Comparison and selection of investment projects: A new standardized risk measure

Standaridized risk measurement terus berusaha diperbaiki. Perhatikan juga topik mengenai semi-variance Markowitz dimana dia mengatakan semi-variance theoretically more robust than variance (atau saudaranya, standard deviasi).


On the basis of drawdown criterion and investors’ target return, a new risk measure called the standardized risk measure is presented in this paper. It is a relative quantity, which defined as the expected loss of value in the required target return divided by the maximum expected loss of value when the probability of failure is guaranteed under the standardized distribution of investment return. The definition is promising and superior to the traditional risk measure in that it readily makes the comparison of different investments be fairly easy and convenient regardless of their currency denominations, given that their return distributions belong to a category with identical standardized distribution.


The risk quantity of a project is a critical index for risk investment. Whether it is measured properly is fairly important. The traditional method of measuring risk is to define it as the fluctuation of investment return at its expectation. In other words, if the investment return R is a random variable, the variance of R is given as [sigma]^sup 2^(R) = E[(R-E(R))^sup 2^], or the standard deviation [sigma](R) is used as the risk measure. This measure has three drawbacks. First, risk generally means that when an investment is taken, the return target required by investors may not be reached. The traditional method unreasonably fixes the mean of return E(R) as the target. Second, the traditional method uses the “fluctuation” to measure risk. Thus the upward fluctuation that is actually what the investors want is calculated into the risk as the downward fluctuation is done. This is not reasonable as well (see [2], [3]). Third, investments are often multi-period. For example, suppose that initial price of a financial security is P (O), holding this asset until the nth period, its price in nth period is P (n). Then the relative return of investment over n periods is

Suppose R^sub i^ = P (i)/P(i-1)(i = 1,2, …, n) is the return of ith period relative to its earlier period, random variables R^sub 1^, R2 …, R^sub n^ are i.i.d. with mean E(R^sub 1^) = [mu]^sub 1^ > 1 and variance [sigma]^sup 2^ (R^sub 1^) = [sigma]^sub 1^^sup 2^ , then ^sup [4]^ we have

E(R^sub 1^) = [mu]^sup n^^sub 1^,

[sigma]^sup 2^(R) = [[mu]^sup 2^^sub 1^+[sigma]^sup 2^^sub 1^]^sup n^ – [mu]^sup 2n^^sub 1^

For the effective return, we have

E(R^sub e^) = [mu]^sup n^^sub 1^ – 1,

[sigma]^sup 2^ (R^sub e^) = [sigma]^sup 2^ (R) = [[mu]^sup 2^^sub 1^+[sigma]^sup 2^^sub 1^]^sup n^ – [mu]^sup 2n^^sub 1^

From Eqs. (4) and (6), we can see that [sigma]^sup 2^(R) or [sigma]^sup 2^(R^sub e^)are increasing functions of n. Therefore, when standard deviation is used for measuring risk, an investment risk will increase as n increase. This is not consistent with the practice. Generally, an investment is not deemed riskier over a long investment horizon since higher volatility only means that there will be more bad moments. But as the increasing of the investment horizon there will also be more good moments to be provided to the investors. In other word, in the case that other conditions are the same, the investors with longer investment horizon can bear larger risk than with shorter horizon. Thus standard deviation does not fit to measuring risk for multi-period investment.

It was in 1952 that Markowitz [8] contributed the seminal paper of mean-variance analysis in which variance is used as the risk measure. Since then many authors had pointed out the shortcomings of variance and proposed the constructive amendments. The use of the semi-variance rather than variance as the risk-measure, was already suggested by Markowitz [9] himself. He stated that semi-variance is theoretically the most robust measure. Markowitz did not adopt semi-variance as a measure of risk because of purported computational problems associated with the semi-variance statistic. As following, there are much more discussions to be provided to semi-variance by Van Horne [17], Mao [7], Zinn, Lesso and Molaged (see [1]), Markowitz ([10], [11]) and Chen [2], Ogryczak and Rusgy…ski [14] demonstrated that the standard semi-deviation (square root of the semi-variance) defined as the risk measure makes the mean-risk model consistent with the second-degree stochastic dominance. And they, with which similar results are obtained as to the case with semi-variance, considered another risk measure, the absolute semi-deviation. Basing on Ogryczak and Rusgy… ski’s work, Gotoh and Konno [6] used lower-semi-skewness to substitute semi-variance and showed that the mean-risk model is efficient in the sense of third degree stochastic dominance.

In addition to above considerations, a group of discussions focus on the target-value. Porter [15] showed that the mean-risk model using a fixed-target semi-variance as the risk measure is consistent with stochastic dominance. Fishburn ([4], [5]) extended this approach to more general risk measures associated with outcomes below some fixed target. Buck and Askin [1] proposed the definitions of partial means which unifies and subsumes many kinds of measures in the economic risk analysis of projects, such as loss integrals Schlaifer [16], Morris [13], Fishburn’s risk measure, etc.

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