### Introduction

In this article [1] I want to discuss the dynamics of Ampleforth [2,3], which is a crypto currency with its twist. Instead of issuing a fixed number of units, Ampleforth monetary supply expands and contracts dynamically to keep one Ampleforth valued at around 1 USD. The expansion and contraction is executed in a pro-rata manner, meaning that they do not impact the percentage of overall supply an investor holds. On first sight this looks like a stable coin, but on second sight it is clear that this is not quite the case – it is a regular crypto coin whose price dynamics has been moved to play out in the volume space. What I mean with this is that when Ampleforth market cap goes up / down 10% then investors will feel the impact of that – not by a change in price, but by the fact that they suddenly hold more / less currency units.

This article is organised as follows: I first recall a few basic concepts from finance and derivatives pricing. In particular I introduce the concept of a numeraire or tradeable asset (which is the same thing), and I discuss how to model variable interest rates. I then apply the discussion to Ampleforth, showing that effectively it is a variable interest rate model on steroids. I then discuss the details of the adjustment mechanics and its economic implications, and I show that Ampleforth in essence is an alt coin that simply exhibits unusual accounting conventions. Before wrapping up I construct a “reverse Ampleforth” based on Bitcoin, and I discuss whether or not the death spiral dynamics according to Hillion and Vermaelen applies.

Those sections are written with the non-technical reader in mind, but ultimately by the nature of those the issues discussed. I therefore end with a non-technical summary and conclusion, and non-technical readers are invited to directly skip from here to this part if they prefer.

### Derivatives pricing

Before we look at Ampleforth, let’s look at the formalism of derivatives pricing in a more traditional setting. In this respect the easiest context is non-dividend paying stocks. Those are usually modelled as a stochastic process of their stock price in USD terms $S(t)$. In a Black Scholes world $S$ follows a geometric Brownian motion, ie $dS/S = r dt + \sigma dW$ where $\sigma$ is the volatility, W is a standard Brownian motion, and $r$ is the USD interest rate (I am slightly simplifying here by placing myself in the risk neutral world, not the real world, but that’s just how it is done). Any financial instrument whose prices depends on $S$ – say a call option – can be priced as expected discounted value of its final payoff. Now the price is nice, but what Black Scholes really gave us was a hedging strategy. Now Black Scholes really was made for convex products like options but it also works for linear products, notably forwards. Using those techniques we find that the forward price of a stock is its spot price divided by the discount factor to the forward date. Why? Because Black Scholes shows us an arbitrage strategy. If we have to deliver one unit of a (non dividend paying) stock in a year from now then now we

• borrow $S$ dollars today fixed for a year,
• use the borrowed money to buy the stock

At maturity we

• deliver the stock
• receive the agreed forward price
• pay back the loan

It is easy to see that we break even if and only if the agreed forward price is exactly what we need to pay back that loan so the fair forward is the repayment amount of the loan, which is well known to be the loan amount divided by the discount factor to the maturity date.

Now let’s move to interest rate derivatives. Those are a bit more tricky conceptually, and also have much more uncertainty as we no longer have one random variable, but hundreds or thousands of them – literally every day is in principle an independent variable. The key to understanding here is that the underlying reference asset – the numeraire as it is known – can no longer be the USD. Why? Because a numeraire must be a tradeable asset, and the USD is not one. That sounds a bit surprising at first sight, but it is actually quite easy: what is a tradeable asset is the USD plus accrued interest. In a deterministic rates world world this is the same as the USD, you just have to move discount factors around. However, in a stochastic rates world this is no longer the case: it makes a difference whether you keep your USD today in an overnight rate savings account, or whether you put it into a one-year zero bond, or whether your fix it for three months at a time.

