FM12 Ch 06 Tool Kit - Chapter 6. Tool Kit for Risk and Return PDF

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Chapter 6. Tool Kit for Risk and Return, Chapter 6. Tool Kit for Risk and Return...

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F 4/11/2010

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Chapter 6. Tool Kit for Risk and Return

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RETURNS ON INVESTMENTS (Section 6.1)

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Figure 6-1. Probability Distributions for Sale.Com and Basic Foods Inc.

26 27 28 29 30 31 32 33

Demand for the Probability of this Company's Products Demand Occurring

Amount invested Amount received in one year Dollar return (Profit) Rate of return = Profit/Investment =

$1,000 $1,100 $100 10%

STAND-ALONE RISK (Section 6.2) The relationship between risk and return is a fundamental axiom in finance. Generally speaking, it is totally logical to assume that investors are only willing to assume additional risk if they are adequately compensated with additional return. This idea is rather fundamental, but the difficulty in finance arises from interpreting the exact nature of this relationship (accepting that risk aversion differs from investor to investor). Risk and return interact to determine security prices, hence it is of paramount importance in finance.

PROBABILITY DISTRIBUTION A probability distribution is a listing of all possible outcomes and their corresponding probabilities.

Strong Normal Weak

0.30 0.40 0.30 1.00

Rate of Return on Stock if this Demand Occurs Sale.com Basic Foods 90% 45% 15% 15% −60% −15%

A 34 35 36 37 38 39 40 41 42

C

D

E

F

EXPECTED RATE OF RETURN The expected rate of return is the rate of return that is expected to be realized from an investment. It is found as the weighted average of the probability distribution of returns.

Figure 6-2. Calculation of Expected Rates of Return: Payoff Matrix Demand for the

Probability of this

Company's Products Demand Occurring (1) (2)

43 44 45 46 47

Strong Normal Weak

48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78

B

Sale.com Rate of Return (3)

0.3 0.4 0.3 1.0 Expected Rate of Return = Sum of Products =

Basic Foods

Product (2) x (3) = (4)

90% 15% −60%

27.0% 6.0% −18.0%

r^ =

15.0%

Rate of Return (5)

Product (2) x (5) = (6)

45% 15% −15%

13.5% 6.0% −4.5%

r^ =

15.0%

MEASURING STAND-ALONE RISK: THE STANDARD DEVIATION The standard deviation is a measure of a distribution's dispersion. Figure 4. Probability Distributions of Sale.com's and Basic Foods' Rates of Return Panel a. Sale.com Probability of Occurrence

Panel b. of Basic Foods Probability Occurrence

0.40

0.40

0.30

0.30

0.20

0.20

0.10

0.10

0.00 -75

-60

-45

-30

-15

0

15

30

Expected Rate of Return

45

60 Rate 75 of 90 Return (%)

0.00 -75

-60

-45

-30

-15

0

15

Expected R of Retur

A 79 80 81 82 83 84 85 86 87 88

B

C

D

E

F

Calculating Standard Deviation Here are the steps used to calculate the standard deviation. First, find the differences of all the possible returns from the expected return. Second, square those differences. Third, multiply the squared numbers by the probability of their occurrence. Fourth, find the sum of all the weighted squares. Finally, take the square root of that number. Here are the calculations for Sale.com and Basic Foods.

Figure 6-5. Calculating Sale.com's and Basic Foods' Standard Deviations

89 90

Sale.com

Panel a.

92 93 94

Probability of Occurring (1) 0.3 0.4 0.3

95

1.0

91

Rate of Return on Stock Expected Return (2) (3) 90% 15% 15% 15% −60% 15%

96 97 98

99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117

Deviation from Squared Expected Deviation Return (2) − (3) = (4) (4)2 = (5) 75.0% 56.25% 0.0% 0.00% −75.0% 56.25%

Sq. Dev. × Prob. (5) x (1) = (6) 16.88% 0.00% 16.88%

Sum = Variance =

33.75%

Std. Dev. = Square root of variance =

58.09%

Basic Foods

Panel b. Probability of Occurring (1) 0.3 0.4 0.3

Rate of Return on Stock Expected Return (2) (3) 45% 15% 15% 15% −15% 15%

1.0

Deviation from Squared Expected Deviation Return (2) − (3) = (4) (4)2 = (5) 30.0% 9.00% 0.0% 0.00% −30.0% 9.00%

