Archive for April, 2007

April 30, 2007

Monday, April 30th, 2007
Note that these indices are experimental; the absolute and relative daily values are expected to change in the final version. In this version, index values are based at 1,000.0 on 2006-6-30
Index Mean Current Yield (at bid) Mean YTW Mean Average Trading Value Mean Mod Dur (YTW) Issues Day’s Perf. Index Value
Ratchet 4.56% 4.61% 42,404 16.40 2 -0.6230% 972.0
Fixed-Floater 5.48% 4.52% 122,596 16.47 6 +0.1112% 942.8
Floater 4.56% -19.27% 60,029 4.18 4 -0.0196% 1,062.0
Op. Retract 4.74% 3.12% 83,261 2.59 17 -0.0409% 1,032.5
Split-Share 5.04% 4.34% 177,507 4.01 12 -0.1518% 1,044.5
Interest Bearing 6.50% 4.87% 61,407 2.25 5 -0.2148% 1,047.8
Perpetual-Premium 5.07% 4.53% 219,572 6.15 54 -0.0271% 1,051.2
Perpetual-Discount 4.58% 4.61% 936,660 16.20 12 -0.2193% 1,054.1
Major Price Changes
Issue Index Change Notes
CM.PR.R OpRet Hit a 52-week low of 25.52 on volume of 500 – count ’em, 500 – shares, closing at 25.75-00, 2×6. Geez, this market is getting sloppy! Now with a pre-tax bid-YTW of 4.50% based on a bid of 25.75 and a softMaturity 2013-4-29 at 25.00.
Volume Highlights
Issue Index Volume Notes
RY.PR.G PerpetualDiscount 839,800 Recent new issue. Now with a pre-tax bid-YTW of 4.62% based on a bid of 24.46 and a limitMaturity.
BNS.PR.L PerpetualDiscount 133,900 Now with a pre-tax bid-YTW of 4.53% based on a bid of 24.90 and a limitMaturity.
GWO.PR.I PerpetualDiscount 57,558 Now with a pre-tax bid-YTW of 4.58% based on a bid of 24.75 and a limitMaturity.
BMO.PR.J PerpetualPremium 26,920 Now with a pre-tax bid-YTW of 4.59% based on a bid of 25.00 and a limitMaturity. Goes ex-Dividend 5/2
BAM.PR.B Floater 22,211 Pays 70% of Canadian Prime on par value. Closed at 24.80-89.

There were eleven other $25-equivalent index-included issues trading over 10,000 shares today.

New Issue : S Split Corp 5.25% Retractibles

Monday, April 30th, 2007

I had a look at the prospectus, as promised, and have added this issue to the HIMIPref™ Universe.

Maturity is 2014-12-1. There are no intervening redemptions.

Dividend is 5.25% p.a., payable monthly, par value $10.00. There will be no distributions to the Capital Unitholders if this would result in asset coverage for the prefs falling below 1.65:1.

Underlying security is shares of BNS. The manager is covering the cost of issue – in exchange, the manager gets a fee (payable by the retracting shareholder) if units are retracted prior to maturity. Hence, asset coverage will initially be (very close to) 2.5:1. Since BNS yields approximately 3.12% p.a., income coverage at issue will be in excess of 1:1.  

Downside is: DBRS rating of Pfd-2(low). I suspect that the issue lost a notch due to a very high concentration risk on BNS, but that’s for DBRS to say and for the rest of us to guess. Additionally, the maximum issue size is only $100-million. If they can get that high, it will be a respectable size as far as split-shares go, but trading may be expected to require patience.

A nice little issue, worthy of consideration as part of a DIVERSIFIED portfolio. Did I mention that a portfolio containing this issue should be DIVERSIFIED? Don’t come running to me if you have to sell 1,000 shares in two years and the price moves a buck. Or if BNS finds out they’ve made a little arithmetic error on their commodities trading and there won’t be any dividends for the next few years. Or whatever.

And while the preferred will be offered separately, there is no guarantee that they will be offered to YOU separately. The prefs are quite attractive enough that they should trade at a premium to par immediately upon issue.

The preIssue securityCode for this issue is P71400.

CU.PR.A & CU.PR.B Dividends Estimated

Monday, April 30th, 2007

It seems the folks at Canadian Utilities are so excited about the recent new issue and upcoming redemption that they have forgotten that they have other outstanding issues!