This is an important and at first sight confusing point, so let’s look at this at some practical examples. I have said above “USD is not a numeraire” but what I really should have said “there are many numeraires that can claim to be the USD, and they are all different”. The standard numeraire which I want to denote as $Q$, is the savings account. It follows the equation $dQ = Q r dt$, ie every night it accrues interest at the then prevailing interest rate. Another important group of numeraires are the zero coupon bond numeraires, which I want to denote as $Z$. $Z^T$ for example would be the numeraire of the zero coupon bond maturing at time $T$ (or, in other words, it is the numeraire whose value is fixed at 1 USD at time $T$. Normally we don’t use this numeraire for $t>T$ but to avoid confusion let’s just assume that after time T the dollar is invested overnight, ie it continues as $Q$ for $t>T$. It is somewhat beyond this tour de force for formally prove this, but it should be intuitively clear that in a world where rates don’t move (meaning short rates follow the forward rates to be precise) all those different measures are equivalent – they only differ because at the reinvestment point rates have moved.

### Ampleforth

In this section we are here looking at a slightly idealised version of Ampleforth. We will discuss the differences with real existing Ampleforth slightly below. The central property of Ampleforth is that

price volatility is converted into volume volatility

For example if the price would be USD 1.10 the money supply would be increased by $10%$ (and distributed pro-rata to the current holders) which should bring the price of one unit back to USD 1.00. Note that none of the investors’ wealth has changed – they now each old 10% more units than they held before, so it nets out. Mutatis mutandis the same happens if a unit trades at USD 0.90. In this case 10% ($1/0.9 - 1$ to be precise) of the total supply is burned, again pro-rated to current holdings. The key insight here is to understand that this is essentially a variable interest rate environment, albeit one that allows for negative rates. Also if normal interest rates are a pleasant ride on horseback through the park then Ampleforth interest rates are more like a rodeo – fundamentally the same, but the underlying movements are much more violent.

If we want to analyse Ampleforth our instinct probably tells us to start with looking at one unit. However, we have seen above the 1 USD is not a traded asset in a variable interest world, and neither is 1 Ampleforth. And remember the rodeo metaphore: whilst 1 USD is not a traded asset, for reasonable time frames it is almost a traded asset in that the different numeraires that can be based upon the USD will all be very similar – hence our intuitive feeling that the USD is a trades asset except for those little interest rate differences. For Ampleforth this is no longer the case - we do not know yet what the interest rate volatility will be, but it can be very big, and in this case the different possible numeraires will also be very different.

So we’ve said that 1 Ampleforth is not a traded asset, but it is easy to actually identify a traded asset here: all Ampleforth: if you own all Ampleforth then it does not matter whether for accounting purposes they are split into 1m or 1b units. So if we look at the market capitalisation in USD $C(t)$ of Ampleforth then this is a proper tradeable asset and numeraire. Obviously any fixed fraction of a numeraire is a numeraire as well. Say you like the number $21m$, ie you define your numeraire is $1/21m$ of all Ampleforth supply then this is a perfectly normal numeraire and tradeable asset, and in fact Ampleforth would in this case look exactly like BTC once all supply has been mined.

However, this is not how Ampleforth works – in Ampleforth the money supply $M(t)$ changes over time to ensure that one unit of Ampleforth is always worth exactly 1USD. It is easy to see that for this to hold we need $m(t)=C(t)$ is in this case the price of 1 Ampleforth is

$$P(t) = C(t) / m(t) = 1 [USD]$$

Now on first sight this looks like Ampleforth is a stable coin, ie a traded asset whose price is fixed against the USD, but this is not the case, the assumption just falls over at a surprising point. As opposed to other not-quite stable coins the issue here is not that the Ampleforth price is not stable against the USD – it is – but that the Ampleforth is not a tradeable asset.

