 - loss + (19 * 100/75)) / 19

return(list(loss_percent = loss,
            employee_contribution_percent = E,
            accrual_reciprocal = 100/A,
            DC_employer_rate_below_55.55k = c1,
            DC_employer_rate_above_55.55k = c2))

The above function will run in base R.

Here are three examples of its use (copied from an interactive session in R):

###  Specify 4% loss level, 
###  still using the current USS DB accrual rate

> flatcurve(4.0, A = 100/75)
[1] 4

[1] 11

[1] 75

[1] 14.33333

[1] 9

###  This time for a smaller (2%) loss, 
###  with specified employee contribution

> flatcurve(2.0, E = 10.5)
[1] 2

[1] 10.5

[1] 70.80745

[1] 16.33333

[1] 11

### Finally, my personal favourite:
### --- status quo with just the "match" lost

> flatcurve(1, A = 100/75)
[1] 1

[1] 8

[1] 75

[1] 17.33333

[1] 12

© David Firth, March 2018

To cite this entry:
Firth, D (2018). Simple maths of a fairer USS deal. Weblog entry at URL

USS proposals: Tail wagging the dog?


Update on 16 March: There’s now a follow-up post to this one, which gives more detail on how (mathematically) to achieve a fairer sharing-out of whatever level of USS member pain might ultimately be deemed necessary.  See Simple maths of a fairer USS deal (but ideally only after reading the necessary background, below!).

In response to my previous post, “Latest USS proposal: Who would lose most?“, someone asked me about doing the same calculation for the USS JNC-supported proposals from January.  For a summary of those January proposals and my comments about their fairness, please see my earlier post “USS pension scheme and fairness“.

Anyway, the calculation is quite simple, and it led to the following graph.  The black curve is as in my previous post, and the red one is from the same calculation done for the January USS proposal.

lost-comparisonThe red curve shows just over 5% effective loss of salary for those below the current £55.55k USS threshold, and then a fairly sharp decline to less than 2% lost at the salaries of the very highest-paid professors, managers and administrators.  Under the January proposals, higher-paid staff would contribute proportionately less to the “rescue package” for USS — less, even, than under the March proposals.  (And if the salary axis were to be extended indefinitely, the red curve would actually cross the zero-line: that’s because in the January proposals the defined-contribution rate from employers would actually have increased from (max) 13% to 13.25%.)

In terms of unequal sharing of the “pain”, then, the January proposal was even worse than the March one.

At the bottom here I’ll give the R code and a few words of explanation for the calculation of the red curve above.

But the main topic of this post arises from a remarkable feature of the above graph! At the current USS threshold salary of £55.55k, the amount lost is the same — it’s 5.08% under both proposals.  Which led me to wonder: is that a coincidence, or was it actually a (pretty weird!) constraint used in the recent UUK-UCU negotiations?  And then to wonder: might the best solution (i.e., for the same cost) be to do something that gives a better graph than either of the two proposals seen so far?

Tail wagging the dog?

The fact that the loss under the March proposal tops out at 5.08%, exactly (to 2 decimals, anyway) the same as in the January proposal, seems unlikely to be a coincidence?

If it’s not a coincidence, then a plausible route to the March proposal, at the UUK-UCU negotiating table, could have been along the lines of:

How can we re-work the January proposal to

  • retain defined benefit, up to some (presumably reduced) threshold and with some (presumably reduced) accrual rate,

while at the same time

  • nobody loses more than the maximum 5.08% that’s in the January proposal
  • the employer contribution rate to the DC pots of high earners is not reduced below the current standard (i.e., without the “match”) level of 12%


Those constraints, coupled with total cost to employers, would lead naturally to a family of solutions indexed by just two adjustable constants, namely

  • the threshold salary up to which DB pension applies (previously £55.55k)
  • the DB accrual rate (previously 1/75)

— and it seems plausible that the suggested (12 March 2018) new threshold of £42k and accrual rate of 1/85 were simply selected as the preferred candidate (among many such potential solutions) to offer to UUK and UCU members.

But the curve ought to be flat, or even increasing!

The two constraints listed as second and third bullets in the above essentially fix the position of the part of the black curve that applies to salaries over £55.55k.  That’s what I mean by “tail wagging the dog”.  Those constraints inevitably result in a solution that implies substantial losses for those with low or moderate incomes.

Once this is recognised, it becomes natural to ask: what should the shape of that “percentage loss” curve be?

The answer is surely a matter of opinion.

Those wishing to preserve substantial pension contributions at high salary levels, at the expense of those at lower salary levels, would want a curve that decreases to the right — as seen in the above curves for the January and March proposals.

For myself, I would argue the opposite: The “percent lost” curve should either be roughly constant, or might reasonably even increase as salary increases.  (The obvious parallel being progressive rates of income tax: those who can afford to pay more, pay more.)

I had made a specific suggestion along these lines, in this earlier post:

The details of any solution that satisfies the “percent loss roughly constant, or even increasing” requirement clearly would need to depend on data that’s not so widely available (mainly, the distribution of all salaries for USS members).

But first the principle of fairness needs to be recognised.  And once that is accepted, the constraints underlying future UUK-UCU negotiations would need to change radically — i.e., definitely away from those last two bullets in the above display.

