# Stake

Stake is a unique investment instrument that allows one to benefit from the growth of the underlying asset: Cash. It has many of the advantages of classical instruments that are equity or debt based, while avoiding their substantive shortcomings.

Stake has an intrinsic value as it receives a cash-flow, from a fraction of the interest generated by Cash. There is a maximum of 1 million units of Stake in existence, and each new unit is only created when it is bought for a price higher than the last. So that the amount of Stake in existence grows only when it trades above its high water mark.

Due to reserves, the market value of Cash in circulation is rigidly tied to the demand for Cash. Naturally, the cash-flow paid across all Stake is an increasing function of the value of Cash in circulation. Together with the restrained Stake issuance, this makes the value of Stake exceptionally well tied to fundamentals, ie. the demand for Cash.

## Issuance​

The prices and quantities at which Stake is issued correspond to offers on a limit order book, where Stake is created for each set of offers lifted. In this representation, there are one thousand units of Stake offered within each one cent increment in price from 0 to 10 US Dollars.

Compared to selling Stake through an auction, early investors can buy at much lower prices - indeed offer prices start all the way down from zero. Compared to the classical venture capital funding round model, the valuation for newly issued Stake is predetermined (with no down round). This removes the friction of negotiating valuation, ensures all parties receive the same terms, and makes arbitrary dilution impossible.

## Cash flow​

The total Cash paid out to Stake holders is $C \times r$ per year, where $C$ is the amount of Cash in circulation, $r$ is the Stake rate. The Stake rate $r$ varies as follows:

Value of Cash in circulationStake rate
1mm (106 )1% (10-2 )
1bn (109 )0.1% (10-3 )
1tr (1012 )0.01% (10-4 )

In general, $r$ is calculated as the value of Cash in circulation (in units of US$) to the power of $-\frac{1}{3}$, bounded to stay between 0.01 (ie. 1%) and 10-4 (1 basis point). The quantity $C r$ is time varying, therefore at each point in time $C r$ measures flow of Cash, and the total Cash paid in a year is the time average of $C r$ over the year. Having a payout which is a predetermined increasing function of fundamentals (the demand for Cash) is superior to both equity and debt, as equity does not ensure any cash-flow, while debt cash-flows only have a known cap, with no prospect of growth. ## Term structure​ At any point in time, defining the high water mark $h$ as the highest traded price of Stake (in US$), capped at 10, the amount of Stake in existence at that time is $10^5 h$. The flow of Cash to a holder of one Stake unit is then at a rate of $\frac{C r}{10^5 h}$ per year. This Cash is paid in exchange for Stake, and the rate at which Stake is exchanged for Cash is 9% of each Stake unit per year1.

As such, the Stake cash-flow acts like a bond principal repayment or share buyback, where the number of instruments in circulation is reduced with a cash payout. It is unlike a share dividend or bond coupon, where the price of the instrument drops by an amount equal to the cash-flow, in response to an upcoming payment. Reducing circulation is preferable because it specifies a time horizon over which the future demand for Cash determines the value of Stake today.

## Valuation​

A benefit of the Stake instrument is that it converts into Cash in a formulaic manner. This implies it can be priced in terms of discounted cash-flows. The expected value of these cash-flows informs the Stake market price, while market price moves can drive issuance, which impacts the cash-flow per unit of Stake. It is possible nonetheless to unravel this cycle, and compute a lower bound for the present value of Stake, valid under any market price action.

