By using the relationship between standard deviation and variance, and the definition of correlation , market beta can also be written as , where is the correlation of the two returns, and , are the respective volatilities. This equation shows that the idiosyncratic risk () is related to but often very different to market beta. If the idiosyncratic risk is 0 (i.e., the stock returns do not move), so is the market-beta. The reverse is not the case: A coin toss bet has a zero beta but not zero risk. Attempts have been made to estimate the three ingredient components separately, but this has not led to better estimates of market-betas. Adding an asset to the market portfolio Suppose an investor has all his money in the market and wishes to move a small amount into asset class . The new portfolio is defined by The variance can be computed as For small values of , the terms in can be ignored, Using the definition of this is This suggests that an asset with greater than 1 increases the portfolio variance, while an asset with less than 1 decreases it if added in a small amount. Beta as a linear operator Market-beta can be weighted, averaged, added, etc. That is, if a portfolio consists of 80% asset A and 20% asset B, then the beta of the portfolio is 80% times the beta of asset A and 20% times the beta of asset B. Choice of market portfolio and risk-free rate In practice, the choice of index makes relatively little difference in the market betas of individual assets, because broad value-weighted market indexes tend to move closely together. Academics tend to prefer to work with a value-weighted market portfolio due to its attractive aggregation properties and its close link with the CAPM. Practitioners tend to prefer to work with the S&P500 due to its easy in-time availability and availability to hedge with stock index futures. A reasonable argument can be made that the U.S. stock market is too narrow, omitting all sorts of other domestic and international asset classes. Thus another occasional choice would be the use of international indexes, such as the MSCI EAFE. However, even these indexes have returns that are surprisingly similar to the stock market. A benchmark can even be chosen to be similar to the assets chosen by the investor. For example, for a person