There is no data available from the company or the TSX regarding the dividends payable June 1. I have estimated the ex-date to be 5/8, record 5/10.

Split Share Discount

Saturday, April 28th, 2007

On the thread for April 27, Drew asked:

The YTW of split shares and perpetual premium shares seems to have risen over the last month substantially more than that of perpetual discount shares. My impression is that the bond yield curve has not flattened like this. Am I correct and, if so, do you have a theory?

Well, first off, let’s look at the index data: March 30:

Note that these indices are experimental; the absolute and relative daily values are expected to change in the final version. In this version, index values are based at 1,000.0 on 2006-6-30
Index Mean Current Yield (at bid) Mean YTW Mean Average Trading Value Mean Mod Dur (YTW) Issues Day’s Perf. Index Value
Op. Retract 4.72% 3.04% 85,479 2.16 17 -0.0828% 1,034.0
Split-Share 5.01% 3.14% 158,951 3.31 14 +0.0234% 1,052.8
Perpetual-Premium 5.02% 3.56% 219,123 5.15 53 -0.0031% 1,059.8
Perpetual-Discount 4.53% 4.54% 762,721 15.37 10 -0.0157% 1,066.8

…and for April 27:

Note that these indices are experimental; the absolute and relative daily values are expected to change in the final version. In this version, index values are based at 1,000.0 on 2006-6-30
Index Mean Current Yield (at bid) Mean YTW Mean Average Trading Value Mean Mod Dur (YTW) Issues Day’s Perf. Index Value
Op. Retract 4.73% 3.22% 84,115 2.38 17 -0.0108% 1,033.0
Split-Share 5.03% 4.29% 179,611 4.02 12 +0.1756% 1,046.1
Perpetual-Premium 5.07% 4.50% 222,579 6.25 54 -0.1567% 1,051.4
Perpetual-Discount 4.57% 4.59% 924,984 16.22 12 -0.0112% 1,056.4

From these indications, we see huge apparent changes in the yield of split shares. There are, as always, details of the analysis that must be understood before we pat ourselves on the back, however.

Consider the April 27 Split Share Index. Well, it looks like one thing that’s going to happen soon is that MUH.PR.A and ASC.PR.A will be moved to the “Scraps” index, on grounds of insufficient averageTradingValue, but never mind that.

One thing we notice is that DFN.PR.A & FFN.PR.A have much higher YTWs than FTN.PR.A, thanks to the recently approved term extensions on the former two issues. Be sure to write a thank-you note to your friendly neighborhood capital unit holder for the gift! Another thing we notice when looking at the index table is that the Split-Share index has been hit a lot harder than the Operating-Retractible index. This effect is due, I think, to a lack of understanding in the marketplace in general as to the nature of a split-share corporation. For example, one commenter on Financial Webring Forum stated that he was “not interested in … split shares that mature at NAV”.

Well, the preferred share component of a split share corp does not mature at NAV, absent default. The last two words are very important, because as I showed in the article Are Floating Prefs Money Market Vehicles?, Split Shares have, historically, been more susceptible to credit downgrades than other classes of share. However, readers who have read Using Credit Ratings When Buying Preferreds and Split Shares will know how to watch for the signs of an imminent downgrade. It seems to me that DBRS has been tightening its standards for Split Share credit ratings in the past year or two; as well, while the nature of a split share makes the rating more volatile, it also makes credit analysis a lot easier! So, while you have to watch them, so what? You have to watch everything in this uncertain world.

Some institutional investors, as well, don’t like Split Shares: one reasonably good reason is that not only are issue sizes relatively small, but they are rarely available as a new issue bought en bloc unless you also buy the Capital Units. One relatively bad reason is that many institutional guys don’t understand them either, another is that buying them might give the impression that they are sub-contracting asset management to the Split-Share’s sponsor, or at least have to explain to clients why that is not a fair characterization.

So in the end, Split Shares become not just a playground for retail, but for a relatively small component of the retail preferred share buying populace at that. This makes them much more susceptible to volatility and what I currently believe is contagion from the continuing woes of BCE.

