# 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: 

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

**In the beginning**, $$r=1$$ **nd will be reduced daily according to the** *Expansion Rate*. **As can be seen from**, **given** $$R$$ and $$r$$ the system has two degrees of freedom, $$P$$ and $$S$$ that will be calculated as described below.   <br>

#### **Equation 2: Buy/Sell Function & Current Price(**[**Bancor Formula**](https://storage.googleapis.com/website-bancor/2018/04/01ba8253-bancor_protocol_whitepaper_en.pdf) **for Automated Market Making)**

Current Price function:&#x20;

#### $$Price = \frac{Reserve~~Balance}{Smart~~Token's~~Total~~Supply\~ \cdot ~~Reserve~~Ratio}$$&#x20;

Buy function:&#x20;

$$Tokens\~Issued = Supply \cdot ((1+\frac{Connected ~~Tokens~~Paid}{Balance}) ^ {CW} -1)$$&#x20;

Sell function:&#x20;

$$Connected~~Tokens~~Paid~~Out = Balance \cdot (\sqrt\[CW]{(1+\frac{Tokens~~Destroyed}{Supply})}  -1)$$&#x20;

####

#### Equation 3: *Expansion Rate* Formula&#x20;

$$q=\frac{r\_t}{r\_{t-1}}$$&#x20;

#### Equation 4: Mint G$ from Declining Reserve Ratio  - **Once a day the reserve ratio is reduced while all other parameters are unchanged.**&#x20;

The GoodDollar Reserve 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:

$$\Delta S=S \cdot \frac{-\Delta r}{r+\Delta r}=S \cdot \frac{r\_{t-1}-r\_{t}}{r\_{t}}$$&#x20;

From Equation 2  we get  $$q=\frac{r\_t}{r\_{t-1}}$$ and thus: $$S \frac{r\_{t-1}-r\_{t}}{r\_{t}}=S \frac{1-q}{q}$$&#x20;

#### Equation 5: Mint G$ from Deposit Formula - **The number of new tokens based on the daily interest deposited in the Reserve.**&#x20;

Z - Value deposited into the reserve&#x20;

E - Newly minted tokens&#x20;

The formula does not change the price or reserve ratio: $$E =  \frac {\frac{R + Z}{Rr} - P \cdot S}{P}$$&#x20;

### Monetary Example

This is an illustrative example with no relation to real parameters.

* 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% &#x20;

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

1. Minting derived from deposite of daily interest to the GoodReserve
   * Deposit of US$2,736 to GoodReserve
   * Calculated as follows:
     1. $$S \cdot P=\frac{R}{Rr}$$&#x20;
     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&#x20;
     5. X = 3,420
   * 3,420 G$ are minted
   * 2,736 G$ are allocated back to the 10 supporters
   * 684 G$ are distributed as basic income
2. Mint from Reserve Ratio Decline&#x20;
   1. Lower the Reserve Ratio from 80% to 79%
   2. Calculated as follows:
      1. $$S \cdot P=\frac{R}{Rr}$$&#x20;
      2. (1,253,420+X)\*1= 1,002,736/0.79
      3. 1,253,420 + X = 1,269,286
      4. X = 15,866&#x20;
   3. 15,866 G$ are distributed as UBI

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


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