So what are the tradeable assets / numeraires? The most important ones are those we have already discussed in the interest rate world: $Q$ and $Z^T$. Let’s start with the (Ampleforth) savings account $Q$ where you start with 1 Ampleforth at $t=0$ and then adjust in line with the interest rates (which, as a reminder, are very volatile and might well be negative). It is easy to see that $Q$ is simply the constant proportion of the total money supply numeraire because this is the way Ampleforth “interest” works. So this savings account looks in practice very much like BTC – if you invest in $1/21m$ of the total money supply at $t=0$ then this is the equivalent of buying one BTC and hodling. Your account statement looks a bit different in that the price of the Ampleforth you hold does not change and instead the volatility in position value is driving by the change in number of units you hold, but this is really only an accounting trick.

Where things get interesting is when we look at the $Z^T$ numeraires. As a reminder, the $Z^T$ numeraire is a “zero coupon bond”, ie the promise to pay 1 Ampleforth at time $T$. In the interest rate case this numeraire was different from the $Q$ numeraire, but not usually by much, the difference being what you can earn investing at overnight rates versus what you can get on a fixed deposit / zero coupon bond. In this case $Q$ tracks the value of Ampleforth as a whole whilst $Z^T$ is designed to be worth exactly 1 Ampleforth, and therefore 1 USD, at time $T$.

### Idealised Ampleforths and AmpleQuoins

Let’s now look at our numeraires in more practical terms. In the previous section we introduced $Q$ and the different $Z^T$ (different in terms of maturity $T$) and we now want to give them names.

Let’s start with our savings account $Q$: we define 1 “AmpleQuoin” (short AMQ) as $1/21m$ of the total supply of Ampleforth at any given point in time. The $Z^T$ are our regular “Ampleforth” except that in order to make them a tradeable asset we need to add a maturity date $T$ - so AMP^T is the tradeable asset that will be settled into a Ampleforth at time $T$ (and it will continue as $Q$ thereafter; but this does not generally matter).

So what happens when we buy Ampleforths? What do we actually buy? Spoiler: we don’t buy Ampleforths, we buy AmpleQuoins. But let’s take this slowly. Let’s say at $t0$ I buy 100 USD worth of Ampleforth, which by construction buys me exactly 100 AMP^t0. Those 100 AMP^t0 correspond to a certain percentage of the total Ampleforth supply (or a certain percentage of 1 $Q$ which is equivalent). Ampleforth mechanics is such that from that point onwards my proportion of the overall Ampleforth supply does not change unless I transact. In other words, I own a certain number of AMQ. Now the system does not quite show it that way because instead of telling me that I own “X Qtoshi” with X being constant over time, every day my holding of AMQ is translated for accounting purposes in that day’s AMP^t. It’s a bit like me having a EUR account at a bank, but the bank balance being displayed in USD. Weird but possible. Or like having a brokerage account, and everyday the balance being displayed in USD instead of in shares. That is actually not weird at all – brokerage statements tend to focus on the USD number, and the number of shares is typically disclosed in a much smaller font. In any case, the important takeaway: even though it might look like I am buying AMP^t0 I am in fact buying X AMP^t0 worth of AMQ. The asset I am actually holding is AmpleQuoins which are very much like any altcoin, and which are not stable at all. At least there is not reason for it to be stable.

So having looked at Ampleforth as store of value – and having found that the real store of value is AMQ which is pretty much like BTC or any alt coin – let’s now look at unit of account. If I am a restaurant and need to put prices on my menu I probably want to put them in USD given my costs are in USD. If I put prices in AMQ or BTC then I am exposed to AMQ or BTC price volatility and this is nice neither on the upside nor on the downside: on the downside my margins are compressed or I am losing money, and on the upside I lose my customers because I am suddenly much more expensive then the restaurant next door. Ampleforth is quite nice here in that I can actually use it for pricing here: if for every day $t0$ I express my prices in terms of AMP^t0 then I in fact operate USD pricing. I just have to make sure that I sell all the AMP^t0 that I receive immediately because as we have seen above once I have received them they are actually AMQ and they start exhibiting volatility against the USD.

