1. USD Reserves Cannot Be Used to Back a USD cryptocurrency
Abstract
This article analyzes the incentive structure of dual cryptocurrency systems,
where the price of one cryptocurrency price is pegged to an external asset and
the price of the other cryptocurrency is allowed to oat. Dual cryptocurrency
systems are often compared to a bank, in which owners of the oating cryp-
tocurrency act as bank shareholders and owners of the pegged cryptocurrency
act as bank depositors. The comaprison is highly misleading. In the event of
insolvency, bank shareholders are legally obligated to liquidate the bank's tan-
gible assets and use the revenue to repay depositors. In a dual cryptocurrency,
on the other hand, shareholders may choose to repay depositors, but are not
bound by any legal obligation to do so. I show that, under the assumption that
shareholders behave rationally, this implies that pegged cryptocurrencies cannot
be backed by tangible assets. I then discuss some implications of this result.
Preliminaries: 100% Reserve System vs. 0% Reserve System
Let's begin by considering the repayment of nubits holders in the nubits
custodial system. Two possible methods of supporting the the nubits peg are
a) repurchase of nubits using newly-issued nushares and b) repurchase of nubits
using USD Reserves. In the extreme, a custodial system supported entirely by
nubits burning would operate without holding USD reserves at all. At the other
extreme, a custodial system relying on USD reserves could maintain a reserve
of 1 USD for every nubit in circulation. The two cases are shown in Table 1.
100% Reserve System 0% Reserve System
USD Revenue from Nubits Sales Held as Reserves Paid to Shareholders
External Assets Backing Nubits USD Reserve Fund None
How Could These Systems Operate?
There are many possible ways of organizing a 0% reserve system and prob-
ably multiple options for a 100% reserve system as well. My conclusions apply
universally to all such systems. However, since it may help to clarify things for
the reader, I will explain how both systems could be incorporated within the
nubits custodial system.
Operations of a 100% Reserve System
Under a 100% reserve system, nushares holders maintain a reserve fund
containing 1 USD for each nubit in circulation. They use this reserve fund
to support the price of nubits at 1 USD per nubit. This process works as
follows. Shareholders vote to allocate newly-minted nubits and USD reserves to
custodians. Custodians use these resources to place bids and asks on exchanges
around a price of 1 USD per nubit. Whenever a custodial ask is accepted, 1
nubit is released into circulation and a custodian receives 1 USD. Whenever a
custodial bid is accepted, 1 nubit is retired from circulation and the custodian
pays out 1 USD. At the expiration of a custodial contract, all USD in possession
of the custodian are returned to the USD reserve fund. In addition, all nubits
in possession of the custodian are destroyed.
Operations of a 0% Reserve System
Under a 0% reserve, nushares holders do not hold any form of reserve assets
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2. at all. In lieu of a USD reserve fund, shareholders use newly-minted nushares to
support the price of nubits at 1 USD per nubit. This process works as follows.
Shareholders vote to allocate newly-minted nubits and newly-minted nushares
to custodians. Custodians sell their nushares allocations for USD. Custodians
then place bids and asks on exchanges around a price of 1 USD per nubit.
Whenever a custodial ask is accepted, 1 nubit is released into circulation and
a custodian receives 1 USD. Whenever a custodial bid is accepted, 1 nubit is
retired from circulation and the custodian pays out 1 USD. At the expiration
of a custodial contract, USD in possession of the custodian are converted into
peercoin and paid out as dividends to nushares holders. In addition, all nubits
in possession of the custodian are destroyed.
A 0% Reserve System Can Become Insolvent. A 100% Reserve
System Can Never Become Insolvent
At a supercial level, the 100% reserve system appears to protect nubits
holders from riTsk associated with a decline in the price of nushares. Under a 0%
reserve, for example, a complete collapse of nushares prices would leave nushares
holders without sucient resources to repurchase all outstanding nubits. In
other words, a decline in the price of nushares can cause a 0% reserve system to
become insolvent. On the other hand, under a 100% reserve, nushares holders
always control sucient USD resources to repurchase all outstanding nubits.