Calculation of the red curve

In the previous post I gave R code for the black curve.  Here is the corresponding calculation behind the red curve:

sacrifice.Jan <- function(salary) { # salary in thousands
    old_threshold <- 55.55
    s <- salary

## sacrifice arising from income up to old_threshold
    s2 <- min(s, old_threshold)
    r2 <- s2 * (19/75 + 1/100 - (13.25 + 8)/100)

## sacrifice (max) arising from income over the old threshold
## -- note that this is negative
    r3 <- (s > old_threshold) * (s - old_threshold) * 
                (13 - 13.25)/100

    return(r2 + r3)

## A vector of salary values up to £150k
salaries <- (1:1500) / 10

## Compute percent of salary that would be lost, 
## at each salary level
sacrifices <- 100 * sapply(salaries, sacrifice.Jan) / salaries

In essence:

  • salary under £55.55k would lose the defined benefit (that’s the 19/75 part) and the 1% “match”, and in its place would get 21.25% as defined contribution.  The sum of these parts is the computed loss r2.
  • salary over £55.55k would gain the difference between potential 13% employer contribution and the proposed new rate of 13.25% (that’s the negative value r3 in the code).

© David Firth, March 2018

To cite this entry:
Firth, D (2018). USS proposals: Tail wagging the dog?. Weblog entry at URL


Latest USS proposal: Who would lose most?


Update on 16 March: After reading this post, you might perhaps be interested in these follow-ups:

Update, 14 March: Some details in the original post yesterday were not quite right, and so the graph/numbers that appear in the now-corrected version below are different in detail from yesterday’s.  But the overall picture is unchanged.  (If you really want to know about those changes in the detail, please see my note in Appendix 2 at the bottom of the post about that.)

Yesterday (March 12th) the UUK/UCU negotiations at ACAS concluded with an agreement document.

In this post I’ll look at the numbers in those proposed interim changes to the Universities Superannuation Scheme, to work out how much money would effectively be lost by USS members at each salary level.

This is inevitably a fairly rough calculation, but its results don’t really demand more precision.  The picture is very clear: the cost of “saving” USS would be felt most by USS members with low or moderate incomes.

The effective marginal rates at which money is lost by members are (as calculated below):

  • 4.7% on salary up to £42k
  • 6.3% on salary between £42k and the current USS threshold salary of £55.55k
  • 1.0% (at most) on salary over £55.55k

This translates into the following relationship between salary and the percentage of total salary lost:


The two “kinks” in that graph reflect the discontinuities in marginal rates, at £42k and at £55.55k.

The vertical lines drawn in green are current full-time pay grades at a typical university (with no London allowance or other extras): grade 6 is the pay of many Research Associates and Teaching Fellows, for example; grade 7 is the pay of most Lecturers; grade 8 is the pay of Senior Lecturers and Readers; and grade 9 is the pay of Professors and other senior staff.  (I have mentioned only academic and research staff here, but the same grades apply also to administrative and technical staff in UK universities.)

The long decay to the right continues indefinitely, ultimately approaching an asymptote at 1% lost, i.e., for those with absolutely stratospheric salaries (if such people are actually members of USS, still, that is — though I would guess that many are not).

In the rest of this post I’ll give the details of the calculation that leads to the above numbers and graph.  (For people who prefer a list of numbers to a graphical display, I have also added the numbers as an Appendix at the bottom of this post.)

Just here, though, let me again comment on how unfair this “remedy” would be.  The unfairness should be obvious from the above graph: those who are paid most, and would stand to benefit most from being in USS, would contribute least, in percentage terms, in this proposed move towards the future sustainability of USS.  For a more general view on this unfairness, see also my previous two posts in this “USS” category:

The calculation

It suffices to consider salaries in three distinct bands.  In each salary band, we can calculate how much is lost, per unit of salary.

The following code in R reproduces the graph drawn above.  A brief explanation is then given, beneath the displayed code.

## This code runs in base R.

## Function to compute the amount that would be lost annually (£k)
## at any given salary level
sacrifice <- function(salary) { # salary in thousands
    old_threshold <- 55.55
    new_threshold <- 42
    s <- salary

## sacrifice arising from income up to the new threshold
    r1 <- min(s, new_threshold) * ((8.7 - 8)/100 +
                                    19 * (1/75 - 1/85) +

## sacrifice arising from income between the thresholds
    s2 <- (s > new_threshold) * (min(s, old_threshold) - 
    r2 <- s2 * ((8.7 - 8)/100 + (19/75 - (12 + 8.7)/100) + 1/100)

## sacrifice (max) arising from income over the old threshold
    r3 <- (s > old_threshold) * (s - old_threshold) * (1/100)

    return(r1 + r2 + r3)

## A vector of salary values up to £150k
salaries <- (1:1500) / 10

## Compute percent of salary that would be lost, 
## at each salary level
sacrifices <- 100 * sapply(salaries, sacrifice) / salaries

## Plot the result
svg(file = "lost.svg", width = 8, height = 4)
plot(salaries, sacrifices, type = "l",
 xlab = "salary (thousands)", ylab = "percent lost",
 main = "Percent of salary lost under UUK-UCU agreement 2018-03-12")
abline(v = c(29, 39, 48, 61), col = "green")
text(x = c(34, 44, 54, 75), y = 2.8,
 labels = c("6", "7", "8", "9"), col = "green")

Band 1: Salary up to £42k

Most contributions from this part of salary go to the “defined benefit” part of USS. The new proposal would see 8.7% of member’s salary up to £42k going in to this, as opposed to 8.0% at present. The return (i.e., the value of the defined-benefit pension) can readily be calculated using the standard HMRC formula, the one that is used for Annual Allowance purposes. Under current USS, the value of this part is 19 times (s/75), where s is either £42k or the member’s salary if the salary is less than £42k. Under yesterday’s proposals, the value of this part would fall to 19 times (s/85). Under yesterday’s proposals, USS members would also lose the possibility to add 1% “matching” employer contribution to an additional, defined-contribution pension pot. The amount lost to each member, relating to salary in this first band, is then the sum of the additional contribution made and the amount of pension value lost: that is r1 in the above code.

