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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.


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×rC \times r per year, where CC is the amount of Cash in circulation, rr is the Stake rate. The Stake rate rr 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, rr is calculated as the value of Cash in circulation (in units of US$) to the power of 13-\frac{1}{3}, bounded to stay between 0.01 (ie. 1%) and 10-4 (1 basis point). The quantity CrC r is time varying, therefore at each point in time CrC r measures flow of Cash, and the total Cash paid in a year is the time average of CrC 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 hh as the highest traded price of Stake (in US$), capped at 10, the amount of Stake in existence at that time is 105h10^5 h. The flow of Cash to a holder of one Stake unit is then at a rate of Cr105h\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.


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 VV_{\ast} beyond tt_{\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 VtV_t be defined as the value in Dai of all Cash in circulation at time tt (also known as the Cash market capitalisation). Let ItI_t be the total value (in Dai) of Cash flowing to all Stake holders, instantaneously at time tt. So It=Vt  r(Vt)I_t = V_t \; r(V_t), where r(Vt)r(V_t) is the Stake rate at time tt. By definition then

r(V)={102if V106V13if 106V1012104if V1012It={Vt100if Vt106Vt23if 106Vt1012Vt104if Vt1012r(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 ItI_t represents the cash-flows in USD terms.

Let ptp_t be the last trade price of Stake at time tt, then the high water mark hth_t is defined as ht=supst  ps10h_t = \underset{s \leqslant t}{\sup} \; p_s \wedge 10

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

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

PVhold=E[0It  0.91tt0  Dt105  htdt]E[0It  0.91t  Dt106dt]()\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 t0t_0 is the first time Cash will enter circulation, or t0=0t_0 = 0 if this has already happened.

Obtaining a better lower bound would require hth_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 t\ell_t, can achieve this. With such a strategy, the present value of Stake is

PVlimit=E[0TIt  0.91tt0  Dt105  htdt+T  0.91(Tt0)+DT]\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 TT is the time at which Stake is sold. As such, T=inf{t0:ptt}T = \inf \{ t \geqslant 0 : p_t \geqslant \ell_t \}, with partial order fills represented such that whenever a fraction πi\pi_i of the order is filled at times TiT_i, then TT equals TiT_i with probability πi\pi_i for each ii.

General lower bound


s=max{vshs,vs} where vs=E[sIt  0.91ts105DtDsdt  |  Fs]\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]


PVlimitE[0TIt  0.91t  Dt105  htdt+1hTE[TIt  0.91t  Dt105dt  |  FT]]=E[0It  0.91t  Dt105  htTdt]\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 PVlimitPVhold\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 h0v0h_0 \leqslant \sqrt{v_0} then, using the fact that htsupstvsh_t \leqslant \underset{s \leqslant t}{\sup} \sqrt{v_s} for all t<Tt < T (proof below), also

PVlimitE[0TIt  0.91t  Dt105  supstvsdt+1vTE[TIt  0.91t  Dt105dt  |  FT]]E[0It  0.91t  Dt105  supstTvsdt]()\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 hh. Alternatively if h0>v0h_0 > \sqrt{v_0}, then h0>0h_0 > \ell_0, which during the initial issuance from pre-orders implies T=0T = 0, and then bound ()(\dagger) holds anyway.

AssumingB non-negative forward rates at time zero, so that DtD_t is decreasing in tt, then

supstvs10.91t  Dt  supst  E[sIτ  0.91τ  Dτ105dτ  |  Fs]12\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 VtV_t, this last bound equals

10.91t  Dt(0Iτ  0.91τ  Dτ105dτ)12=v00.91t  Dt\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}}


PVlimit0It  (0.91t  Dt)32105v0dt\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 v0p010\sqrt{v_0} \leqslant p_0 \leqslant 10 then v0h0p02p010=p0\frac{v_0}{h_0} \leqslant \frac{p_0^2}{p_0 \wedge 10} = p_0 so that 0p0\ell_0 \leqslant p_0 and T=0T = 0. In this case PVlimit=0p0\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 htsupstvsh_t \leqslant \underset{s \leqslant t}{\sup} \sqrt{v_s} for all t<Tt < T, whenever h0v0h_0 \leqslant \sqrt{v_0}:
For t<Tt < T have pt<tp_t < \ell_t so one of pt<vthtp_t < \frac{v_t}{h_t} or pt<vtp_t < \sqrt{v_t} must hold.
If pt<vthtp_t < \frac{v_t}{h_t} then (pt10)2pt(pt10)<vththt=vt(p_t \wedge 10)^2 \leqslant p_t (p_t \wedge 10) < \frac{v_t}{h_t} h_t = v_t
If pt<vtp_t < \sqrt{v_t} then (pt10)2pt2<vt(p_t \wedge 10)^2 \leqslant p_t^2 < v_t
Therefore ht2=supst(ps10)2h0sup0stvsh_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 htsupstvsh_t \leqslant \underset{s \leqslant t}{\sup} \sqrt{v_s} for all t[0,T)t \in [0, T)

Target size model

Suppose VtVV_t \geqslant V_{\ast} for all ttt \geqslant t_{\ast} then define I=V  r(V)I_{\ast} = V_{\ast} \; r(V_{\ast}) and set s\ell_s be the constant

=0I  0.91t  Dt105dt\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 DtDsDts\frac{D_t}{D_s} \leqslant D_{t - s} whenever 0st0 \leqslant s \leqslant t, implies

2=sI  0.91ts  Dts105dt10.91s  DssI  0.91t  Dt105dt\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 s0s \geqslant 0. If ht<h_t < \ell for all t<Tt < T, as holds during the initial issuance from pre-orders, then

PVlimitE[0TIt  0.91t  Dt105  dt+1TI  0.91t  Dt105dt]E[0(ItI)  0.91t  Dt105  dt]tI  0.91t  Dt105  dt()\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 PVhold\mathrm{PV}_{\mathrm{hold}} as in ()(*), the top strategy can be defined satisfying

tI  0.91t  Dt105  (10)dtPVtop={PVholdif 10PVlimitif <10()\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)


2=tI  0.91tt  Dtt105dtsupτ0DτDτ+t0.91ttI  0.91t  Dt105dt\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

PVlimitI0.91t  infτ0Dτ+tDτt0.91t  Dt105dt\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=0t_{\ast} = 0 this is simply PVlimit\mathrm{PV}_{\mathrm{limit}} \geqslant \ell, and the 00.91tDtdt\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.