Sum = Variance = Std. Dev. = Square root of variance =

Sq. Dev. × Prob. (5) x (1) = (6) 2.70% 0.00% 2.70% 5.40% 23.24%

If Sales.com's and Basic Foods' stock return distributions are from normal distributions, then we can find confiden 0.6826 Sale.com Basic Foods

Expected Return 15% 15%

Std. Deviation 58.09% 23.24%

USING HISTORICAL DATA TO MEASURE RISK

1-s s range around expected return -43.09% to 73.09% -8.24% to 38.24%

A 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148

B

C

D

E

Figure 6-7. Standard Deviation Based On a Sample of Historical Data Year 2008 2009 2010 Average =AVERAGE(D122:D124) = Standard deviation =STDEV(D122:D124) =

Realized return 15.0% −5.0% 20.0% 10.0% 13.2%

MEASURING STAND-ALONE RISK: THE COEFFICIENT OF VARIATION The coefficient of variation indicates the risk per unit of return, and it is calculated by dividing the standard deviation by the expected return.

Sale.com Basic Foods

Std. Dev. 58.09% 23.24%

Expected return 15% 15%

RISK IN A PORTFOLIO CONTEXT

CV 3.87 1.55

(Section 6.3)

Portfolio Expected Return The expected return on a portfolio is simply a weighted average of the expected returns of the individual assets in the portfolio. The weights are the percentage of total portfolio funds invested in each asset. Consider the following portfolio and the hypothetical illustrative returns data.

Figure 6-8. Expected Returns on a Portfolio of Stocks

149 150 Stock 151 152 153 154 155 156

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Southwest Airlines Starbucks FedEx Dell Total investment =

Amount of Investment

Portfolio Weight

$300,000 $100,000 $200,000 $400,000 $1,000,000

0.3 0.1 0.2 0.4 1.0

157 158 159 Portfolio Standard Deviation

Expected Return 15.0% 12.0% 10.0% 9.0%

Portfolio's Expected Return =

Weighted Expected Return 4.5% 1.2% 2.0% 3.6%

11.3%

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160 161 162 163 164 165 166 167

Portfolios of stocks are created to diversify investors from unnecessary risk. The diversifiable, or idiosyncratic, risk is eliminated as more stocks are added. Diversification effects are strongest when combining uncorrelated assets. The following figures illustrate how creating two-stock portfolios with different correlations between the stocks affects the expected return and risk of various fictional portfolios.

Figure 6-9. Portfolio Risk: Perfect Negative Correlation

168 169 170 171

Return

Return

Stock W

Return

Stock M

40%

40%

40%

30%

30%

30%

20%

20%

20%

10%

10%

10%

Portfolio WM

172 173 174 175 176 177 178 179

0%

180 181

-10%

2010

0%

-10%

2010

0%

2

-10%

182 Stock W Stock M 183 Weights 0.5 0.5 184 185 Portfolio WM Stock W Stock M Year 186 2006 40% -10% 15% 187 2007 -10% 40% 15% 188 2008 35% -5% 15% 189 2009 -5% 35% 15% 190 2010 15% 15% 15% 191 Average return = 15.00% 15.00% 15.00% 192 22.64% 22.64% 0.00% 193 Standard deviation = 194 Correlation coefficient = -1.00 195 196 197 198 CONCLUSION: When two stocks are perfectly negatively correlated, diversification is its strongest, and in this 199 case the portfolio return is a certain (no risk) 15%. Of course, this situation is very rare. 200 201 202 203 Figure 6-10. Portfolio Risk: Perfect Positive Correlation

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B

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204 205 206 207

Return

Stock W

Return

Return

Stock W'

40%

40%

40%

30%

30%

30%

20%

20%

20%

10%

10%

10%

Portfolio WW'

208 209 210 211 212 213 214 215

0%

216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239

0%

2010

-10%

2010

-10%

0%

2

-10%

Weights

Stock W 0.5

Stock W' 0.5

Year 2006 2007 2008 2009 2010 Average return = Standard deviation =

Stock W 40% -10% 35% -5% 15% 15.00% 22.64%

Stock W' 40% -10% 35% -5% 15% 15.00% 22.64%

Portfolio WW'

40% -10% 35% -5% 15% 15.00% 22.64%

Correlation coefficient =

1.00

CONCLUSION: When two stocks are perfectly positively correlated, diversification has no effect, and the portfolio is a weighted average of its individual stocks' risks. Note that in this graph only the portfolio returns are visible, bu realize that the stocks' returns follow an identical path.

