- Describe and evaluate the low-risk anomaly of asset returns.
- Define and calculate alpha, tracking error, the information ratio, and the Sharpe ratio.
- Explain the impact of benchmark choice on alpha, and describe characteristics of an effective benchmark to measure alpha.
- Describe Grinold's fundamental law of active management, including its assumptions and limitations, and calculate the information ratio using this law.
- Apply a factor regression to construct a benchmark with multiple factors, measure a portfolio's sensitivity to those factors and measure alpha against that benchmark.
- Explain how to measure time-varying factor exposures and their use in style analysis.
- Describe issues that arise when measuring alphas for nonlinear strategies.
- Compare the volatility anomaly and beta anomaly, and analyze evidence of each anomaly.
- Describe potential explanations for the risk anomaly.

- Video Lecture
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- PDFs
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- List of chapters

- Introduction
- GM Asset Management And Martingale
- Active Management
- Benchmark Selection For Alpha
- Active Management
- Characteristics Of Ideal Benchmarks
- Factor Benchmarks
- Size And Value Growth Benchmarks
- Adding Momentum
- Time-Varying Factor Exposures
- Style Analysis With No Shorting
- Non-Linear Payoffs
- Volatility Anomaly
- Beta Anomaly
- Low Risk Anomaly
- Low Risk Anomaly Explanations

- Alpha-the average return in excess of a benchmark-tells us more about the set of factors used to construct that benchmark than about the skill involved in beating it.
- A positive alpha under one set of factors can turn negative using a different set.
- Whatever the benchmark, alpha is often hard to detect statistically, especially when adjustments for risk vary over time.
- The risky anomaly-that stocks with low betas and low volatilities have high returns-appears to be a strong source of alpha relative to standard market-weighted benchmarks and value-growth, momentum, and other dynamic factors.

- Alpha is often interpreted as a measure of skill. That is debatable. It is first and foremost a statement about a benchmark. Alpha is the average return in excess of a benchmark. Thus, the concept of alpha requires first defining a benchmark against which alpha can be measured.

- The excess return, \(r_t^{ex}\) , as the return of an asset or strategy in excess of a benchmark can be

defined as

\(r_t^{ex} = r_t โ r_t^{bmk}\)where

๐_{t} is the return of an asset or strategy and

\(r_t^{bmk}\) is the benchmark return. \(r_t^{ex}\), is referred to as active returns. This terminology assumes that the benchmark is passive and can be produced without any particular investment knowledge or even human intervention.

- Common passive benchmarks are market weighted portfolios, like the S&P 500 or the Russell 1000, which investors can track by buying low-cost index funds.
- Alpha is computed by taking the average excess return

where

there are ๐ observations in the sample.

- Tracking error is the standard deviation of the excess return; it measures how disperse the managerโs returns are relative to the benchmark:

- Tracking error constraints are imposed to ensure a manager does not stray too far from the benchmark. The larger the tracking error, the more freedom the manager has. If the benchmark is risk adjusted, then academics like to call tracking error โ
**idiosyncratic volatility**โ - The information ratio is the ratio of alpha to tracking error:

- Alpha by itself could be produced by a manager taking large amounts of risk. The information ratio divides the alpha by the risk taken, so it is the average excess return per unit of risk. Information ratios above one are not common-although many hedge funds trying to raise money claim to have them. (Since the financial crisis, information ratios on many funds and strategies have come down substantially.)
- A special case occurs when the benchmark is the risk-free rate, \(r_t^f\). Then, the alpha is the average return in excess of the risk-free rate, \(\alpha = \overline{r_t โ r_t^f}\), and the information ratio coincides with the Sharpe ratio,

where ๐ is the volatility of the asset.

- Martingaleโs low volatility strategy is based on the Russell 1000 universe of large stocks, so naturally, the Russell 1000 is the benchmark. A combination of assets or asset classes can also serve as a benchmark.
- Martingaleโs low volatility strategy has a high tracking error, of 6.16%, relative to the Russell 1000. The tracking error is high because Martingaleโs product has a low beta of 0.73 relative to the same index. Beta is measured by regressing excess returns of the fund (using T-bills as the risk-free-asset) on excess returns of the Russell 1000(which is the regression implied by the CAPM)

where

R1000 represents the return of the Russell 1000, and

\(\epsilon_t\) is the residual of the regression.

