# Conventional gilt auctions: errors in the results; and an information leakage

Julian D. A. Wiseman

Abstract: in April 2016 there was a small change in practice for auctions of conventional gilts (non-inflation-linked), which leaks a little information.

Publication history: only at www.jdawiseman.com/papers/finmkts/gilt_auction_info_leakage.html. Usual disclaimer and copyright terms apply.

Contents: • Introduction; • Calculations; • 1H26 on 05 May 2016; • DMO’s arithmetic error?; • Public policy considerations: what to do; • Conclusion; • Afterwords (10th August 2016, FT).

## Introduction

In an auction of a conventional gilt, there is a fixed amount for sale, such as £2½bn. Some is sold in a low-information way at the average of the accepted competitive bids, GEMMs being limited as to the amount for which they can bid this way, and retail players being allowed up to £½ million. Competitive bids are for a quantity of whole numbers of millions of pounds. Soon after the auction the DMO publishes the minimum accepted price. Bids above this minimum are filled in full, at the price bid. Bids below this minimum are rejected. Bids at this minimum are typically accepted in part, are ‘scaled down’ such that what is sold is the correct total = whole auction minus non-competitives. The scaledown is a percentage, strictly greater than zero, and at most 100%.

Until March 2016, this percentage was quoted to one decimal place. For example, at the auction of 3T52 on 8 March 2016, the scaledown was quoted as “approximately 23.5%”. But from the auction of of 1H21 on 5 April 2016, the scale down has been quoted to 6 decimal places: “58.3980%”.

This is a small leakage of information.

## Calculations

As an initial example, let’s start with the auction of 0H22 on 2 August 2016. We know that the lowest, average, and highest accepted prices were 99.830, 99.767, and 99.745. Assume that the unrounded average price was 99.7665, implying that at most 74.7% of the auction was sold at the lowest price. Amount for sale was £2½bn; the non-competitive bids were £374.4m from the GEMMs, and £0.089m from retail ⇒ £2125.511m was to be sold competitively. Competitive bids at the minimum accepted price (indeed at any price) were for an integer number of millions. The amount remaining for sale, after removing the quantity sold at higher prices, was £n+0.511 million, n integer. And the scaledown was 71.0439%.

Putting all this together narrows it down to only three possibilities (all quantities shown in £millions):

For sale
at min
Bids
at min
Ratio
987.511139071.0439568%
1198.511168771.0439241%
1409.511198471.0439012%

Of these, none are impossible, but the first seems most likely. And an interesting sign of an efficient market, that 987.511÷2125.511 ≈ 46.5% was sold at the lowest accepted price.

The same calculation follows for all the conventional auctions since the start of April 2016:

Auction
Date
GiltAmount
For Sale
Non-Comp
GEMM
Non-Comp
Other
Total
bids
ScaledownHighest
accept.
Average
accept.
Lowest
accept.
Max
Propn
Possibilities at the min. price:
ForSale ÷ Bids ≈ Scaledown  [Propn]
Tue 05 Apr 20161H212750412.80.0015516.80158.3980%103.313103.282103.26768.5%1006.199 ÷ 1723 ≈ 58.398085%  [43.05%]
1232.199 ÷ 2110 ≈ 58.398057%  [52.72%]
1458.199 ÷ 2497 ≈ 58.398038%  [62.39%]
Thu 07 Apr 20161H262500374.40.0584905.45782.9740%99.91699.86899.85174.6%952.542 ÷ 1148 ≈ 82.974042%  [44.81%]
1337.542 ÷ 1612 ≈ 82.974069%  [62.93%]
Wed 13 Apr 20163H451750262.40.0133640.41339.7935%124.121124.062123.97641.0%79.587 ÷ 200 = 39.793500%  [05.35%]
426.587 ÷ 1072 ≈ 39.793563%  [28.68%]
Wed 04 May 20161H212750412.80.0025806.80024.2077%102.696102.672102.65661.3%200.198 ÷ 827 ≈ 24.207739%  [08.57%]
391.198 ÷ 1616 ≈ 24.207797%  [16.74%]
475.198 ÷ 1963 ≈ 24.207743%  [20.33%]
559.198 ÷ 2310 ≈ 24.207706%  [23.93%]
… and 11 others.
Thu 05 May 20161H262500374.40.1014473.50066.8421%98.56698.54398.52658.7%— No solutions ⇒ data contradictory. —
Wed 18 May 20164Q361500225.60.0322575.6322.3000%134.640134.493134.36052.7%0.368 ÷ 16 = 2.300000%  [00.03%]
23.368 ÷ 1016 = 2.300000%  [01.83%]
44.368 ÷ 1929 ≈ 2.300052%  [03.48%]
46.368 ÷ 2016 = 2.300000%  [03.64%]
Wed 01 Jun 20161H212750412.80.0284396.82717.6371%102.906102.889102.86239.8%112.172 ÷ 636 ≈ 17.637107%  [04.80%]
218.172 ÷ 1237 ≈ 17.637187%  [09.33%]
321.172 ÷ 1821 ≈ 17.637122%  [13.74%]
427.172 ÷ 2422 ≈ 17.637159%  [18.28%]
… and 5 others.
Tue 07 Jun 20164Q461500225.60.0132060.61351.2900%148.560148.383147.95029.1%15.387 ÷ 30 = 51.290000%  [01.21%]
Tue 05 Jul 20161H212500374.00.0144492.01292.1450%105.090105.055105.03059.2%— No solutions ⇒ data contradictory. —
Thu 07 Jul 20161H262250337.60.0865244.68321.7268%105.641105.627105.61148.3%137.314 ÷ 632 ≈ 21.726899%  [07.18%]
366.314 ÷ 1686 ≈ 21.726809%  [19.16%]
444.314 ÷ 2045 ≈ 21.726846%  [23.23%]
522.314 ÷ 2404 ≈ 21.726872%  [27.31%]
… and 5 others.
Wed 20 Jul 20164Q391500225.60.0442250.64428.7120%150.274150.104149.64227.0%14.356 ÷ 50 = 28.712000%  [01.13%]
Tue 02 Aug 20160H222500374.40.0895709.48871.0439%99.83099.76799.74574.7%987.511 ÷ 1390 ≈ 71.043957%  [46.46%]
1198.511 ÷ 1687 ≈ 71.043924%  [56.39%]
1409.511 ÷ 1984 ≈ 71.043901%  [66.31%]