Figure 6-11. Portfolio Risk: Imperfect (Partial) Correlation

240 241 242 243

Return

Stock W

Return

Stock Y

Return

40%

40%

40.00%

30%

30%

30.00%

244 245 246

Portfolio WY

A 247

B

C

D

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20%

20%

20.00%

10%

10%

10.00%

F

248 249 250 251

0%

252 253

0%

2010

-10%

2010

-10%

0.00%

2

-10.00%

254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281

Weights

Stock W 0.5

Stock Y 0.5

Year 2006 2007 2008 2009 2010 Average return = Standard deviation =

Stock W 40% -10% 35% -5% 15% 15.00% 22.64%

Stock Y 40% 15% -5% -10% 35% 15.00% 22.64% Correlation coefficient =

Portfolio WY

40.00% 2.50% 15.00% -7.50% 25.00% 15.00% 18.62% 0.35

CONCLUSION: In the case where two stocks are somewhat correlated, diversification is effective in lowering portfolio risk. Here, the portfolio return is an average of the stock returns and risk is reduced from 22.64% for the individual stocks to 18.62% for the portfolio. Notice that the portfolio's return is always between that of the two stocks. If more similarly-correlated stocks were added, risk would continue to fall, but as we shall see, there is a limit to how low risk (the portfolio's SD) can go.

Contribution to Market Risk: Beta The beta coefficient measures the amount of risk that a stock contributes to a well-diversified portfolio. It also reflects the tendency of a stock to move up and down with the market. Shown below in the chart and in the table are the returns for three stocks and for the stock market.

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282 Figure 6-13. Relative Returns of Stocks H, A, and L 283 284 Returns on Stocks H, A, and L

285 286

40.0%

287

Stock H: b = 1.5

288 289

Stock A: b = 1.0

290 291 292

Stock L: b = 0.5

293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323

-40.0%

0.0%

0.0%

40.0%

Return on the Market

-40.0%

Year 1 2 3 Average = Standard deviation = Beta =

Market 19.0% 25.0% -15.0% 9.7% 21.6%

Historical Returns Stock H 26.0% 35.0% -25.0% 12.0% 32.4% 1.5

Stock A 19.0% 25.0% -15.0% 9.7% 21.6% 1.0

Stock L 12.0% 15.0% -5.0% 7.3% 10.8% 0.5

Note: These three stocks plot exactly on their regression lines. This indicates that they are exposed only to market risk. Portfolios that concentrate on stocks with betas of 1.5, 1.0, and 0.5 have patterns similar to those shown in the graph. Standard deviation is calculated with the Excel STDEV function because the data come from an historical sample.

Probability Distributions for H, A, and L

324 325 Notice that Stock L has the lowest average return, but it also has the tightest distribution. On the other hand, 326 Stock H has the highest average return, but the widest distribution. 327

A B 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 -80.0% -60.0% -40.0% -20.0% 348 349 350 351 Calculating Beta for H, A, and L 352 353 First, calculate correlation and covariance. 354 Correlation of stock 355 with Market, ri,M 356 357 358 359 360 361 362 363 364 365 366 367

C

D

E

Stock L

Stock A

Stock H

0.0%

20.0%

40.0%

60.0%

1.00

1.00

1.00

Covariance of stock with Market, COVi,M

6.98%

4.65%

2.33%

Method 1: bi = ri,M (si / sM)

1.5

1.0

0.5

Method 2: bi = COVi,M / (sM)2

1.5

1.0

0.5

1.5

1.0

0.5

Method 3: Slope of regression Beta =

F

80.0%

100.0%

A B C 368 369 CALCULATING BETA COEFFICIENTS (Section 6.4) 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394

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Now we show how to calculate beta for an actual company, General Electric. Step 1. Retrive Data We downloaded stock prices and dividends from http://finance.yahoo.com for General Electric, using its ticker symbol GE, and for the S&P 500 Index ( symbol ^SPX), which contains 500 actively traded large stocks. For example, to download the GE data, enter its ticker symbol in the upper left section and click Go. Then select Historical Prices from the upper left side of the new page. After the daily prices come up, click monthly prices, enter a start and stop date, and click "Get Prices." When presenting monthly data, the date shown is for the first date in the month, but the data are actually for the last day of trading in the month, so be alert for this. Note that these prices are "adjusted" to reflect any dividends or stock splits. Scroll to the bottom of the page and click "Download to Spreadsheet." The downloaded data are in csv format. Convert to xls by opening a new Excel worksheet, copying the date and adjusted index price data to it, and saving as an xls file. Then repeat the process to get the S&P index data. At this point you have returns data for GE and the S&P Index, as we show below. Step 2. Calculate Returns Next, calculate the percentage change in adjusted prices (which already reflect dividends) for GE and the S&P to obtain returns, with the spreadsheet set up as shown below. At this point, we are ready to calculate some statistics and to find GE's beta coefficient. This is shown below the data.