I’ve uploaded a graph of the yieldCurvePremiumRetractible and the yieldCurvePremiumSplitShareCorp. On April 27, these values stood at -0.44% and +0.40%, respectively, changing from -0.42% and +0.34%, respectively, on March 30. So, yeah, Split Share spreads have widened quite noticeably over the past month. I’ve also uploaded a graph of the core yield curves at year-end, March month-end and now, for your inspection. All these curves and spreads, I hasten to note before I forget, are AFTER TAX.

Malachite Aggressive Preferred Fund currently has a relatively high exposure to Split Shares, so I could be accused of talking up my inventory. I could also be accused of putting my money where my mouth is. Take your pick – you have been warned!

 

April 27, 2007

Saturday, April 28th, 2007
Note that these indices are experimental; the absolute and relative daily values are expected to change in the final version. In this version, index values are based at 1,000.0 on 2006-6-30
Index Mean Current Yield (at bid) Mean YTW Mean Average Trading Value Mean Mod Dur (YTW) Issues Day’s Perf. Index Value
Ratchet 4.50% 4.51% 43,165 16.46 2 -1.3722% 978.1
Fixed-Floater 5.49% 4.50% 121,041 16.50 6 +1.1272% 941.7
Floater 4.56% -19.25% 57,731 4.19 4 -0.0194% 1,062.2
Op. Retract 4.73% 3.22% 84,115 2.38 17 -0.0108% 1,033.0
Split-Share 5.03% 4.29% 179,611 4.02 12 +0.1756% 1,046.1
Interest Bearing 6.49% 4.42% 61,507 2.26 5 -0.1175% 1,050.1
Perpetual-Premium 5.07% 4.50% 222,579 6.25 54 -0.1567% 1,051.4
Perpetual-Discount 4.57% 4.59% 924,984 16.22 12 -0.0112% 1,056.4
Major Price Changes
Issue Index Change Notes
BCE.PR.S Ratchet -2.9757% Exchange/Reset date is 2011-05-1 (exchanges with BCE.PR.T, which pay 4.502% of par until then). A sale of 1,000 shares by Scotia took the price from 23.86 to 23.36; later, Nesbitt bought 10,000 from National Bank at 23.25. Closed at 23.15-25, 1×8, compared to their exchangers at 23.00-50, 15×1. The relative prices don’t make a lot of sense, unless you assume that the “S” will not pay 100% of Prime (a decline of prime to 4.5% works too) so this looks like a good pairs opportunity provided that, like Nesbitt, you can buy 1,000+ shares without moving the price fifty cents.
W.PR.H PerpetualPremium -1.3642% Now with a pre-tax bid-YTW of 4.74% based on a bid of 26.03 and a call 2013-2-14 at 25.00.
BNS.PR.K PerpetualPremium -1.0465% When you start seeing solid issues like W.PR.H & BNS.PR.K on this list without a good explanation, you know the market’s getting a little sloppy. Now with a pre-tax bid-YTW of 4.48% based on a bid of 25.53 and a call 2014-5-28 at 25.00
SLF.PR.D PerpetualDiscount -1.0221% Blame it on Scotia’s wild-man clients, as their sale of 16,500 shares in four tranches (briefly interupted by a sale of 1,000 by Anonymous) took the price from 24.40 (Buyer = Scotia) to the final tranche of 14,300 at four minutes to the bell (Buyer = Scotia). Now with a pre-tax bid-YTW of 4.63% based on a bid of 24.21 and a limitMaturity.
BCE.PR.A FixFloat +1.0305% Yep, that’s a plus sign by the number – and there’s more to come! Exchange/Reset date is 2007-09-01 (with series ‘AB’, not issued); until then, pay 5.03% of par. Closed at 23.25-36, 16×3.
BCE.PR.I FixFloat +1.1468% Exchange/Reset date is 2011-8-1 (Exchange with series ‘AJ’, not issued); until then pay 4.65% of par. Closed at 22.05-50, 6×1, on good volume of 10,256 shares.
LBS.PR.A SplitShare +1.1505% Almost, but not quite, undoing yesterday’s swoon, closing at 10.55-57, 60×1 (a good sized bid for this issue). Now with a pre-tax bid-YTW of 4.34% based on a bid of 10.55 and a hardMaturity 2013-11-29 at 10.00
BCE.PR.G FixFloat +1.4052% Exchange/Reset date is 2011-5-1 (exchange with BCE.PR.H); until then pay 4.35% of par. Today’s return, by the way, is based on the closing bid … the closing price was down on the day, which just goes to show … something or other. I pay little attention to closing price … it’s far more volatile than the quotes. Usually. Anyway, the closing quote was 21.65-94, 8×4, while the BCE.PR.H were at 23.40-00. The Gs still look cheap relative to the Hs, but the Hs look expensive relative to the Ss, so it’s all very complicated.
BCE.PR.C FixFloat +2.3344% Exchange/Reset date is 2008-03-01 (exchange with series ‘AD’, not issued); Until then pay 5.54% of par. Closed at 23.60-82, 3×3.
Volume Highlights
Issue Index Volume Notes
CM.PR.G PerpetualPremium 411,025 Scotia crossed 28,100 at 26.50; RBC crossed 82,900 at 26.52. Now with a pre-tax bid-YTW of 4.52% based on a bid of 26.37 and a call 2014-5-31 at 25.00.
RY.PR.G PerpetualDiscount 519,113 Recent new issue. Now with a pre-tax bid-YTW of 4.62% based on a bid of 24.42 and a limitMaturity.
BMO.PR.J PerpetualPremium 158,550 Now with a pre-tax bid-YTW of 4.58% based on a bid of 25.02 and a limitMaturity.
BAM.PR.K Floater 151,300 RBC crossed 150,000 at 24.87. Pays 70% of Prime on par value.
TD.PR.O PerpetualPremium 137,245 National Bank crossed 25,000 at 26.25, then another 85,700 at the same price. Now with a pre-tax bid-YTW of 4.12% based on a bid of 26.21 and a call 2014-11-30 at 25.00.