This calculation is laid out in the next section. The resulting lower bounds for the present value of Stake are shown in the tool below. There are two models considered: the target size model supposes that the Cash market capitalisation (in US$) exceeds $V_{\ast}$ beyond $t_{\ast}$ years after the Stake purchase; the reference asset model supposes that Cash market capitalisation follows the same trajectory as a chosen crypto asset. ## Price calculations​ This section requires knowledge of financial mathematics. Let $V_t$ be defined as the value in Dai of all Cash in circulation at time $t$ (also known as the Cash market capitalisation). Let $I_t$ be the total value (in Dai) of Cash flowing to all Stake holders, instantaneously at time $t$. So $I_t = V_t \; r(V_t)$, where $r(V_t)$ is the Stake rate at time $t$. By definition then $r(V) = \left\{\begin{array}{ll} 10^{-2} & \text{if } V \leqslant 10^6\\ V^{-\frac{1}{3}} & \text{if } 10^6 \leqslant V \leqslant 10^{12}\\ 10^{-4} & \text{if } V \geqslant 10^{12} \end{array}\right. \quad I_t = \left\{\begin{array}{ll} \frac{V_t}{100} & \text{if } V_t \leqslant 10^6\\ V_t^{\frac{2}{3}} & \text{if } 10^6 \leqslant V_t \leqslant 10^{12}\\ \frac{V_t}{10^4} & \text{if } V_t \geqslant 10^{12} \end{array}\right.$ It is assumed throughout that the price of Dai is exactly$1, so that $I_t$ represents the cash-flows in USD terms.

Let $p_t$ be the last trade price of Stake at time $t$, then the high water mark $h_t$ is defined as $h_t = \underset{s \leqslant t}{\sup} \; p_s \wedge 10$

The stochastic processes $V_t$ and $p_t$ are adapted to a filtration $\mathcal{F}_t$. Time $t$ is in units of years (by definition equal to 365.2425 days), and by convention time $t=0$ refers to the present.

Let $D_t$ be the USD discount factor to time $t$, ie. the present value of receiving \$1 in $t$ years from now. The present value of holding one unit of Stake is then

$\mathrm{PV}_{\mathrm{hold}} = \mathbb{E} \left[ \int_0^{\infty} \frac{I_t \; 0.91^{t - t_0} \; D_t}{10^5 \; h_t} \,\mathrm{d} t \right] \geqslant \mathbb{E} \left[ \int_0^{\infty} \frac{I_t \; 0.91^t \; D_t}{10^6} \,\mathrm{d} t \right] \qquad (*)$

where $t_0$ is the first time Cash will enter circulation, or $t_0 = 0$ if this has already happened.

Obtaining a better lower bound would require $h_t$ to remain below a known level (other than 10) while Stake is held. A strategy which maintains an offer for the full Stake held, at some limit price $\ell_t$, can achieve this. With such a strategy, the present value of Stake is

$\mathrm{PV}_{\mathrm{limit}} = \mathbb{E} \left[ \int_0^T \frac{I_t \; 0.91^{t - t_0} \; D_t}{10^5 \; h_t} \,\mathrm{d} t + \ell_T \; 0.91^{(T - t_0)^+} D_T \right]$

where $T$ is the time at which Stake is sold. As such, $T = \inf \{ t \geqslant 0 : p_t \geqslant \ell_t \}$, with partial order fills represented such that whenever a fraction $\pi_i$ of the order is filled at times $T_i$, then $T$ equals $T_i$ with probability $\pi_i$ for each $i$.

### General lower bound​

Setting

$\ell_s = \max \left\{ \frac{v_s}{h_s}, \sqrt{v_s} \right\} \text{ where } v_s = \mathbb{E} \left[ \int_s^{\infty} \frac{I_t \; 0.91^{t - s}}{10^5} \frac{D_t}{D_s} \,\mathrm{d} t \;\middle|\; \mathcal{F}_s \right]$

gives

\begin{aligned} \mathrm{PV}_{\mathrm{limit}} & \geqslant \mathbb{E} \left[ \int_0^T \frac{I_t \; 0.91^t \; D_t}{10^5 \; h_t} \,\mathrm{d} t + \frac{1}{h_T} \mathbb{E} \left[ \int_T^{\infty} \frac{I_t \; 0.91^t \; D_t}{10^5} \,\mathrm{d} t \;\middle|\; \mathcal{F}_T \right] \right] \\ & = \mathbb{E} \left[ \int_0^{\infty} \frac{I_t \; 0.91^t \; D_t}{10^5 \; h_{t \wedge T}} \,\mathrm{d} t \right] \end{aligned}

so that $\mathrm{PV}_{\mathrm{limit}} \geqslant \mathrm{PV}_{\mathrm{hold}}$, meaning this limit strategy has an expected return at least that of the holding strategy.

If $h_0 \leqslant \sqrt{v_0}$ then, using the fact that $h_t \leqslant \underset{s \leqslant t}{\sup} \sqrt{v_s}$ for all $t < T$ (proof below), also

\begin{aligned} \mathrm{PV}_{\mathrm{limit}} & \geqslant \mathbb{E} \left[ \int_0^T \frac{I_t \; 0.91^t \; D_t} {10^5 \; \underset{s \leqslant t}{\sup} \sqrt{v_s}} \,\mathrm{d} t + \frac{1}{\sqrt{v_T}} \mathbb{E} \left[ \int_T^{\infty} \frac{I_t \; 0.91^t \; D_t}{10^5} \,\mathrm{d} t \;\middle|\; \mathcal{F}_T \right] \right] & \\ & \geqslant \mathbb{E} \left[ \int_0^{\infty} \frac{I_t \; 0.91^t \; D_t} {10^5 \; \underset{s \leqslant t \wedge T}{\sup} \sqrt{v_s}} \,\mathrm{d} t \right] & (\dagger) \end{aligned}

which does not depend on $h$. Alternatively if $h_0 > \sqrt{v_0}$, then $h_0 > \ell_0$, which during the initial issuance from pre-orders implies $T = 0$, and then bound $(\dagger)$ holds anyway.

AssumingB non-negative forward rates at time zero, so that $D_t$ is decreasing in $t$, then

$\underset{s \leqslant t}{\sup} \sqrt{v_s} \leqslant \frac{1}{\sqrt{0.91^t \; D_t}} \; \underset{s \leqslant t}{\sup} \; \mathbb{E} \left[ \int_s^{\infty} \frac{I_{\tau} \; 0.91^{\tau} \; D_{\tau}}{10^5} \,\mathrm{d} \tau \;\middle|\; \mathcal{F}_s \right]^{\frac{1}{2}}$

In case moreover a deterministic model is used for $V_t$, this last bound equals

$\frac{1}{\sqrt{0.91^t \; D_t}} \left( \int_0^{\infty} \frac{I_{\tau} \; 0.91^{\tau} \; D_{\tau}}{10^5} \,\mathrm{d} \tau \right)^{\frac{1}{2}} = \frac{\sqrt{v_0}}{\sqrt{0.91^t \; D_t}}$

therefore

$\mathrm{PV}_{\mathrm{limit}} \geqslant \int_0^{\infty} \frac{I_t \; (0.91^t \; D_t)^{\frac{3}{2}}}{10^5 \sqrt{v_0}} \,\mathrm{d} t$

On the other hand, if $\sqrt{v_0} \leqslant p_0 \leqslant 10$ then $\frac{v_0}{h_0} \leqslant \frac{p_0^2}{p_0 \wedge 10} = p_0$ so that $\ell_0 \leqslant p_0$ and $T = 0$. In this case $\mathrm{PV}_{\mathrm{limit}} = \ell_0 \leqslant p_0$ so that it is not profitable to invest in Stake now with this strategy.

Proof that $h_t \leqslant \underset{s \leqslant t}{\sup} \sqrt{v_s}$ for all $t < T$, whenever $h_0 \leqslant \sqrt{v_0}$:
For $t < T$ have $p_t < \ell_t$ so one of $p_t < \frac{v_t}{h_t}$ or $p_t < \sqrt{v_t}$ must hold.
If $p_t < \frac{v_t}{h_t}$ then $(p_t \wedge 10)^2 \leqslant p_t (p_t \wedge 10) < \frac{v_t}{h_t} h_t = v_t$
If $p_t < \sqrt{v_t}$ then $(p_t \wedge 10)^2 \leqslant p_t^2 < v_t$
Therefore $h_t^2 = \underset{s \leqslant t}{\sup} (p_s \wedge 10)^2 \leqslant h_0 \vee \underset{0 \leqslant s \leqslant t}{\sup} v_s$ and $h_t \leqslant \underset{s \leqslant t}{\sup} \sqrt{v_s}$ for all $t \in [0, T)$