Band 2:  Salary between £42k and £55.55k

Now, for salaries greater than £42k, let s2 be the smaller of (salary minus £42k) and (£55.55k minus £42k). Then current USS has members contributing 8% of s2 in the defined-benefit part, for a return of 19 times s2/75. Yesterday’s proposal would change the contribution to 8.7% of s2, for a return of s2 times (12% + 8.7%). And again, the possibility of 1% matching employer contribution to the defined-contribution pot would be lost. The amount lost to each member, relating to salary in this second band, is again just the sum of the additional contribution made and the amount of pension value lost: that is r2 in the above code.

Band 3: Salary over £55.55k

Relating to salary above the current £55.55k threshold, the loss would be limited to loss of the 1% matching employer contribution.  This is computed as r3 in the above code. (In practice this will be an upper bound on what is lost.  Those USS members with the very highest salaries are likely also to face issues relating to the HMRC Annual Allowance and Lifetime Allowance limits, in which case the loss of the matching employer contribution could be worth substantially less than 1% to them.)


I have reproduced the full calculation here, with code, because I found the result of the calculation so shocking!  If anyone reading this thinks I have made a mistake in the calculation, please do let me know. If it is correct — and right now I have no reason to suspect otherwise — then I confess I’m alarmed that this is actually being proposed as a potential solution, even as an interim solution for the next 3 years, to the perceived problems with USS.  It shakes my faith in those who have been involved in negotiating it.  With seemingly intelligent people on both sides of the table, how could they possibly come up with something as bad as this?

© David Firth, March 2018

To cite this entry: Firth, D (2018). Latest USS proposal: Who would lose most?  Weblog entry at URL

Appendix 1: A tabular view of what’s in the graph

## Make a table for anyone who wants more detail than the graph
salary <- c(10:55, 55.55, 56:100, 150)
percent_lost <- round(100 * sapply(salary, sacrifice) / salary, 2)
salary <- 1000 * salary
my_table <- data.frame(salary, percent_lost)

That’s the code for making a little table, showing the same numbers as those in the above graph.

Here is the resulting table:

salary    %
 10000 4.68 -- I started the table at £10k for no good reason
 11000 4.68
 41000 4.68
 42000 4.68 -- the proposed new threshold
 43000 4.72
 44000 4.76
 45000 4.79
 46000 4.82
 47000 4.86
 48000 4.89
 49000 4.92
 50000 4.94
 51000 4.97
 52000 5.00
 53000 5.02
 54000 5.05
 55000 5.07
 55550 5.08 -- current USS threshold, highest % of salary lost
 56000 5.05
 57000 4.98
 58000 4.91
 59000 4.84
 60000 4.78
 61000 4.72
 62000 4.66
 63000 4.60
 64000 4.54
 65000 4.49
 66000 4.44
 67000 4.39
 68000 4.34
 69000 4.29
 70000 4.24
 71000 4.19
 72000 4.15
 73000 4.11
 74000 4.07
 75000 4.02
 76000 3.98
 77000 3.95
 78000 3.91
 79000 3.87
 80000 3.84
 81000 3.80
 82000 3.77
 83000 3.73
 84000 3.70
 85000 3.67
 86000 3.64
 87000 3.61
 88000 3.58
 89000 3.55
 90000 3.52
 91000 3.49
 92000 3.47
 93000 3.44
 94000 3.41
 95000 3.39
 96000 3.36
 97000 3.34
 98000 3.31
 99000 3.29
100000 3.27
150000 2.51 -- possibly there are even some salaries this high?!

Appendix 2: Details of the update made on 14 March

Many thanks to all who gave feedback on the original posting, yesterday (13 March).

In response to that feedback, I made two substantive changes to the calculation.  This Appendix gives details of those changes, for those who are interested (and for the record).

Neither change affects the story qualitatively: only the detailed numbers have changed a bit.

Change 1: Use of HMRC multiplier 19 rather than 23

The HMRC calculations for Annual Allowance and Lifetime Allowance purposes are different in detail: the former uses a multiplier of 19 times pension to value USS defined benefits, while the latter uses 23 (i.e., in place of 19).  In yesterday’s post I had used 23.  The updated figures calculated above use multiplier 19 instead.

Mainly I decided to use the smaller figure as it’s a bit more conservative, in relation to the value lost through the proposed reduction of defined benefits.  (I certainly don’t want to be accused of bias in the other direction, through having picked the larger multiplier.)

The effect on the calculated numbers is mainly to reduce the height of the “spike” that appears in the graph, around the £55k salary level.  The spike is still there; it’s just a bit smaller.

My friend Jon commented that the actual value of a defined-benefit pension is harder to quantify than the HMRC formula would suggest — and that it’s likely to be dependent on age and perhaps other factors.  This is undoubtedly true, and certainly I would not suggest that anyone should use the above numbers for their own financial planning!  Rather, the aim here was (only) to show through a simple, transparent calculation how the losses arising from current proposals would differ — in rough, average terms — between pay levels.

Since writing my post yesterday I found that I am not alone in having done a calculation like this: see also (and maybe there are others too?).

Change 2: Inclusion of the USS “Match” at all salary levels

Several people pointed out to me that the USS “Match” possibility is available at all salary levels.  So it’s a benefit that would be lost at all salary levels, under the 12 March agreement.  In yesterday’s post I had taken it into account only at salaries over £55.55k: that (relatively minor) error is now corrected, in the revised figures shown above.


