Rate of return and yield during a release schedule

According to the quantity theory of money, when new tokens are issued, the price of the token should decrease accordingly. We assume an inflation rate \(J\) and also expect the market cap to grow with a rate \(G\)^[1]. Then the price at the end of the first year is equal to

\[ P(J, G) = P_0\left(\frac{1+G}{1+J}\right) \]

where \(P_0\) is the initial price.

We measure measure rate of return and yield by comparing the dollar values of a validator's holdings at the beginning and end of the first year. We take all validators as a single group

\[ Q_{0,v} = Q_0KS \]

where \(Q_0\) is the initial total supply (which is partially released), \(K\) is the ratio of the initial sale and \(S\) is the stake percentage.

At the end of the first year, the validators will have received all of the seigniorage which is minted at a rate \(I\):

\[ Q_{v} = Q_0KS + Q_0I \]

Note: Seigniorage rate \(I\) != Inflation rate \(J\)

Rate of return function

With the definitions above, we can define a generalized function that can be used to calculate returns for various price trajectories. We calculate how much the dollar value of validator holdings increase

\[ R(J, G) = \frac{P(J,G)Q_{v}}{P_0Q_{0,v}} - 1 = \frac{P(J,G)}{P_0}\left(\frac{I}{KS}+1\right)-1 \]

Real Rate of Return

To calculate the annual real rate of return, we substitute \(I/K\) and \(G\) as the inflation and growth rates respectively

\[ R_\text{real} = R(I/K, G) = \frac{(1+G)(I+KS)}{S(I+K)}-1 \]

We factor in the initial sale percentage in the inflation rate to because the release schedule exacerbates the devaluation caused by seigniorage.

Nominal Rate of Return

To calculate the annual nominal rate of return, we keep growth, but ignore the devaluing of the token due to seigniorage

\[ R_\text{nom} = R(0, G) = \frac{GKS+GI+I}{KS} \]

Real Yield

Assumes no price growth, i.e. \(G=0\).

\[ Y_\text{real} = R(I/K, 0) = \frac{I(1-S)}{S(I+K)} \]

Nominal Yield

Assumes price growth is equal to seigniorage with the release ratio factored in, i.e. \(G=I/K\).

\[ Y_\text{nom} = R(G, G) = \frac{I}{KS} \]

Examples

For \(I=3\%\), \(S=80\%\) and \(K=10\%\)

Metric	Value
Real yield	5.77%
Nominal yield	37.5%

Implementation

from sympy import *

P0 = Symbol('P_0', positive=True)
S = Symbol('S', positive=True)
K = Symbol('K', positive=True)
Q0 = Symbol('Q_0', positive=True)
I = Symbol('I', positive=True)
J = Symbol('J', positive=True)
G = Symbol('G', positive=True)

P = lambda J,G: P0*(1+G)/(1+J)

Qv_0 = Q0*K*S
Qv = Q0*K*S + Q0*I

real_ror = lambda J, G: simplify(P(J, G)/P0*Qv/Qv_0-1)

vals = (
    (I, 0.03),
    (K, 0.1),
    (S, 0.8),
)

print('Real rate of return')
pprint(real_ror(I/K, G))
print()

print('Nominal rate of return')
pprint(real_ror(0, G))
print()

print('Real yield')
pprint(real_ror(I/K, 0))
pprint(real_ror(I/K, 0).subs(vals))
print()

print('Nominal yield')
pprint(real_ror(G, G))
pprint(real_ror(G, G).subs(vals))

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1. \(J\) and \(G\) are defined as annual rates. ↩︎