Appendix

Mathematical Equations

Equation 1: Reserve Ratio Equation

Let:

$P$

- the price of one 1G$ in Supported Currency;
$R$

- the amount of Supported Currency in the reserve;
$S$

- total G$ coins in circulation. Then

$r$

- the reserve ratio - is defined as:

$r=\frac{R}{P \cdot S}$

At launch,

$r=1$

and overtime, will computationally decline at a daily rate in accordance with the Expansion Rate. In Equation 1, given $R$

and $r$

the system has two degrees of freedom, $P$

and $S$

that will be calculated as described below.
Current Price function:

$Current Price = \frac{Reserve~Balance}{Smart~Token's~Total~Supply~ \cdot ~Reserve~Ratio}$

Buy function:

$Tokens~Issued = Supply \cdot ((1+\frac{Connected ~Tokens~Paid}{Balance}) ^ {CW} -1)$

Sell function:

$Connected~Tokens~Paid~Out = Balance \cdot (\sqrt[CW]{(1+\frac{Tokens~Destroyed}{Supply})} -1)$

Equation 3: Expansion Rate Formula

$q=\frac{r_t}{r_{t-1}}$

Equation 4: Mint G$ from Declining Reserve Ratio - Once a day the reserve ratio is reduced while all other parameters are unchanged.

The GoodReserve contract mints an amount of G$ to satisfy Equation 1

$P=\frac{R}{r \cdot S}$

. If P and R do not change and the daily change in r=𝚫r and the corresponding change in S is 𝚫S, then $r s=(r+\Delta r)(S+\Delta S)$

and from here we can see that:

$\Delta S=S \cdot \frac{-\Delta r}{r+\Delta r}=S \cdot \frac{r_{t-1}-r_{t}}{r_{t}}$

From Equation 2 we get

$q=\frac{r_t}{r_{t-1}}$

and thus: $S \frac{r_{t-1}-r_{t}}{r_{t}}=S \frac{1-q}{q}$

Equation 5: Mint G$ from Deposit Formula - The number of new tokens based on the daily interest deposited in the Reserve.

Z - Value deposited into the reserve

E - Newly minted tokens

The formula does not change the price or reserve ratio:

$E = \frac {\frac{R + Z}{Rr} - P \cdot S}{P}$

Monetary Example

This is an illustrative example, with simple rounded numbers, aimed at trackability:

10 Supporters have staked a total value of US$10 million in supported crypto to a third-party protocol

Protocol annual interest rate of 10%

Daily interest of US$2,736

G$ Price = US $1

GoodReserve = US $1,000,000

Supply of G$ = 1,250,000

G$ Market Capitalization = US$ 1,250,000

G$ Reserve Ratio = 80%

Daily Expansion Rate = 1%

Every day G$ minting occurs from two methods: From the deposit of interest to the GoodReserve and from the decline in reserve ratio.

1.

Minting derived from deposite of daily interest to the GoodReserve

1.

Deposit of US$2,736 to GoodReserve

2.

Calculated as follows:

1.

$S \cdot P=\frac{R}{Rr}$

2.

(1,250,000 +X)*1=(1,000,000 + 2736)/0.8

3.

1,250,000 + X = 1,002,736/0.8

4.

1,250,000 + X = 1,253,420

5.

X = 3,420

3.

3,420 G$ are minted

4.

2,736 G$ are allocated back to the 10 supporters

5.

684 G$ are distributed as basic income

2.

Mint from Reserve Ratio Decline

1.

Lower the Reserve Ratio from 80% to 79%

2.

Calculated as follows:

1.

$S \cdot P=\frac{R}{Rr}$

2.

(1,253,420+X)*1= 1,002,736/0.79

3.

1,253,420 + X = 1,269,286

4.

X = 15,866

3.

15,866 G$ are distributed as UBI

In total, on that day 16,550 G$ were distributed as basic income to Claimers.

Last modified 1yr ago

Export as PDF

Copy link