Since the groundbreaking efforts of Treynor , Sharpe , and Jensen , portfolios have been evaluated on the basis of how favorable they compare with an expected (or required) rate of return. In Sharpe's scheme, that return is an increasing function of the portfolio's total risk.
[k.sub.s] = [R.sub.f] + [SD.sub.p] ([R.sub.m] - [R.sub.f])/[SD.sub.m] (1)
where: [k.sub.s] = Sharpe's expected portfolio return
[R.sub.f] = riskfree return
[SD.sub.p] = portfolio's standard deviation
[R.sub.m] = expected market return
[SD.sub.m] = market's standard deviation
By contrast, Treynor  and Jensen  limit the portfolio's expected return to systematic risk.
[K.sub.TJ] = [R.sub.f] + [COV.sub.i,m] ([R.sub.m] - [R.sub.f])/[VAR.sub.m] (2)
where: [K.sub.TJ} = Treynor's and Jensen's expected portfolio return
[COV.sub.i,m] = covariance of returns from stock "i" and market
[VAR.sub.m] = variance of returns from market
For portfolios that are completely diversified, the two methods lead to the same expected returns.
Further divergence of opinion emerged a decade later when Hogan and Warren  and Bawa and Lindenberg  agreed with Treynor and Jensen that only market risk is important but defined that risk as
where: [CLPM.sub.i,m] = co-lower partial moment of returns from stock "i" and market
[LPM.sub.m] = lower partial moment of returns from market
The (intuitive) argument that risk is best measured by the lower partial moment (or semivariance) was empirically tested by Nantell and Price . They found that systematic risk in the equation (3) sense led to required returns not unlike those based on the more conventional definition of market risk: [COV.sub.i,m]/[VAR.sub.m].
Price, Nantell, and Price  also compared the two approaches to systematic risk within the framework of the capital asset pricing model (CAPM). They reported that how market risk is defined is an unimportant detail only when the investment possesses average risk. For assets with either above or below average risk, the CAPM-VAR model generated a return greater than that produced by the newer CAPM-LPM. In the absence of more recent research, previous studies of expected portfolio returns can be partitioned as follows:
[TABULAR DATA OMITTED]
It is tempting to develop a downside-deviations-by-total-risk measure if only to "complete" Figure I. However, there are at least four more meaningful incentives. One, the intuitive appeal of the semivariance requires continuing investigation. The one common thread in the post-1960s' studies cited in this paper is the "gut" feeling on the part of the authors that risk has less to do with total deviations than with the negative ones. Two, there is growing evidence that the semivariance more closely conforms to the rule that risk and return are highly positively correlated than does the more common standard deviation. Most recently, Kochman  found that the 1-2-3-4 order of four popular proxies for the market portfolio based on the average annual return for the 1965-84 period was duplicated by the semivariance -- i.e., the index with the highest return had the highest semivariance, the index with the second-highest return had the second-highest semivariance, and so on -- but not by the standard deviation where the index with the lowest average return was rated the second-riskiest.
A third reason to fill cell 2,2 in Figure I is the reality that the co-lower partial moment of returns (CLPM) in equation (3) is neither obvious nor easily calculated. There is even some disagreement among its proponents over whether the measure should be restricted to returns below the asset's risky mean or below the riskfree rate. Finally, the kind of diversification that makes only systematic risk relevant may be more theoretical than actual. …