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*Learning objectives: 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.*

**Questions:**

704.1. Below is the regression output of a portfolio's excess returns against its benchmark's excess return over the last three months (n = 60 trading days). Excess return is defined as return above the risk-free rate.

Key output from the regression includes:

- The sample size is 60 trading days
- With respect to the portfolio, its average excess return is 3.29% (in excess of the riskfree rate) with volatility of 3.67%
- With respect to the benchmark, its average excess return is 0.98% (in excess of the riskfree rate) with volatility of 1.99%
- The average difference in return between the portfolio and the benchmark, avg(P-M), is 2.31%; this is also called the
*active return* - The regression intercept is 0.0180 and the regression slope is 1.5231 (as displayed on plot)
- The tracking error (standard error of the regression) is 2.10%

**nearest**to the information ratio (IR) if we measure the IR as

*residual return per unit of residual risk*?

a. 0.357

b. 0.630

c. 0.857

d. 1.102

704.2. The low-risk anomaly is a combination of each of the following three true effects

**EXCEPT**which is false (and not technically included in the low-risk anomaly)?

a. Both contemporaneous and lagged volatility are inversely (aka, negatively) related to returns

b. Contemporaneous beta is inversely (aka, negatively) related to raw returns

c. Lagged beta is inversely (aka, negatively) related to risk-adjusted returns

d. Minimum variance portfolios do better than the market

704.3. Peter the aspiring FRM candidate is estimating the alpha for his firm's (Martingale's) new low-volatility fund. His naive benchmark is the Russel 1000 large-cap index. He has collected the following (ex ante) statistics over the historical sample where the period returns are monthly:

- The regression slope coefficient, β, is 0.40
- The portfolio's average excess return is 3.15% per month
- The Russel index's average excess return is 0.65% per month

**TRUE**?

a. The portfolio's alpha is about +289 basis points

b. He should assume a beta (aka, slope) coefficient, β, of 1.0 such that the portfolio's alpha is about +250 basis points; this is a lower alpha due to the implicit risk-adjustment

c. The Russel is NOT an appropriate benchmark because the low beta, β, implies that the Russel 1000 cannot be combined with another asset in order to generate a market-adjusted portfolio

d. The Russel is NOT an appropriate benchmark because it represents a tradeable, low-cost alternative but the firm's low-volatility fund is active and charges high fees; an ideal benchmark charges comparable fees

**Answers here:**

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