Now on the face of it this sounds quite useful, and to some extent it is. However, the alternative would be to simply price the menu in USD and allow customers to pay in AMQ (or BTC for that matter) converted at spot rate. Economically this is essentially the same – including the need to convert the crypto received into fiat asap – except that this is much easier to understand for the average customer who will probably take a while getting used to how this currency works.

### Real existing Ampleforth and AmpleQuoins

In the last section we introduced AmpleQuoins (AMQ) – which are a constant proportion of the outstanding supply of Ampleforth rather than a fixed number of it, and which we found behave similar to BTC or an altcoin – and Ampleforths (AMP^T) to which we needed to add a maturity date to make them a tradeable asset. We have seen that the AMP^T make a decent unit of account. For example if a restaurant expresses prices in AMP (meaning AMP^t on day $t$) then those are essentially USD prices. Similarly, if a business agrees to provide a service at time $T$ then fixing the AMP^T amount is essentially the same as agreeing a price in USD. We have also seen however that the AMP^T received become AMQ for $t>T$, meaning that the recipient should really convert them into USD unless they want to take the AMQ vs USD currency risk.

In the idealised world that we were looking at the supply $M$ was adjusted instantenously (as being equal to the market cap $C$) as to keep the price of one Ampleforth constant at 1 USD. We now look at real existing Ampleforths that operate on a discrete supply adjustment schedule and we also will need to analyse whether our assumption that the market cap $C$ of Ampleforth is independent of $M$ holds.

#### Market cap and supply changes

Let’s start with the second question: will the market capitalisation $C$ of Ampleforth remain constant when the money supply $M$ is adjusted? The traditional finance answer is “yes” because changing the number of shares should not change the valuation of an asset. This would indeed be (mostly) true if Ampleforth was backed by an asset. Let’s assume that Ampleforths were liabilities of a company whose assets were an investment portfolio in publicly quoted shares, and there would be some mechanism to ensure that assets and liabilities remain roughly in line, eg because the share portfolio will eventually be sold and the proceeds will be distributed to Ampleforth holders. In this case the value of the share portfolio is some intrinsic number that is independent of what happens on the Ampleforth side. Granted, the market capitalisation $C$ of all Ampleforth issued might not always exactly match the market value of the portfolio $PV$, but by and large it will be similar. So if something happens on the Ampleforth side – say, supply doubles – then $C$ might slightly move because of that, but as the possible range of C is contraint by $PV$ and $PV$ is not influence by anything that happens on the Ampleforth side $C$ must indeed remain constant’ish as a reaction to changes in Ampleforth supply $M$.

However, Ampleforth is not backed by an equity portfolio. In fact, Ampleforth is not backed by anything. Ampleforth just is. So let’s assume just before readjustment Appleforth trades at USD 1.10, and every holder is issued another 10% worth of tokens. Assuming everyone is rational (but not too rational) and attentive then it might trade at USD 1.00 afterwards. But maybe people are not attentive and they miss it, so it continues to trade at the old price. Or they don’t understand that the price should have changed. Or they understand that there is really no reason other than common belief that the price should change. Long story short: base case scenario is that indeed the price moves, but this is far from sure. Also there is somewhat of a path dependency here. Let’s assume thus far it worked and market cap remained constant’ish at each volume adjustment, so chances are it will work again because everyone believes it works. However, imagine there is this one time where it does not work, whatever the reason. From this point onward people understand that market cap might change when the supply changes, so they’ll be much more cautious and it is much more likely that it will happen again. At one point people might even expect it to happen, in which case it will most likely happen. So, in conclusion for this question, whilst market cap remains constant is likely to be a decent assumption for a while there is actually no good reason why this must happen. Markets move, and maybe the move in line with changes of supply.

#### Ampleforth and AmpleQuoins

Having talked about the issue that because of the lack of asset backing there is really no guarantee that prices behave as they should when the monetary supply is adjusted – or, to say it differently, there is no reason that would guarantee the market cap $C$ to be even approximately constant during an adjustment of the money supply $M$ – we will ignore this for now and assume that $C$ remains constant during adjustments of $M$ which is still our base case.

