The document discusses the uncovered interest rate parity condition, which states that expected returns from investing in two currencies should be equal when interest rates and exchange rates adjust to equilibrium. It presents the basic equation and explains how it relates the interest rate differential to expected exchange rate changes. The condition is examined under fixed exchange rates, where it suggests capital flows will pressure a central bank defending a peg. Risk and liquidity premia are introduced to account for why investors currently hold a volatile currency like nubits.

1. Section 1: The Interest Rate Parity Condition
RUSD =
Et["t+1]
"t
(1 + RNB) (1)
Equation 1 shows the basic expression for the uncovered interest parity con-
dition. Economists use this equation to explain the relationship between interest
rates and current exchange rates. The uncovered interest parity condition is an
arbitrage condition for investment in risk-free assets. The basic form of the
equation focues on entirely on investment as a source of demand for currency
and also ignores risk. We will consider risk and alternative sources of demand
later in the document. Here, we just focus on an investor choosing between two
risk-free investment assets.
To understand the Equation 1, consider an investor who is deciding whether
to invest 1 USD in a USD-denominated bank deposit or invest 1 USD in nubits.
The investor plans to spend USD one year from now, so regardless of which
option he chooses, he will need USD in the future. The investor is assumed
to choose whichever option yields the highest expected return. If the investor
decides to choose USD, then he will receive 1+RUSD USD one year from now.
If the investor picks nubits, he will exchange his 1 USD for 1
"t
Nubits, where
"t is the current USD/Nubits exchange rate measured in terms of USD per
Nubit. He will then hold his 1
"t
Nubits for one year, yielding 1
"t
(1 + RNB). He
is not certain of what the exchange rate one year from now will be, but expects
that on average this exchange rate will be Et["t+1]. This expression, Et["t+1],
denotes the exhange rate investors making decisions at time t expect to obtain
one year from now at time t+1. Based on this expected exchange rate, an
investor choosing nubits will expect to obtain Et["t+1]
"t
(1 + RNB) USD when he
converts his 1
"t
(1 + RNB) nubits back into USD next year.
The investor chooses whichever option yields the highest return. Therefore,
if 1 + RUSD > Et["t+1]
"t
(1 + RNB), then USD yield a higher expected return
than nubits and the investor should choose the USD deposit. If 1 + RUSD <
Et["t+1]
"t
(1 + RNB), then Nubits yield a higher expected return and the investor
choose Nubits. As long as investors are free to choose between the two assets,
market forces will tend to equalize returns between the new assets, so that
1 + RUSD = Et["t+1]
"t
(1 + RNB). To see why, suppose that all investors prefer
USD to Nubits. If this is the case, then demand for USD will exceed supply
of USD at the current exchange rate "t. Demand for USD comes from people
seeking to sell nubits. Supply of USD comes from people seeking to buy nubits.
To match people seeking to sell nubits with people seeking to buy nubits, the
current exchange rate "t will have to fall, i.e. Nubits will have to depreciate. If
we examine Equation 1, we can see that a fall in "t increases the expected return
on investments in Nubits. Through this mechanism, the current exchange rate
adjusts to a level where If 1 + RUSD = Et["t+1]
"t
(1 + RNB). At this equilibirum
exchange rate, both investment strategies yield the same expected return and
investors are indierent between the two assets.
To highlight some key points, it is useful to rewrite Equation 1 as shown
1

2. in Equation 2. Here, I have just used algebra to rewrite Et[t+1]
t
(1 + RNB) as
RNB + Et[t+1]t
t
+ RNB
Et[t+1]t
t
:
RUSD = RNB +
Et[t+1] t
t
+ RNB
Et[t+1] t
t
(2)
The right-hand side of Equation 2 contains three additive terms. The

3. rst
term is the interest rate oered on nubits. All other things equal, an increase
in the interest rate on nubits will cause nubits to appreciate right now. That is,
if RNB increases, then t will have to rise in order for the equation to continue
to hold with equality. The second additive term,Et[t+1]t
t
is the expected
appreciation of nubits against the USD. If we expect nubits to appreciate over
time, we will be willing to invest in nubits even if they oer a lower interest
rate than USD deposits. The third additive term captures interactions between
the interest rate and expected appreciation. As long as the level of expected
appreciation (or depreciation) is small and the interest rate is low, the third
term tends to be very small in magnitude. To simplify things, the interest
parity condition is often approximated by dropping the third term as shown in
Equation 3. I will use this approximation in the remainder of the document.
Keep in mind that the approximation breaks down for very high interest rates
and very high levels of expected appreciation (or depreciation).
RUSD = RNB +
Et[t+1] t
t
(3)
Section 2: The Interest Rate Parity Condition under Fixed Ex-
change Rates
Taken literally, the uncovered interest parity condition says that exchange
rates will adjust instaneously. In practice, investors take some time to reallocate
their portfolios, so that we should not expect Equations 1,2, and 3 to obtain
instataneously, i.e. there may be some time delay before the exchange rate fully
responds to a movement in interest rates or shift in expectations. This time
delay is useful for thinking about