- The ฮฑ = 3.44% per year in this CAPM regression is the average excess return of the low volatility strategy relative to a market-adjusted portfolio. The CAPM regression can be rewritten using a benchmark portfolio of a risk-free asset and 0.73 of the Russell 1000:

Here, \(0.2728r_t^f + 0.7272r_t^{R1000}\) = \(r_t^{bmk}\)ย

- The low volatility strategy outperforms this benchmark by 3.44% per year. The information ratio of the low volatility strategy with this risk adjusted benchmark is a very high 0.78.
- If the benchmark is taken as just the Russell 1000, it is falsely assumed that the beta of the low volatility strategy is one (when in fact it is 0.73).
- With the Russell 1000 benchmark

Here, \(r_t^{R1000}\) = \(r_t^{bmk}\)ย ย

- So the alpha is 1.50% per year.
- The information ratio of the low volatility strategy relative to the naive Russell 1000 benchmark is just 0.24.
- This is not the correct risk adjusted benchmark because the beta of the low volatility strategy is not one.
- Even with a simple Russell 1000 portfolio, failing to adjust the benchmark for risk can make a huge difference in the alpha!

**Well defined โ**It should be verifiable and free of ambiguity about its contents. Thus it should be able to properly define the โmarket portfolioโ.**Tradeable**โ Alpha must be measured relative to tradeable benchmarks, otherwise the computed alphas do not represent implementable returns on investment strategies. So the benchmark should be a realistic, low-cost alternative for the asset owner.**Replicable โ**Both the asset owner and the fund manager should be able to replicate the benchmark. Certain benchmarks canโt be replicated by the asset owner because they are beyond the asset ownerโs expertise. Such nonreplicable benchmarks are not viable choices and make it difficult or impossible to measure how much value a portfolio manager has added because the benchmark itself cannot be achieved by the asset owner. There are some benchmarks, like absolute return benchmarks, that canโt even be replicated by the fund manager.**Adjusted for risk**โ It should be adjusted for risk as it makes a big difference in the alpha. Unfortunately, most benchmarks used in the money management business are not risk adjusted.

- A portfolio manager creates alpha relative to a benchmark by making bets that deviate from that benchmark. The more successful these bets are, the higher the alpha.

- Grinoldโs (1989) โfundamental lawโ of active management makes this intuition formal. It states that the maximum information ratio attainable-since it ignores transactions costs, restrictions on trading, and other real-world considerations-is given by: ๐ผ๐ โ ๐ผ๐ถ ร ๐ต๐

where

๐ผ๐ is the information ratio

๐ผ๐ถ is the information coefficient, which is the correlation of the managerโs forecast with the actual returns (how good the forecasts are), and

๐ต๐ is the breadth of the strategy (how many bets are taken). Breadth is the number of securities that can be traded and how frequently they can be traded.

- High information ratios are generated by a manager finding opportunities-and many of them- where she can forecast well. Grinold and Kahn (1999) state that โit is important to play often (high breadth, ๐ต๐ ) and to play well (high ๐ผ๐ถ).
- There are two very important limitations of the fundamental law.
- The first is that ๐ผ๐ถs are assumed to be constant across ๐ต๐ .
- Second, it is difficult to have truly independent forecasts in ๐ต๐ .

- Consider the CAPM applied to asset (or strategy or fund) ๐:

- Factor benchmarks are a combination of investment portfolios, or factors, on the right-hand side that give the same return as the asset on the left-hand side. The factor benchmark describes the systematic components of asset ๐โs return.
- Berkshire Hathway v S&P

Coefficient | T-stat | |
---|---|---|

Alpha | 0.72% | 2.02 |

Beta | 0.51 | 6.51 |

Adj R^{2} |
0.14 |

- The ๐๐๐ต and ๐ป๐๐ฟ factors are long-short factors. They are mimicking portfolios that consist of simultaneous $1 long and $1 short positions in different stocks. That is,

And so SMB is designed to capture the outperformance of small companies versus large companies.