Even if nushares prices dropped to zero, a 100% reserve system could never
become insolvent.
But Solvency Doesn't Count For Squat in a Decentralized System
In the absence of third-party legal enforcement, solvency is not a very mean-
ingful condition. Since shareholders are not bound by law to repurchase nubits,
shareholders could allow the nubits peg to break even when they have the nan-
cial capacity to support it. In my view, we should only expect nubits repurchases
to occur when repurchases align with shareholders' economic interests. Rather
than vote for repurchases, shareholders could just as easily vote to abandon
a 100% reserve system and distribute reserve funds as shareholder dividends.
If we allow shareholders to vote on the use of reserve funds and assume that
they vote according to self-interest, the argument that a 100% reserve oers
support for the peg falls apart. To show this, I describe a simple economic
model of shareholder voting in the next section. The model implies the fol-
lowing: If supporting the peg requires shareholders to deplete reserves, then
rational shareholders will allow the peg to collapse. Conversely, if shareholders
can support the peg without depleting reserves, then rational shareholders will
choose to support the peg. The implication is that value held in reserves does
not provide any support for the peg whatsoever.
Model
In the model, I assume negative market shocks put pressure on the peg.
Furthermore, I assume shareholders' intangible assets are tied to maintaining
the peg, but that the value of tangible assets is independent of the peg. If the
peg collapses, shareholders face a total loss on their intangible assets, but no
loss on tangible assets. Shareholders have two resources they can use defend
the nubits peg: a) nubits burning b) depletion of USD reserves. Shareholders
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3. vote to use these resources, so as to maximize shareholder value. The model
implies that value held as USD reserves has no inuence on the likelihood that
the peg will collapss whatsoever. Essentially, shareholders will support the peg
whenever it would be possible to do so through nubits burning alone. As long as
shareholders choose to support the peg, they are indierent between providing
this support through nubits burning, depletion of USD reserves, or any mixture
of the two. If the peg cannot be supported solely through nubits burning, then
shareholders will choose to allow the peg to break. In this case, shareholders
will not be willing to sacrice any USD reserves to support the peg.
The model follows and may be too technical for many readers. If you like,
you can skip over to the section labelled discussion.
Note about models
Multi-period economic models have three types of variables.
1) Variables that are known at the beginning of a period. These variables
are called state variables.
2) Variables that are randomly determined during the period. These vari-
ables are called random variables.
3) Varaibles that are determined by agent decisions. These variables are
called decision variables or choice variables.
Multi-period economic models follow a specic timeline describing events
within a period.
initial state in period t - realization of random variables - agents observe
initial state and realizations of random variables - agents choose decision vari-
ables - initial state in period t+1
Summary of Model Variables
State Variables
Nt the stock of oustanding nubits at the beginning of period t
Kt−1 the USD market valuation of nushares' intangible assets during period
t-1
Rt the stock of USD reserves held at the beginning of period t
θt−1 the equilibrium USD price of a nubit during period t-1
Random Variables
wt the change in aggregate demand for nubits during period t
pt the USD price of nushares' intangible assets in period t divided by the
price of nushares' intangible assets in period t-1
Decision Variables
τt the USD amount of nubits repurchases funded through sales of intangible
capital during period t
ρt the USD amount of nubits repurchases funded through depletion of USD
reserves during period t
Timeline of Model
1) The values of state variables Nt, Kt−1, Rt, and θt−1 are taken as given.
Period t begins.
2) The random variable wt is drawn from the distribution g(wt|Nt, Kt−1) on
support wt∈(-∞,Nt].The realization of wt determines the state variable Nt+1.
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4. 3) The random variable pt is drawn from the distribution h(pt|wt, Nt, Kt−1, θt−1)
on support pt∈[0,∞]. The realization of pt determines the state variable Kt.