4. xed exchange rate regimes.
Consider a Nubits central bank that holds reserves of Nubits and USD in
a vault. The Nubits central bank promises to maintain a 1 to 1 exchange rate
perpetually. To back up this promise, the central bank oers to use its reserves
to trade Nubits and USD at the pegged exchange rate. If investors, expect the
central bank to keep this promise, then they will believe that Et[t+1] = t = 1:
If this is the case, expected appreciation is equal to 0 and investment decisions
are based entirely on whichever currency oers a higher nominal interest rate.
If Nubits oers a higher interest rate, then all investors will want to trade their
USD for nubits. Since anyone holding nubits will want to keep them, the only
party willing to satisfy this demand will be the central bank. Accordingly,
investors will take their USD to the central bank and trade them for Nubits.
The central bank will accumulate USD reserves and will release Nubits into
circulation.
2

5. Importantly, this process cannot be kept forever. The central bank can
earn some interest on its USD assets at a rate RUSD: However, the central
banks liabilities will grow at a faster rate, RNB: Eventually, the central bank's
liabilities will grow so large relative to its assets that investors will begin to
question the central bank's solvency. In particular, they will note that if a bank
run occurs, the central bank will not be able trade all outstanding nubits for USD
at a 1 to 1 exchange rate peg. This means that investors will begin to anticipate a
future devaluation of nubits. Rather than believing Et[t+1] = t = 1; they will
come to expect Et[t+1] t = 1. To avoid losses, they will want to withdraw
money from nubits now before the devaluation occurs. Referring to Equation
3, expectations of a devaluation drive down expected returns to investments in
nubits and encourage investors to
ee to USD. To stem the
ight from USD, the
central bank can increase the interest rate on nubits even further. Ultimatelly,
however, this just postpones the inevitable. If investors are con

6. dent that a
collapse will eventually occur, they will not view increases in the interest rate as
credible, i.e. any increase in the interest rate will be oset by an accompanying
decrease in Et[t+1]: I will elaborate on this issue further in a subsequent section
of the document.
Section 3: The Risk and Liquidity Premia
Equations 1,2, and 3 are not entirely satisfactory because they ignore issues
of risk and liquidity. In general, investors will demand an interest rate premium
on risky assets such as Nubits. To capture this interest rate premium, we modify
Equation 3 to incorporate a variable in Equation 4, . We refer to as the risk
premium on nubits. If is positive, then investors view USD as safer than
nubits and will only be willing to hold nubits if they oer a higher interest rate
than investments in USD.
RUSD + = RNB +
Et[t+1] t
t
(4)
If we look at Equation 4, the current willingness of investors to hold Nubits
would seem puzzling. Since Nubits currently trade at parity, investors cannot
expect a depreciation. Indeed, it is far more likely that investors believe a future
devaluation of nubits is a possible outcome. To state this more explicitly, let's
say that investors believe that Nubits parity will hold (t+1 = t) until next
year with probability (1-p) and that nubits value will drop to 0 (t+1 = 0) with
probability p. If this is the case, we can rewrite expectations of future exchange
rates as shown in Equation 5.
Et[t+1] = (1 p)t + p(0) = (1 p)t (5)
If we substitute Equation 5 into Equation 4, we get Equation 6.
RUSD + = RNB p (6)
Equation 6 is no more helpful in explaining demand for nubits. If investors
place some positive probability on a future devaluation, then nubits will need
3

7. to oer an even larger interest rate premia to convince investors to hold nubits
over USD deposits.
To explain demand for nubits, we need to introduce demand for currency as
a means of txns. Nubits allow for online exchanges that would be impossible
or much more dicult to perform using value held in USD bank accounts. To
incorporate this demand, we can introduce a liquidity premium that is a function
of the level of demand for nubits. This liquidity premium can be expected to
grow as the number of business accepting nubits increases and the reputation
of nubits as a stable unit of account improves. I denote this liquidity premium
as l(D), where D represents demand for Nubits txns, f is the level of txn fees,
and l(D; f; v) is the liquidity premium as a function of demand, txn fees, and
historical volatility of the nubits/USD exchange rate. The more demand for
Nubits txns, the higher the liquidity premium. The lower txn fees on nubits are,
the higher the liquidity premium. If nubits fail to hold to 1 to 1 peg with the
USD, then volatility will increase and nubits will lose their distinguish property.
This would cause the liquidity premium to become negative. In Equation 7, I
add the liquidity premium to Equation 6.
RUSD + = RNB p + l(D; f; vnubits) (7)
In Equation 7, the liquidity premia justi