Mathematical operations with the Normal distribution [a re-blog from corp.ling.stats]


Following a recent Twitter exchange with Ian Dryden, I was thinking I’d write something about the risk of default for the USS pension scheme. But then I came across this other new post at corp.ling.stats — which has already done what I was intending!


This post is a little off-topic, as the exercise I am about to illustrate is not one that most corpus linguists will have to engage in.

However, I think it is a good example of why a mathematical approach to statistics (instead of the usual rote-learning of tests) is extremely valuable.

Case study: The declared ‘deficit’ in the USS pension scheme

At the time of writing nearly two hundred thousand university staff in the UK are active members of a pension scheme called USS. This scheme draws in income from these members and pays out to pensioners. Every three years the pension is valued, which is not a simple process. The valuation consists of two aspects, both uncertain:

  • to value the liabilities of the pension fund, which means the obligations to current pensioners and future pensioners (current active members), and
  • to value the future asset value of the pension fund…

View original post 2,753 more words

Future USS: Robin Hood can help?


Update on 16 March: After reading this post, you might perhaps be interested in some of these follow-ups:

In this post I’ll follow up on the previous one, USS pension scheme and fairness, in the light of this week’s UCU proposals (proposals which provide a basis for renewed talks with Universities UK about the future of USS).

My purpose here is to suggest that a fairer solution could also be more affordable — in that it would help fund the defined-benefit section of USS, but would also reduce (and might perhaps even eliminate?) the upward pressure on employer and employee contribution rates.

1. The current UCU proposals

The following summary excerpt is taken from


The University and College Union (UCU) deserves all of the many congratulations it is currently getting, for having produced such a proposal this week — in particular, a proposal that has at last persuaded the employers (Universities UK) to take part in further talks about the future of USS.

That said: I should admit to feeling disappointed by the 3rd and 4th bullet points above!  USS members would all be getting a slightly worse pension, in return for an appreciable increase in the contribution rates (both employee and employer contribution rates).

More crucially — if I have understood it correctly —  the implied loss of pension (as a percentage of contributions paid) would be greatest for those USS members whose salary is £55k or lower.  Higher-paid members of USS would be required to make a smaller sacrifice — substantially smaller, in the case of those with the very highest salaries — relative to the overall size of their pensions.

Recognising this unfair distribution of the pain, and thinking about how best to fix it, leads naturally to an additional device that could make the proposed changes fairer and more easily affordable.

2.  Robin Hood to the rescue

The current USS setup has a slightly progressive aspect, which is that the 18% employer contribution for salary over the £55k threshold pays only 12% into the member’s defined-contribution (DC) pension pot.  The remaining 6% therefore helps to support the running costs of USS, and (mainly, it seems safe to presume) the defined-benefit part of USS.

The suggestion I want to make here is that future USS should become more progressive — that is, USS should move further in that same, progressive direction.  (As mentioned in the previous post, the current USS proposal would increase that 12% figure to 13.25%, thereby making the scheme appreciably less progressive.  And as I have argued in that previous post, that seems completely indefensible.)

The specific suggestion:

  1. For salary between the current £55k threshold and some specified higher threshold, pay x% from the employer contribution into the employee’s DC pot.  The value of x could be left at 12, for example.
  2. For salary above the higher of the two thresholds, pay y% from the employer contribution into the employee’s DC pot, where y is smaller than x.

Even if x remains at 12, such a device would allow (through suitable choice of the higher threshold and the value of y) substantial savings to be made from the employer contributions on higher salaries — savings which could then be used directly to support the threatened, defined-benefit part of USS.

The precise arithmetic on this becomes possible only with detailed knowledge (not available to me) of the distribution of USS-member salaries over £55k.  Some general considerations on the choice of the higher threshold, and of y, are:

  • The higher threshold clearly should not be so high as to make the resultant savings too small to be of much consequence.
  • The value of y probably needs to be at least as high as the employee contribution rate (currently 8%), otherwise it could become unattractive for those with the highest salaries to remain in USS.

Illustrative example:

Just to give a sense of what might be possible through this device.  Let’s suppose:

  • salary between £55k and £75k gets 12% into the DC pot, from the employer contribution — as now.
  • salary over £75 gets 8% (i.e., the current employee contribution rate) instead of 12%.

Then the amount saved, which could then be used to support the defined-benefit part of USS, would be 4% of all member salaries over £75k.

3. Conclusion

It is unclear to me, in absence of enough data to determine it, whether the ‘Robin Hood’ device just described would be enough to completely eliminate the need for a change in the accrual rate and/or increased contributions (as proposed by UCU in their points 3 and 4 above).

What is clear to me, though, is that such a device would help eliminate the unfairness described above.  And it would, at least, reduce the need for any changes in accrual or contribution rates, even if such need is not completely eliminated.

© David Firth, March 2018

To cite this entry:
Firth, D (2018). Future USS: Robin Hood can help?  Weblog entry at URL

Postscript: about Robin Hood

The legend of Robin Hood, ‘feared by the bad, loved by the good’, will already be known to most people who have grown up in England.  There are countless stories of Robin Hood ‘persuading’ the rich to part with their money, for the benefit of poorer folk.

For anyone interested, there’s a lot to read about it here:

One aspect that I particularly like is that my home city of Wakefield is one of the (many!) places that lay claim to Robin Hood as one if its own townspeople.  But then there’s all that romantic Sherwood Forest nonsense…

USS pension scheme and fairness


Update on 16 March: After reading this post, you might perhaps be interested in some of these follow-ups:

The Universities Superannuation Scheme (USS) is among the largest pension schemes in the UK.  It provides pensions for academic and other staff in most of the UK’s universities — and right now it is the subject of strike action in around 60 universities.  The strike relates to substantial proposed changes to the pension scheme.

