# Calculating Expected Return from Historical Data Alone

Say that my returns are normally distributed. I have the historical returns data for 3 index funds and nothing else.

I want to compute expected return and standard deviation solely from this data. What are some methods for doing this besides just taking the average return for example? Are there some non-obvious strategies that might enhance the results, or does the normality assumption remove any need for this?

I'll break the question down into two questions:

1. Can expected returns be calculated from historical returns: If you make the assumption that expected returns are constant over time (a prevalent view in the 1980s but has since seen challenged), then you could take a sample with relatively neutral valuation (e.g., bond yield at roughly same level at the beginning and end of period, or P/E at roughly same levels), calculate the realized return during this period, and use it as the long-term expected return. If you don't believe expected return is constant over time, then it's not possible to get expected returns out of returns series alone.

2. Can expected returns be calculated from historical data: If you can use historical data beyond returns, then your opportunity expands significantly. For example, 10-year bond yield provides an excellent forecast for expected bond returns over the next 10-years (even if you're rolling the bond monthly instead of holding it to maturity). Likewise, dividend yield, earnings yield, etc. can be used to proxy for equity expected returns. More sophisticated econometric models of course can be used as well.

Ilmanen's book Expected Returns is the best literature I've read and discusses this topic from both an academic and practitioner's perspective.

One could do a simple linear regression which would be easy, but would have a large margin for confidence intervals. Just multiply the beta co-efficient by the periods you want to calculate in advance. This will also have some possible lag.

You could also take the Correlation between a given trading period's return, and the prior period's return. Just multiply the previous periods's return by the beta co-efficient and you can forecast future returns that way with less lag for a linear relationship. I advice doing this on a longer time frame, forecasting possibly a week, month or even quarter depending on what works best for you.

Standard Deviation could be forecasting with a GARCH Model: https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity