The Fundamental Law of Active Management in Finance 3.0

Continuous Concepts

The process of estimating the value of continuous probability functions is taught to students in calculus 101 using integrals and the Riemann hypothesis, which for the latter involves summing the area of drawn rectangles under a probability curve. The task of estimating a continuous probability function is related to measuring alpha in finance, as the foundation for finding alpha is the capital asset pricing model (CAPM). The CAPM model analyzes expected return of assets in equilibrium based on a mean-variance framework. Gaussian concepts in probabilistic statistics such as normal distributions, Z-scores and correlations are foundational for mean-variance techniques. Within the area of financial analysis, CAPM is a key method for explaining investment strategy, asset class and security outperformance, as CAPM sets a baseline for measuring financial statistics such as systematic risk, beta and alpha.

Figure 1

Figure 1 shows left and righthand Rieman sums used to the compute area under a none-linear graph. The final answer is always going to be an approximation, as it is impossible to exactly measure a curve with a straight line. Conceptually, the same goes for measuring a managers alpha, as knowing if a manager generated enhanced returns due to skill or luck is impossible.

Idiosyncratic and Systematic Risk

All securities in the market are impacted by both idiosyncratic and systematic risk. The former risk is geographic, asset class, industry and security specific while the latter involve general risks impacting all asset classes. If a manager optimizes a portfolio according to the CAPM framework, then that portfolio diversifies away all idiosyncratic risks. However, the CAPM portfolio fails to secure itself against systemic risks. The S&P 500 and Russel 2000 indices represent diversified market portfolios, as both of these indices include many securities from lots of different industries. Important to note is that both S&P and Russel are market cap weighted. If the airline industry is under attack because of a pandemic, then perhaps technology companies such as Amazon or Apple outperforms, as people rely on ordering food online instead of grocery shopping. The market cap weight for an index constituent is calculated by multiplying the number of shares with the price of each share. If Google consists of 1000 shares and the price of each share is 10$, then the market cap of Google is 10,000$. A market cap weighted index divides each constituents market cap by the summed market cap of all constituents represented in the index. In S&P, Facebook, Amazon, Microsoft and Apple represents 20% of the total market cap, which limits the diversification benefit of holding S&P. An undiversified index increases idiosyncratic risks and in S&P, technology specific risks dominate the index. Traditional long only US equity portfolios are often measured against S&P. As a result, long only portfolios often have a high equity beta against the S&P. The equity beta means that the US long only portfolio will move in the same direction of S&P regardless of market direction.

Figure 2

Figure 2 demonstrates that the standard deviation of a stock portfolio decreases with the number of stocks added. That relationship holds true until the all of the unsystematic/idiosyncratic risk has been diversified away.

Beta

If the long only portfolio outperforms S&P over a long-time span, then investors should probably purchase shares of the fund rather than investing directly in S&P. The paths to outperforming an index are many, but most common is holding a portfolio of stocks tilted away from S&P exposure. For example, investment strategies that caps total portfolio exposure at 3% for single securities automatically remove risk from large technology giants in a market cap weighted index like S&P. As a result, the annual return for such a strategy will most likely vary less than S&P, as technology stocks tend to have a relatively high beta. If technology stocks instead had a low beta, the conclusion would reverse. Now, Adaptive Alph defines beta as a statistical coefficient measuring the volatility of a portfolio of stocks or an individual stock relative to the systematic risk of a market portfolio such as S&P. The coefficient beta basically measures a stock or portfolio’s response to changes in the market as compared to S&P. If the portfolio beta equal 1.0, then the portfolio is expected to go up and down proportionally with the benchmark. If the beta ranges from 0 to 1 that portfolio is expected to increase or decrease in the same direction, but less than the benchmark. If the beta is greater than 1, the portfolio moves up and down more than the benchmark. Finally, if the beta is less than 0, the portfolio moves in opposite direction of the market. A greater negative number means a greater move in the opposite direction.

Figure 3

Figure 3 depicts a simple calculation of beta. All securities can have a beta toward any type of benchmark so it is up to the observer to determine the value of the betas being analyzed.

What types of Betas exist?