- The ๐ป๐๐ฟ factor picks up the outperformance of value stocks versus growth stocks:

- The Fama-French benchmark can be estimated by running the following regression:

- Estimating the Fama-French regression for Berkshire Hathaway yields the following coefficients:

Coefficient | T-stat | |
---|---|---|

Alpha | 0.65% | 1.96 |

MKT Loading |
0.67 | 8.94 |

SMB Loading |
-0.50 | -4.92 |

HML Loading |
0.38 | 3.52 |

Adj R^{2} |
0.27 |

- The benchmark implied by the Fama-French regression estimates is:

\[ (1 โ 0.67) = 0.33 \text{ in T-bills} \] \[ + 0.67 \text{ in the market portfolio} \] \[ โ 0.50 \text{ in small caps} \] \[ + 0.50 \text{ in large caps} \] \[ + 0.38 \text{ in value stocks} \] \[ โ 0.38 \text{ in growth stocks} \]

In addition to this benchmark, Buffet is generating

+ 0.65% (๐๐๐โ๐) ๐๐๐ ๐๐๐๐กโ.

- A momentum factor, ๐๐๐ท, constructed by taking positions in stocks that have gone up minus stocks that have gone gown, is added to the Fama-French benchmark:

- Estimating the Fama-French regression after adding the momentum effect for Berkshire Hathaway yields the following coefficients:

Coefficient | T-stat | |
---|---|---|

Alpha | 0.68% | 2.05 |

MKT Loading | 0.66 | 8.26 |

SMB Loading | -0.50 | -4.86 |

HML Loading | 0.36 | 3.33 |

UMD Loading | -0.04 | -0.66 |

Adj Rยฒ | 0.27 |

For completeness, the mimicking portfolio implied by this Fama-French plus momentum benchmark is:

\[ (1 โ 0.66) = 0.34 \, \text{in T-bills} \] \[ + 0.66 \, \text{in the market portfolio} \] \[ โ 0.50 \, \text{in small caps} \] \[ + 0.50 \, \text{in large caps} \] \[ + 0.36 \, \text{in value stocks} \] \[ โ 0.36 \, \text{in growth stocks} \] \[ โ 0.04 \, \text{in past winning stocks} \] \[ + 0.04 \, \text{in past losing stocks} \]Buffett is also adding: +0.68% (๐๐๐โ๐) ๐๐๐ ๐๐๐๐กโ

- William Sharpe introduced a powerful framework called โ
**style****analysis**โ to handle time-varying benchmarks in 1992. - To illustrate time-varying factor exposures, consider four funds:
**LSVEX**: LSV Value Equity**FMAGX**: Fidelity Magellan**GSCGX**: Goldman Sachs Capital Growth**BRK**: Berkshire Hathaway

Here are the Fama-French and momentum factor regressions using constant factor weights:

LSVEX | FMAGX | GSCGX | BRK | |
---|---|---|---|---|

Alpha | 0.00% | -0.27% | -0.14% | 0.22% |

t-stat | 0.01 | -2.23 | -1.33 | 0.57 |

MKT Loading | 0.94 | 1.12 | 1.04 | 0.36 |

t-stat | 36.9 | 38.6 | 42.2 | 3.77 |

SMB Loading | 0.01 | -0.07 | -0.12 | -0.15 |

t-stat | 0.21 | -1.44 | -3.05 | -0.97 |

HML Loading | 0.51 | -0.05 | 0.34 | 0.34 |

t-stat | 14.6 | -1.36 | 2.57 | 2.57 |

UMD Loading | 0.2 | 0.02 | 0.00 | -0.06 |

t-stat | 1.07 | 0.17 | -0.17 | -0.77 |

- Style analysis seeks to rectify two potential shortcomings of the analysis done so far:
- The Fama-French portfolios are not tradeable.
- The factor loadings may vary over time.

- Style analysis tries to replicate the fund by investing passively in low-cost index funds. The collection of index funds that replicate the fund is called the โstyle weightโ.
- The main idea with style analysis is that actual tradeable funds are used in the factor benchmark.
- To illustrate, letโs take the following index ETFs:
**SPY**: SPDR S&P 500 ETF, which is designed to mimic the S&P 500;**SPYV**: SPDR S&P 500 Value ETF, which tracks the S&P 500 value index; and**SPYG**: SPDR S&P 500 Growth ETF, which replicates the S&P 500 growth index.