Most of the current arguments are about the long-term affordability of USS in its present form.  I do not consider myself sufficiently expert to add much that would be useful on that (crucial) aspect of USS — so that’s not the subject of this post.  Instead, here I will take a look at the current USS scheme in regard to aspects of fairness; and I’ll examine the new proposals and their implied direction of travel, in relation to fairness.

Why am I writing this?  Well, I needed to find out enough about USS and the new proposals to be able to explain — to friends, family, colleagues in other countries, etc. — what all the current argument is about.  One thing I found — quite separately from the arguments about affordability of a defined-benefit pension scheme — was that the new proposals seem to have the wrong direction of travel in relation to (at least my notions of) fairness.

At the end I’ll make a specific suggestion towards improving the currently proposed changes to USS.

1.  A bit of history

Back in the mists of time, when I first joined USS, the university that appointed me was (rightly) very keen to stress how great the pension scheme was.  I would need to pay 6.35% of salary, and the university would contribute an additional 18.55% of my salary, to provide a guaranteed pension based on whatever salary level I had achieved at retirement.  This was regarded as an important part of the deal: our salaries were considerably lower than we could be paid elsewhere, but the promise of a decent pension would compensate to some extent.

I remember well, a few years later in 1997, the widespread feeling of outrage and dismay when the universities collectively decided to reduce their contributions from 18.55% to 14%.  In 1997 the USS pension fund was substantially in surplus, and the universities succumbed to the natural temptation of a ‘contributions holiday’.  (It turned out to be a very long holiday: the universities’ contribution rate remained at 14% until 2011.)

Right now, in 2018, employer contribution is 18%, and USS members themselves pay 8% of salary into the scheme (with an option to increase that to 9% in return for a matching 1% increase of the employer contribution to the USS member’s “defined contribution” account with USS).

In relation to this history, let me mention here two aspects relating to fairness:

  1. The ‘final salary‘ basis for determining the amount of pension payable was (in my opinion!) blatantly unfair.  As someone who stands to benefit from it, I think I am in a good position to say this.  Members’ life-long contributions to USS were being used disproportionately to pay the pensions of those who were paid the highest salaries, and especially those whose salaries became high relatively late in their careers.
  2. The long contributions holiday taken by universities between 1997 and 2011 is clearly connected with whatever problems USS has today.  The universities ought to be prepared to substantially increase their contributions to the USS pot when the historic surplus has been run down partly as a result of the earlier (substantial!) ‘holiday’.  It would be completely unfair to expect future USS members to pay the cost of the ‘holiday’.

2.  Current USS

This excerpt from gives a very rough summary of the current USS setup:


In relation to fairness of the current setup, I want to note three things:

  1. The unfair final salary aspect has now gone (following the changes to USS that were made in 2016).  Instead, we now have ‘Career Revalued Benefits’ (CRB), based on contributions from annual salary up to the ‘threshold’ amount of around £55k.  This results in lower pensions for many (perhaps most) USS members, especially those whose salaries increase substantially in mid-late career.  But CRB does seem a fairer basis than ‘final salary’, for determining the relationship between life-long contributions and the level of pension ultimately provided.  (The 1/75 multiplier is arguably not high enough; but that relates more to affordability than to fairness.)
  2. The current USS setup has an appealing, progressive aspect: the employer contribution of 18% supports only a 12% contribution to the ‘USS Investment Builder’ pension pot that relates to to annual salary in excess of the £55k threshold.  The implication of this is that some (maybe most?) of the remaining 6% of employer contribution, for salaries over £55k, gets used in support of the defined-benefit CRB scheme that applies to the first £55k of every member’s salary.  As a principle, this seems right and fair: a priority for USS should be to ensure that even the least well-paid members get a decent pension.  Appropriateness of the precise details (12% versus perhaps some lower figure or a decreasing schedule for higher salaries; and the £55k level of the salary ‘threshold’) is of course open to debate, still.
  3. The previous two points were about ways in which USS currently is fair.  For balance, let me mention one aspect of current USS that does seem unfair, still.  The USS scheme has always, as far as I know, included specific pension provision for the surviving spouse/partner and/or other dependants of a member who dies.  While this clearly is a benefit to those members who have got such dependants, it implies some level of subsidy from the contributions — whether made directly, or by the employer — of any member without such dependants.  (Again I write as someone who does benefit from this; but I don’t regard it as fair!)

3.  The USS proposals for change

The following excerpt is from


My thoughts on the specific numbers there, in terms of fairness, are as follows.  (The first item in the following list is by far the most important of the three; the other two are quite minor by comparison.)