There were twenty-one other $25-equivalent index-included issues trading over 10,000 shares today.

Boy, these daily summaries are taking a long time to write nowadays! I’m going to have to start charging you guys extra for this.

Two New Split Corps!

Friday, April 27th, 2007

There are two new split-share corporations in the distribution pipelines:

S Split Corp has been announced by Mulvihill. These will be backed by shares of BNS – the Bank of Nova Scotia. The news release says that DBRS has assigned a preliminary rating of Pfd-2(low) and that the final prospectus has been filed with SEDAR – but it’s not available there right now. They may have it on-line Monday. Mulvihill has an impressive distribution pipeline and all the usual suspects are in the underwriting syndicate for this one. Depending on the terms of the issue, I may be adding this one to the HIMIPref™ universe.

ML Split Corp has been announced by Quadravest. These will be backed by shares of Merrill Lynch. A Preliminary Prospectus is available – I will not be adding these to the HIMIPref™ universe as the company is not having the “Priority Equity Shares”, as they call the preferred-sort-of-equivalent components of the split, rated by any agency.

RY.PR.G Splatters onto Market

Friday, April 27th, 2007

The new issue of Royal Bank 4.5% perpetuals announced April 17 settled today and met a very poor reception, trading in a range of 24.48-60 and closing at 24.49-50, 20×12.

I’m at a bit of a loss to understand this and can only speculate that the continuing BCE debacle has caused a little nervousness amongst retail, while institutional buyers may be filled up on Royal after their string of new issues:

RY Issues Tracked by HIMIPref™
Ticker Listing Date Shares
RY.PR.K 1998-4-27 12,000,000
RY.PR.W 2005-01-31 12,000,000
RY.PR.A 2006-04-04 12,000,000
RY.PR.B 2006-07-20 12,000,000
RY.PR.C 2006-11-01 8,000,000
RY.PR.D 2006-12-13 10,000,000
RY.PR.E 2007-01-19 10,000,000
RY.PR.F 2007-03-14 8,000,000
RY.PR.G 2007-04-26 10,000,000

RY.PR.K is retractible – all the others are perps. 

However, it might not matter a lot whether the market is fed up with the name or not! Examining the figures for Royal’s tier one capital limits, we see that they had room to issue preferred of $520-million on February 6 (after the issuances of RY.PR.C, RY.PR.D & RY.PR.E and redemption of RY.PR.O) and with the 18-million shares issued since then have used up $450-million of that. That leaves a mere $70-million in issuance room and they might not be willing to go to market for such a paltry amount.