8. es a positive demand for nubits
even though they are relatively risky and low-return asset. As demand for
nubits grows, one would expect the liquidity premium on nubits to increase as
well. In the long-run, this may lead to a situation where people are willing to
hold nubits even if they pay a negative interest rate or require txn fees. Both txn
fees and negative interest rates are potential mechanisms for generating revenue
from nubits that could be paid to holders of nushares.
Section 4: Application of Liquidity Premia to understand why
BitUSD consistently trade at a discount relative to USD and Nubits
BitUSD dier from nubits in that they are not backed by central bank inter-
vention. Or at least not explicitly. Unlike Nubits, the exchange rate on BitUSD
oats according to market demand. Historically, we have seen BitUSD consis-
tenty trading below USD parity. Our equation is eective in explaining this. In
Equation 8, I add the liquidity premium to Equation 4 to illustrate this point.
RUSD + = RBitUSD +
Et[t+1] t
t
+ +l(D; f; vbitUSD) (8)
Our goal here is to explain why the market price of bitUSD is consistently
below 1 USD. Speci

9. cally, why are people willing to purchase Nubits at par and
in larger volumes, while bitUSD trade at a discount and in smaller volumes.
The key point here is that the current exchange rate of bitUSD, t,
oats.
Because of these exchange rate movements, the liquidity premium on bitUSD
is likely much lower than that on Nubits. BitUSD are simply not that useful
as a means of exchange. Investors, however, may hope that this situation is
temporary, and that eventually bitUSD will stablize at parity. To capture this,
let's suppose that investors believe that t+1 = 1 with probablity 1-p and that
4

10. bitUSD will collapse to 0 with probability p, so that Et[t+1] = (1)(1p) = 1p:
If we substitute this into Equation 8, we get Equation 9.
RUSD + = RBitUSD +
(1 p) t
t
+ +l(D; f; vbitUSD) (9)
Examining this equation, let's suppose that the risk premium on bitUSD and
nubits are the same, and that the percieved probability of collapse of the two
systems is the same. We can also note that they both oer almost 0 interest,
so that interest rates are the same as well. Based on these assumptions, we
can substitute Equation 7 into Equation 9, cancel terms, and obtain Equation
10. In Equation 10, the exchange rate is the current price of nubits in terms of
USD.
p + l(D; f; vnubits) =
(1 p) t
t
+ +l(D; f; vbitUSD) (10)
Note that l(D; f; vnubits) l(D; f; vbitUSD) implies that we must have (1p)t
t
+
p = (1p)(1t)
t
0; which is only possible if t1. In other words, since bitUSD
are less useful for txn purposes than nubits but oer similar yields and expose
investors to similar risks, they must trade at a discount relative to both nubits
and USD. In order for bitUSD to reach USD parity, the bitUSD will have to
either use intervention to reduce volatility as in Nubits, or alternatively raise
interest rates to make investments in bitUSD more attractive. As discussed pre-
viously, the latter mechanism is costly and dicult to sustain for a prolonged
period. The upshot of this is that the use of an active stabilization mechanisms
gives Nubits a strong competitive advantage over bitUSD.
Section 5: Losing Control: The relationship between the level of
reserves, the risk premium, and the expected exchange rate
In section 2, I discussed how excessive accumulation of central bank liabilities
can lead to expectations of a devaluation and a collapse in the exchange rate.
Here, I model these ideas more formally by showing how the ratio of central bank
assets to liabilities aects the risk premium and the expected future exchange
rate.
To start o, let's consider assets and liabilities of the nubits central bank. I
show these in Table 1. Assets are for our purposes anything the central bank
can sell to holders of nubits to support the nubits price. I am going to suppose
that the nubits central bank has authority to issue nushares at will to repurchase
nubits, so that the USD market cap of nushares is one of these assets. The second
asset that the nubits central bank can use is USD held in trust by custodians.
These USD can also be used to repurchase nubits. Both types of repurchases
are referred to as open market operations in central bank speak.
Table 1
Central Bank Assets Central Bank Liabilities
USD Market Capitalization of Nushares Number of Outstanding Nubits
USD held in reserve by Custodians
5

11. Nushares and USD held in reserves are very dierent in character. Essen-
tially, as long as 1 USD is held in reserve for every nubit, there is no risk of a
collapse in the system. On the other hand, maintenance of a large USD reserve
is also unpro

12. table. To simplify things, I am going to net out USD reserves from
assets and describe central bank net liabilities as the nubits less USD reserves.
I incorporate this modi

13. cation in Table 2. [Note: We should keep in mind that
these USD could be stolen by an exchange or misappropriated by a custodian.
I ignore this risk in this document.]
Table 2
Central Bank Assets Central Bank Net Liabilities
USD Market Capitalization of Nushares Outstanding Nubits Net of USD Reserves
To capture these concepts, I am going to de