- These low-cost index ETFs are tradeable, unlike the Fama-French portfolios. They belong to the SPDR (pronounced โspiderโ) family of ETFs sponsored by State Street Global Advisors. The benchmark factor regression for fund ๐ is

where

๐ฝ_{SPF,t} + ๐ฝ_{SPFV,t} + ๐ฝ_{SPFG,t} = 1 and all the ๐ฝs are positive

- The weights are estimated using information up to time ๐ก. The return of the fund over the next period, ๐ก+ 1, is equal to the replicating portfolio formed at the beginning of the period at time ๐ก plus a fund-specific residual, s
_{t+1}, and the fund alpha, ฮฑ_{t}for that period. The weights can change over time. The returns of the fund have to match the varying investments made in ๐๐๐, ๐๐๐๐ and ๐๐๐๐บ. So, ๐ฟ๐๐ fund can be constructed as a combination of the ๐ธ๐๐นโ๐ ๐๐๐ and ๐๐๐๐. Similarly, other funds can be replicated as a combination of these ๐ธ๐๐นโ๐ . - The excess return for ๐ก + 1 is the return of the fund at the end of the period, ๐ก to ๐ก + 1, minus the benchmark portfolio formed using the weights at time ๐ก:

- If shorting is allowed while trading in the ETFโs, then this factor regression is used.

\( r_{t+1} โ r_{f,t+1} = \alpha_{i,t} + \beta_{SPY,t}(SPY_{t+1} โ r_{f,t+1}) + h_t(SPYV_{t+1} โ SPYG_{t+1}) + \epsilon_{t+1} \)

- This is the โ๐ธ๐๐น versionโ of the Fama-French (1993) without the SMB factor, except that the factor loadings are allowed to change over time. The ๐๐๐๐ โ ๐๐๐๐บ is an investment that goes long the value ๐๐๐๐ ๐ธ๐๐น and simultaneously shorts the growth ๐๐๐๐บ ๐ธ๐๐น. Thus, it is analogous to the ๐ป๐๐ฟ factor.
- The factor loadings show the strong value bias of LSV; with a positive โ loading on the ๐๐๐๐ โ ๐๐๐๐บ factor. Magellan becomes more of a growth fund over time, with increasingly negative โ loadings, as does Goldmanโs growth fund. Berkshire Hathawayโs changing factor loadings from value to growth to value can be seen in its negative โ loadings during 2008 and 2009.
- Allowing shorting does not much change the cumulated excess returns. But allowing shorting, not surprisingly, reduces the alphas.

- With alphas and information ratios, any manager can appe to have talent when he actually doesnโt.
- Alphas are computed in a linear framework. There are many nonlinear strategies, especially those involving dynamic option strategies, that can masquerade as alpha. To give an extreme example, consider this Figure. It is produced by selling put options on the market portfolio in a small sample, just using Black-Scholes prices. The returns on these put options are recorded. A CAPM regression is the run with these simulated returns. The โalphaโ appears to be positive. But it is known that in the Black-Scholes world there is no extra value created in puts or calls. The alpha is purely illusory. No nonlinear strategy can be adequately captured in a linear framework. This is a serious problem because many common hedge fund strategies, including merger arbitrage, pairs trading, and convertible bond arbitrage, have payoffs that resemble nonlinear option volatility strategies.

- Dynamic, nonlinear strategies yield false measures of alpha because buying and selling options-or any dynamic strategy-changes the distribution of returns. Static measures, like the alpha, information, and Sharpe ratios, capture only certain components of the whole return distribution. Often, short volatility strategies can inflate alphas and information ratios because they increase negative skewness. These strategies increase losses in the left-hand tails and make the middle of the distribution โthickerโ and appear to be more attractive to linear performance measures. Skewness and other higher moments are not taken into account by alphas and information ratios.
- There are two ways to account for nonlinear payoffs.
- Include Tradeable Nonlinear Factors
- Examine Non-tradeable Nonlinearities

- Volatility anomaly is the risk that volatility is negatively related to returns. But the traditional view has always been that volatility is positively related to returns.
- Haugen and Heins (1975) use data from 1926 to 1971 and also investigate the relation between beta and volatility risk measures and returns. They reported that-
*The results of**our empirical effort do not support the conventional hypothesis that risk-systematic**or otherwise-generates a special reward. Indeed, our results indicate that, over the long run, stock portfolios with lesser variance in monthly returns have experienced greater average returns than โriskierโ counterparts.*- Ang, Hodrick, Xing, and Zhang found that the returns of high-volatility stocks were โabysmally low.โ So low that they had zero average returns. Particularly notable was the robustness of the negative relation between both idiosyncratic and total volatility with returns. They employed a large number of controls for size, value, leverage, liquidity risk, volume, turnover, bid-ask spreads, co-skewness risk, dispersion in analystsโ forecasts, and momentum. They also did not find that aggregate volatility risk explained our result-even though volatility risk is a pervasive risk factor. In subsequent work, they showed that the volatility effect existed in each
**G7**country and across all developed stock markets. They also controlled for private information, transactions costs, analyst coverage, institutional ownership, and delay measures, which recorded how fast information is impounded into stock prices. Skewness did not explain the puzzle.