4) Shareholders vote to determine the decision variables, ρt and τt. These
decision variables determine the state variables Rt+1 and θt.
5) Market Opens. Exchanges of USD reserves, nushares, and nubits take
place.
6) Period t is completed. Values for state variables in period t+1 are given
by Nt+1, Kt, Rt+1, and θt.
Model Assumptions
1) Let Nt be the stock of oustanding nubits at the beginning of period t.
Let Kt−1 be the USD market valuation of nushares' intangible assets during
period t-1. During period t, demand for nubits is aected by a random shock
wt. I assume that wt is a random variable with the probability density function
g(wt|Nt, Kt−1) on the support wt∈(-∞,Nt]. The starting stock of outstanding
nubits and the realization of the random shock determine the stock of nubits at
the beginning of period t+1, as shown in Equation 1. Note that, if wt is positive,
then aggregate demand for nubits falls during t. Likewise, if wt is negative, then
aggregate demand for nubits rises during period t.
Nt+1 = Nt − wt
wt∈(-∞,Nt]
wt ∼ g(wt|Nt, Kt−1)
(1)
2) Let be pt is the price of a claim on intangible assets during period t
divided by the price of a claim on intangible assets in period t, and let θt−1
be the USD price of nubits in period t-1. By denition, Kt = ptKt−1. I
assume that pt is a random variable on support pt∈[0,∞) and allow pt to
be correlated with wt, Nt, Kt−1, and θt−1. I denote the probability den-
sity function of pt as h(pt|wt, Nt, Kt−1, θt−1). I assume that pt reects a fair
market valuation of intangible capital based on available information, so that
E[pt+1|pt, Nt, Kt−1, wt, θt−1] = 1I also assume that the price of intangible capi-
tal drops to 0 if the peg was not maintained during the previous period, that is
Pr(pt+1 = 0|θt̸=1) = 1. The assumptions are summarized in Equation 2.
Kt = ptKt−1
pt∈[0,∞)
pt ∼ h(pt|wt, Nt, Kt−1, θt−1)
E[pt+1|pt, Nt, Kt−1, wt, θt−1] = 1
Pr(pt = 0|θt−1̸=1) = 1
(2)
3) After observing the realizations of pt and wt, shareholders vote on ex-
penditures on nubits repurchases. In the event that wt0, shareholders must
repurchase (i.e. burn) exactly wt nubits to avoid a fall in the nubits USD
price, θt. If they fail to do so, θt will fall below parity, and this will cause the
intangible value of nushares to fall to zero in the next period (See assumption
2). Shareholders have two resources available that they can use to support the
nubits price. These resources are claims on nushares' intangible assets and USD
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5. reserves. Let Rt denote shareholders' stock of USD reserves at the beginning
of period t, let ρt denote the USD value of nubits repurchased through the
depletion of USD reserves during period t, let τt denote the USD value of nu-
bits repurchased through sales of intangible assets during period t. Values of
ρt and τt must obey the shareholders budget constraint, so that τt∈[0,Kt] and
ρt∈[min(0,wt),Rt]. Negative values of ρt represent the accumulation of addi-
tional reserves. As shown in Equation 3, the equilibrium price of nubits, θt, is
a function of τt,ρt, and wt, and Rt+1 is a function of Rt and ρt.
τt∈[0,Kt]
ρt∈[min(0,wt),Rt]
θt = wt
τt+ρt
Rt+1 = Rt − ρt
(3)
Shareholder Objective Function
Let At+1 denote the USD value of incumbent shareholders' assets at time
t+1. Shareholders vote for values of τt and ρt that maximize the expected value
of At+1 based on information available after wt and pt are realized. I write At+1
as a function of wt,ρt,τt,pt+1,Kt, and Rt in Equation 4.
At+1 = pt+1(Kt − τt) + Rt − ρt (4)
In Equation 5, I write the expected value of At+1 at time t, making use of
the assumption that E[pt+1|wt = ρt + τt] = 1 and E[pt+1|wt̸=ρt + τt] = 0.
E[At+1] = Kt − τt + Rt − ρt if wt = ρt + τt
E[At+1] = Rt − ρt if wt ̸= ρt + τt
(5)
Optimal Choice of Decision Variables
Shareholders can obtain the maximize E[At+1] if they choose values of ρt
and τt that satisfy Equation 6.
{(ρt, τt) : wt = ρt + τt, ρt∈[min(0,wt),Rt], τt∈[0,Kt]} if wt ≤ Kt
{(ρt, τt) : ρt = 0, τt∈[0,Kt]} if wt Kt
(6)
To see this, refer to Equation 5 and observe that if wt ≤ Kt, shareholders
obtain a weakly higher payo when wt = ρt + τt, since together these conditons
imply that Kt − τt≥0. In words, as long as the demand shock wt is less than
the market value of intangible capital, Kt, it is worthwhile for shareholders
to support the nubits price through repurchases of wt nubits. In this case,
shareholders are indierent as to how they nance repurchases. Repurchases
could be nanced entirely through sales of intangible capital, entirely through
depletion of USD reserves, or through some combination of the two.
If wt Kt, shareholders obtain a payo strictly less than Rt if wt = ρt + τt
and a payo less than or equal to Rt if wt ̸= ρt +τt. In this case, shareholder can
attain the maximum possible payo of Rt when ρt = 0. In words, if the demand
shock wt exceeds the market value of intangible capital, Kt, shareholders will
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6. prefer to allow the peg to collapse. If they plan to allow the peg to collapse,
they will not want to waste any USD reserves supporting the peg. Instead, they
will choose to retain the entire stock of USD reserves for future distribution as
a nal shareholder dividend.
Discussion
The model tells us that shareholders' nancial commitment to the peg is
bounded by the market's valuation of nushares' intangible capital. Nushares'
intangible capital is a measure of the value that the peg creates for sharehold-
ers. If the intangible value associated with the peg is greater than the cost of
supporting the peg, shareholders benet from supporting the peg. If the intan-
gible value asociated with the peg is less than the cost of supporting the peg,
shareholders benet from allowing the peg to collapse.
Furthermore, the model tells us that the strength of the peg is not related
to value held in USD reserves. In the model, the peg fails when the event shown
in Equation 7 occurs.
wt
pt−1
Kt−1 (7)
The probability of this event is completely independent of the stock of USD
reserves, Rt. In words, the expected lifespan of the nubits peg depends solely on
the ratio of outstanding nubits,Nt , to USD value of nushares' intangible capital,
Kt−1. If Nt is kept small relative to Kt−1, then the peg should be expected to
persist for a long period, perhaps indenately. If Nt grows to large relative to
Kt−1, then the peg's days will be numbered.
Implications
1) The capacity of dual currency systems to store USD value is fundamentally
limited by the intangible value of the oating price currency. Reserves are costly
to maintain and oer zero benet in terms of enhanced stability.
2) The only way of ensuring the long-term stability in these systems is cap-
ping the ratio of value held in nubits to intangible value held in nushares.
3) Not all nubits use cases add value.
When a user stores value in nubits, he imposes costs on other users in the
form of an increased probability of collapse.
When a user generates fees, he increases the intangible value of nushares
and generates benets for other users in the form of a reduced probability of
collapse.
To the extent that users store value in nubits, but do not generate fees, they
subtract value from the system.
4) Collection of user fees from nubits holders should be based primarily on
negative interest rates, not txn fees.
The key nite resource in dual currency systems is capacity to store USD
value, not blockchain space. Negative interest rates charge people according to
resource usage. This discourages wasteful uses that do not generate a sucient
benet to justify the external costs they impose. Txn fees charge people based
on use of blockchain space. This encourages wasteful uses that generate little
benet, but impose substantial external costs on other users.
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