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02 OQ - Portfolio Performance Evaluation
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Consider the rate of return of stocks ABC and XYZ.
Year rABC rXYZ
1 20% 30%
2 12% 12%
3 14% 18%
4 3% 0%
5 1% -10%
Calculate the arithmetic average return on these stocks over the sample period.
Arithmetic average:
rABC = 10%; rXYZ = 10%
Consider the rate of return of stocks ABC and XYZ.
Year rABC rXYZ
1 20% 30%
2 12% 12%
3 14% 18%
4 3% 0%
5 1% -10%
Which stock has greater dispersion around the mean return?
Dispersion:
σABC = 7.07%; σXYZ = 13.91%
Stock XYZ has greater dispersion.
Consider the rate of return of stocks ABC and XYZ.
Year rABC rXYZ
1 20% 30%
2 12% 12%
3 14% 18%
4 3% 0%
5 1% -10%
Calculate the geometric average returns of each stock. What do you conclude?
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.
Consider the rate of return of stocks ABC and XYZ.
Year rABC rXYZ
1 20% 30%
2 12% 12%
3 14% 18%
4 3% 0%
5 1% -10%
lf 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%.
Consider the rate of return of stocks ABC and XYZ.
Year rABC rXYZ
1 20% 30%
2 12% 12%
3 14% 18%
4 3% 0%
5 1% -10%
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%.
Consider the rate of return of stocks ABC and XYZ.
Year rABC rXYZ
1 20% 30%
2 12% 12%
3 14% 18%
4 3% 0%
5 1% -10%
Given your answers above, 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.
XYZ's stock price and dividend history are as follows:
Yr / Beginning Yr Price / Dividend Paid Yr-End
2018 $100 $4
2019 $120 $4
2020 $90 $4
2021 $100 $4
An investor buys three shares of XYZ at the beginning of 2018, buys another two shares at the beginning of 2019. sells one share at the beginning of 2020, and sells all four remaining shares at the beginning of 2021.
What are the arithmetic and geometric average time-weighted rates of return for the Investor?
Time-weighted average returns are based on year-by-year rates of return:
Year;
Return = (Capital gains + Dividend)/Price;
2018 − 2019
[($120 - $100) + $4]/$100 = 24.00%
2019 - 2020
[($90 - $120) + $4]/$120 = -21.67%
2020 − 2021
[($100 - $90) + $4]/$90 = 15.56%
Arithmetic mean: (24% - 21.67% + 15.56%)/3 = 5.96%
Geometric mean: (1.24 × 0.7833 × 1.1556)^1/3 - 1 = 0.0392 = 3.92%
XYZ's stock price and dividend history are as follows:
Yr / Beginning Yr Price / Dividend Paid Yr-End
2018 $100 $4
2019 $120 $4
2020 $90 $4
2021 $100 $4
An investor buys three shares of XYZ at the beginning of 2018, buys another two shares at the beginning of 2019. sells one share at the beginning of 2020, and sells all four remaining shares at the beginning of 2021.
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, 2018, to January 1, 2021. If your calculator cannot calculate internal rate of return, you will have to use trial and error.)
1/1/18, -$300, Purchase of three shares at $100 each
1/1/19, -$228, Purchase of two shares at $120 less dividend income on three shares held
1/1/20, $110, Dividends on five shares plus sale of one share at $90
1/1/21, $416, Dividends on four shares plus sale of four shares at $100 each
Date:
1/1/18 -> -300
1/1/19 -> -228
1/1/20 -> 110
1/1/21 -> 416
Dollar-weighted return = Internal rate of return = -0.1607%
(CF 0 = -$300; CF 1 = -$228; CF 2 = $110; CF 3 = $416; Solve for IRR = 16.07%.)
A manager buys three shares of stock today and then sells one of those shares each year for the next three years. His actions and the price history of the stock are summarized below. The stock pays no dividends.
Time Price Action
0 $90 Buy 3 shares
1 $100 Sell 1 share
2 $100 Sell 1 share
3 $100 Sell 1 share
Calculate the time-weighted geometric average return on this "portfolio".
Time, Cash Flow, Holding Period Return
0: 3×(-$90) = -$270; -
1: $100; (100-90)/90 = 11.11%
2: $100; 0%
3: $100; 0%
Time-weighted geometric average rate of return = (1.1111 × 1.0 × 1.0)^1/3 - 1 = 0.0357 = 3.57%
A manager buys three shares of stock today and then sells one of those shares each year for the next three years. His actions and the price history of the stock are summarized below. The stock pays no dividends.