- Beta anomaly is the risk that the relation between beta and returns are negative. But CAPM implies that the relation between beta and returns are positive.
- Black, Jensen, and Scholes (1972), found the relation between beta and returns to be โtoo flatโ compared with what the CAPM predicted, but at least the relation was positive.
- Fama and French wrote a major paper in 1992 that struck at the heart of the CAPM. While their main results showed that size and value effects dominated beta in individual stocks, they noted that โbeta shows no power to explain average returns.โ In fact, their estimated relation between beta and returns was statistically insignificant. Worse, the point estimates indicated that the relation between beta and returns was negative.
- Ang explained that beta anomaly does not mean that stocks with high betas have low returns. Instead it meant that stocks with high betas have high volatilities. Hence the Sharpe ratios of high beta stocks were lower than that of low beta stocks. This was clearly observed when the relation between lagged or past beta and future returns was examined.
- Betas obtained from information other than past betas seem to have positive risk relations.
- Buss and Vilkov (2012) estimated betas from options and found out that they were better predictors of future betas as compared to betas estimated from past returns. There was a positive riskโreturn relation in these betas exhibited.
- Cosemans et al. (2012) used valuation information from accounting balance sheets to calculate betas along with past returns and showed that there is a positive relation between betas and returns.
- Therefore, from the research conducted so far, low beta anomaly cannot be linked to the idea that beta does not help in explaining returns. Instead, it is difficult to predict future betas with past betas.

- The low-risk anomaly is a combination of three effects, with the third a consequence of the first two:
- Volatility is negatively related to future returns;
- Realized beta is negatively related to future returns; and
- Minimum variance portfolios do better than the market.

- The combined effect of four factors can explain the risk anomaly to some extent.

**Data****Mining**

In Angโs research, there are a few concerns about data mining. There is some sensitivity in the results to different portfolio weighting schemes and illiquidity effects. Even though data mining is an issue, but low-risk effect is seen in many other contexts. It has been shown that the effect appears during recessions and expansions and during stable and volatile periods. It has also been shown that it takes place in international stock markets. Frazzini and Pedersen (2011) showed that low-beta portfolios have high Sharpe ratios in U.S. stocks, international stocks, Treasury bonds, corporate bonds cut by different maturities and credit ratings, credit derivative markets, commodities, and foreign exchange. Cao and Han (2013) and Blitz and de Groot (2013) showed that the low-risk phenomenon even shows up in option and commodity markets, respectively.

**Leverage****Constraints**

Some investors want to take on more risk by obtaining more leverage, but they might not be able to do so because of some constraints. So, they go for high beta stocks with ย โbuilt-inโ leverage. Hence, the price of high-beta stocks go up to a point where there are overpriced and give lower returns. According to this explanation, institutions which are having leverage constrains should be attracted to high-risk stocks. But empirically, institutional investors have been observed to give less weight to high-risk stocks. Also, the leverage constraint factor explains only the overpricing of high beta stocks. It offers no explanation for underpricing of low beta stocks relative to the market portfolio.

**Agency****Problems**

Unconstrained investors keep buying stocks which are having a higher actual return relative to the CAPM return until their excess returns disappear. Similarly, they sell stocks with lower actual returns relative to CAPM. But if investors who are long-only face tracking error constraints (i.e. have limits on how much they can deviate from benchmark), the requirement of following the market-weighted benchmarks itself may lead to the low volatility anomaly.

**Preferences**

Sometimes there is a plain preference for high-volatility and high-beta stocks by asset ย owners. This can be a result of their extra bullish capital market expectations, where they want to amplify their returns. This leads to the investors buying the higher-beta investments. This leads to an increase in their prices to the point where future returns will be much lower. Such group of investors who do want to go for โsafeโ and โboringโ lower-volatility stocks. Hou and Loh (2012) explain low risk anomaly mainly with lottery preferences. Hong and Sraer (2012) show that when disagreement is low and all investors take long-only positions, the CAPM holds but when preferences vary and when restraints are placed for short selling big beta stocks become overpriced such that their returns start decreasing.