Figure 6-14. Stock Return Data for GE and the S&P 500 Index

395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414

D

Month March 2009 February 2009 January 2009 December 2008 November 2008 October 2008 September 2008 August 2008 July 2008 June 2008 May 2008 April 2008 March 2008 February 2008 January 2008 December 2007 November 2007 October 2007 September 2007

Market Level (S&P 500 Index) at Month End 797.87 735.09 825.88 903.25 896.24 968.75 1,164.74 1,282.83 1,267.38 1,280.00 1,400.38 1,385.59 1,322.70 1,330.63 1,378.55 1,468.36 1,481.14 1,549.38 1,526.75

GE Adjusted Stock Price at Market's Month End Return 8.5% $10.11 -11.0% $8.51 -8.6% $11.78 0.8% $15.74 -7.5% $16.37 -16.8% $18.60 -9.2% $24.30 1.2% $26.43 -1.0% $26.61 -8.6% $25.10 1.1% $28.57 4.8% $30.42 -0.6% $34.43 -3.5% $30.83 -6.1% $32.59 -0.9% $34.17 -4.4% $35.00 1.5% $37.62 3.6% $37.84

GE's Return 18.8% -27.8% -25.2% -3.8% -12.0% -23.5% -8.1% -0.7% 6.0% -12.1% -6.1% -11.6% 11.7% -5.4% -4.6% -2.4% -7.0% -0.6% 7.2%

415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458

A August 2007 July 2007 June 2007 May 2007 April 2007 March 2007 February 2007 January 2007 December 2006 November 2006 October 2006 September 2006 August 2006 July 2006 June 2006 May 2006 April 2006 March 2006 February 2006 January 2006 December 2005 November 2005 October 2005 September 2005 August 2005 July 2005 June 2005 May 2005 April 2005 March 2005

B

C 1,473.99 1,455.27 1,503.35 1,530.62 1,482.37 1,420.86 1,406.82 1,438.24 1,418.30 1,400.63 1,377.94 1,335.85 1,303.82 1,276.66 1,270.20 1,270.09 1,310.61 1,294.87 1,280.66 1,280.08 1,248.29 1,249.48 1,207.01 1,228.81 1,220.33 1,234.18 1,191.33 1,191.50 1,156.85 1,180.59

Description of Data Average return (annual): Standard deviation (annual): Minimum monthly return: Maximum monthly return: Correlation between GE and the market: Beta: bGE = rGE,M (sGE / sM) Beta (using the SLOPE function): Intercept (using the INTERCEPT function): R2 (using the RSQ function):

D 1.3% -3.2% -1.8% 3.3% 4.3% 1.0% -2.2% 1.4% 1.3% 1.6% 3.2% 2.5% 2.1% 0.5% 0.0% -3.1% 1.2% 1.1% 0.0% 2.5% -0.1% 3.5% -1.8% 0.7% -1.1% 3.6% 0.0% 3.0% -2.0% NA

-8.5% 15.9% -16.8% 8.5%

E $35.29 $35.19 $34.75 $33.87 $33.22 $31.87 $31.47 $32.24 $33.28 $31.32 $31.17 $31.34 $30.02 $28.81 $29.05 $29.97 $30.26 $30.43 $28.76 $28.44 $30.44 $30.80 $29.24 $29.03 $28.79 $29.55 $29.68 $31.05 $30.82 $30.70

F 0.3% 1.3% 2.6% 2.0% 4.2% 1.3% -2.4% -3.1% 6.3% 0.5% -0.5% 4.4% 4.2% -0.8% -3.1% -1.0% -0.6% 5.8% 1.1% -6.6% -1.2% 5.3% 0.7% 0.8% -2.6% -0.4% -4.4% 0.7% 0.4% NA

-22.9% 28.9% -27.8% 18.8% 0.76 1.37 1.37 -0.01 0.57

459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503

A B C D E Step 3. Examine the Data Using the AV...