Time Price Action
0 $90 Buy 3 shares
1 $100 Sell 1 share
2 $100 Sell 1 share
3 $100 Sell 1 share
Calculate the time-weighted arithmetic average return on this portfolio.
Time, Cash Flow, Holding Period Return
0: 3×(-$90) = -$270; -
1: $100; (100-90)/90 = 11.11%
2: $100; 0%
3: $100; 0%
Time-weighted arithmetic average rate of return = (11.11% + 0 + 0)/3 = 3.70%
The arithmetic average is always greater than or equal to the geometric average; the greater the dispersion, the greater the difference.
A manager buys three shares of stock today and then sells one of those shares each year for the next three years. His actions and the price history of the stock are summarized below. The stock pays no dividends.
Time Price Action
0 $90 Buy 3 shares
1 $100 Sell 1 share
2 $100 Sell 1 share
3 $100 Sell 1 share
Calculate the dollar-weighted average return on this portfolio.
Time, Cash Flow, Holding Period Return
0: 3×(-$90) = -$270; -
1: $100; (100-90)/90 = 11.11%
2: $100; 0%
3: $100; 0%
Dollar-weighted average rate of return = IRR = 5.46%
Based on current dividend yields and expected capital gains, the expected rates of return on portfolios A and B are 12% and 16%, respectively. The beta of A is .7, while that of B is 1.4. The T-bill rate is currently 5%, whereas the expected rate of return of the S&P 500 index is 13%. The standard deviation of portfolio A is 12% annually, that of B is 31 %, and that of the S&P 500 index is 18%. If you currently hold a market-index portfolio, would you choose to add either of these portfolios to your holdings?
The alphas for the two portfolios are:
αA = 12% - [5% + 0.7 × (13% - 5%)] = 1.4%
αB = 16% - [5% + 1.4 × (13% - 5%)] = -0.2%
Ideally, you would want to take a long position in Portfolio A and a short position in Portfolio B.
Based on current dividend yields and expected capital gains, the expected rates of return on portfolios A and Bare 12% and 16%, respectively. The beta of A is .7, while that of B is 1.4. The T-bill rate is currently 5%, whereas the expected rate of return of the S&P 500 index is 13%. The standard deviation of portfolio A is 12% annually, that of B is 31 %, and that of the S&P 500 index is 18%. lf instead you could invest only in T-bills and one of these portfolios, which would you choose?
If you will hold only one of the two portfolios, then the Sharpe measure is the appropriate criterion:
s(A) = (0.12 - 0.05)/0.12 = 0.583
s(B) = (0.16 - 0.05)/0.31 = 0.355
Using the Sharpe criterion, Portfolio A is the preferred portfolio.
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.
Index model regression estimates:
1% + 1.2(rM-rf); 2% + 0.8(rM- rf)
R-square:
0.576; 0.436
Residual standard deviation, σ(e)
10.3%; 19.1%
Standard deviation of excess returns
21.6%; 24.9%
Calculate the following statistics for each stock:
i. Alpha
ii. Information ratio
iii. Sharpe ratio
iv. Treynor measure
(i) Alpha = regression intercept
1.0%; 2.0%
(ii)
Information ratio = alpha(p)/(sigma(e)(p))
0.0971; 0.1047
(iii)
*Sharpe measure = (r(p) - r(f))/sigma(p)
0.4907; 0.3373
(iv)
†Treynor measure = (r(p) - r(f))/Beta(p)
8.833; 10.500
* To compute the Sharpe measure, note that for each stock, (rP - rf ) can be computed from the right-hand side of the regression equation, using the assumed parameters rM = 14% and rf = 6%. The standard deviation of each stock's returns is given in the problem.
† The beta to use for the Treynor measure is the slope coefficient of the regression equation presented in the problem.
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.
Index model regression estimates:
1% + 1.2(rM-rf); 2% + 0.8(rM- rf)
R-square:
0.576; 0.436
Residual standard deviation, σ(e)
10.3%; 19.1%
Standard deviation of excess returns
21.6%; 24.9%
Which stock is the best choice under the following circumstances?
i. This is the only risky asset to beheld by the investor
il. 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
Evaluate the market timing and security selection abilities of four managers whose performances are plotted in the accompanying diagrams.
(see diagrams)
We need to distinguish between market timing and security selection abilities. The intercept of the scatter diagram is a measure of stock selection ability. If the manager tends to have a positive excess return even when the market's performance is merely "neutral" (i.e., has zero excess return), then we conclude that the manager has on average made good stock picks. Stock selection must be the source of the positive excess returns.
