**Expected Returns**

The application uses a forward model to estimate future returns. Rather than using raw historical returns to predict future returns, the model uses risk/return relationships to calculate the estimate long term expected annual returns, as described below.

The default assumptions are provided as a starting point. You should review the assumptions and change them if needed to reflect your views.

For each security, the expected return is calculated in three steps:

- Calculate the historical Beta of the security (based on the past three years)
- Calculate the forward Beta, using the following formula:
forward Beta = historical Beta * 0.666 + 0.333(The forward Beta is the historical Beta, adjusted for the fact that beta is not persistent over time and tends to reverse to the mean [1].)
- Calculate the Expected return for the security as:
return = risk-free rate + forward Beta * equity risk premium

where:

- risk free rate = 2% (long term historical average.)
- equity risk premium = 5% (long term historical average)

The equity risk premium is the amount of return that equity investors are expecting as compensation for the risk they are taking. In this model, note that adding risk-free rate and the equity risk premium, gives the expected market return:

expected market return = risk-free rate + equity risk premium

**Standard deviation and correlations**

The standard deviation and correlations used in the optimization are shrinkage estimates based on 36 trailing months [2]. These numbers are not editable. Note that because of shrinkage, these estimates may vary slightly from the numbers presented in the portfolio Performance tab.

[1] A detailed explanation of beta adjustment may be found in CFA publication "Quantitative methods for investment analysis", DeFusco, page 628. [https://www.amazon.com/Quantitative-Methods-Investment-Analysis-Richard/dp/1932495088]

[2] "Improved estimation of the covariance matrix of stock returns with an application to portfolio selection", Ledoit, Wolf [https://web.archive.org/web/20141205062053/http://www.econ.uzh.ch/faculty/wolf/publications/jef.pdf]