14. ne a ratio of assets to liabilities,
as the 'reserve ratio', s = USDMarketCapitalizationofNushares
OutstandingNubitsNetofUSDReserves :
Note here that a ratio of s=1 here would represent a highly unstable situa-
tion. If s=1, then any decline in the price of nushares would make it impossible
to repurchase all outstanding net liabiliites even if the central bank was to print
up an in

15. nite quantity of nushares. Moreover, since massive central bank sales
of nushares would greatly depress the USD price of nushares, it probably be
impossible to repurchase all outstanding net liabilities at a higher ratio as well,
such as s=2.
Now let's consider a much higher ratio, such as s=21. In this case, we can
imagine the central bank printing up nushares equal to 5% of the outstanding
quantity of nushares. At the initial market price of nushares, this would yield
21*0.05 = $1.05 of nushares for every net USD liabilitity. Printing of new
nushares issues equal to 5% of the outstanding volume would dilute existing
shares by 5% and lead to a nushares price drop of approximately 5%, provided
the sale conducted slowly. After adjusting for the price drop, printing these
nushares would yield enough revenue to purchase all outstanding net liabilities.
Accordingly, a very high reserve ratio such as s=21 could would ensure parity
of the USD.
Now that we've de

16. ned the reserve ratio, s. Let's consider incorporating how
we can incorporate it in Equation 7. The

17. rst place I think we can incorporate
it is in p, the market's percieved probability of a future collapse in the exchange
rate to 0. We can think of p as a function of s, p(s). The percieved probability
of collapse is also likely to increase in the interest rate, particularly when s is
low. To see why s matters for eects of changes in the interest rate, suppose
that s=100 and nubits decides to oer 100% annual interest. Even though this
is a very high interest rate, interest payments could be supported for a very
long time through sales of nushares. Therefore, the probability of collapse will
not be very sensitive to the interest rate when s is high. When s is low however,
high interest payments could shift the system into insolvency over a short time
scale. Therefore, you would expect p to be extremely sensitive to the interest
rate when s is low, e.g. say s=2. In mathmatical terms, we write the probability
of collapse p(RNB,s), as function of the interest rate RNB and the reserve ratio
s. The assumed behavior of this function is shown in Equation 11.
6

18. @p(RNB,s)
@s
0;
@p(RNB,s)
@RNB
0;
@d2p(RNB,s)
@RNB@s
0 (11)
The second place we can think of incorporating s, is in the risk premia, .
The risk premium essentially depends on the amount of risk of collapse, p. Risk
increases as p goes from 0 to 0.5, so for all practical purposes we can also think
of the risk premium as decreasing in s and increasing in RNB: We can write
the risk premium as (p((RNB; s))): In Equation 12, I substitute both of these
expressions into Equation 7.
RUSD + (p((RNB; s))) = RNB p(RNB; s) + l(D; f; vnubits) (12)
Equation 12 implies that an increase in the interest rate can actually de-
crease demand for nubits under certain circumstances. When the reserve ratio
s is very low, the perceived probability of collapse can be very sensitive to the
interest rate. Due to this sensitivity, an increase in the interest rate could cause
negative eects on expectations that outweigh the attraction of a higher interest
rate. On the other hand, an increase in the interest rate is most eective at
attracting investors to nubits when the reserve ratio s is very high. In this case,
investors do not need to fear near-term insolvency of the nubits system even if
a very high interest rate is oered.
Some Implications:
1) The system can degenerate into an unrecoverable state if the reserve ratio
falls to too low of a level. Interest rate changes become completely ineective
in this case.
2) Nubits should target an explicit healthy reserve ratio range. This is just
as important to Nubits viability as maintaining USD parity. In fact it can be
more important. A fall from USD parity would likely be a temporary situation.
Once the reserve ratio falls to an excessively low level, it could be very dicult
for nubits to recover.
3) Interest rates increases are most eective at supporting prices of nubits
when the reserve ratio is quite high. You might want to use interest rates to
oset a temporary shortfall in deman for nubits. Such a temporary shortfall
would not have much eect on the price of nushares, but could lead to pressure
for nubits to fall below parity. Interest rate increases are appropriate in this
case
4) Interest rates are also useful as a temporary incentive to get new users
to try out nubits. Provided the reserve ratio is high enough to support this. A
temporarily high level of interest on nubits might be a good means of expanding
the user base. For example, when paypal

19. rst formed they oered new users 10
USD for free to expand market share. Nubits could achieve this by oering new
users high interest rates as a temporary promotion. As long as high interest rates
are temporary and the reserve ratio is monitored carefully, such an incentive
would not compromise the sustainability of nubits.
I'm stopping here for now. Later, I will add to this to consider the role of
txn fee policy in maintaining a stable reserve ratio and a stable exchange rate.
7

20. Before doing that, however, I want to get some comments on the document.
8