  1. For salary amounts over the current threshold of £55k, the proposal would actually increase the employers’ pension-pot contributions (from the current 12%, to 13.25%).  This runs counter to the ‘progressive’ aspect, mentioned above, of the current USS setup.  Moreover, it implies that less of the employer contribution on salary amounts over £55k — that is to say, less than now — would actually be used to service the defined-benefit commitments of USS.  This seems absurd.  It represents a direction of travel that’s completely unfair: it would systematically shift the future cost of those defined-benefit commitments, away from the higher-paid members, onto those whose salary is mostly or entirely below the current £55k threshold.
  2. The suggested flexibility to adjust member contributions downwards to 4%, while still getting 13.25% employer contributions, looks good on the face of it.  But who would it benefit most?  Most members with low to middling incomes will probably want or need to contribute 8%, still, in order to (aim to) ensure that they have a decent pension at retirement.  Probably the main beneficiaries of this flexibility would be those members with the highest salaries, who could use it to reduce or eliminate potential breaches of the HMRC allowances (either the Annual Allowance for pension contributions, or the Lifetime Allowance for total pension-pot size).
  3. That last bullet-point in the above excerpt looks relatively minor.  It suggests shifting some of the cost of USS death benefits away from employers, to members — if I have understood it correctly.  The details of this are not yet clear, at least to me; but it could imply that the cost of pensions for widows/partners and other dependants — which I have argued above is already unfair — would be carried more directly than before by members, i.e., through their own direct contributions rather than their employers’.

4.  Conclusions

My comments in this post really are quite separate from the on-going arguments about affordability of the defined-benefit part of USS.  But…

If those arguments about affordability do result in substantial changes being made, then I really hope that considerations of fairness will play a major role.  As things stand at present, the proposed USS changes would appear to be quite neutral or even beneficial to those members with the very highest salaries, but clearly detrimental to the majority.

A more positive take on the points made above would be that there is plenty of room for manoeuvre, towards the negotiation of a future USS that could work better for all.  The universities plainly are not currently at the absolute limit of what is affordable.  The proposal to contribute more to the pension pots of the highest paid, rather than less, makes this abundantly clear (as indeed does the USS history, as mentioned above).  A future USS that’s designed to be more progressive than the present version — i.e., with the highest paid receiving gradually lower marginal rates of employer contribution to their pension-pots — could perhaps make the defined-benefit component of USS work out to be (even more) clearly affordable?

Update (4 March 2018): The follow-up post Future USS: Robin Hood can help? formulates a concrete suggestion along the lines just indicated above — a simple device that would make USS both fairer and more affordable.


© David Firth, February 2018

To cite this entry:
Firth, D (2018). USS pension scheme and fairness. Weblog entry at URL

Exit poll for June 2017 election (UK)



It has been a while since I posted anything here, but I can’t resist this one.

Let me just give three numbers.  The first two are:

  • 314, the number of seats predicted for the largest party (Conservatives) in the UK House of Commons, at 10pm in Thursday (i.e., before even a single vote had been counted) from the exit poll commissioned jointly by broadcasters BBC, ITV and Sky.
  • 318, the actual number of seats that were won by the Conservatives, now that all the votes have been counted.

That highly accurate prediction changed the whole story on election night: most of the pre-election voting intention polls had predicted a substantial Conservative majority.  (And certainly that’s what Theresa May had expected to achieve when she made the mistake of calling a snap election, 3 years early.)  But the exit poll prediction made it pretty clear that the Conservatives would either not achieve a majority (for which 326 seats would be needed), or at best would be returned with a very small majority such as the one they held before the election.  Media commentary turned quickly to how a government might be formed in the seemingly likely event of a hung Parliament, and what the future might be for Mrs May.  The financial markets moved quite substantially, too, in the moments after 10pm.

For more details on the exit poll, its history, and the methods used to achieve that kind of predictive accuracy, see Exit Polling Explained.

The third number I want to mention here is

  • 2.1.0

That’s the version of R that I had at the time of the 2005 General Election, when I completed the development of a fairly extensive set of R functions to use in connection with the exit poll (which at that time was done for BBC and ITV jointly).  Amazingly (to me!) the code that I wrote back in 2001–2005 still works fine.  My friend and former colleague Jouni Kuha, who stepped in as election-day statistician for the BBC when I gave it up after 2005, told me today that (with some tweaks, I presume!) it all works brilliantly still, as the basis for an extremely high-pressure data analysis on election day/night.  Very pleasing indeed; and strong testimony to the heroic efforts of the R Core Development Team, to keep everything stable with a view to the long term.

As suggested by that kind tweet reproduced above from the RSS President, David Spiegelhalter: Thursday’s performance was quite a triumph for the practical art and science of Statistics.  [And I think I am allowed to say this, since on this occasion I was not even there!  The credit for Thursday’s work goes to Jouni Kuha, along with John Curtice, Steve Fisher and the rest of the academic team of analysts who worked in the secret exit-poll “bunker” on 8 June.]


About heading soccer balls, and memory loss


This new research paper, by a group led from the University of Stirling, made the national news in the UK last week:

Di Virgilio, T.G., et al., Evidence for Acute Electrophysiological and Cognitive Changes Following Routine Soccer Heading, EBioMedicine (2016),

My declaration of interest:

I will write some notes below about my reading of the paper.  But first I should make clear that I am not a completely disinterested scientist when it comes to this topic.  For quite some years now, my son and I have been avid supporters of West Bromwich Albion FC, where calls for better research on the long-term effects of heading footballs have been made following the death of former Albion and England centre forward Jeff Astle in 2002 (aged 59).  Jeff Astle was a prolific goalscorer for West Brom, well known for his outstanding ability as a header of the ball.  The Coroner’s verdict in 2002 was “death by industrial disease”, and his report on Jeff Astle’s death included the comment that “The trauma caused to the front of his brain [by heading the ball] is likely to have had a considerable effect on the cause of death.”  There was even an adjournment debate in the House of Commons on this subject, following Jeff Astle’s death.

The background to my notes below is that I strongly support the case for better research on the long-term effects of heading a football: it seems clear that not enough is known about the health risks, and such questions as whether heading the ball is safer now that footballs have become lighter.