So the difference we have against the idealised case is that instead of having a continous adjustment of $M$ it is only adjusted at discrete points. There are multiple possibilites for that, and we discuss two:

Periodic adjustment. Let’s start with the periodic adjustment, and to avoid too much confusing mathematical symbols let’s agree on a specific schedule even though it is clear that this generalises very easily. So we assume that the supply is adjusted on a monthly schedule, with the adjustment date being the 1st of every month. If need be we also assume that every month has 30 days, because why not?

So the first thing to note is that we have much fewer numeraires in this scenario. Instead of having one numeraire $Z^T$ for every instant $T$ we only have numeraires for every month. During the month the respective $Z^T$ simply follows the savings account $Q$, ie its proportion of the overall market capitalisation $C$ remains constant, and its value therefore changes in line with C. So again AMQ is our AmpleQuion, ie $1/21m$ of the overall outstanding Ampleforth supply. AMP^Jan20 is the Ampleforth corresponding to the fixing on 1/Jan 2020, so we know that 1 AMP^Jan20 was worth 1 USD on 1/Jan 2020. We also know that from this point onwards is progressed in time as AMQ, so its value at time $t > t0 = 1 Jan 2020$ is

$$C(t)/C(t0) = Q(t)/Q(t0) = AMQ(t)/AMQ(t0)$$

where the equality between $C, Q, AMQ$ follows simply from the fact they those are all constant proportions of the overall money supply, whatever the money supply. Note that there is no reason to restrict $t$ to January 2020 in this case: AMP^Jan20 will run as numeraire forever, it is just that after 1/ Feb it will no longer be “on the run”, ie people will most likely prefer to use the most recent fixing.

