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:

@MayankSeksaria @RomanchukBrian Think this old post works OK as a response: https://t.co/Fuf1139Gcm

— Nick Rowe (@MacRoweNick) July 25, 2016

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