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Sunday, July 24, 2016

Seriously, Money Is Not A Zero Coupon Perpetual

Money needs to be abolished -- from economic theory. Too much importance is attached to it, and it skews people's abilities to analyse it properly. For example, people make fundamental errors, such as thinking of it as a "zero coupon perpetual bond." (UPDATE: I respond to Nick Rowe's response...)

Background

A perpetual bond is a bond that pays a fixed coupon on a fixed schedule (for example, annually, or semi-annually), but has no fixed maturity date. For example, we could have a perpetual bond that pays $1 on every December 1st (with the standard correction for weekends). These show up a lot in financial and economic theory, but are rare in practice. (The British government issued perpetual bonds called consols. Consol is often used as a synonym for perpetual, since it sounds cooler).

It should be noted that companies issue perpetual preferred shares that look like consols, but their valuation is skewed by the fact that they are relatively junior in the capital structure of the issuer.

Valuation

If a perpetual bond does not have any embedded options, and there is no default risk (see below), we can scale the face value/coupon by any amount, and it does not really matter. For example, the following two ownership structures exhibit the exact same cash flows:
  1. Holding perpetual bonds with a face value of $100 and a 2% annual coupon,
  2. Holding perpetual bonds with a face value of $200 and a 1% annual coupon.
In both cases, the bonds pay $2 per year, and there are no other possible cash flows. (If there is a default, the face value might matter for recovery calculations; I discuss the issue of embedded options below.)

The usual convention in mathematical finance is to assume that the bonds pay $1 per year, and there is a notional 1% coupon. This works fine if the issuer does manage to auction the bond at a 1% yield; otherwise the issuer is auctioning the bond away from a par value. Since these bonds tend to only appear in mathematical models, the institutional problems with auctioning bonds away from par does not really matter.

For such a standard structure, if we are conveniently pricing the bond on the coupon date (ex-coupon), the pricing formula is:  p = 1/r, with p being the dollar price, and r the quoted yield.

Examples:
  • If r = 0.01 (1%), p = $100.
  • If r = 0.005 (0.5%), p = $200.
It can be seen that the price tends to infinity as the quoted yield goes to zero. This well-known relationship has caused some commentators to make incorrect claims about bond market duration as bond yields went to zero; for finite maturity bonds, the maturity date caps the Macaulay duration of the bond.

Embedded Put Option

Some bonds have an embedded put option (redemption option) -- owners of the bond can demand immediate payment on the bond ahead of maturity (following some contractual limitations).

This turns the bond valuation into an exciting exercise in fixed income derivative pricing. I do not want to go into that subject, instead, I will give a lower bound for the value of the bond.

If we have an embedded put option, we have a possible cash flow that is not a coupon payment. In this case, the face value matters.

We will assume the perpetual is described by:

  • $100 face value.
  • 1% annual coupon.
  • Puttable to the issuer (the government) at par ($100) on any coupon date (after the owner receives the coupon). (On non-coupon days, the bond is puttable at a price that takes into account accrued interest on the coupon. My analysis here ignores that issue; the time axis only considers annual coupon dates.)
If we had a non-puttable perpetual trading at a yield of r, it can be easily shown that the lower bond of the price of a $100 face value puttable bond is:
  • $100 if r > .01,
  • 1/r otherwise. (Note: this is greater than $100).
If the volatility of interest rates is non-zero, the value of the bond with the embedded option is going to be greater than the lower bound given above. The extra value is the "time value" of the perpetual (American) put option. (The option value would be greatest near the strike of the redemption option, which is at r=1%.)

It is not too hard to see that any perpetual bond that is putable at par will have a lower bound for its fair value at par, no matter what the coupon rate is.

Money - A Zero Coupon Putable Perpetual

We can now see how people thinking that money is a zero coupon perpetual go wrong.

The argument that money is a zero coupon perpetual looks at a sequence of contract structures, where the coupon as a percentage of face value declines. It is not hard to see that setting the coupon rate to zero causes problems in an option-free perpetual.

This is not the case for a putable perpetual. It always has a floor value of 100% of face, regardless of the coupon. Setting the coupon rate to zero just ensures that the value remains at 100% of face, there is no upside (assuming rational market pricing).

Is money putable? Yes it is. You can redeem it for face value to meet tax obligations. Furthermore, private debts are payable with government or bank money at par (due to legal tender laws), and so you can invoke the put option against your lenders. You cannot do this with a bond (under normal circumstances; one can imagine an issuer allowing taxpayers to pay by returning bonds using some redemption price). The money instrument cancels out the tax/debt obligation at a 1:1 ratio.

Even if you do not personally pay taxes (such as "you" being a pension fund), you just need to find somebody who does. You would be a sucker to sell your monetary instrument at below face value to such an entity.

(UPDATE: added, in response to comments elsewhere.) If we invoke the principle that there is one price for assets in a market, it does not matter that I do not have the right to always put (redeem) all of my money holdings. (A putable bond would have such a right.) I would be exposing myself to an arbitrage if I sold my monetary instruments for less than par (somehow), to the advantage of entities need to raise money. Since the option to redeem is also perpetual, the option value includes the possibility of redeeming at a future date. The opportunity cost of waiting to redeem is equal to the cost of holding a non interest-bearing asset, as one would expect.

Money With Interest

An additional wrinkle is that money as it appears in economic models pays 0% interest. This is because writers have currency (notes and coins) in mind when talking about "money." In the real world, there are a lot of financial instruments that are part of "monetary aggregates" (since money-fixated economists need to keep patching the aggregates so that they give the answers that the economists want them to give).

Even if currency pays no interest, it can be exchanged at par by banks for settlement balances at the central bank. Those settlement balances may pay a rate of interest, and so we can view them as a perpetual floating rate instrument. Such an instrument could be theoretically swapped into a fixed-rate instrument, and so has non-zero value (since the swaps have a net present value of zero).

Since the central bank must redeem currency for settlement balances, we can see that currency has a redemption option into a perpetual floating-rate security.

How Does Money Price Versus Real Stuff?

The discussion above was purely a fixed income pricing argument, where we are reducing future cash flows to their present value. "Money" has a price of $1 in a Net Present Value (NPV) calculation by definition; in order to conceive of a NPV away from $1, we have to conceptualise a new unit account that is somehow not "money." If we move away from the abstract concept to actual real-world instruments (which is what I assert that economic theory needs to do), such a new unit of account is more easily understood. For example, deposits in a failing bank are not going to trade at par in the absence of deposit insurance.

Does any of this tell us about the value of money versus goods and services in the real world (that is, the price level)? No, it doesn't. When we are discussing NPV calculations, we are in a closed mathematical world of money and forward money; whether that money can buy anything in the real world is unknown. What determines the price level depends upon your economic belief system. (Once again, showing that a degree in anthropology is probably more useful for understanding economic discussion than a degree in economics.)
  • If you believe that mainstream economics is even slightly close to correct, if you pursue the mathematics, you end up with the Fiscal Theory of the Price Level. The punchy summary of the Fiscal Theory of the Price Level is that expected fiscal surpluses (that is, taxes) drives the price level (also known as the value of money).
  • If you are a post-Keynesian, what you think about this topic might depend upon your tribal affiliation. The neo-Chartalist wing (for example, Modern Monetary Theory) argues that taxes drive money, which as a verbal formulation sounds similar to the Fiscal Theory of the Price Level.
  • I have a hard time characterising other schools of thought. For example, some Austrians dispute the very existence of the concept of a price level.
In any event, saying that the value of money is driven by taxes ties in exactly with the redemption option discussed above.

Concluding Remarks

This essay was triggered by the article "Some Simple Basic Money for Finance People" by Nick Rowe. His analysis was premised on perpetuals that have no embedded option, and thus he ended up with the wrong answer.

UPDATE (2016-07-25) Nick Rowe's Response

Professor Nick Rowe responded on twitter:

The linked post did not respond to this article, and raises other issues.

