Finance

The scatter diagram shows the relationship between standard deviation and portfolio return, the shape of which is like a “bullet”. Also, the portfolio frontier is symmetric, which means that with one standard deviation, there will be at least one portfolio return.

Illustration:

Suppose there is a portfolio composed by asset A and B, and a proportion denoted by is invested in asset A, and the remainder 1- denoted by is invested in asset B. The expected rate of return on the portfolio is a weighted average of the expected returns on the component asset, with the same portfolio proportions as wights.

Therefore,

The variance of the rate of return on the two-asset portfolio is:

where is the correlation coefficient between the returns on asset A and asset B, which determines the shape of the portfolio frontier. If the correlation is small or negative, this will reduce portfolio risk.

Assume =1, which means that asset A and B are perfectly positively correlated.

then

Therefore, Whatever the proportions of asset A and asset B, both the portfolio mean and the standard deviation are simple weighted averages. Figure X.1 shows the opportunity set with perfect positive correlation - a straight line through the component assets.

With any correlation coefficient less than 1.0(), there will be a diversification effect, the portfolio standard deviation is less than the weighted average of the standard deviations of the component securities. Therefore, there are benefits to diversification whenever asset returns are less than perfectly correlated.

Having ranged from very attractive diversification benefits () to no benefits at all . For within this range, the benefits will be somewhere in between.

0

Standard Deviation

Expected Return

asset B

asset A

0

Standard Deviation

Expected Return

asset B

asset A

Figure X.1 Investment opportunity sets for asset A and asset B with various correlation coefficients

Negative correlation between a pair of...