### Casper FFG: CAPM & Validation Yield

**tl;dr** While the CAPM model and Sharpe ratio have major limitations, there are concrete takeaways for designing Casper incentivization. Primarily, **the more we can limit the standard deviation of validation yield, the lower the required returns of the validators**. That allows for either lower issuance/dilution—or for any given level of issuance, a higher risk-adjusted return—which will make participation in the network more compelling for the same level of issuance.

#### Introduction to CAPM

E(R_i) = R_f + \beta_i (R_m - R_f)

E(R_i) - R_f = \beta_i (R_m - R_f)

In other words, the risk premium of a given asset (such as ETH validation stake) should be the (a) relative volatility of the asset vs the market times (b) the market premium of the asset.

- The risk premium (E(R_i) - R_f) is defined as the expected return of the asset in excess of the risk-free rate (e.g. 3 month US Treasury bill)
- The beta of the asset (B_i) is the standard deviation of the asset returns divided by the standard deviation of market returns (\sigma_i/\sigma_m). This measures the relative volatility of the asset returns to the market.
- The excess market returns R_m - R_f is the returns of a given “market” in excess of the risk-free return.

The two large factors for this analysis is:

- What is the correct & reasonable selection of the relevant “market returns” that this asset class is under?
- How the reward/penalty parameters will affect \sigma_i and therefore the \beta_i of the asset.
- The more we can limit the standard deviation of the asset returns (make it more predictable, the less we have to reward the validators. i.e. less issuance / dilution of ether value. or higher excess return for same level of issuance).

To take it one step further: there are three real drivers of the assets required returns E(R_i). The required returns of the asset will be greater when:

- The market returns are higher.
- The standard deviation of the asset returns are higher.
- The standard deviation of the market returns are lower.

#### Conclusion

The main takeaway here is that, **there is a direct cost to ETH holders for having high standard deviation validator returns**. So for a given level of “economic security,” we should strive to minimize the standard deviation of validator returns. That will allow for (1) additional “resources” to increase penalties / cost of attack by increasing TD, (2) decreasing issuance and enhancing value of ETH, or (3) provide additional excess risk-adjusted returns to validators (attracting a broad set of validators).

*Also related: Sharpe ratio and its cousin Sortino ratio*