  1. His original thesis was that "finance people" could not convincingly model the NPV of money as a zero coupon perpetual. This article shows that his suggested structure was wrong. and that finance people can create a structure that mimics the NPV characteristic of money.
  2. The thesis in his linked article is either revolutionary, or incorrect. His basic assumption is that economic models cannot relate stock and flow variables; however, almost all of them do (including Market Monetarist assertions that GDP can controlled by changing the monetary base). The Fiscal Theory of the Price Level (FTPL) offers what appears to be a mathematically coherent framework that relates fiscal policy and the real value of government liabilities, and it respects accounting identities. Professor Rowe would need to demonstrate the mathematical errors within the FTPL to prove his thesis. The mathematical framework of the FTPL can be adapted to Chartalism, although the Chartalists would then throw out many of the behavioural assumptions (but keep the accounting identities).
  3. We cannot just look at the tax demand for money. Private bank money is pegged at a 1:1 ratio to government money, and the amount of private debt that is rolled over within a week dwarfs national income and the monetary base. Private entities need that money to roll those debts, and the "law of one price" ensures that entities without an immediate need for money would not give it away for free.

(c) Brian Romanchuk 2016

28 comments:

  1. "Private bank money is pegged at a 1:1 ratio to government money, and the amount of private debt that is rolled over within a week dwarfs national income and the monetary base."

    Is that 1 to 1 ratio both ways?

    Does the price inflation target apply to the lender of last resort function of the central bank to the commercial banks?

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    1. The pegging is the result of a lot of legal and regulatory institutions; and the pegs effectively work both ways. Legal tender laws are not there to prevent new currencies from being used (the gold enthusiast complaint), rather requiring that government/bank money be accepted at par.

      There are a lot of things we take for granted which did not exist in the 19th century (in the US, at least) - cheques are cleared at par, the government accepts bank cheques for taxes, banks accept government money/cheques at par, deposit insurance, etc.

      There are not a lot of ways in which bank/government money can end up below par; one of the few exceptions is when you go beyond deposit insuran limits, and the bank goes bust.

      This parity does not apply to the instruments in wider "monetary aggregates"; M3, etc. However, we may note that the US Federal government did backstop money market funds voluntarily during the crisis; and there is a very large "too big to fail" belief with regards to non-guaranteed bank instruments (outside the euro area).

      The price inflation target is distinct from the issue of different monetary instruments trading at par from each other. There is no guarantee that a particular amount of money can buy anything; inflation targeting is just a suggestion. Lending of last resort operations effectively keep private money at par, and might be "relatively inflationary". However, if we look at the Financial Crisis, "relatively inflationary" may just mean "preventing a debt deflation." (I am not going to try to guess what would have happened if the central banks just let the banking system try to save itself.)

      I got involved in the argument er debate on Worthwhile Canadian Initiative, and wrote more about the pegging issue than I did here.

      Delete
    2. "the pegs effectively work both ways."

      And, "The price inflation target is distinct from the issue of different monetary instruments trading at par from each other."

      That is not what nick rowe says. He says the monetary base is medium of account. Demand deposits of the commercial banks are medium of exchange, but not medium of account.

      If the peg is both ways (currency goes up, demand deposits can go up AND demand deposits go up, currency can go up), then there is a fixed exchange rate both ways. Currency and demand deposits are BOTH medium of account and medium of exchange. That means the price inflation target does not apply to the lender of last resort function of the central bank to the commercial banks.

      Those two systems are COMPLETELY different from each other.

      Nick's system means demand deposits can prevent price deflation but cannot cause too much price inflation.

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    3. Nick calls his system "asymmetric redeemability". I do not think asymmetric redeemability exists. I think it is both ways.

      I say the Bank of Canada also makes a 1 to 1 promise for demand deposits to currency as long as the bank is solvent. If the bank is not solvent, then there is deposit insurance.

      This is an extremely, extremely, extremely important point about how the system actually works.

      I am pretty sure nick thinks the price inflation target applies to the lender of last resort function of the central bank to the commercial banks.

      Delete
    4. He's looking at some aspects of the relationship (the Bank of Canada has a lot more clout than a private bank); but in practice, the legal obligations seem symmetric to me.

      The price target theoretically applies, but there's not a lot of evidence that the lender-of-last-resort operations have much of an effect on the price level. There was some concern to that effect in the 1970-80s, which is probably what he has in mind. (There were a series of financial crises that helped stop the central bank from tightening policy.) It's a fairly theoretical worry in this environment.

      Delete
  2. I will try to show an example.

    On odd days, entities use all demand deposits. On even days, they use all currency (using lender of last resort from the central bank).

    Assume the velocity of demand deposits and currency are the same.

    Every entity spends what they earn. Price inflation is on target.

    This goes on for days. Now alter the scenario.

    Entities start borrowing from commercial banks and spending raising the price inflation above target that day. What happens the next day?

    I say the central bank allows the conversion, meaning price inflation in terms of currency goes above target. The central bank then hopes market interest rates rise and/or raises the overnight rate it controls.

    I believe nick would say the central bank would deny the conversion because price inflation goes above target. Something else now needs to happen so the number of demand deposits goes back down so price inflation goes back down.

    I believe it is an interest rate system. Nick thinks it is an asset buying system.

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    1. The amount of currency in circulation is actually relatively small; big swings in usage would probably overwhelm bank machines. That was one of the worries about Y2K (New Year's Eve 1999); if "everyone" took out an extra $20, it would have swamped the normal bank currency stockpiles (or something like that; I forget the exact number that was floating around ahead of the event).

      Although unlikely, let's imagine that the central bank did need to intervene to deal with an unusual demand for currency. If spending did not rise, there is no implication for monetary policy.

      If there is was simultaneously an increase in spending, the central bank would have to raise rates and simultaneously lend to banks against illiquid assets. Yes, those operations work at cross purposes, but allowing the banking system to crater is a really bad idea. If the central bank is giving banks an emergency lifeline, they would have considerable clout to keep the quantity of lending under control.

      The more realistic potential problems would resemble those that arose in the 1970s. Systemic fragility made the Fed skittish about raising interest rates. The issue was not the lending operations, which were small and self-liquidating; it was the unwillingness to tighten policy for fear of causing wider problems. I think Minsky wrote about them; I am unsure where I read it.

      Delete
  3. "If there is was simultaneously an increase in spending, the central bank would have to raise rates and simultaneously lend to banks against illiquid assets. Yes, those operations work at cross purposes, but allowing the banking system to crater is a really bad idea."

    That sounds like an interest rate system, not an asset buying system. It also looks like you are on the side of demand deposits being not just MOE, but also MOA.

    If you are willing, you should try to describe that to nick. My guess is that he will *vehemently* say that is not right.

    The MM's (including nick) believe the central bank is in control of "money". If the central bank uses lender of last resort function unconditionally to the commercial banks (other than solvency), then the central bank is not in control of "money".

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    1. The lender of last resort is technically a solvency operation. It could be used as cover for loose monetary policy. It's an issue that is pretty much dead right now, so I have not really worried about it.

      My view is the standard one that the central bank can set the policy rate; one can debate about the rest of the yield curve. I will actually be putting out an article on one aspect of the MM view of interest rates tomorrow.

      Whether or not the central bank can control the monetary base is a quite old debate. There was a flare up around 2012 (Krugman contra Keen) that was a real spectacle for econblogging. (Before my blog was up and running; I would have been muck raking with the rest of them.) At this point, debating the issue seems futile. I will have to have a short section in my next book about it, mainly because it is too big to ignore, but other than the MM holdouts, no one really believes the "central bank sets the monetary base" line (other than in some mystical expectations fairy sense).

      Delete
    2. I will leave this one for reference:

      http://worthwhile.typepad.com/worthwhile_canadian_initi/2016/04/on-central-bank-lending-to-government.html

      "John: Yep. That's the danger of a central bank acting as *unconditional* lender of last resort, whether to a commercial bank or to a government. Suppose the Bank of Montreal (BMO) gets into trouble, and the Bank of Canada steps in as LOLR. If it does so unconditionally (as opposed to ordering BMO to get its act together), then the BoC is now responsible for maintaining the fixed exchange rate between BMO dollars and BoC dollars. Alpha and beta have changed places. BMO is now the new central bank, and can create as many BMO dollars as it likes, knowing the BoC must follow along."

      Nick does not say what get in trouble means. I am pretty sure he thinks having demand deposits cause too much price inflation is *one* (not the only) way the BMO could get in trouble. Having demand deposits cause too much price inflation causes a run on the bank. In that scenario, the central bank will not act as a lender of last resort and therefore limit the amount of monetary base. I do not agree with that.

      Delete
    3. Here is another one about "money" to think about:

      http://worthwhile.typepad.com/worthwhile_canadian_initi/2015/12/money-demand-and-supply-in-a-redgreen-world.html

      ""Green" money has positive value; "red" money has negative value. (An overdraft in your chequing account is red money.) We live in a red/green world with both types of money, so we ought to start thinking about money supply and demand in a red/green world. The red/green world is the real world."