Note I will admit that I am somewhat at a loss to reconcile their Preferred Share Tier One Capital of $1,345-million at year end with their list of issues outstanding. The figure of $1,345-million is referred to in the MD&A on page 66 of the Annual Report – this table contains no mention of any preferred shares in Tier Two Capital, which is where I would expect to find the retractible issue RY.PR.K. Note 18 on Page 130 of the Report shows $1,050-million perpetuals, and $298-million “Preferred Share Liabilities”, specifically including RY.PR.K (Series N). So I guess that, somehow or other, they were able to include RY.PR.K in Tier 1 Capital.

So, given that the RY.PR.O has been redeemed, their Tier One Capital preferred situation now looks like this:

Tier 1 Capital / Preferreds / Royal Bank
Item Value (million)
Outstanding, year-end 1,345
Redeemed (150)
Issuance 1,150
Current Total 2,345
Preferred Limit, as of Year-End 2006 2,415

All in all, they’re very close to their limit now, unless they boost their equity capital in other ways, like hiking ATM fees. But fear not! The RY.PR.K becomes redeemable at par 2007-08-24 (although not retractible by holders until 2008-8-24) and eliminating this issue with open up another $300-million of issuance room.

RY.PR.G & Comparatives
Data RY.PR.G BNS.PR.M BAM.PR.?
Price due to base-rate 22.47 22.49 23.29
Price due to short-term -0.21 -0.21 -0.21
Price due to long-term 1.29 1.29 1.27
Price to to Cumulative Dividends 0.00 0.00 0.00
Price due to Credit Spread (2) 0.00 0.00 -0.62
Price due to Liquidity 1.53 1.53 1.48
Price due to error -0.06 -0.06 0.08
Price due to Credit Spread (low) 0.00 0.00 -0.62
Curve Price (Taxable Curve) 25.02 25.04 24.68
Dividend Rate 1.125 1.125 1.1875
Quote 4/26 24.49-50 24.89-92 25.00 Issue
YTW (at bid, after tax) 3.66% 3.61% 3.79%
YTW Date Infinite Infinite 2016-8-30 / Infinite
Credit Rating (DBRS) Pfd-1 Pfd-1 Pfd-2(low)
YTW (Pre-Tax) 4.61% 4.55% 4.76%
YTW Modified Duration (Pre-Tax) 16.23 16.31 15.95
YTW Pseudo-Convexity (Pre-Tax) 1.15 -30.29 -55.80

It is not my habit to include such an incomparable comparable as the BAM new issue, but I just couldn’t resist! BAM has a boatload of preferreds outstanding, and if we can blame overall market tone and angst for today’s RY.PR.G debacle, then the May 9 BAM settlement could prove interesting in the extreme.

The securityCode for RY.PR.G is A45016, replacing the preIssue code of P87500. A reorgDataEntry has been processed.

April 26, 2007

Thursday, April 26th, 2007
Note that these indices are experimental; the absolute and relative daily values are expected to change in the final version. In this version, index values are based at 1,000.0 on 2006-6-30
Index Mean Current Yield (at bid) Mean YTW Mean Average Trading Value Mean Mod Dur (YTW) Issues Day’s Perf. Index Value
Ratchet 4.42% 4.42% 42,977 16.62 2 -0.1605% 991.7
Fixed-Floater 5.50% 4.55% 118,964 16.29 6 -0.3041% 931.2
Floater 4.56% -19.12% 57,071 0.13 4 +0.0884% 1,062.4
Op. Retract 4.73% 3.24% 84,582 2.38 17 -0.0507% 1,033.1
Split-Share 5.04% 4.33% 182,814 4.02 12 -0.0422% 1,044.2
Interest Bearing 6.48% 3.87% 61,798 2.27 5 +0.1972% 1,051.3
Perpetual-Premium 5.07% 4.48% 223,033 6.23 54 -0.1385% 1,053.1
Perpetual-Discount 4.57% 4.59% 935,745 16.23 12 -0.3302% 1,056.6
Major Price Changes
Issue Index Change Notes
BCE.PR.G FixedFloater -2.7335% Exchange/Reset date is 2011-05-1 (exchanges with BCE.PR.H); until then pays 4.35% of par. Another new low today, 21.01. Closed at 21.35-55, 10×2. The BCE.PR.H closed at 23.35-98, 50×3. Let’s see …BCE Pairs… say the “H” pay 6% (=100% of current prime), vs. the 4.35% on the “G”. Difference = 1.65% = $0.4125 p.a. Term to exchange is 4 years. I guess the market is betting that not only will the “H” pay 100% of prime until the exchange date, but that prime’s going to go up, too!
LBS.PR.A SplitShare -1.4178% Now with a pre-tax bid-YTW of 4.54% based on a bid of 10.43 and a hardMaturity 2013-11-29 at 10.00
Volume Highlights
Issue Index Volume Notes
RY.PR.G PerpetualDiscount 519,113 New issue settled today. Now with a pre-tax bid-YTW of 4.61% based on a bid of 24.49 and a limitMaturity.
BNS.PR.M PerpetualDiscount 169,125 Scotia crossed 101,600 at 24.87. Now with a pre-tax bid-YTW of 4.55% based on a bid of 24.89 and a limitMaturity.
BCE.PR.C FixedFloater 166,820 Exchange/Reset date is 2008-3-1 (Exchange with series ‘AD’, not issued). Until then, pay 5.54% of par.
PWF.PR.K PerpetualPremium 52,770 Scotia crossed 50,000 @ 25.88. Now with a pre-tax bid-YTW of 4.44% based on a bid of 25.85 and a call 2014-11-30 at 25.00
CM.PR.I PerpetualPremium 49,125 Now with a pre-tax bid-YTW of 4.62% based on a bid of 25.18 and a call 2016-3-1 at 25.00.

There were eighteen other $25-equivalent index-included issues trading over 10,000 shares today.

HSBC Tier 1 Capital

Thursday, April 26th, 2007

HSBC has released its 2006 Annual Report, so I thought I’d take a look at their regulatory capital to see if any issues can be hoped for from that direction. I’ve discussed Tier 1 Capital before in a mini-series of posts: Part 1, Part 2 & Part 3.

HSBC Tier One Capital  

(From December 31, 2006 Financial Statements)

Total Tier 1 Capital (millions) 3,283
Common Shareholders’ Equity 76.7%
Preferred Shares 10.7%
Innovative Tier 1 Capital Instruments 12.2%
Non-controlling interests in subsidiaries 0.9%
Goodwill -0.5%

So if they wished, they could issue addition non-cumulative perpetuals to a total of 2.1% of capital, or $69-million worth, and have them all included in Tier 1 Capital – unfortunately, such a small issue scarcely seems worthwhile.

And, unfortunately, not only do their two outstanding issues (HSB.PR.C & HSB.PR.D) pay a dividend that’s not a lot more than market rates, but they’re not redeemable until 2010, either.

So … probably not much issuance from that quarter! Unless, of course, Canada’s banking laws triumphantly enter the 19th century, and HSBC developes an appetite for one of the big five!

CL.PR.B : YTW Returns to Positive Territory!

Wednesday, April 25th, 2007

Readers will remember that I have been fascinated by CL.PR.B and its negative yield-to-worst for some time. Apart from very particular portfolio-management factors, there hasn’t seemed to be much rationale for holding it, other than a hope that it would continue to pay an annual dividend of 1.5625 forever.

The chance of this has never seemed too large to me, given that CL (Canada Life) is part of the GWO (Great-West) group of companies which is in turn controlled by PWF (Power Financial) … a conglomerate that has something of a reputation for knowing how many beans make five.

Even four months after I expected them to be redeemed, I am still a little confused, since paying 1.5625 on a perp when new perps are paying 1.125 – and I can’t see a GWO issue having to pay more that 1.30, no matter how sick the Street is of seeing the name – doesn’t make a lot of sense to me. However, as I have mentioned numerous times, the purchase of Putnam still needs to be financed and maybe they’re just delaying a little until they’ve got all that stuff squared away.

Today, however, the bid broke down and while the trading range was 26.31-42 (on volume of 1,950 shares), the closing quotation was 26.09-40, 5×25. So, in celebration, I’ve uploaded a few graphs:

The recent decline in price of CL.PR.B (and consequent increase in yieldToWorst) has had a salutary effect on the calculated mean-average-YTW of the PerpetualPremium index, which now has no members with a negative Yield-to-Worst and consequently a more meaningful mean. Problems with computing the mean – even less meaningful when negatives are included than it is with all positive numbers – has led me to use the median for the official HIMI Preferred Indices … and don’t worry, guys, I’m having scheduling problems at the moment, but will return to those computations in the near future. Unless a piano falls on my head.