Tutorial WEEK 10 PDF

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Chapter 24: 5, 6, 9, 12, CFA-12

5. Consider the rate of return of stocks ABC and XYZ.

a. Calculate the arithmetic average return on these stocks over the sample period.

Arithmetic average: rABC = 10%; rXYZ = 10% b. Which stock has greater dispersion around the mean return?

Dispersion: σABC = 7.07%; σXYZ = 13.91% Stock XYZ has greater dispersion. (Note: We used 5 degrees of freedom in calculating standard deviations.) c. Calculate the geometric average returns of each stock. What do you conclude?

Geometric average: rABC = (1.20 × 1.12 × 1.14 × 1.03 × 1.01)1/5 – 1 = 0.0977 = 9.77% rXYZ = (1.30 × 1.12 × 1.18 × 1.00 × 0.90)1/5 – 1 = 0.0911 = 9.11% Despite the fact that the two stocks have the same arithmetic average, the geometric average for XYZ is less than the geometric average for ABC. The reason for this result is the fact that the greater variance of XYZ drives the geometric average further below the arithmetic average. d. If you were equally likely to earn a return of 20%, 12%, 14%, 3%, or 1% in each year (these are the five annual returns for stock ABC), what would be your expected rate of return?

Your expected rate of return would be the arithmetic average, or 10%. e. What if the five possible outcomes were those of stock XYZ?

Even though the dispersion is greater, your expected rate of return would still be the arithmetic average, or 10%.

f. Given your answers to parts (d) and (e), which measure of average return, arithmetic or geometric, appears more useful for predicting future performance?

In terms of “forward-looking” statistics, the arithmetic average is the better estimate of expected rate of return. Therefore, if the data reflect the probabilities of future returns, 10 percent is the expected rate of return for both stocks.

6. XYZ’s stock price and dividend history are as follows:

An investor buys three shares of XYZ at the beginning of 2016, buys another two shares at the beginning of 2017, sells one share at the beginning of 2018, and sells all four remaining shares at the beginning of 2019. a. What are the arithmetic and geometric average time-weighted rates of return for the investor?

b. What is the dollar-weighted rate of return? (Hint: Carefully prepare a chart of cash flows for the four dates corresponding to the turns of the year for January 1, 2016, to January 1, 2019. If your calculator cannot calculate internal rate of return, you will have to use trial and error.)

9. Consider the two (excess return) index-model regression results for stocks A and B. The risk-free rate over the period was 6%, and the market’s average return was 14%. Performance is measured using an index model regression on excess returns.

A. Calculate the following statistics for each stock: i. Alpha ii. Information ratio iii. Sharpe ratio iv. Treynor measure

B. Which stock is the best choice under the following circumstances? i. This is the only risky asset to be held by the investor. ii. This stock will be mixed with the rest of the investor’s portfolio, currently composed solely of holdings in the market-index fund. iii. This is one of many stocks that the investor is analyzing to form an actively managed stock portfolio.

(i) If this is the only risky asset held by the investor, then Sharpe’s measure is the appropriate measure. Since the Sharpe measure is higher for Stock A, then A is the best choice. (ii) If the stock is mixed with the market index fund, then the contribution to the overall Sharpe measure is determined by the appraisal ratio; therefore, Stock B is preferred.

(iii) If the stock is one of many stocks, then Treynor’s measure is the appropriate measure, and Stock B is preferred.

12. A global equity manager is assigned to select stocks from a universe of large stocks throughout the world. The manager will be evaluated by comparing her returns to the return on the MSCI World Market Portfolio, but she is free to hold stocks from various countries in whatever pro- portions she finds desirable. Results for a given month are contained in the following table:

A. Calculate the total value added of all the manager’s decisions this period.

B. Calculate the value added (or subtracted) by her country allocation decisions

. C. Calculate the value added from her stock selection ability within countries.

D. Confirm that the sum of the contributions to value added from her country allocation plus security selection decisions equals total over- or underperformance.

12. During the annual review of Acme’s pension plan, several trustees questioned their investment consultant about various aspects of performance measurement and risk assessment. A. Comment on the appropriateness of using each of the following benchmarks for perfor- mance evaluation: ∙ Market index. ∙ Benchmark normal portfolio. ∙ Median of the manager universe.

Each of these benchmarks has several deficiencies, as described below. Market index: 

A market index may exhibit survivorship bias. Firms that have gone out of business are removed from the index, resulting in a performance measure that overstates actual performance had the failed firms been included.

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A market index may exhibit double counting that arises because of companies owning other companies and both being represented in the index.

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It is often difficult to exactly and continually replicate the holdings in the market index without incurring substantial trading costs.

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The chosen index may not be an appropriate proxy for the management style of the managers.

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The chosen index may not represent the entire universe of securities. For example, the S&P 500 Index represents 65 to 70 percent of U.S. equity market capitalization.

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The chosen index (e.g., the S&P 500) may have a large capitalization bias.

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The chosen index may not be investable. There may be securities in the index that cannot be held in the portfolio.

Benchmark normal portfolio: 

This is the most difficult performance measurement method to develop and calculate.

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The normal portfolio must be continually updated, requiring substantial resources.

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Consultants and clients are concerned that managers who are involved in developing and calculating their benchmark portfolio may produce an easily- beaten normal portfolio, making their performance appear better than it actually is.

Median of the manager universe: 

It can be difficult to identify a universe of managers appropriate for the investment style of the plan’s managers.

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Selection of a manager universe for comparison involves some, perhaps much, subjective judgment.

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Comparison with a manager universe does not take into account the risk taken in the portfolio.

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The median of a manager universe does not represent an “investable” portfolio; that is, a portfolio manager may not be able to invest in the median manager portfolio.

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Such a benchmark may be ambiguous. The names and weights of the securities constituting the benchmark are not clearly delineated.

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The benchmark is not constructed prior to the start of an evaluation period; it is not specified in advance.

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A manager universe may exhibit survivorship bias; managers who have gone out of business are removed from the universe, resulting in a performance measure that overstates the actual performance had those managers been included.

B. Distinguish among the following performance measures:

∙ The Sharpe ratio. ∙ The Treynor measure. ∙ Jensen’s alpha.

i. Describe how each of the three performance measures is calculated.

ii. State whether each measure assumes that the relevant risk is systematic, unsystematic, or total. Explain how each measure relates excess return and the relevant risk.

The Sharpe ratio assumes that the relevant risk is total risk, and it measures excess return per unit of total risk. The Treynor measure assumes that the relevant risk is systematic risk, and it measures excess return per unit of systematic risk. Jensen’s alpha assumes that the relevant risk is systematic risk, and it measures excess return at a given level of systematic risk....