### Target size model​

Suppose $V_t \geqslant V_{\ast}$ for all $t \geqslant t_{\ast}$ then define $I_{\ast} = V_{\ast} \; r(V_{\ast})$ and set $\ell_s$ be the constant

$\ell = \sqrt{\int_0^{\infty} \frac{I_{\ast} \; 0.91^t \; D_t}{10^5} \,\mathrm{d} t}$

Assuming2 the yield curve is normal at time zero, so that $\frac{D_t}{D_s} \leqslant D_{t - s}$ whenever $0 \leqslant s \leqslant t$, implies

$\ell^2 = \int_s^{\infty} \frac{I_{\ast} \; 0.91^{t - s} \; D_{t - s}}{10^5} \,\mathrm{d} t \geqslant \frac{1}{0.91^s \; D_s} \int_s^{\infty} \frac{I_{\ast} \; 0.91^t \; D_t}{10^5} \,\mathrm{d} t$

for all $s \geqslant 0$. If $h_t < \ell$ for all $t < T$, as holds during the initial issuance from pre-orders, then

\begin{aligned} \mathrm{PV}_{\mathrm{limit}} & \geqslant \mathbb{E} \left[ \int_0^T \frac{I_t \; 0.91^t \; D_t}{10^5 \; \ell} \,\mathrm{d} t + \frac{1}{\ell} \int_T^{\infty} \frac{I_{\ast} \; 0.91^t \; D_t}{10^5} \,\mathrm{d} t \right] & \\ & \geqslant \mathbb{E} \left[ \int_{0}^{\infty} \frac{(I_t \wedge I_{\ast}) \; 0.91^t \; D_t}{10^5 \; \ell} \,\mathrm{d} t \right] & \\ & \geqslant \int_{t_{\ast}}^{\infty} \frac{I_{\ast} \; 0.91^t \; D_t}{10^5 \; \ell} \,\mathrm{d} t & (**) \end{aligned}

Using a simple bound on $\mathrm{PV}_{\mathrm{hold}}$ as in $(*)$, the top strategy can be defined satisfying

$\int_{t_{\ast}}^{\infty} \frac{I_{\ast} \; 0.91^t \; D_t}{10^5 \; (\ell \wedge 10)} \,\mathrm{d} t \leqslant \mathrm{PV}_{\mathrm{top}} = \left\{\begin{array}{ll} \mathrm{PV}_{\mathrm{hold}} & \text{if } \ell \geqslant 10 \\ \mathrm{PV}_{\mathrm{limit}} & \text{if } \ell < 10 \end{array}\right. \qquad (\ddagger)$

Separately

$\ell^2 = \int_{t_{\ast}}^{\infty} \frac{I_{\ast} \; 0.91^{t - t_{\ast}} \; D_{t - t_{\ast}}}{10^5} \,\mathrm{d} t \leqslant \frac{\underset{\tau \geqslant 0}{\sup} \frac{D_{\tau}}{D_{\tau + t_{\ast}} }}{0.91^{t_{\ast}}} \int_{t_{\ast}}^{\infty} \frac{I_{\ast} \; 0.91^t \; D_t}{10^5} \,\mathrm{d} t$

which gives

$\mathrm{PV}_{\mathrm{limit}} \geqslant \sqrt{I_{\ast}} \sqrt{0.91^{t_{\ast}} \; \underset{\tau \geqslant 0}{\inf} \frac{D_{\tau + t_{\ast}}}{D_{\tau}}} \sqrt{\int_{t_{\ast}}^{\infty} \frac{0.91^t \; D_t}{10^5} \,\mathrm{d} t}$

In the case $t_{\ast} = 0$ this is simply $\mathrm{PV}_{\mathrm{limit}} \geqslant \ell$, and the $\int_0^{\infty} 0.91^t D_t \,\mathrm{d} t$ term measures the duration (as in bond duration) of the Stake cash-flow schedule.

1. No exchange takes place before the first unit of Cash has entered circulation however.
2. This condition holds in particular for the US Treasury curve as of Mar-2021, which is used in the pricing tool.