🤑Token Price Dynamics

This page describes how the price of the TurboSwap token is affected by various circumstances

Initial Token Price

In general, the price of the TurboSwap tokens is defined by the following formula:

$$p = \frac{\alpha C}{N}$$

Here$C$is the value of the capital,$N$is the number of tokens in circulation, and$\alpha$the capitalization ratio.

The initial capital value$C_0$could be calculated as:

$$C_0 = m - s$$

Here$m$is the amount of money invested by the pre-sale investors, and$s$is the amount of money spend for the protocol development, marketing, etc, i.e. all the money spent before the launch.

The initial number of tokens$N_0$could be calculated as:

$$N_0 = n + k$$

Here $n$ is the number of tokens issued for the pre-sale investors, and $k$ is the number of tokens issued to the team, advisors, to fund operational costs, etc, i.e. to those who didn't pay for the tokens.

Thus, the initial token price is:

$$p_0 = \alpha \frac{m - s}{n + k}$$

Note, that the average price paid by the presale investors for the tokens is:

$$\bar p_0 = \frac{m}{n}$$

In order to make pre-sale investing profitable, the presale price should be less than the initial token price, thus we have:

$$\bar p_0 < p_0 \\[1em] \frac{m}{n} < \alpha \frac{m - s}{n + k}$$

This limits the number of tokens that could be issued for non-investors ($k$):

$$k < n \left( \alpha - \frac{\alpha s}{m} - 1 \right)$$

The more money were spent before the launch ($s$), the lower the$k$limit is. In order for the limit to be non-negative, the amount of money spent before the launch should fulfill the following constraints:

$$\alpha - \frac{\alpha s}{m} - 1 > 0 \\[1em] s < m \left( 1 - \frac{1}{\alpha} \right)$$

Fees and Interest

The protocol charges fees for most operations, such as deposits, withdrawals, trades etc. Also, in some cases the protocol collects interest on the capital loaned to the borrowers. These fees and interest are added to the capital. In case the protocol collected$\Delta C$worth of fees and interest, the token price increase would be:

$$\Delta p = \frac{\alpha \left( C + \Delta C \right)}{N} - \frac{\alpha C}{N} = \alpha \frac{\Delta C}{N}$$

So, if the capital per token has increased by$x = \frac{\Delta C}{N}$, then the token price increase by$\alpha x$.

Losses

When an amount in default is liquidated, the protocol may take losses. These losses are subtracted from the capital and potentially may even turn the capital negative. In case the protocol took$\Delta C$worth of losses, the token price decrease would be:

$$\Delta p = \frac{\alpha C}{N} - \frac{\alpha \left( C - \Delta C \right)}{N} = \alpha \frac{\Delta C}{N}$$

So, again, if the loss per token is$x$the token price decrease will be$\alpha x$.

Investments and Redemptions

When an investor invests$\Delta C$worth of assets, the amount of new minted tokens issued to him is:

$$\Delta N = N \left( \left( \frac{C + (1 - \phi_m)\Delta C}{C} \right)^{\frac{1}{\alpha}} - 1\right)$$

Here$\phi_m$is the minting fee. The token price increase is:

$$\Delta p = \alpha \frac{C + \Delta C}{N + \Delta N} - \alpha \frac{C}{N} \\[1em] \Delta p = \alpha \frac{C + \Delta C}{N + N \left( \left( \frac{C + (1 - \phi_m)\Delta C}{C} \right)^{\frac{1}{\alpha}} - 1\right)} - \alpha \frac{C}{N} = \frac{\alpha}{N} \left (\frac{C + \Delta C}{\left( \frac{C + (1 - \phi_m)\Delta C}{C} \right)^{\frac{1}{\alpha}}} - C \right)$$

With zero minting fee this would be:

\Delta p = \frac{\alpha}{N} \left (C^\left( \frac{1}{\alpha} \right) \left( C + \Delta C \right)^\left( 1 - \frac{1}{\alpha} \right) - C \right)

When an investors redeems$\Delta N$token, the value of assets released to the investor is:

$$\Delta C = (1 - \phi_b) \cdot C \left( 1 - \left( \frac{N - n}{N} \right)^\alpha \right)$$

Here $\phi_b$ is the burning fee. The token price decrease is:

$$\Delta p = \alpha \frac{C}{N} - \alpha \frac{C - \Delta C}{N - \Delta N} \\[1em] \Delta p = \alpha \frac{C}{N} - \alpha \frac{C - (1 - \phi_b) \cdot C \left( 1 - \left( \frac{N - \Delta N}{N} \right)^\alpha \right)}{N - \Delta N} \\[1em] \Delta p = \alpha \cdot C \left( \frac{1}{N} - \frac{1 - (1 - \phi_b) \left( 1 - \left( \frac{N - \Delta N}{N} \right)^\alpha \right)}{N - \Delta N} \right)$$

Note, that very high burning fee could make the price decrease negative, i.e. redemptions could in theory increase the token price instead of decreasing it.

With zero burning fee the price decrease would be:

$$\Delta p = \frac{\alpha C}{N} \left( 1 - \left( \frac{N - \Delta N}{N} \right)^{\alpha - 1} \right)$$