Token Valuation

Valuation Method

Let $\mathtt{price}(t), \mathtt{token}(t), \mathtt{asset}(t)$ be the token price, token supply, and asset under management (AUM), at time $t$, respectively. Since all the management fee is used for buying back and burning. Using the discounted cash flow (DCF) model, the following relation holds.

$$\mathtt{price}(t) \cdot \mathtt{token}(t) = \int_t^\infty \phi \cdot \mathtt{asset}(x) \cdot e^{-\lambda x} dx$$

where $\phi$ is the annual management fee rate and $\lambda$ is the discount rate or rate of return on risky assets.

Following the staking-reward model and the buyback-and-burn mechanism, we break down the token supply into

$$\mathtt{token}(t) = \mathtt{mint}(t) - \mathtt{burn}(t) \ge 0$$

where $\mathtt{mint}(t), \mathtt{burn}(t)$ are the total tokens minted and burned, respectively.

Assume that $\mathtt{mint}(t)$ is monotonically increasing and bounded over $[M_0, M_\infty] \subseteq \mathcal{R}^+$. That is, $M_0$ represents the initial token supply and $M_\infty$ the maximum token supply. The total tokens burned is given by

$$\mathtt{burn}(t) = \int_0^t \phi \cdot \dfrac{\mathtt{asset}(x)}{\mathtt{price}(t)} dx$$

The equations above give

$$\begin{align*} \mathtt{price}(t) \cdot \Big \{ \mathtt{mint}(t) - \mathtt{burn}(t) \Big \} &= \int_t^\infty \phi \cdot \mathtt{asset}(x) \cdot e^{-\lambda x} dx \\ \mathtt{price}(t) \cdot \Big \{ \mathtt{mint}(t) - \int_0^t \phi \cdot \dfrac{\mathtt{asset}(x)}{\mathtt{price}(t)} dx \Big \} &= \int_t^\infty \phi \cdot \mathtt{asset}(x) \cdot e^{-\lambda x} dx \\ \mathtt{price}(t) \cdot \mathtt{mint}(t) - \int_0^t \phi \cdot \mathtt{asset}(x) dx &= \int_t^\infty \phi \cdot \mathtt{asset}(x) \cdot e^{-\lambda x} dx \end{align*}$$

​The generic model of the token price therefore is

$$\mathtt{price}(t) = \frac{1}{\mathtt{mint}(t)} \left \{ \int_0^t \phi \cdot \mathtt{asset}(x) dx + \int_t^\infty \phi \cdot \mathtt{asset}(x) \cdot e^{-\lambda x} dx \right \}$$

​Expected Supply and Price Function

Assume the following parametric forms.

$$\begin{align*} \mathtt{asset}(t) &= A \cdot e^{\alpha t} \\ \mathtt{mint}(t) &= M_0 \cdot e^{-\alpha t} + M_\infty \cdot (1 - e^{-\alpha t}) \end{align*}$$

where $\alpha > 0$ is the expected (log) growth rate in AUM. Note that the initial and asymptotic conditions already hold.

$$\begin{align*} \mathtt{mint}(0) &= M_0 \\ \lim_{t \to \infty} \mathtt{mint}(t) &= M_\infty \end{align*}$$

To show that $\mathtt{mint}(t)$ is monotonically increasing at a diminishing growth rate.

$$\begin{align*} \dfrac{d}{dt} \mathtt{mint}(t) &= -\alpha \cdot M_0 \cdot e^{-\alpha t} + \alpha \cdot M_\infty \cdot e^{-\alpha t} \\ &= \alpha \cdot (-M_0 + M_\infty) \cdot e^{-\alpha t} \ge 0 \\ \dfrac{d^2}{dt^2} \mathtt{mint}(t) &= \alpha^2 \cdot M_0 \cdot e^{-\alpha t} - M_\infty \cdot \alpha^2 \cdot e^{-\alpha t} \le 0 \\ &= \alpha^2 \cdot (M_0 - M_\infty) \cdot e^{-\alpha t} \le 0 \end{align*}$$

The initial AUM is targeted at 30 million USD and is expected to grow at a rate of 61.80% annually. Hence, $\alpha = \log (1 + 0.6180) \approx 0.4812$. The tokens have an initial supply of 30 million and will be gradually released to reach the maximum supply of 100 million. Hence, in a scale of $10^6$ (million), we have:

$$\begin{align*} \mathtt{asset}(t) &= 30 \cdot e^{0.4812 t} \\ \mathtt{mint}(t) &= 30 \cdot e^{-0.4812 t} + 100 \cdot (1 - e^{-0.4812 t}) \end{align*}$$

Based on the the generic pricing equation and assuming a discount rate of 100.00% annually i.e. $\lambda = \log (1 + 1.0000) \approx 0.6931$, we have

$$\begin{align*} \mathtt{price}(t) &= \dfrac{1}{\mathtt{mint}(t)} \left \{ \int_0^t 0.02 \cdot 30 \cdot e^{0.4812 x} dx + \int_t^\infty 0.02 \cdot 30 \cdot e^{-0.2119 x} dx \right \} \\ &= \dfrac{1}{\mathtt{mint}(t)} \left \{ 1.2469 \cdot (e^{0.4812 t} - 1) + 2.8311 \cdot e^{-0.2119 t} \right \} \\ &= \dfrac{ 1.2469 \cdot (e^{0.4812 t} - 1) + 2.8311 \cdot e^{-0.2119 t} }{ 30 \cdot e^{-0.4812 t} + 100 \cdot (1 - e^{-0.4812 t}) } \end{align*}$$

Visualizing the supply and price function

1. Expected supply over time

The token emission is scheduled based on all the assumptions above. This schedule represents only the emission plan yet does not take into account the burning mechanism.

The following plot shows the token supply function, designated by:

$\mathtt{mint}(t) = 30 \cdot e^{-0.4812 t} + 100 \cdot (1 - e^{-0.4812 t})$

2. Expected price over time with burning mechanism

The expected pricing is based on all the assumptions above and evaluated at a unit price-to-earning ratio (P/E). Note also that we assume that the tokens initially offered to Backers Sale and Avareum Lab are available without any promotion terms and conditions.

The following plot shows the token price function, designated by:

$\mathtt{price}(t) = \dfrac{ 1.2469 \cdot (e^{0.4812 t} - 1) + 2.8311 \cdot e^{-0.2119 t} }{ 30 \cdot e^{-0.4812 t} + 100 \cdot (1 - e^{-0.4812 t}) }$