**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:

[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]

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

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]