Up until the introduction of arbitrage pricing theory (APT) in late 1970s, CAPM was the leading theory among both financial academics and practitioners. These communities considered all non-market risk factors impacting firms as company specific events, which included factors such as CEO changes, accounting issues and earnings misses. Since the introduction of APT by Stephen Rose in late 1970s, investment research has evolved and new beta factors have emerged. Rose’s APT concludes that a multi-factor approach more accurately describes stock market return rather than a single market factor as demonstrated by CAPM. Furthering the conclusions of APT, researchers Fama and French added size and style factors, which are two systematic risk factors other than the market factor that can enhance portfolio returns. The size and style factors demonstrate how the size and valuation of firms drives stock prices. According to Fama-French research, the size factor demonstrates that small-cap firms tend to outperform large-caps while the style factor proves that cheap stocks relative to intrinsic value tend to outperform relatively expensive firms. Since the invention of the Fama-French factors, three other risk factors have been derived by the investment community; volatility, momentum and quality. Building a portfolio that diversifies across these risk factors have the potential of enhancing portfolio returns because it reduces volatility and thus enhances a portfolio’s return per unit of risk. Adding companies with low volatility involves investing in companies that vary less in stock price than the broad market. Low volatility companies often generate stable revenues and their accounting methods are robust without errors. Momentum involves investing in companies performing well over certain lookback periods and short those companies that have underperformed. The momentum premium exists over time because of fear of missing out. When Tesla’s stock price goes to the moon more investors jumps on the rocket ship, which drives the stock price even higher. Lastly, quality is factor debated heavily by the investment community, as defining a company’s quality is qualitative task. However, higher income, lower accruals and low levels of leverage signifies quality companies according to academic research. All of these five factors are betas meaning that they are risk factors shared by many companies and asset classes in the market. Investing in index portfolios involves beta exposure to size, style, momentum, quality and volatility. All indices with slightly unique constituents will have a different risk weighted exposure to these factors.

Figure 4

Figure 4 show 3 additional risk premias that was not mentioned in yield, growth and liquidity. Growth and value tend to be opposites while yield tend to correlate highly with value. Liquidity is a risk premium that many private equity funds take advantage off when purchasing illiquid companies.

Alpha versus Beta

Finding an edge in investing is a complex and time-consuming process. For most individuals the task of portfolio management is overwhelming, as a typical person has a life outside of investing. That is why professionals recommend most individuals to invest in betas such as S&P rather than actively seeking to outperform the market by self-directing investments in individual stocks or actively managed funds. A key concept for these so-called retail investors is diversification, which involves spreading investments across uncorrelated return streams. A classic example is the 60-40 equity-bond portfolio, as represented by MSCI and JP Morgan bond index. The former and latter have a correlation of around 0.4. If the MSCI investment underperforms then the JP index should perform well around 60 percent of the time over long time periods. With factor, currency, geography and asset class diversification a strong portfolio may be created even if the portfolio solemnly relies on beta. That is because beta represents a risk premium, which is additional compensation received by increasing risk. When purchasing a bond that pays a coupon every month for 10 years, there is a risk that the bond defaults or that a similar bond is issued paying twice the coupon as the original bond. As a result, the investor receives a risk premium to compensate against default and liquidity risks. These risk premiums are generally higher for equities than bonds, as insolvent companies return money to all bond holders before equity holders when liquidating assets. A mixed portfolio of different uncorrelated risk premiums will reward investors with a higher return per unit of risk all else equal. If the risk premium is not high enough to satisfy investor appetite then pursuing active managers might be an option. The job of active investors is to earn alpha plus the beta. In other words, alpha involves capturing the risk premium plus additional return by outperforming a benchmark within a sector, asset class, currency, geography or factor.

Figure 5

Figure 5 is a timeline depicting the evolution of risk premias in finance. As new factors emerged, parts of what was considered alpha in the 70s is now considered beta. As a result, only top notch managers are able to deliver alpha to their clients.

Information Ratio

Earning alpha is the holy grail in financial investing. To earn alpha, investors must create an edge. Luckily, edges come in many shapes and sizes such as informational, technological, intellectual, emotional or relational advantages. In the end, alpha quantitively expresses the return generated above a benchmark. The concept of quantitatively explaining alpha through a multi-factor equation was made possible thanks to the single-factor model in CAPM. The return of any market portfolio is directly correlated to the market risk. The goal of many hedge fund managers is therefore to take idiosyncratic risk in equities to outperform the expected return of CAPM. For example, a US market neutral equity hedge fund might use a risk-free rate as a benchmark. Worth noting is that a market neutral hedge fund places long and short stock bets in the market to achieve a beta equal to 0. According to CAPM, a portfolio with 0 beta should have an expected return equal to the risk-free rate. If in our example the risk-free rate consistently has a return of around 3% and the equity fund has a return of 7%, the alpha is equal to 4% (7% -3% = 4%). However, only comparing realized returns between investments is a sub-optimal method, as the volatility of a typical market neutral hedge fund is greater than the realized volatility for the risk-free rate, which is close to 0%. Realized volatility measures the riskiness of an investment. A higher volatility implies higher risk, but not necessarily less return per unit of risk. A market neutral fund generally lowers the volatility well below a typical index investor to achieve a higher information ratio. The simplest definition of information ratio is to subtract benchmark return from portfolio return and divide the resulting difference by the standard deviation of difference in benchmark returns and portfolio returns over a period of time. A consistently high information ratio over long time signifies a manager that either invests with great skill by earning a high alpha or has chosen a terrible benchmark. Having a diversified portfolio with high information ratios and alpha is the holy grail for an active manager.

Figure 6

The top equation on the left side in Figure 6 shows the full fundamental law of active management. The other four equations starting with the information ratio demonstrates how to calculate the sub-components of the fundamental law. The most important sub-component is the information ratio because that is what earns managers their fees.

Fundamental Law of Active Management

The goal of all investment portfolios is to generate attractive returns while minimizing risk. The fundamental law of active management is a mathematical method used to measure the attractiveness of active managers by relating the expected information ratio of actively managed portfolios down to a few key parameters. These four parameters making up the fundamental law is the information coefficient (IC), breadth, transfer coefficient (TC) and benchmark tracking risk. When the parameters are multiplied, the fundamental law equation spots out a number that quantitively demonstrates the value add by the portfolio manager being analyzed. While benchmark tracking risk indicates a manager’s aggressiveness, the IC is defined as the level of correlation between forecasted returns with actual returns and measures a manager’s skill. Larger amounts of independent bets as per the breadth measurement in combination with a higher IC means that a great amount of value is added to the portfolio by the manager. The TC measures how much alpha information that is actually used in the portfolio, which is another way of measuring the success of a manager’s portfolio construction. A higher TC is better, but real-life constraints often lowers the TC. For example, if a portfolio faces short selling restrictions, then that portfolio’s TC is lowered all else equal, as all alpha information cannot be used. Like a short-selling ban’s impact on the TC, some drivers of the fundamental law parameters for a manager is outside of their control, which makes the job of analyzing a manager a difficult yet valuable job in the financial industry. Other factors that can impact a manager’s future performance are regulatory restrictions, transaction costs, and problems with mean-variance optimization. As a result, the realized information ratios tend to be lower than what is expected ex ante. In security selection strategies, the actualized information ratio has empirically been determined by research to be 45-91% lower than what was originally expected. However, investors such as Rentech, Paul Tudor Jones, and Stanley Druckenmiller, have proven that earning alpha is possible with an edge.

Figure 7

Figure 7 is the fundamental law of active management in a triangle. This correlation triangle depicts a relationships between forecasted and realized returns and active security weights.

Conclusion

Quantifying investing success and explaining underlying drivers of asset returns are goals shared by most investment professionals. The first recognized academic theory that achieved quantification and measurement of return drivers was CAPM and this fundamental theory of finance is therefore one of the greatest research leaps in history. CAPM applies gaussian statistics on security returns to define market risk, which helps separating idiosyncratic from systematic risk. In the ensuing decades, new theories such as APT emerged, which expanded the universe of factors impacting asset returns. These new factors include quality, momentum, size and style. As a result of advancements in financial academia over the past 50 years, concepts such as systematic risk, beta and alpha have been dissected further to explain investment performance. These key statistics have now become regular terms thrown around by researchers and portfolio managers on quantitative trading floors. For some investors it is optimal to diversify across beta exposures through investing in mutual funds, ETFs, indices, asset classes, geographies and risk premias. A portfolio strategy applying a diversified beta approach lowers volatility while maintaining a reasonably high investment return. However, some investors seek higher returns than what is provided by risk premias and index funds and that is when active management enters the picture. The goal of active managers is to outperform beta risk premias or offer uncorrelated sources of return to the typical investor portfolio. These active managers earn a higher compensation and need to offer high information ratios in return for fees paid by their clients. Although the fundamental law of active management is far from perfect, achieving a high information ratio with many independent bets is currently the optimal way to measure performance of active managers.

Well done!

//Stay Adaptive

1. https://www.investopedia.com/terms/b/beta.asp 2. https://www.fidelity.com/bin-public/060_www_fidelity_com/documents/fidelity/fidelity- overview-of-factor-investing.pdf

3. 2018 CFA Program Level 2