Some of the news headlines from last week:

Heading footballs ‘affects memory’BBC Scotland, 2016-10-24

Heading footballs affects memory and brain function, study findsITV News, 2016-10-24

Study finds heading a football has immediate effect on the brainThe Guardian, 2016-10-24

Heading a Soccer Ball Affects Memory FunctionWall Street Journal video, 2016-10-24

Calls for more research as football headers linked to memory lossSky News, 2016-10-24.  (This features a short video clip of Dawn Astle, Jeff’s daughter, talking persuasively about the need for thorough, longitudinal research.)

Here are the original press release and an online article by the some of the authors of the original research paper:

Heading a football causes instant changes to the brainNIHR press release, 2016-10-24

How we discovered that heading a football causes impairment of brain functionThe Conversation, 2016-10-24

And on the same day, the story was reported also on the public news website of the UK’s National Health Service:

Heading footballs may cause short-term brain changesNHS Choices, 2016-10-24

My reading of the original research paper

The research reported in the paper is a small, before-and-after experiment.  Data are analysed from 19 amateur footballers who took part in a controlled heading-practice session, with various measurements made before and after the session (immediately before, immediately after, and at three later time-points up to 2 weeks after).

The paper’s main findings are based on before-to-after differences in three of the measurements made, these three having apparently achieved statistical significance in the experiment, with reported p-values less than the pre-assigned threshold of 0.05.  The three “statistically significant” differences found were:

  1. The “primary outcome measure cSP” — a measure of corticomotor inhibition — was found to have increased for 14 of the 19 participants when measured immediately after the heading practice session.  The reported p-value, for the apparent increase in response time that was seen on average, is 0.049.  [Section 3.1 of the paper]
  2. The “Spatial Working Memory” (SWM) test scores showed an increased error rate on average (on the log scale the change was from 0.79 before to 1.00 after the heading session).  The reported p-value for this apparent difference is 0.03.  [Section 3.2 of the paper]
  3. The “Paired Associated Learning” (PAL) test scores also showed an increased error rate on average (on the log scale the change was from 0.38 before to 0.65 after).  The reported p-value for this apparent difference is 0.007.  [Section 3.2 of the paper]

How to interpret those apparent effects and their p-values?

I was prompted to think a bit about this by the reported p-value of 0.049 for the primary outcome measure: that’s only just less than the pre-assigned threshold of 0.05.  So if it’s agreed that p equal to 0.05 is the largest value that can reasonably count as “statistically significant” evidence, the value of p=0.049 found for this apparent increase in cSP time should probably be labelled “almost insignificant”!  (This is in agreement with the “14 out of 19” finding mentioned already above, for the number of subjects whose cSP time had shown any increase at all; a simple sign test is enough to tell us that 14 out of 19 is not quite significant at the 0.05 level.)

But was 0.05 a reasonable threshold to use, anyway?   A computed p-value of 0.05, or even 0.03, should really be considered very weak evidence when quite a large number of different measurements are being recorded and tested, as was the case in this study.    As Table 2 of the paper shows, there were 8 different measurements taken, each done on four occasions after the heading session: that’s a lot of chances to find some “significant” differences.  The much-used threshold of p<0.05, which is designed to limit the chance of a spuriously significant finding to 5% when conducting a single test, is much more likely to throw up spuriously significant results when several hypotheses are being tested.  A crude Bonferroni correction based on 8 tested differences, for example, would result in the threshold of 0.05 being reduced to 0.05/8 = 0.006, as a much more stringent criterion to apply in order to be sure that the chance of a spuriously significant finding is still less than 5%.

Of the paper’s three main findings, then, only the third one — the increased average error rate in the PAL test scores — seems to be an apparent effect that might demand further attention.  The paper mentions [in Section 3.2] that an increased error rate in the PAL test is compatible with reduced long-term memory function.  (But note that if we do take the route of a Bonferroni correction, to allow for the fact that 8 different measurements were tested — while still neglecting the number of occasions on which post-session measurements were made — the reported p-value of 0.007 still would fail to reach significance at the traditional 5% level.)

Some methodological quibbles and questions

Q1.  The big one: Causality

The press release mentioned above, and hence the media coverage of this research paper, reports an apparently causal link found between routine heading of footballs and outcomes such as long-term memory function.  Such a causal link does seem rather plausible, a priori.  But the research reported in this paper does not (for me, at any rate) firmly establish that cause.  The study design leaves open the possibility of an alternative explanation (for an increase in PAL test error scores, for example).  The paper’s authors allude to this problem in their Discussion section [Section 4 of the paper], where they write: “Future work should include a control activity such as body movement without head impact”.  I do agree; and careful design of the control regime is essential if causality is to be established compellingly.

What sort of alternative explanation(s) might there be?  Well, the problem is that heading the football was not the only thing that happened to each of the 19 experimental subjects between the first two sets of measurements.  Some other things that might conceivably have produced an effect are:

  • the passing of time (e.g., time since breakfast?)
  • the order in which measurements were taken (the research paper is not altogether clear on this, actually — there seem to be conflicting statements in Section 2.2 there)
  • the thrill of taking repeated shots at a goal (which might have been just as great had the shots been volley-kicks instead of headers?)

I am not suggesting here that any of these possible alternative causes is the reality; only that if we want to establish that heading the ball is a cause of something, then other potential causes must be eliminated as part of the study design or through external scientific knowledge.

Q2.  Missing data?

In Section 2.1 of the paper it is mentioned that there were originally 23 study participants recruited.  This got reduced to 19 for the analysis, because “Data from one participant could not be analyzed and three more participants withdrew from the study for personal reasons”.  It would have been good to have some more detail on this.  In particular:

  • what was it about one participant’s data that meant they could not be analyzed?
  • at what point in the study did each of the other three participants withdraw?