If restaurants would price their menus in AMP^T they would most likely do it with reference to the on the run AMP, meaning that in January 2020 prices were in AMP^Jan20, in February 2020 they were in AMP^Feb20 etc. Similarly, someone who agreed to provide a service in January 2021 could now agree a price in terms of the then on the run AMP, ie AMP^Feb21. A priori both payer and payee take a price risk from the day of the fixing, ie the first of the month, until the payment is actually made. Given the nature of Ampleforth – in particular its lack of backing with real assets – this price risk is hard to estimate. If we are able to fit say a lognormal model (the gold standard in a Black Scholes world) then our standard deviation scales with $\sigma \sqrt{t}$. If the volatility $\sigma=20%$ then the maximum standard deviation at the end of the month is $20% * \sqrt{1/12} ~ 6%$. At $10%$ vol this number would be $3%$, and at $40%$ vol this would be $12%$. Of course a lognormal model might not at all appropriate in this case. In particular there might be fat tails and jumps, in particular downward jumps. If AMQ say collapses in Jan 2021 and drops 90% then a payment of 100 AMP^Jan21 that was expected to be worth around USD 100 would suddenly only be worth USD 10. Note that this is only and issue in January: on 1/ Feb 2021 the monetary supply $M$ would reset, and 100 AMP^Feb21 would be worth $100 again. If however for some reason AMQ recovers in Feb 2021 and increases by 10x then suddenly those 100 AMP^Feb21 would be worth USD 1,000. Hedging. Having said this, this actually sounds worse than it is, because both parties can hedge. Let’s first assume that you know you will have to pay 1,000 AMP^Jan21 on the 25/Jan 2021. In this case you could wait until the 25/Jan to purchase those AMPs, and therefore take the price risk, or you could purchase them on 1/Jan at par. If you are on the receiving end this is somewhat harder because you’d need to be able to short sell those AMPs on 1/Jan, ie you would need to borrow them from someone and sell them at par, and then when you receive them you can return the loan. This is not always possible, and if it is it might be costly, especially on a high volatility asset. In case you are not entirely sure about your receipts, so because you are running a restaurant, hedging becomes a bit harder. Essentially you have to hedge an expected amount (and here again, subject to the ability to borrow the AMP) and you take on the basis risk of having over or under hedged. Range-driven adjustment. Above we looked at a periodic adjustment and we found that whilst this is easy to described, the risk is essentially unlimited: if during the month there is significant volatility in AMQ then everyone who did not hedge will be subjected to this volatility which is in principle unbounded, especially on the downside. To control for this a fixing could be introduced whenever the on the run AMP is say$5%$or$10%$away from par. In this case contracts would not reference a specific AMP like AMP^Jan21 but rather stipulate that the AMP that is on the run at the time the payment is made should be used. Fixing the USD amount. Having said all the above it is worth considering an alternative way of contracting. Instead of agreeing on a certain amount of on-th-run AMP contracting or menus could simply be done in USD, and the agreement could say that the bill should be settled in on the run AMP, converted at market value. This would by and large remove all basis risk. This would of course also remove the need to use AMP in the first place: once you go for the “a fixed USD amount of ___” solution one might as well get rid of all the AMP and use AMQ (or BTC for that matter, because AMQ in this case is just another alt coin). ### Reverse Ampleforth The construction of Ampleforth uses well-known financial services primitives and is not actually overly complex. It can however be somewhat confusing because everything is defined in terms of the most complex component of the system, ie the on-the-run AMP. I want to here propose an alternative construction that yields exactly that same result, but that is much clearer in terms of the financial services primitives. Instead of starting with the AMP we start with the AMQ. We have previously defined AMQ as$1/21m$of the overall AMP outstanding, so now we turn this around: our reverse Ampleforth has a monetary supply$21m * 100m$“Qtoshi”, with$100m$Qtoshi being 1 AMQ. For simplicity let’s assume that they are ERC20 tokens on Ethereum or POA, and let’s ignore gaz costs (…which is easier on POA than on Ethereum of course…). We now want to construct AMP^T within a periodic adjustment framework, and for simplicity we use the same monthly framework that we used above, ie fixings on the first of every month. Say we were on the fixing day in January 2020, so we created a new ERC20 token called AMP^Jan20. Remember from the discussion above that on 1/Jan 2020 this token must be worth 1 USD, and after that date it simply behaves like the AMQ token. In other words: 1 AMP^Jan20 token is simply a wrapper for AMQ tokens, worth 1 USD on the creation date. Now if we really want to recreate Ampleforth we need to package all AMQs into the on-the-run AMP. So essentially we create a smart contract that holds all AMQ, and that issues AMP against them, the number of AMP token being determined by the market cap of the AMQ. On 1/Feb 2020, and on the first of every month thereafter, this smart contract adjusts the number of AMP outstanding in line with the AMQ market cap, with all investors being impacted pro-rata to their existing holdings. The attentive reader might have spotted one issue with this methodology. I give you a moment to think. Can you find it? So the issue here was that we stuck all our AMQ into our smart contract, so noone can buy or sell AMQ and therefore it is somewhat difficult to imply a market cap. We could of course back out the AMQ market cap from the market cap of the on-the-run AMP, but in this case we really just recreated regular Ampleforth on Ethereum, and not a reverse Ampleforth as the plan was. In particular, if all AMQ tokens are inside the smart contract, why do we need them in the first place? This whole construction becomes more interesting if we do not stick all AMQ into the smart contract. In fact, we don’t even have to use AMQ which after all is just an alt coin, we might as well use BTC instead (wrapped BTC because we are on Ethereum, but that’s a detail). So we can initialise our reverse Ampleforth smart contract with$100m worth of BTC, and we issue 100m AMP tokens against it. Note in this case we do not have the issue that we do not know what happens if AMP supply changes because we exactly know what the wrapped BTC are worth. So after a month we look at the BTC price, and we adjust the AMP supply accordingly, burning or minting tokens pro rate. Rinse and repeat - we recreated Ampleforth but not based on an unproven altcoin, but based on BTC itself.

