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### Subject 9. Applications

Sources of Returns

Active return is the return in excess of the return on the benchmark. It can be either positive or negative.

active return = RP - RB

Sources of Risks

Active risk, also known as tracking risk (TR) or tracking error (TE), is the standard deviation of active returns.

Active risk = s(RP - RB)

Active risk squared can be decomposed as the sum of active factor risk and active specific risk. Both components contribute to the risk of active returns.

Active risk squared = Active factor risk + Active specific risk

• Active factor risk is the contribution to active risk squared resulting from the portfolio's different-than-benchmark exposures relative to factors specified in the risk model. For example, a portfolio manager may under- or overweight some industries in the portfolio relative to its benchmark. The portfolio's industry factor sensitivities will not be the same as those of the benchmark.

• Active specific risk, also known as asset selection risk, is the contribution to active risk squared resulting from the portfolio's active weights on individual assets as those weights interact with assets' residual risk. For example, a portfolio manager may under- or overweight some stocks within a particular industry, and consequently, the portfolio returns may deviate from the benchmark.

Information Ratio

The information ratio (IR) is mean active return divided by active risk. It measures the increment in mean active return per unit of active risk.

• Since this ratio considers the annualized standard deviation of both series (as measures of risks inherent in owning either the portfolio or the benchmark), the ratio shows the risk-adjusted excess return of the portfolio over the benchmark. The higher the ratio, the higher the excess return of the portfolio, given the amount of risk involved, and the better a portfolio manager.
• The information ratio is similar to the Sharpe ratio, but there is a major difference. The Sharpe ratio compares the return of an asset against the return of Treasury bills, but the information ratio compares excess return to the most relevant equity (or debt) benchmark index.

Factor Portfolios

A factor portfolio is a well-diversified portfolio constructed to have a beta of 1.0 on one factor and a beta of zero on any other factors. It represents the risk of that factor only. A portfolio manager can use a factor portfolio to hedge that risk or speculate on it.

Tracking Portfolio

A tracking portfolio is a portfolio with factor sensitivities that match those of benchmark portfolio or other portfolio. The idea is to match the systematic risk of the benchmark portfolio and at the same time to get excess returns through superior stock selection. For example, a portfolio manager can create a portfolio that has the same factor exposures as the benchmark portfolio (e.g. S&P 500). However, the portfolio contains a different set of securities. The portfolio manager may hope to get a better return than the benchmark portfolio.

#### Practice Question 1

Which terminologies refer to the same thing?

I. Active return.
II. Active risk.
III. Tracking risk.
IV. Tracking error.

Correct Answer: II, III and IV

They all refer to the same thing.

#### Practice Question 2

The risk from active factor tilts attributable to deviations of the portfolio's factor sensitivities versus the benchmark's sensitivities to the same set of factors is known as:

A. active factor risk.
B. active specific risk.
C. asset selection risk.

#### Practice Question 3

Factor portfolios are useful for:

A. speculation purposes only.
B. hedging purposes only.
C. both speculation and hedging purposes.

#### Practice Question 4

Assume a portfolio manager wants to hedge against time horizon risk. She should ______ time horizon risk factor portfolio because such a factor portfolio ______ the time horizon risk.

B. sell, is exposed to.

The factor portfolio is assumed to be hedged against all but time horizon risk. That is, it has zero sensitivities for all but time horizon risk.

#### Practice Question 5

The risk from active asset selection attributable to deviations of the portfolio's individual asset weightings versus the benchmark's individual asset weightings, after controlling for differences in factor sensitivities of the portfolio versus the benchmark, is known as:

A. active factor risk.
B. active specific risk.
C. tracking risk.