Q3.  Size of effect, misleadingly reported?

Update, 2016-11-04:  I have now heard back from one of the paper’s authors about this question.  Dr Magdalena Ietswaart has kindly told me that the 67% figure “is based on raw scores rather than log transformed values”.  In which case, my conjecture below about the origin of that figure was false — and for that reason I have now struck through it, and I humbly apologise to the paper’s authors for having guessed wrongly about this particular point.  (The 67% increase still does strike me as startling, though!)

As discussed above, of all the various effect sizes that are considered in the paper it is the increase in the PAL test error rate that might merit further attention, since that is the one effect for which the experimental evidence might be viewed as statistically significant.  In the paper [Section 3.2], it is stated that the error score on the PAL task immediately after heading increased by 67%.  But this rather startling statement of the effect size — which appeared also in much of the press coverage, and indeed is the single number that prompted me to read the paper in full — appears to be wrong.    If my calculations are correct, the increase found in the PAL test error rate is in fact more like 26% (which is still a fairly substantial increase, of course).

The reason for the discrepancy is that the 67% figure appears to have been calculated directly from Table 2 in the paper, where the increase in the PAL test error rate is measured by averaging the logarithms of the raw error rates.  The ratio (0.65 – 0.38)/0.38, calculated from Table 2, is roughly 1.67 to within the effects of rounding error, i.e., it corresponds to a 67% increase in the logarithm.  But it makes no sense at all to calculate a ratio of logarithms — the numerical value of such a ratio in the present context is completely meaningless.  (This will be obvious to the reader who understands how logarithms work, but probably not otherwise!  The key point, mathematically, is that while a percentage increase in error rate — or in anything else, for that matter — does not depend at all on the units of measurement, the same is not the case after logarithmic transformation.  A ratio of logarithms will depend in a completely arbitrary way on the original units of measurement used, and so will be meaningless.)

How did I get the 26% figure mentioned above as the correct percentage increase?  Well it’s actually something of a guess, because I do not have access to the raw data.  (I have asked the paper’s authors if they will share the data with me; but right now I only have Table 2 to work from.)  It’s probably not such a bad guess, though.  I made the working assumption that the distributions underlying the figures in Table 2 are normal, which seems reasonable given the rationale for logarithmic transformation that is given in Section 2.8 of the paper.  With that assumption, the ratio of means is calculated (by standard properties of the log-normal distribution) as

       exp(0.65 + ((0.29)^2 / 2)) / exp(0.38 + (0.41^2 / 2)) = 1.26

I should emphasise, though, that this is largely guesswork.  In particular, it is possible that I have misunderstood how the 67% figure, quoted in the Section 3.2 of the paper, was arrived at.


Let me restate here what I said near the top: I strongly support the case for better research on the long-term effects of heading a football.  With that in mind, I am disappointed that the paper that I have read and discussed here does not provide very convincing evidence to help our understanding.

One rather positive aspect of the paper is that it did get a lot of media coverage, which helped to bring the issue (and accompanying memories of Jeff Astle!) to wider public attention, at least for a day or two.

But, as Dawn Astle so eloquently argued in the Sky News interview that’s linked above: there is still a clear need for good research on the matter of “routine”, but potentially long-lasting, brain injuries in football.

© David Firth, November 2016

To cite this entry:
Firth, D (2016). About heading soccer balls, and memory loss. Weblog entry at URL

RSS discussion paper on model-based ranking of journals, using citation data


This paper has been around on arXiv for quite some time.  Now, having survived various rounds of review — and having grown quite a bit as a result of reviewers’ requests! — it will be discussed at an Ordinary Meeting of the Royal Statistical Society on 13 May 2015 (just follow this link to the recent Allstat announcement, for instructions on how to contribute to the RSS discussion either in person or in writing).

Here is the link to the preprint on

Statistical modelling of citation exchange between statistics journals by Cristiano Varin, Manuela Cattelan and David Firth.

(Note that the more ‘official’ version, made public at the RSS website, is an initial, uncorrected printer’s proof of the paper for JRSS-A.  It contains plenty of typos!  Those obviously will be eliminated before the paper appears in the Journal.)

The paper has associated online supplementary material (zip file, 0.4MB) comprising datasets used in the paper, and full R code to help potential discussants and other readers to replicate and/or experiment with the reported analyses.


Figure 4 from the paper (a ranking of statistics journals based on the Bradley-Terry model)

The paper’s Summary is as follows:

Rankings of scholarly journals based on citation data are often met with skepticism by the scientific community. Part of the skepticism is due to disparity between the common perception of journals’ prestige and their ranking based on citation counts. A more serious concern is the inappropriate use of journal rankings to evaluate the scientific influence of authors. This paper focuses on analysis of the table of cross-citations among a selection of Statistics journals. Data are collected from the Web of Science database published by Thomson Reuters. Our results suggest that modelling the exchange of citations between journals is useful to highlight the most prestigious journals, but also that journal citation data are characterized by considerable heterogeneity, which needs to be properly summarized. Inferential conclusions require care in order to avoid potential over-interpretation of insignificant differences between journal ratings. Comparison with published ratings of institutions from the UK’s Research Assessment Exercise shows strong correlation at aggregate level between assessed research quality and journal citation ‘export scores’ within the discipline of Statistics.

Facts checked: The UK general election, 7-way TV debate between Westminster party leaders


I was part way through doing some of this myself, when I found that has done a great job already with checking some of the main numbers quoted in last night’s televised debate: