The design and analysis of investment portfolios were for ever changed with the development of the Capital Asset Pricing Model (CAPM) and Modern Portfolio Theory (MPT). Since the early 1970’s, academics, consultants and investors have used statistics to quantify expected or realized reward and the variability or dispersion of these rewards. Investment performance attribution models built using the statistical technique of linear regression are widely used to measure the potential sources of investment performance for portfolio managers. Services such as Morningstar, Lipper Reuters and eVestments report statistics beautifully packaged to give an investor the impression that there is one size that fits the needs of all investors.
We all know that different investors have different aspirations for investment rewards and tolerance for potential loses. The efficient frontier from MPT is an attempt to provide different portfolios that define for all investors the possible tradeoffs of expected reward and volatility of those rewards. Is there any way that an investor can impose his or her target return goals and say that the real tradeoff is by seeking rewards above this target and controlling or minimizing the variability of returns below the target?
Economists and statisticians in the 1980’s look for a way of expressing these views. A principal result has been the concept of partial moments as asymmetric counterparts to the mean and variance of returns. Partial moments are sometimes referred to one-sided moments. All partial moments have a target or threshold return r and an integer degree m. Bernd Scherer in his book Portfolio Construction & Risk Budgeting offers a thoughtful discussion of partial moments. We will distill the concepts he presents into illustrations to bring to light the effect of the recent credit crisis by measuring these moments in the time period 2005 to 2010.
Target returns depend on the investor’s objective. A zero return target is suitable for investors seeking capital preservation. The inflation rate is appropriate for endowments and foundations seeking to protect purchasing power. Return targets that are moving benchmarks are appropriate for investors with relative performance objectives.
The lower partial moment (LPM) of degree m measures the average value for the difference between the target return and investment return raised to the power m and for the returns below the target return. If the degree is 0, it is called the short fall or downside probability of underperforming the target return. If the degree is 1, the LPM is called the target shortfall and it is the mean loss below the target return. If the degree is 2, the LPM is called the target lower variance and it measures the variability or dispersion of returns below the target return. The square root of this target lower variance is a lower standard deviation.
The upper partial moment (UPM) of degree m measures the average value for the difference between the investment return and the target return raised to the power m and for the returns above the target return. The degree 0 UPM is called the upside probability of outperforming the target return. If the degree is 1, the UPM is called the target upside and it is average outperformance of the investment over the target return. If the degree is 2, the UPM is called the target upper variance and it measures the variability of returns above the target return. The square root of this target upper variance is an upper standard deviation.
We illustrate the calculate of partial moments using the quarterly returns for long term U.S. government bonds and U.S. large stocks derived from the monthly returns in the Ibbotson and Associates database for the time period 1Q 2005 through 4 Q 2010. Tables 1 and 2 summarize the returns. The standard symmetric first moments are the mean returns which for the government bonds and large stocks are 1.64% and 1.07%, respectively. The corresponding variances of returns for these asset classes are 0.004298 and 0.008062. The associated standard deviations are 6.56% and 8.98%. This time period is characterized as one in which government bonds outperformed stocks while the returns of stocks was noticeably more volatile than those of bonds. The two assets were also strongly negatively correlated with a correlation coefficient of -0.5356.
Tables 3 and 4 illustrate the calculation of LPM and UPM for degrees 0, 1, and 2 for government bonds and large stocks, respectively. The target return is 1%. Each table is organized into two sets of four columns, one set for LPM and the other set for UPM. For the LPM set, the column labeled Indicator value is 1 if the corresponding quarterly return is less than the target return and zero otherwise. For the UPM set, the Indicator value is 1 if the corresponding quarterly return is greater than the target return and zero otherwise. The Moment 0 column contains the product of the Indicator column and the Target Return minus the Asset Class return raised to the power 0. The Moment 1 column contains the product of the Indicator column and the Target Return minus the Asset Class return. The Moment 2 column contains the product of the Indicator column and the Target Return minus the Asset Class return raised to the power 2. The row labeled Moments has the average values in each of the Moment columns. The square root of the Moment 2 column average is the lower and upper standard deviation.
Figure 1 shows how upper mean and lower standard deviation change when you vary the target return from 0% to 5%. Note that at all target return levels, large stocks had higher levels of lower standard deviations than government bonds with comparable upper mean returns.
Traditional MPT looks to compare the mean returns and volatilities of portfolios. Figure 2 shows a typical investment opportunity set obtained by varying the exposure of stocks from 0% to 100% and bonds from 100% to 0% respectively. The minimum risk portfolio that assumedly would be held by all investors would be that with a 39.92% exposure to stocks. Figure 3 shows a dramatically different view of the opportunity sets when the tradeoff is between upper mean and lower standard deviation for the different target returns. The minimum lower standard deviation in this simple case is fairly close for all target returns although at higher target returns the exposure to stocks is greater.
The story that emerges when you look at investment opportunities from a partial moments point of view is that there is no one size fit all solution to the investment problem. Investors need to give careful consideration to selecting target returns in light of their investment objectives.
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