This note updates our analysis we previously produced, which looked at the potential impact of pledge on staking returns. Since the exact values of important parameters were not set at the time, we had to look at the range of possible values for these parameters, while fixing others to analyze the outcomes for both stakeholders and pools. On June 24, 2020 IOHK revealed several system parameters for the upcoming Cardano Shelley release.Chief among them were:

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1. rho=0.22% the rate at which the reserves are reduced; it affects the total staking returns
2. k=150…250 the number of desired pools; it determines the pool saturation point
3. a=0.3 introduces differences in rewards for pools as a function of the pledge amount
4. min. operating costs=ADA340 sets a minimum cost per epoch for each pool to avoid the race to zero
5. tau=5% the treasury tax rate. It reduces total rewards, redirecting them to a treasury account for later use

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The release of these parameters effectively fixes the range of possible outcomes for returns on staking (ROS), such that we can now predict with greater certainty the minimum and maximum boundaries for ROS. Knowing these boundaries, along with how they will evolve over time is an important step in planning pool operations and will help stakeholders make intelligent staking decisions. Here are our key points:

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1. At rho=0.22 Shelley will generate returns in the range of 4.56% to 5.93% after the 5% treasury tax deductions and before pool fees. The exact amount of rewards for each pool will depend on their pledge levels. These numbers exclude transaction fees which are likely to grow significantly after the launch of Goguen. It is hard to predict their volume, so we excluded those in our calculations. Effectively the return numbers above constitute a floor range for the entire system as they assume transaction fees are zero.
2. The k factor will start at 150 and will increase to 250, eventually reaching 1000. The timing of this increase is unknown, but the adding more pools to the system will decrease the saturation point, making it more likely for smaller pools to earn maximum returns, all else equal.
3. The pledge influence factor of 0.3 makes the role of pledge in ROS less significant at lower levels of k, but as the system adds more pools the absolute amount of pledge will start making a difference. Small differences in ROS can translate into significant differences in pool fees.
4. One thing that we forgot to mention is the block rewards. In year one these are expected to be around ADA1.2K per block. Since rewards for pools are counted per block produced, the target pool size should be enough to produce at least 1 block per epoch.That number is 1/21600 * total stake (31.5B) = 1.4M. At ADA544K the pool will produce 1 block every 13 days which means the decelerators will receive their rewards minus pool fees, but the pool will run a deficit for epochs it did not produce a block.Thanks to marcelklammer on reddit for pointing this out.

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In sections below we will examine the impact of each of these parameters on pool operators and stakeholders.

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How Do We Calculate Return on Staking?

Cardano’s reward calculations are based on the reward formula below. It first appeared in the IOHK paper Reward Sharing Schemes for Stake Pools and is now part of the design specification for the Cardano mainnet. We use this formula to determine the level of rewards for each pool before treasury taxes. Think of it as a systemic level returns influenced by a range of systemic parameters. It is important to note that this formula is not used forDaedalus pool ranking.

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• R=total stake
• k=desired number of pools [a system constant]
• a=pledge influence factor [a system constant]
• ơ'=pool’s share of the total stake, capped at 1/k
• ƛ’=share of pledge in total stake, capped at 1/k
• b=maximum pool size = 1/k

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Total rewards are a function of total stake, the desired number of pools the pledge influence factor and pledge levels among other things. The interaction of all these elements determines the resulting reward level for each pool.

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Note:

1. Our staking return calculations do not take into consideration the pool fees which can vary from pool to pool. Instead they give you the range of returns, which are influenced by system parameters. The stakeholders’ return before pool fees is affected by the size of the pledge which is why we do not have a single number for returns.
2. Also, we exclude transaction fees from our analysis (except for one instance) as we think they are hard to estimate at this point, but we recognize that these fees can make up a significant portion of the total reward pot for stakeholders down the road.

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What Does rho=0.22% Mean?

Systems with fixed token supply like Cardano rely on disbursements from reserves to economically sustain their networks. Contrary to inflation rates, which represent the rate by which the “available uncapped supply” grows, the reward distributions on Cardano are bounded by the size of the reserves and follow a deflationary token emission policy.

Rho is the rate of emission per epoch for undistributed reserves. Cardano has around 13.5B ADA of undistributed reserves, which will decrease every epoch. Rho is the rate by which these reserves will decrease every epoch. With each passing epoch the size of rewards decreases with the reduction in the reserves. With rho=0.22%, reserves will be gradually cut in half by the end of year 4. Mathematically the outstanding reserves for any year can be found using the formula below:

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𝑂𝑢𝑡𝑠𝑡𝑎𝑛𝑑𝑖𝑛𝑔 𝑅𝑒𝑠𝑒𝑟𝑣𝑒𝑠 𝑖𝑛 𝑦𝑒𝑎𝑟 𝑛 = 𝐼𝑛𝑖𝑡𝑖𝑎𝑙 𝑅𝑒𝑠𝑒𝑟𝑣𝑒𝑠 * (1 − 0. 0022)𝑛*73

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Why rho is important: Knowing rho, we can see how much ADA exactly will be issued each year, which ultimately allows us to determine the return on staking. In the graph below, we see that Cardano will issue roughly 2B ADA inits first year of operation.

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Due to the nature of reserve emissions, the first year will produce the highest number of ADA distributed from reserves. The number will gradually decrease with time. Since rewards are just the flip side of remaining reserves, we can easily see how much ADA will be issued each year.

The chart below illustrates the impact of rho on staking returns. The blue and orange columns represent total returns on staking based solely on reserve emissions which is conservative, as they exclude transaction fees. The grey line adds some guesstimate for future transaction fees. This line illustrates one of the many potential ways the transaction fees could kick in and support the network returns, especially with the introduction of Goguen and the later implementation of Hydra.

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For 2020-2021, the Cardano network will generate a maximum ROS of around 5.93% per year after treasury taxes for a saturated, fully pledged pool and a minimum ROS of around 4.56% for a small pool with no pledge. Sections below corroborate these findings within the context of other system parameters.

Note: These ROS numbers in this note represent the total network staking returns and can be thought of as staking returns a pool can achieve, excluding the transaction fees. Individual stakeholder ROS will be lower as stakeholder returns are affected by (i) exact pledge amounts (ii) pool saturation levels (iii) pool performance and (iv) pool fees.

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How Does the k Parameter Impact Returns?

IOHK has set the initial k parameter at 150 and released a schedule where this parameter will gradually increase to 250. We expect the k parameter to increase to 1000 with time. It is important to realize that the k parameter does not limit the creation of pools. Rather it determines the eventual number of pools once the system reaches economic and game-theoretic equilibrium. The big unknown is how fast the k parameter will be increased with time and what would influence that decision.

The k-parameter does not affect the level of returns. Rather, it defines the saturation level, which does affect the total staking returns. The highest returns are earned by pools at their saturation levels. At k=150 the saturation level will be around ADA209M. As the k=parameter increases the amount of ADA required to reach a saturation point decreases, in other words the maximum return becomes more attainable. Going from 150 to 1000 pools can gradually push the total stake from large, saturated pools into smaller pools. Increasing the k parameter is also good for network security.

The table below shows the relationship of staking returns to the desired number of pools in the network. As the network decentralizes the maximum return “migrates” toward smaller and smaller sized pools as these represent the new saturation points in the system at the new k level.

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Pledge Factor=0.3: What Should You Know?

In our previous note we discussed the importance of the pledge factor on staking returns in greater detail. In that note we stated that IOHK is likely to set the pledge factor somewhere between 0.15 and 0.75.

IOHK ran millions of simulations to arrive at a pledge factor level of 0.3. Knowwing this allows us to analyze its impact on returns without too much ambiguity. The table below illustrates how the return on staking changes based on the level of pledge. Given the relatively small magnitude of changes in ROS between a fully pledge pool5.93% and a small pool with no pledge 4.56%, running a small stake pool becomes a real possibility for those who do not have a large pledge.

At k=150, it also means that pools will require a large amount of pledge (ADA30M+) to capture any significant difference in returns. For example there is almost no difference in staking returns between running a pool with zero pledge and a pool with say ADA665K pledge as shown below.

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Regardless of this fact, pledging may still be an economically attractive move for pool operators because of the expected network decentralization. As the k increases, it will reduce the saturation point for pools. At higher k levels, the amounts of pledge considered too small to create any difference in returns, will be producing noticeable differences in ROS. At k=1000 for example, a pool with ADA665K pledge has the potential to produce4.59% in returns compared to a pool with no pledge, which would produce 4.56%.

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Don’t let these optics fool you. These 0.03% in ROS translate to 0.65% in pool fees, which means that a pool withADA665K pledge can charge 0.65% more in pool fees and still be competitive with the zero pledge pool, all else equal.

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Min. Operating Costs=ADA340 Per Epoch

This parameter is new and is most probably was introduced after seeing the ITN pools declare zero fixed operating costs and make all their returns from variable fees.Fixed operating costs are an essential part of the ranking mechanism in Cardano. Their presence raises the minimum income for pool operators and improves the network security by ensuring that the network is not overtaken by cheap nodes running on unreliable infrastructure. These costs have a couple of implications.

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1. they raise the minimum compensation for pool operators, which attracts capable pool operators into the ecosystem
2. they introduce an effective entry point for delegators as pools now must show that they have enough pledge/delegation to cover their fixed operating cost before stakeholders consider delegating to them. The table below shows total pool rewards based on pool size.

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Conclusion

As we approach the launch of Shelley, understanding how the incentives work becomes important for both the pool operators and the stakeholders. With some of the most important system parameters set, it is now easier to chart the course for the overall reserve distributions and assess the factors that influence returns.

We now have an unambiguous way of calculating reserve distributions, which helps us understand the level of expected returns for the foreseeable future. At rho=0.22% the reserves will be cut in half in around 4 years.Cardano will produce a minimum range of returns before pool fees of 4.56% to 5.93% for the first year.

At k=150 the pools with significant pledge amounts will earn higher returns. Small amounts of pledge will not produce significant difference in ROS but will generate significant enough difference in returns as we move the k factor toward 1000 pools.

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EDIT: One thing that we forgot to mention is the block rewards. In year one these are expected to be aroundADA1.2K per block. Since rewards for pools are counted per block produced, the target pool size should beenough to produce at least 1 block per epoch. That number is 1/21600 * total stake (31.5B) = 1.4M. At ADA544K the pool will produce 1 block every 13 days which means the decelerators will receive their rewards minus poolfees, but the pool will run a deficit for epochs it did not produce a block. Thanks to marcelklammer on redditfor pointing this out.