Timing ability is indicated by the curvature of the plotted line. Lines that become steeper as you move to the right along the horizontal axis show good timing ability. The steeper slope shows that the manager maintained higher portfolio sensitivity to market swings (i.e., a higher beta) in periods when the market performed well. This ability to choose more market-sensitive securities in anticipation of market upturns is the essence of good timing. In contrast, a declining slope as you move to the right means that the portfolio was more sensitive to the market when the market did poorly and less sensitive when the market did well. This indicates poor timing.
We can therefore classify performance for the four managers as follows:
Selection Ability/Timing Ability
A:
Bad / Good
B:
Good / Good
C:
Good / Bad
D:
Bad / Bad
Consider the following information regarding the performance of a money manager in a recent month. The table represents the actual return of each sector of the manager's portfolio in column 1, the fraction of the portfolio allocated to each sector in column 2, the benchmark or neutral sector allocations in column 3, and the returns of sector indices in column 4.
Act. Return Act. Weight Benchmark Weight IndexReturn
Equity 2% 0.70 0.60 2.5% (S&P500)
Bonds 1 0.20 0.30 1.2 (Barcley's Aggregate)
Cash 0.5 0.10 0.10 0.5
What was the manager 's return in the month? What was her overperformance or underperformance?
Actual: (0.7x2%) + (0.2x1%) + (0.1x0.5%) = 1.65%
Benchmark: (0.6x2,5%) + (0.3x1.2%) + (0.1x0.5%) = 1.91%
1.65% < 1.91% => Underperformance (0.26%).
Consider the following information regarding the performance of a money manager in a recent month. The table represents the actual return of each sector of the manager's portfolio in column 1, the fraction of the portfolio allocated to each sector in column 2, the benchmark or neutral sector allocations in column 3, and the returns of sector indices in column 4.
Act. Return Act. Weight Benchmark Weight IndexReturn
Equity 2% 0.70 0.60 2.5% (S&P500)
Bonds 1 0.20 0.30 1.2 (Barcley's Aggregate)
Cash 0.5 0.10 0.10 0.5
What was the contribution of security selection to relative performance?
Security Selection:
Delta of Return:
Equity: -0.5%
Bonds: -0.2%
Cash: 0
Contribution to performance
Equity: -0.5% x 0.7 = - 0.35%
Bonds: -0.2% x 0.2 = -0.04%
Cash: 0 x 0.1 = 0%
=> Total: -0.39%
Consider the following information regarding the performance of a money manager in a recent month. The table represents the actual return of each sector of the manager's portfolio in column 1, the fraction of the portfolio allocated to each sector in column 2, the benchmark or neutral sector allocations in column 3, and the returns of sector indices in column 4.
Act. Return Act. Weight Benchmark Weight IndexReturn
Equity 2% 0.70 0.60 2.5% (S&P500)
Bonds 1 0.20 0.30 1.2 (Barcley's Aggregate)
Cash 0.5 0.10 0.10 0.5
What was the contribution of asset allocation to relative performance? Confirm that the sum of selection and allocation contributions equals her total "excess" return relative to the bogey.
Asset Allocation:
Delta weight:
Equity: 0.1
Bonds: -0.1
Cash: 0
Contribution to performance
Equity: 0.1 x 2.5% = 0.25%
Bonds: -0.1 x 1.2% = -0.12%
Cash: 0 x 0.5% = 0%
=> Total: 0.13%
Contribution of security selection = -0.39%
Contribution of asset allocation = 0.13%
Total = -0.26% == Underperformance -0.26%
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 proportions she finds desirable. Results for a given month are contained in the following table:
Country; Weight in MSCI Index; Manager's Weight; Manager's Return in Country; Return of Stock Index for that Country
U.K. 0.15 0.30 20% 12%
Japan 0.30 0.10 15% 15%
U.S. 0.45 0.40 10% 14%
Germany 0.10 0.20 5% 12%
Calculate the total value added of all the manager's decisions this period (i.e. return difference between manager portfolio and Bogey).
Manager: (0.30 × 20%) + (0.10 × 15%) + (0.40 × 10%) + (0.20 × 5%) = 12.50%
Bogey: (0.15 × 12%) + (0.30 × 15%) + (0.45 × 14%) + (0.10 × 12%) = 13.80
Added value: -1.30%
A global equity manager is assigned to select Stocks from a universe of !arge 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 proportions she finds desirable. Results for a given month are contained in the following table:
Country; Weight in MSCI Index; Manager's Weight; Manager's Return in Country; Return of Stock Index for that Country
U.K. 0.15 0.30 20% 12%
Japan 0.30 0.10 15% 15%
U.S. 0.45 0.40 10% 14%
Germany 0.10 0.20 5% 12%
Calculate the total value added (or subtracted) by her country allocation decisions.