## 1H26 on 05 May 2016

There is no solution for the auction on 05 May. (Nor on 05 July, but the former is the example used here.) Why? It cannot be a typing error by the DMO, because these press notices are produced entirely by machine. Perhaps it is an error in my assumptions, or perhaps it is an error in my algorithm? It is easier to check the latter.

The press notice announcing the result of this auction says that the GEMMs’ non-competitive amount was something and 374.4 million, and the “Others” non-competitive was 0.101 million. Competitive bids above the minimum must have been whole numbers of millions. So the quantity sold at the minimum must be integer+.499 million. The average price is near the middle of the lowest and the highest, so it seems believable that not more than 58.7% of the auction could have been sold at the minimum. Scaledown of about two-thirds means that bids at the minimum can’t have exceeded (2500-374.4-0.101) × 58.7% ÷ 66.8421% < £1.9bn. For each of the integer number of bids up to this limit, multiply by the scaledown, and round to the nearest .499. Divide one by the other to compute a ratio. Those closest to 66.8421% are shown in the next table, starting with the smallest absolute errors.

SoldBidRatio≥ Scaledown ?
1206.499180566.8420499%False
1079.499161566.8420433%False
952.499142566.8420351%False
825.499123566.8420243%False
698.499104566.8420096%False
571.49985566.8419883%False
444.49966566.8419549%False
317.49947566.8418947%False

Observe that the first line just rounds the wrong way: rounding to nearest would take this to 66.8420% rather than the required “66.8421%”. But the error is greater than this: the scaledown would be the ratio rounded down, never rounded up (the DMO would not sell more than the quoted “£2,500 million”; instead the DMO would sell a mite less, and have a scrap left on its own books). And since the scaledown must be this ratio rounded down, the error is greater—the failure to find a solution more thorough. Hence, for this auction, there is no valid solution.

## DMO’s arithmetic error?

There is one candidate type of error. The press notice of 05 May 2016 says that the “Total bids received” is “£4,473.500 million”. This can’t be just competitive bids, as they are in whole millions. This can’t be just competitive bids + GEMMs’ non-comps, as the latter has a fractional part of .4. This can’t be competitive bids + all non-comps, as the latter has a fractional part of .400 + .101 = .501 rather than .500. This can’t be rounding at display, because even retail non-comps must be whole numbers of thousands (¶75 of Information Memorandum of March 2016: “must be for a multiple of £1,000 nominal of Stock”). Indeed, on 07 Jul 2016 the error was £3k: it really can’t be rounding. So perhaps the computer slightly mis-reported the retail bids? If it did—and I don’t like the explanation, but have you better?—that would fix the scaledown problems:

Auction
Date
GiltAmount
For Sale
Non-Comp
GEMM
Non-Comp
Other
Total
bids
ScaledownHighest
accept.
Average
accept.
Lowest
accept.
Max
Propn
Possibilities at the min. price:
ForSale ÷ Bids ≈ Scaledown  [Propn]
Thu 05 May 20161H262500374.40.1004473.50066.8421%98.56698.54398.52658.7%63.500 ÷ 95 ≈ 66.842105%  [02.99%]
190.500 ÷ 285 ≈ 66.842105%  [08.96%]
317.500 ÷ 475 ≈ 66.842105%  [14.94%]
444.500 ÷ 665 ≈ 66.842105%  [20.91%]
… and 6 others.
Tue 05 Jul 20161H212500374.00.0124492.01292.1450%105.090105.055105.03059.2%93.988 ÷ 102 ≈ 92.145098%  [04.42%]
398.988 ÷ 433 ≈ 92.145035%  [18.77%]
703.988 ÷ 764 ≈ 92.145026%  [33.11%]
1008.988 ÷ 1095 ≈ 92.145023%  [47.46%]

