Active Management Evaluation with Partial Moments

By Fred Novomestky, Ph.D.

In my previous blog, I showed how different the world of investment return and risk when viewed through the prism of partial moments. Your personal target determines the performance of capital markets and the efficient frontier from which you select desirable portfolios.

Services that offer performance evaluation services on traditional investment funds such as mutual funds have traditionally focused on distilling reported performance history into statistical measures such as average return, the volatility or standard deviation of historical returns, systematic risk or market beta, and, in the case of U.S. equity funds, style analysis derived from exposures to multiple common factors such as market, size, book to market and momentum. It is difficult to see the extent that a manager’s performance is dominated by upside or downside reward and risk.

We begin with two simple examples of evaluating a manager’s performance relative to a zero target level. The manager selected is a Multi-Cap Core Fund manager with reported returns starting in January 1972. Specifically, we examine the realized returns for the 24 month period ending March 2011. Table 1 shows the standard and partial moments for the nominal returns. The average monthly return was 2.5621% with a realized volatility of 4.6678%. The columns labeled LPM 0, LPM 1 and LPM 2 correspond to downside probability, downside loss and downside volatility. The columns labeled UPM 0, UPM 1 and UPM 2 are associated with upside probability, upside reward and upside volatility On the basis of nominal returns, the investor would be pleased to see that the upside probability of 0.7917 overwhelms the downside probability of 0.2083 and that the upside reward of exceeds the downside loss by a factor of three. There is also more upside volatility than downside volatility.

A different picture emerges when you look at the active returns of the manager relative to its benchmark. I have taken the Russell 3000 to be the benchmark for the manager and Table 2. The active fund on the basis of average return performed like the benchmark. Looking at upside and downside volatility, the active reward or loss is about the same that you would get by flipping a fair coin with similar amounts and rewards. I suspect that, given this information, the prospective investor might be a bit concerned about consistency of active returns from this manager.


Table 1 Nominal Return Moments Click to enlarge

Table 2 Active Return Moments Click to enlarge

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Partial Moments – the Up and Down of Performance and Risk

By Fred Novomestky, Ph.D.

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.

Click on the figures below for larger viewing

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What to look for in a mutual fund

By Andy Lawson, Ph.D.

The short answer: Positive risk-adjusted returns, stable risk exposures, long management tenure, diversification and moderate turnover.

• Positive risk-adjusted returns, or alphas. A fund with positive alphas adds value by delivering returns in excess of those expected from the fund given the fund's risk exposures. In contrast, a fund with zero alphas behaves like an index fund by earning returns exactly commensurate with the amount of risk it is taking on. A fund with negative alphas underperforms by earning returns which don't compensate investors for the risk they are exposing themselves to by investing in the fund and therefore a negative-alpha fund should be generally be avoided.

• Stable risk exposures (consistent investment style). A fund's risk is generated by its exposures to risk factors in stock and bond markets. Drift or shift in a fund’s risk exposures over time may indicate an inconsistent investment strategy. Also, a fund with drifting or shifting risk exposures tends to contribute more risk to an investor's overall portfolio than a fund with consistent exposures.

• Management tenure. AlphaFunds evaluates a fund using at least 36 months of historical data. We can expect the evaluation to be more informative if the fund's management team has not changed over the evaluation period (after all, how much faith would we have in an assessment of a fund based on the fund's track record over the prior 5 years if the management team changed last quarter?). This suggests requiring a fund's management tenure to be at least 36 months.

• Diversification. A diversified fund has exposure only to market risk while an undiversified fund has exposure to both market risk and non-market risk. Since exposure to non-market risk is not compensated, a fund with exposure to only market risk (a fund which is diversified) tends to be less risky than a fund with exposure to both market and non-market risk. One way to increase the chance of diversification is to require funds to have at least 50 positions.

Note that the more funds your portfolio contains, the less important it is for each fund to be diversified. This is because the other funds in your portfolio will diversify away the non-market risk in any undiversified fund. This is the case even if all funds in the portfolio are undiversified. So if you are building a portfolio of just one or two funds, your portfolio will not be diversified unless each of the funds is diversified. However, if you are building a portfolio with eight funds, your portfolio will probably be diversified even if each of the funds in it is undiversified.

• Net assets. Funds with a larger amount of net assets have a higher chance of being well-established than funds with a smaller amount of net assets. It therefore may be prudent to require funds to have at least $100 million in net assets.

• Turnover. For portfolios which are not tax-sheltered, funds which frequently realize returns by selling stocks or bonds tend to be less tax-efficient than funds which sell less frequently. One way to identify funds which are more tax-efficient is to require funds to have a turnover rate which is below the stock or bond fund median.

Asset Allocation, Diversification Now Passé?

By Fred Novomestky, Ph.D.

A surprising notion is currently gaining traction on Wall Street: asset allocation and diversification are now passé. Just pick the asset classes you like and ride the market.

Really? Let’s look at this more closely.

Brinson Partners has generated significant investment value for its clients through a quantitative, disciplined approach to global asset allocation. Early in his career, Gary Brinson, one of the founders of Brinson Partners, published an important paper(1) based on empirical research that he performed with Brian Singer and Gilbert Beebower on the impact of asset allocation and the effect of active management on realized returns.

They found that more than 90 percent of the portfolio returns comes from the asset allocation, with the balance due to active management. Others have repeated his analysis, based on more recent data, and have reached the same conclusion.

The fall 2010 issue of the Journal of Wealth Management contains an article by two professors of finance from Indiana University Northwest, Bala G. Arshanapalli and William B. Nelson, with the intriguing title “Yes Virginia, Diversification Is Still a Free Lunch”(2). To see if diversification really matters, they used changing correlations to show that diversification is still the tool that enhances risk-adjusted performance. They also note, however, that economic turbulence makes it necessary to include more asset classes.

We take the view presented by Andrew Worthington in a study that he published in 2009 (3) on the extent to which Australian households have investments and the diversification of those investments. He presented a number of useful measures of diversification. He called “perfect diversification” an equally weighted portfolio of investments.

At the other end of the spectrum, he used the term “perfect concentration” for portfolios with almost all the weight given to very few assets. I like the measure he called the “Shannon entropy index” which I have modified to be on the scale of 0 percent to 100 percent. Zero percent corresponds to perfect concentration where as 100 percent is perfect diversification.

I collected the asset allocation policies for six different institutional investors which are summarized below:

  

The index proxies are used to simulate portfolios that are rebalanced on a quarterly basis and have corresponding returns that are the index returns. The performance of these portfolios is evaluated over three time periods, January 2000 to December 2004, January 2005 to December 2009 and the entire 10-year period. For each of the investors and time periods, I also calculate the Shannon diversification index.

The following charts summarize the results. Bottom line: the long-term record shows that good diversification leads to superior performance results. Innovative investors such as Stanford University, University of Chicago, Harvard University and Yale University benefit significantly from the inclusion of more asset classes with greater exposure across all the asset classes.

So much for fashionable notions on Wall Street.

(1)  Brinson, Gary P.,  Brian D. Singer & Gilbert L. Beebower. (May/June 1991). Determinants of Portfolio Performance II: An Update,  Financilal Analysis Journal, Vol. 47, No. 3, pp 40-48.

(2) Arshanapalli, Bala G. & William B. Nelson. (Fall 2010). Yes Virginia, Diversification Is Still A Free Lunch, Journal of Wealth Management, Vol. 13, No. 2, pp 34-40.

(3) Worthington, Andrew C. (June 17, 2009). Household Asset Portfolio Diversification: Evidence from the Household, Income and Labour Dynamics in Australia (Hilda) Survey, Griffith University - Department of Accounting, Finance and Economics.

Leveraged Bonds – Déjà Vu All Over Again

By Frederick Novomestky, Ph.D

“Those who do not remember the past are condemned to repeat it.”

-- George Santayana

It has been only a few years since August 2007, the beginning of the global recession and the credit market turbulence. The highly publicized problems facing institutional investors, high-net-worth individuals, investment advisers and the financial institutions that provide services to these market participants suggest a more conservative and risk focused approach to wealth management is necessary. Unfortunately, history appears to be repeating itself.

The bond geeks and derivatives mechanics are at it again, attempting to sell a concept that flies in the face of what investors have learned and practiced over 50 years. The concept is called leveraged bonds and was recently described in very cautionary terms by Rodney N. Sullivan in Pensions and Investments and in the Financial Times as an alternative to equity investing.

These are concentrated bets that ignore the benefits of diversification across multiple investment opportunities. Modern finance has demonstrated that the only free lunch in town as far as risk reduction goes is diversification and the Nobel Prize of 1990 to Markowitz and Sharpe is a testimony to that.

The kraken or ‘crack’ of leverage is being unleashed on or dangled before investors to increase bond returns while ignoring potential losses. It is one thing to engage in pairs trading like statistical arbitrage traders or long-short hedge funds. It’s another thing to expect to get equity-like returns without being exposed to comparable risks. Stressful market events such as the technology bubble burst, the 9/11 terrorist attacks, and the most recent credit crisis have dramatically shown the high risk of concentrated investments in any security or asset class.

An example of the effect of a leveraged bonds and asset allocation is for a U.S. investor in U.S. large cap stocks and U.S. long term government bonds. Our investor uses the historical realized quarterly returns of these asset classes derived from the Ibbotson Associates database. The following table shows the Sharpe ratio, a risk adjusted return measure, for three different strategies:

Bond Leverage                        None           1.2              1.4              1.6              1.8 

All bonds                               0.1886         0.1777         0.1668         0.1561         0.1455

40% stocks 60% bonds          0.1757         0.1930         0.2011         0.2033         0.2021

60% stocks 40% bonds          0.0377         0.0651         0.0901         0.1120         0.1305

Suppose that our investor begins with $100,000 and purchases a 1.2 leveraged bond. This means that the investor in effect has borrowed $20,000 to get $120,000 of exposure. The quarterly return to this bond is 1.2 times the unleveraged bond investment. Since this return is derived from an underlying bond investment, the leveraged bond is a derivative security. It is also a credit derivative because the underlying security is a credit market investment. You could also do the same thing with mortgage-backed securities and we know what happened to that market.

If we didn’t use leverage, then the story for the five year period 2005 to 2009 is to go with the government bonds because it has the highest Sharpe ratio. But see what happens when you add leverage to bonds alone ---- the more leverage you add the lower the Sharpe ratio is.

The best alternative for the investor who still wants to use leveraged bonds is a portfolio that is rebalanced quarterly to 40% stocks and 60% bonds. Yet at higher leverage, the portfolio begins to suffer. Interestingly, the 60% stock and 40% bond portfolio begins to look good once you use higher leveraged bonds. It is the effect of diversification that is providing these more tasty possibilities.

If you don’t think that you would experience these results, then heed the words of Santayana. Look out for Nassim Taleb’s black swans of extreme events. Or as Mr. Micawber of Charles Dickens’ "David Copperfield" said, “Annual income, 20 pounds. Annual expenditure, 20 pounds and six. Result, misery. Blossom is blighted. Relief is withered. You are, in short, flattened.”

As Rodney Sullivan suggests, don’t under-estimate the risk management importance of diversification.

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