One final point to address: in order to ensure that the AMP market cap matches the market cap of the BTC in the smart contract we need to link those two. The best way of doing this is to structure the reverse Ampleforth contract as an open ended fund: AMP holders can always redeem their AMPs against the corresponding amount of (wrapped) BTC, and people can create and receive new AMPs by sending (wrapped) BTC into the smart contract.

### Death spiral dynamics

The reason why I had been initially drawn to Ampleforth was that I had thought that it would exhibit a death spiral dynamics as described by Hillion and Vermaelen. As a reminder, the original death spiral dynamics is related to convertibles that convert at (a discount to) market price, and that also pay coupons in kind. To give an example, assume an investor purchasing USD 1m of such convertibles at face value, with a payment in kind (PIK) rate of $10%$. Before conversion this means that once a year, when the coupon is “paid”, the face value of the convertible is written up by USD 100k. Other than that, nothing happens. So let’s assume we are now at the one year anniversary date of the issuance, so the investor now owns USD 1.1m face value of convertible, and she decides to convert. This means she gets to buy USD 1.1m worth of stock at (a discount to) current market value.

Why are those securities called “death spiral” convertibles? One of the issue here is that if the stock price is on a downwards dynamics then the convertible holders will own a bigger and bigger share of the company when they convert, and once a critical threshold is breached shareholders will just try to get out rather than losing everything to dilution. Moreover, holders of those convertibles have an incentive to short the stock when it is in the critical area. This allows them pick up additional profits, and it is relatively low risk as they can always cover their shorts by converting, provided the stock did not bounce back too much.

Ampleforth with its rebalancing of supply and the artificial maintainance of a value at par has, on the face of it, similarities to the death spiral dynamics. However, as we have seen above the AMP is not a tradeable asset, only the “Ampleforth savings account” AMQ and the different “Ampleforth forwards” AMP^T are. Looking at the Ampleforth dynamics from an AMQ angle – and this is a completely valid point of view – AMQ does not look different from BTC or any alt coin. It simply is a fixed supply asset that is not backed by anything than its scarcity. There is no hint of a death spiral dynamics from this point of view, and this means that simply there is not one. This does not change the fact however that, because the value of Ampleforth like the value of BTC is not anchored in anything, it can be subject to a death spiral of a different kind when everyone is losing faith in the asset and is heading to the door. This is a meltdown / death-spiral scenario, but this is not the death spiral scenario as described by Hillion and Vermaelen.

### Non-technical summary and conclusion

On first sight, Ampleforth looks like a cryptocurrency with a twist, and possibly even a stable coin, but our analysis above has revealed that it is neither and that it is mainly an alt coin with an unusual accounting mechanism. The basic arguments in this respect is as follows:

In financial theory a tradeable asset are the proceeds of a trading strategy. For example “invest 1 USD overnight and grow it in line with interest” is a tradeable asset, and so is “1 USD to be paid at 1/Jan 2030” but “1 USD” is not. A stock is a tradeable asset if and only if it does not pay any dividends. However, “stock X with all dividends reinvested” is a tradeable asset, and so is “stock X with all dividends moved to a savings account” but those two assets are not the same. By and large, tradeable assets and their prices matter, everything else does not.

We found that the Ampleforth dynamics can be very well described within an stochastic interest framework on steroids: if supply is adjusted upwards then this corresponds to an interest payment, and if supply is adjusted downwards this is simply negative interest. Note that negative interest is more a psychological and operational problem – finance can handle negative interest rates without much problems.

Just as “1 USD” is not a tradeable asset, “1 Ampleforth” is not a tradeable asset – “1 Ampleforth with accumulated positive and negative interest” is. We above denoted that particular asset as $Q$ and we noted that at every time it corresponds to a fixed percentage of the outstanding Ampleforth supply. We then denoted AMQ is $1/21m$ of the outstanding Ampleforth supply and we found that AMQ is essentially like BTC or any other alt coin a fixed supply asset – there will always be exactly 21m. It is important to note that this really concludes the economic analysis of Ampleforth – economically it simply is a fixed supply altcoin, nothing more, nothing less.

So what about Ampleforth coins? It turns out that they are essentially just a fancy way of accounting. As pointed out above – you can not invest in 1 AMP: once you own it, it automatically gets applied positive or negative interest, so what you own is really AMQ. You can however think about AMP as something akin to zero coupon bonds. For example, if you agree to deliver a product or service in 1 year’s time you could specify the prices in Ampleforth. It is important to note that those Ampleforth have a date attached, so you might want to denote them as say AMP^Jan2021. Note however that there is no trading strategy based on AMQ that guarantees you to have 1 AMP^Jan2021 in January 2021. It simply is an independent asset, and arguably “1 USD” is the closes possible replication strategy. That is of course to be expected: there is not BTC trading strategy either that guarantees you 1 USD in Jan 2021 – the best way to ensure you have 1 USD then is to buy a USD zero coupon bond / make a fixed term deposit. The secondbest way is to buy a bit less than 1 USD now and invest it overnight. Investing in BTC or AMQ is pretty low on the list of possible investments if you want to end up as close as possible to 1 USD at maturity.

One way of thinking about Ampleforths is as on-the-run assets. Above I discussed the hypothetical example of rebalancing on the 1st of every month. This creates 12 independent-but-correlated asset per year, that we denoted as AMP^Jan21, AMP^Feb21 etc. An agreement to pay 100 AMP in February 2021 essentially means that you agree to deliver 100 units of the on-the-run AMP which at this date will be the AMP^Feb21. It must be stressed again however that the AMP^Feb21 is an asset independent of AMP^Jan21 and AMQ, and that by design the best replication strategy for an AMP^Feb21 before February 2021 is simply a USD term deposit account.

Let me repeat it because this is important. Whilst there is a decent chance that the on-the run AMP will clock in at around 1 USD when the time comes the best replication strategy for a future AMP^T is not an Ampleforth now (what we called the “Ampleforth savings account” AMQ) but a USD term deposit.

In this sense, all the procedures about rebasing and expanding and reducing the money supply are simply accounting. AMP itself does not exist as a tradeable asset, there is only AMQ – the AMP savings account – and there are the various future on-the-run AMP^T. When people talk about AMP what they mean is the current on-the-run AMP. What happens at every change of money supply is effectively a roll over from the previous on-the-run AMP to the now current on-the-run AMP. By design just after the roll the on-the-run AMP is meant to be worth 1 USD, so the conversion rate, and therefore the post-roll holdings will reflect this. We could have expressed someone’s holdings as percentage of the overall money supply – that’s essentially AMQ – and this would have remainded constant over the roll. Instead we chose to convert the holdings in the on-the-run AMP (which of course after the conversion is effectively AMQ as it will expand and contract in line with overall money supply) meaning that the numbers changed. However, economically everything is the same, hence it is only an accounting change.

Conclusion. In conclusion, on the face of it Ampleforth is doing what it is saying on the tin: it is creating an asset that is always worth about 1 USD and that can therefore be used more easily in normal commerce than a regular cryptocurrency. However, it is doing it in a way that is not necessarily very useful because the AMP forwards are not traded assets – only the current on-the-run AMP is, and this one is essentially an AMQ. The best replication strategy for a future AMP is fixed deposit USD account, which somewhat negates the need to hold Ampleforth. Last but not least, whilst it is useful to be able to denote contracts in AMP and reduce volatility there is an even easier option for this: simply denote the contract in USD, and agree that payment will be made in AMQ (or BTC for that matter) converted at spot rate. Ultimately Ampleforth is simply a fixed supply alt coin that is backed by nothing than limited supply. Whether or not the psychology of the volume adjustments will make it more stable over time than a regular alt coin remains to be seen. But this is outside the realm of what economic analysis can provide.