Indeed, the totals seem to be slightly wrong for several recent auctions: Thu 07 Apr 2016, Wed 04 May 2016 (by £2k), Thu 05 May 2016, Wed 01 Jun 2016, Tue 05 Jul 2016 (by £2k), Thu 07 Jul 2016 (by £3k), and Tue 02 Aug 2016.

So there seems to be at least one error in the DMO’s software; and there might or might not also be one in my assumptions.

## Public policy considerations: what to do

What should the authorities do?

In theory, it could be that the DMO wants to release this subtle information about the distribution of bids. In theory. But having everybody compute the meaning has two societal costs: the computation is done many times; and it privileges those who have employed STEM graduates to do this—whom the Treasury now wants employed by manufacturers. So if the DMO wants the information released, please could it be released cleanly rather than cryptically.

But what if, as is more plausible, the DMO doesn’t want to release this information? The best thing to do would be to fix the multiple problems with the current auction mechanism. Regular readers of jdawiseman.com will surely be expecting me to remark that the current rules entail multiple needless risks for both issuer and bidders, and these can be fixed. For more start at An Open Letter to the Commercial Secretary to the Treasury: Trouble Coming, Easily Avoided.

Let’s assume that isn’t yet happening. So we need to observe that the problem has an extra difficulty. Even if the press releases revert to the old format (“approximately 23.5%”), those GEMMs with a bid at the minimum would, by inspecting the amount actually delivered, still be able to deduce the precise scaledown. Hence this might not even be a new problem, it might be a very old problem which I, and perhaps others, had failed to notice.

A good algorithm is as follows. Compute the precise scaledown. In the press release round it to the nearest multiple of 10%: “approximately 20%”. For each GEMM compute the precise scaledown × bid size. Round down to a multiple of £1 million. Each GEMM will be filled in at least this size. Pick a GEMM at random. How much remains to be assigned? If zero, stop. If ≥1 million, that GEMM is changed from rounded down to rounded up (which, rarely, will make no difference), and is not picked again. If positive by <1 million, add that amount to the GEMM’s fill, and stop. And repeat this picking of a GEMM until all sold.

Consider a GEMM bidding for £1bn at the minimum. Then it could estimate the scaledown to within ±0.1%. That level of inaccuracy would mean that there are hundreds of possible bid quantities satisfying that scaledown: it’s effectively information-free.

## Conclusion

The DMO should fix or explain its apparent arithmetic errors.

Please could others check my reasoning, and, if the DMO doesn’t, explain what happened on 05 May 2016.

The DMO should consider whether it really wants to release this information about the distribution of bids, and if not, how to blur it (ideally with a much better auction mechanism).

— Julian D. A. Wiseman, London
Written, 3rd August 2016
Shown to DMO, 3rd August 2016
Published, 8th August 2016
www.jdawiseman.com

## Afterwords

### 10th August 2016, FT

The FT’s Alexandra Scaggs briefly discusses this in How much auction information is too much? It quotes the DMO:

“We regard the decision to announce the scaling ratio to 4 decimal points (from 1 decimal point) as a means of providing more detailed information about the auction statistics. We think that is a (minor) benefit to all interested parties rather than an information ‘leakage’.”

If the DMO really intended to reveal a small set of possibilities for the quantity of bids received at the minimum, that’s fine. But the quoted DMO statement doesn’t provide full reassurance about this.

She then goes on to quote “Wiseman’s recommendations for the UK” (except that they are recommendations for all large sovereigns). And:

It’s an interesting proposal, for sure, and does seem to solve both informational issues (of too much and too little). But we still kind of think a Treasury auction pit would be fun.

Yes, I agree with all that. But is “fun” really the central desideratum?

### 17th August 2016, subsequent auctions

Subsequent auctions, with the others’ non-comp adjusted to remove the small extras the DMO seems to add.

Auction
Date
GiltAmount
For Sale
Non-Comp
GEMM
Non-Comp
Other
Total
bids
ScaledownHighest
accept.
Average
accept.
Lowest
accept.
Max
Propn
Possibilities at the min. price:
ForSale ÷ Bids ≈ Scaledown  [Propn]
Wed 17 Aug 20164Q551250187.20.012
0.009
2124.20999.7597%195.000194.677194.35049.8%86.791 ÷ 87 ≈ 99.759770%  [08.17%]