Curved Token Bonding is a set of protocol designs first documented by Simon de la Rouviere in his Medium articles. Ethers are bonded to tokens, which are minted or burned by the smart contract on buy and sell demands respectively. Every buy and sell changes the price of the token according to a predefined formula. The “curves” here represent the relationship between the price of a single token and the token supply, as defined by said formula. The result is an ether-backed token that rewards early adopters.

Inside a transaction, the price paid/received per token is not constant and depends on the amount that is bought or sold. This complicates the calculations.

Let’s say for an initial supply of $S_0$, we want to buy $T$ tokens which are added to the new supply $S_1=S_0+T$. The ether E that must be spent for this purchase is defined as

which is illustrated in the figure above. If one wanted to sell $T$ tokens, the upper limit for the integral would be $S_0$ and the lower $S_0-T$, with E corresponding to the amount of ether received for the sale.

# Linear Curves

A linear relationship for the bonding curves are defined as

where $P_0$ is the initial price of the token and $I_p$ is the price increment per token.

Let us have E ethers which we want to buy tokens with. Substituting $P$ into the integral above with the limits $S_0\to S_0+T$, we obtain $E$ in terms of the tokens $T$ that we want to buy:

where $S$ is the supply before the purchase. Solving this for $T$, we obtain the tokens received in a purchase as a function of the supply and ethers spent:

## Selling Tokens

Let us have T tokens which we want to sell for ethers. Selling $T$ tokens decreases the supply from $S_0$ to $S_0-T$, which we apply as limits for the above integral and obtain:

# Breaking Even in PoWH3D

PoWH3D is one of the applications of curved token bonding with a twist: 1/10th of every transaction is distributed among token holders as dividends. When you buy tokens with $E$ ethers, you receive $9/10 E$ worth of tokens and $1/10 E$ is distributed to everybody else in proportion to the amount they hold.

This means you are at a loss when you buy P3D (the token used by PoWH3D). If you were to sell immediately, you would only receive 81% of your money. Given the situation, one wonders when exactly one can break even with their investment. The activity in PoWH3D isn’t deterministic; nonetheless we can deduce sufficient but not necessary conditions for breaking even in PoWH3D.

See Bitses.org for a report on PoWH3D.

## Sufficient Purchase

Let us spend $E_1$ ethers to buy tokens at supply $S_0$. The following calculations are done with the assumption that the tokens received

are small enough to be neglected, that is $T_1 \ll S_0$ and $S_1 \approx S_0$. In other words, this only holds for non-whale purchases.

Then let others spend $E_2$ ethers to buy tokens and raise the supply to $S_2$. The objective is to find an $E_2$ large enough to earn us dividends and make us break even when we sell our tokens at $S_2$. We have

Our new share of the P3D pool is $T_1/S_2$ and the dividends we earn from the purchase is equal to

Then the condition for breaking even is

This inequality has a lengthy analytic solution which is impractical to typeset. The definition should be enough:

and

$E^{\text{suff}}_2$ can be obtained from the source of this page in JavaScript from the function sufficient_purchase. Note that the function involves many power and square operations and may yield inexact results for too high values of $S_0$ or too small values off $E_1$, due to insufficient precision of the underlying math functions. For this reason, the calculator below is disabled for sensitive input.

The relationship between the initial supply and sufficient purchase is roughly quadratic, as seen from the graph above. This means that the difficulty of breaking even increases quadratically as more people buy into P3D. As interest in PoWH3D saturates, dividends received from the supply increase decreases quadratically.

The relationship is not exactly quadratic, as seen from the graph above. The function is sensitive to $E_1$ for small values of $S_0$.

## Sufficient Sale

Let us spend $E_1$ ethers to buy tokens at supply $S_0$ and receive $T_1$ tokens.

Then let others sell $T_2$ P3D to lower the supply to $S_2$. The objective is to find a $T_2$ large enough to earn us dividends and make us break even when we sell our tokens at $S_2$. We have

Our new share of the P3D pool is $T_1/S_2$ and the dividends we earn from the purchase is equal to

Then the condition for breaking even is

Similar to the previous section, we have

and

$T^{\text{suff}}_2$ can be obtained from the function sufficient_sale.

The relationship between $S_0$ and $T^{\text{suff}}_2$ is linear and insensitive to $E_1$. Regardless of the ethers you invest, the amount of tokens that need to be sold to guarantee your break-even is roughly the same, depending on your entry point.

## Calculator

Below is a calculator you can input $S_0$ and $E_1$ to calculate $E^{\text{suff}}_2$ and $T^{\text{suff}}_2$.

$S_0$
$E_1$
$E^{\text{suff}}_2$
$T^{\text{suff}}_2$

For the default values above, we read this as:

For 100 ETH worth of P3D bought at 3,500,000 supply, either a purchase of ~31715 ETH or a sale of ~3336785 P3D made by other people is sufficient to break even.