Added value from country allocation:
Country / Excess Weight (Manager-Benchmark) / Index Return minus Bogey:
UK / 0.15 / -1.8%
Japan / -0.20 / 1.2%
US / -0.05 / 0.2%
Ger / 0.10 / -1.8%
Contribution of country allocation = -0.70%
A global equity manager is assigned to select Stocks from a universe of !arge 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 proportions she finds desirable. Results for a given month are contained in the following table:
Country; Weight in MSCI Index; Manager's Weight; Manager's Return in Country; Return of Stock Index for that Country
U.K. 0.15 0.30 20% 12%
Japan 0.30 0.10 15% 15%
U.S. 0.45 0.40 10% 14%
Germany 0.10 0.20 5% 12%
Calculate the total value added from her stock selection ability within countries.
Added value from stock selection:
(1) Differential Return within Country (Manager - Index)
(2) Manager's Country weight
(3) Contribution to Performance [(3) = (1) x (2)]
Country / (1) / (2) / (3)
UK / 0.08 / 0.30% / 2.4%
Japan / 0.00 / 0.10 / 0.0
US / -0.04 / 0.4 / -1.6
Ger / -0.07 / 0.20 / -1.4
=> Contribution of stock selections = -0.6%
Summary:
Country allocation = -0.70%
Stock selection = -0.60
A global equity manager is assigned to select Stocks from a universe of !arge 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 proportions she finds desirable. Results for a given month are contained in the following table:
Country; Weight in MSCI Index; Manager's Weight; Manager's Return in Country; Return of Stock Index for that Country
U.K. 0.15 0.30 20% 12%
Japan 0.30 0.10 15% 15%
U.S. 0.45 0.40 10% 14%
Germany 0.10 0.20 5% 12%
Confirm that the sum of the contributions to value added from her country allocation plus security selection decisions equals total over- or underperformance.
Added value from stock selection:
(1) Differential Return within Country (Manager - Index)
(2) Manager's Country weight
(3) Contribution to Performance [(3) = (1) x (2)]
Country / (1) / (2) / (3)
UK / 0.08 / 0.30% / 2.4%
Japan / 0.00 / 0.10 / 0.0
US / -0.04 / 0.4 / -1.6
Ger / -0.07 / 0.20 / -1.4
=> Contribution of stock selections = -0.6%
Summary:
Country allocation = -0.70%
Stock selection = -0.60
Betrachten Sie das Beispiel zur Performance Attribution aus der Vorlesung in den Tabellen 24.6 - 24.9.
a) Weisen Sie mit den Renditen für die Sektoren nach, dass die Rendite des S&P500 gleich 5,81% ist.
Das gemanagte Aktienportfolio erzielt für die in Tabelle 24.8 aufgeführten Sektoren folgende Renditen: 7,04% (Rohmaterialien), 7,142%, 4,30%, 9%, 10,2%, 5,2%, 2,7%, 0,4% (Technologie).
b) Berechnen die Rendite des gemanagten Aktienportfolios.
c) Weisen Sie mit den Überrenditen für die Sektoren nach, dass der Beitrag der Wertpapierselektion 0,18% ist.
a) Aus der Tabellen 24.8 kann man die Gewichte der Sektoren im S&P500 und die Renditen der Sektoren entnehmen. Die gewichtete Summe der Renditen ist
0,083·6,9%+0,041·7,0%+0,078·4,1%+0,125·8,8%+0,204·10,0%+0,218·5,0%+0,142·2,6%+0,109·0,3%=5,81%
b) Aus der Tabelle 24.8 kann man die Gewichte der Sektoren im Portfolio entnehmen und in der Aufgabenstellung angegeben sind die Renditen des Portfolios für die Sektoren. Die gewichtete Summe der Renditen ist
0,0196·7,04%+0,0784·7,142%+0,0187·4,3%+0,0847·9%+0,4037·10,2%+0,2401·5,2%+0,1353·2,7%+0,0195·0,4%=7,28%.
c) Der Beitrag der Wertpapierselektion ist die mit den Gewichten der Sektoren im Portfolio gewichtete Summe über die Renditedifferenzen.
Die Renditedifferenzen sind für Rohmaterial 0,14% (= 7,04% - 6,9%), für Dienstleistungen 0,142%, für Kapitalgüter 0,2%, für zyklische Konsumgüter 0,2%, für nichtzyklische Konsumgüter 0,2%, für Kredite 0,2 %, für Energie 0,1%, für Technologie 0,1%.
Die gewichtete Summe ist 0,14%·0,0196 + ... + 0,1%·0,0195 = 0,18% = Performancebeitrag zur Wertpapierselektion
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