      I say we live in a green world only. An "overdraft" of a checking account is not "red money". An overdraft of a checking account is just another type of loan.

      Delete
    4. The "get in trouble" phrasing for a bank would normally imply making bad loans, and lenders do not want to buy the bank's debt securities. (Banks cannot fund themselves 100% with deposits; their customers want to buy bonds as well, and so deposits are lost to the bond market.) This is a problem for the bank, but might not a general macro issue unless the entire banking system generated a bubble of some sort. But once the bubble has burst, there is unlikely to be an inflation problem; the inflationary pressure would have been felt during the bubble expansion.

      An argument that lender-of-last-resort gives the private bank free rein is dubious. A competent central bank is going to breath down the neck of the back, forcing it to get its act together. A bank is not going to go off on a lending rampage when it is crawling out from under bad loans.

      Did not read the red/green article (yet). An overdraft is a form of a loan. You could try to treat them as a negative deposit, but you end up hurting your brain and getting not much in the way of an insight. I do not pay much attention to the details of bank regulation, but I assume that they need to hold capital against overdrafts, like any other loan. If banks need to hold capital against overdrafts, they are indistinguishable from a loan in terms of their economic impact.

      Delete
    5. "The "get in trouble" phrasing for a bank would normally imply making bad loans"

      True, but normal does not apply to a lot of economists. Nick would say creating too many demand deposits leading to violating the price inflation target would "get the bank in trouble", not just insolvency.

      "An argument that lender-of-last-resort gives the private bank free rein is dubious."

      For insolvency (bad loans), yes. For violating the price inflation target, I would say no.

      "An overdraft is a form of a loan. You could try to treat them as a negative deposit, but you end up hurting your brain and getting not much in the way of an insight."

      I would say an overdraft is a type of loan, period. There is no such thing as "red money". There is "green money". There are "green bonds". An overdraft is nothing more than an exchange of "green money" and "green bonds", just like other types of loans.

      Delete
    6. "True, but normal does not apply to a lot of economists. Nick would say creating too many demand deposits leading to violating the price inflation target would "get the bank in trouble", not just insolvency."

      I'm not sure that Nick Rowe would say that. The Bank of Canada is supposed to respect the inflation target; not the Bank of Montreal. If the BoC tightens monetary policy, that's not supposed to be the concern of a well-run bank; it's a problem for the bank's customers. (If those customers go bust, then it's the bank's problem.)

      "'I would say an overdraft is a type of loan, period. There is no such thing as "red money"'"

      I have not read his article (still), but I did write a draft primer on banking for my next book. (Which may or may not make it into the book as an appendix.) In principle, we could try to account for overdrafts as a negative deposit, but thinking about it made my head hurt. (A liability with a negative value? Huh?) I think I dropped the subject, as I did not want to have to get my hands on a banking accounting textbook to explain it properly. But yes, an overdraft is equivalent to a loan, and should act the same within a model.

      I don't really care Whether this messes up "money supply" numbers, since I think those numbers have almost no real world significance. In other words, the distinction is cosmetic, and not worth arguing about.

      http://www.bondeconomics.com/2016/06/pragmatism-and-mmt.html

      Delete
    7. Wrong place earlier.

      "I'm not sure that Nick Rowe would say that. The Bank of Canada is supposed to respect the inflation target; not the Bank of Montreal."

      Would you be willing to ask him? That is the impression I get from his posts.

      "If the BoC tightens monetary policy"

      Is that a reference to the overnight interest rate?

      "But yes, an overdraft is equivalent to a loan, and should act the same within a model."

      So does an overdraft and a car loan have the same accounting (the actual bond/loan terms could be different)?

      Delete
  4. Money is the buffer stock of customary means of payment held as a precaution when purchasing power of money is declining (inflation) and held as an investment when purchasing power of money is rising (deflation). Beyond that understanding one must observe how money is created and destroyed as the residual interaction between customers and financial intermediaries under different economic conditions. Hyman Minsky understood money as a buffer stock and recognized how it is generated or destroyed by government and market finance activity.

    ReplyDelete
  5. "I'm not sure that Nick Rowe would say that. The Bank of Canada is supposed to respect the inflation target; not the Bank of Montreal."

