While there are many flavors of Proof of Stake out in the wild, they all share a common theme: network nodes, usually called validators, deposit—or stake—tokens in order to obtain the right to carry out work for the network’s users, and receive revenue in return. If they behave well, they make a profit by receiving a share of token issuance and transaction fees. If they don’t, they incur a loss by not receiving any revenue and having a part of their deposit slashed.

Moreover, users can give out—delegate—their tokens to validators for them to stake on their behalf. This way, users can earn tokens without having to run nodes, and the validators who actually run them take out a commission in return. Delegation is a win-win situation for both parties if the delegatee performs well.

How should prospective validators and delegators make decisions on

• how much to invest,
• whom to invest in,
• which commission rate to choose?

The first thing everybody looks at is the performance metrics. The following sections outline how to derive them from network parameters.

## Total Supply, Issuance and Price

In most Proof of Stake networks, new tokens are issued at a predetermined rate. Denoting the issuance rate by $I$, the token supply can be calculated as a function of time as

where $Q_0$ is the initial supply.

According to the quantity theory of money, when new tokens are issued, the price of the token should decrease accordingly. We also expect the market cap to grow with a rate $G$. Then the price is in a tug of war between issuance and growth, defined by

where $P_0$ is the initial price.

Note: $I$ and $G$ are defined as annual rates.

## Performance Metrics

#### 1. Real Rate of Return

Let $Q_\ast$ be the amount of tokens of a participant invested in the network, either a validator or a delegator. The annual real rate of return for the corresponding participant is calculated as

It is an approximation, because it assumes that $Q_\ast$ remains constant for that duration. The error should be negligible for small values of $I$.

Other performance metrics are derived from the real rate of return:

#### 2. (Nominal) Rate of Return

Ignores issuance, i.e. $I=0$.

#### 3. Yield

Assumes that the price remains constant, i.e. $G=I$. The yield is then simply calculated as

## No Delegation

Without delegation, validators receive all the issued tokens. If the initial stake percentage is $S_0$, staked tokens are calculated at any time as

assuming no exchange of tokens between validators and users take place. Then the formulas for real rate of return, rate of return and yield for all validators become

$\tilde{R}$ converges to $Y$ for small enough values of issuance and stake percentage, i.e. $I\to 0$, $S\to 0$. If stake percentage is high enough, however, the lack of price growth can lead to a considerable difference between metrics. Let $I=7\%$ and $S = 80\%$ as an example. In that case, $Y = 8.75\%$ whereas $\tilde{R} \approx 1.64\%$. Therefore yield might not be a very good metric for economics decisions in a PoS network, especially if growth is not expected.

Also interestingly, real rate of return is equal to the growth rate when all tokens are staked (and yield too):

## With Delegation

Let $S$ be the percentage of tokens staked by validators themselves, $D$ be the percentage of tokens delegated by the users, and $C$ be the percentage of the commission validators receive. We have the constraint

Tokens owned by validators and delegators are expressed as

respecively. Then the corresponding real rates of return are defined as

It is easy to see that validators and delegators receive the same rate of return ($R_{s}=R_{d}$) for zero commission $C=0$.

Corresponding yields are calculated as

Furthermore, these simplify to the same equations as in the previous section if we assume no delegation, i.e. $D=C=0$.

Below is a figure showing how the yield to validators change for constant total stake with decreasing number of delegators. The curves correspond to different commission percentages, and eventually converge to $I/(S+D)$ as delegation approaches zero.

Although this section assumes a single commission percentage for all validators and delegators, the same formulas can also be used to calculate the returns on average, given the average percentage of commission.

### Generalization to $N$ pairs with different commission percentages

Let there be $N$ pairs of validators and corresponding delegators, where each pair has its own stake, delegation and commission percentages $\{(S_i, D_i, C_i)\}_{i=1}^{N}$. The parameters are constrained by

Similar to the previous section, the tokens owned by validators and delegators in each pair $i$ is defined as

respectively. The main difference is the denominator which contains the aggregate amount of staked and delegated tokens.

An important observation is that the rates of return in a pair depends on the stake and delegation percentages of other pairs, but not the commission percentages. In other words, your revenue as a validator or delegator doesn’t directly depend on how the others are sharing their revenue.

Let’s assume that you are in the position to either become a validator or a delegator. Given the existing network conditions, how much should you invest and which commission percentage should you choose? You can use the following formulation to make a well informed decision.

Regardless of whether you are a validator or a delegator, let the stake, delegation and commission percentages of a given pair be $S$, $D$ and $C$ respectively. Also, let the total stake and delegation percentages of the remaining pairs be $\bar{S}$ and $\bar{D}$.

Then you can calculate the real rates of return to validators and delegators of the given pair as

respectively.

TBD