    Would you be willing to ask him? That is the impression I get from his posts.

    "If the BoC tightens monetary policy"

    Is that a reference to the overnight interest rate?

    "But yes, an overdraft is equivalent to a loan, and should act the same within a model."

    So does an overdraft and a car loan have the same accounting (the actual bond/loan terms could be different)?

    ReplyDelete
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    1. I have been in enough tangles with Nick Rowe recently; not sure if I need to open more arguments... Even if he did say that, he would be the only person who would. Private sector entities are just supposed to assume that the central bank hits the target; it is the central bank's problem to get it to target. Whether or not a private entity is exposed to central bank tightening depends entirely on how it is positioned, so we cannot make any generalisations that a private entity would be in trouble because of monetary tightening.

      When I write "tightening policy," I mean "increase the overnight rate"; I think it would be safe to say that Nick Rowe would that it means "shrink the monetary base." However, the exact way in which the central bank shrinks the monetary base involves expectations, and the discussion gets extremely arcane. It's an old argument, which I do not want to reopen. (The "endogenous money debate".) I will eventually write about it, but only because it's too big a topic to ignore.

      I am unsure how overdrafts are accounted for. You could do it a number of ways; I would need to find a textbook on bank accounting to get the "correct answer." Within economic models, we do not have to follow GAAP, and so we can chose whatever is convenient. The sensible way would be to treat it pretty much like an auto loan. (So long as it stays on the bank's balance sheet, which is getting rare in practice. Furthermore, if a loan is floating versus a fixed coupon does matter, but most economic models do not worry about that level of complexity. Some of the Stock-flow Consistent models do worry about effects like that; it is related to how inflation affects behaviour.)

      Delete
  6. Brian and Anonymous: this has been a good comment thread. I don't have much (useful) to add at the moment.

    Brian: I would love you to read mys stuff on red/green money.

    Here is the simplest post, where I started out. I imagined two different worlds: one with green money only; and one with red money only. (The real world has both).
    http://worthwhile.typepad.com/worthwhile_canadian_initi/2014/02/negative-money.html

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    1. Hello Nick,

      I will try to take a look (been hit with a bunch of random stuff). Like I said, I was doing a basic banking system primer, and I realised what a nightmare overdrafts are from an accounting perspective. I moved on to other topics (that banking primer may not make the next book due to size concerns), so I forget what state I left that section in.

      I fear that I would have to get the GAAP rules on this, since we could account for them either way in an economic model. (I normally only worry about "model accounting," but in this case, I believe I need to actually verify how they show up in the data. The U.K. is (or at least was) big on overdrafts; but I believe that they are a residual in North America.)

      Delete
    2. Brian: I barely kept hold of my sanity writing that post on negative money. Trying to do the accounting would flip me over the edge. Be careful!

      Delete
    3. The accountants in R'lyeh would have no problems with this, of course.

      Ph'nglui mglw'nafh Cthulhu R'lyeh wgah'nagl fhtagn!

      Delete
    4. OK, I read it. I got sucked in. I had a few articles planned, but I think I will end up writing about this (with a bit of a different spin on the article).

      Delete
  7. Based on my research in the flow of funds data in the US, and applying some basic rules for debit and credit mechanics (rules of accounting), an overdraft at the central bank should be the same as an overnight loan at the Fed discount window. The Fed debits loans to banks (an asset) and credits bank reserves (its liabilities). The bank debits reserves (an asset) and credits loans payable to Fed (its liabilities). This creates the overdraft via expansion of the central bank and aggregate bank balance sheets. When the bank repays the loan the debits and credits are reversed so there is a corresponding contraction of the respective balance sheets.

    ReplyDelete
    Replies
    1. The issue is for private overdrafts; do we count them as negative money?

      Although it might work in an economic model (Nick Rowe describes how it would), there are some embedded assumptions in those economic models. I will be turning this into an article, but I think I will only be publishing at after a couple others that I had queued up.

      Delete
    2. Reference:

      Basics of Banking: Loans Create a Lot More Than Deposits

      http://www.cnbc.com/id/100497710

      Delete
  8. Reference:

    How is a short term bank loan recorded?

    http://www.accountingcoach.com/blog/short-term-bank-loan-recorded

    ReplyDelete

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