Al Bundy is evaluating a new advertising program that could increase shoe sales Possible outcomes and probabilities of the outcomes are shown below. Compute the coefficient of variation Additiona Possible outcomes Sales in Units Probabilities Ineffective campaign 40 Normal response emely Solution: Al Bundy Coefficient of variation()=standard deviation/expected value D=∑DP DP 20 8 50 30 140 42 √∑D-DP (D-D)(D-D)2 P (D-D)2P 4080 40 1600 20 320 400 200 1408 +60 1.080 1.600 600=40 40 50 80 CopyrightC 2005 by The McGray-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-482 13-3. Al Bundy is evaluating a new advertising program that could increase shoe sales. Possible outcomes and probabilities of the outcomes are shown below. Compute the coefficient of variation. Possible Outcomes Additional Sales in Units Probabilities Ineffective campaign 40 .20 Normal response 60 .50 Extremely effective 140 .30 Solution: Al Bundy Coefficient of variation (V) = standard deviation/expected value. D = DP D 40 60 140 P .20 .50 .30 DP 8 30 42 80 = D (D D) P 2 = − D D (D− D) 2 (D − D) P (D D) P 2 − 40 80 –40 1,600 .20 320 60 80 –20 400 .50 200 140 80 +60 3,600 .30 1,080 1,600 .50 80 40 V 1,600 40 = = = =
Possible outcomes for three investment alternatives and their probabilities of occurrence are given below Alternative 1 Alternative 2 Alternative 3 Outcomes Probability Outcomes Probability Outcomes Probability Acceptable 160 Successful 120 244 200 20 Rank the three alternatives in terms of risk from lowest to highest(compute the coefficient of variation Solution: Alternative 1 Alternative 2 Alternative 3 DⅹP=DP DⅹP=DP DⅹP=DP $502$10$903$27$804$32 80432 160580 200 100 120448 200240400140 D=$90 D=$147 D=$172 Standard Deviation alternative 1 D (D-D)(D-D D-D)P $50$90$-40 $1.600 $320 10 100 4 40 12090+30 900 $720 720=$2683=a by The M
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-483 13-4. Possible outcomes for three investment alternatives and their probabilities of occurrence are given below. Alternative 1 Alternative 2 Alternative 3 Outcomes Probability Outcomes Probability Outcomes Probability Failure 50 .2 90 .3 80 .4 Acceptable 80 .4 160 .5 200 .5 Successful 120 .4 200 .2 400 .1 Rank the three alternatives in terms of risk from lowest to highest (compute the coefficient of variation). Solution: Alternative 1 D x P = DP Alternative 2 D x P = DP Alternative 3 D x P = DP $ 50 .2 $10 80 .4 32 120 .4 48 D = $90 $ 90 .3 $ 27 160 .5 80 200 .2 40 D = $147 $ 80 .4 $ 32 200 .5 100 400 .1 40 D = $172 Standard Deviation Alternative 1 D D (D− D) 2 (D − D) P (D D) P 2 − $ 50 $90 $–40 $1,600 .2 $320 80 90 –10 100 .4 40 120 90 +30 900 .4 360 $720 720 = $26.83 =
13-4. Continued Alternative 2 ①D-D)(D-D) D-D)P $90$147$-57 $3,249 3$974.70 160147 13 169 84.50 200147 53 2,809 561.80 √1621=$40.26=a Alternative 3 D D-D (D-D (D-D)P $80$172$-92 $8464 4$3,38560 200172 28 784 9200 400172+228 51984 15.19840 8,97600 8,976=$9474=σ Rank by Coefficient of Variation Coefficient of Variation(V)=Standard Deviation/Expected value 40.26 Alternative 2 =.274 147 $26.83 Alternative 1 =.298 94.74 Alternative 3 172 CopyrightC 2005 by The McGray-Hill Companies, Inc
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-484 13-4. Continued Alternative 2 D D (D− D) 2 (D − D) P (D D) P 2 − $ 90 $147 $–57 $3,249 .3 $ 974.70 160 147 +13 169 .5 84.50 200 147 +53 2,809 .2 561.80 $1,621.00 1,621 = $40.26 = Alternative 3 D D (D− D) 2 (D − D) P (D D) P 2 − $ 80 $172 $–92 $ 8,464 .4 $3,385.60 200 172 +28 784 .5 392.00 400 172 +228 51,984 .1 5,198.40 $8,976.00 8,976 = $94.74 = Rank by Coefficient of Variation Coefficient of Variation (V) = Standard Deviation/Expected Value V Alternative 2 .274 147 40.26 = Alternative 1 .298 90 $26.83 = Alternative 3 .551 172 94.74 =
Five investment alternatives have the following returns and standard deviations of returns Alternative Returns: Standard Expected value Deviation ABCDE 4.000 600 4,000 8.000 3,200 10.000 900 Using the coefficient of variation, rank the five altematives from lowest risk to highest risk Solution: Coefficient of variation (V)=standard deviation/mean return Ranking from lowest to highest A$1,200$5,000=24 E(.09) B$600/$4.000=.15 B(15) C$800/$4,000=20 C(20) D$3,200/$8,000=40 A(24) E$900/$10.000=.09 D(40 13-6 In problem 5, if you were to choose between Alternative B and C only, would you need to use the coefficient of variation? why? Solu Since b and c have the same expected value they can be evaluated based on their standard de ofreturn C has a I deviation and so is riskier than b for the same expected return S-485 by The M
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-485 13-5. Five investment alternatives have the following returns and standard deviations of returns. Alternative Returns: Expected Value Standard Deviation A................. $ 5,000 .............. $1,200 B................. 4,000 .............. 600 C................. 4,000 .............. 800 D................. 8,000 .............. 3,200 E ................. 10,000 .............. 900 Using the coefficient of variation, rank the five alternatives from lowest risk to highest risk. Solution: Coefficient of variation (V) = standard deviation/mean return Ranking from lowest to highest A $1,200/$5,000 = .24 E (.09) B $600/$4,000 = .15 B (.15) C $800/$4,000 = .20 C (.20) D E $3,200/$8,000 = .40 $900/$10,000 = .09 A (.24) D (.40) 13-6. In problem 5, if you were to choose between Alternative B and C only, would you need to use the coefficient of variation? Why? Solution: Since B and C have the same expected value, they can be evaluated based on their standard deviations of return. C has a larger standard deviation and so is riskier than B for the same expected return
Tom Fears is highly risk-averse while Sonny Outlook actually enjoys taking a a. Which one of the four investments should Tom choose? Compute the coefficients of variation to help you in your choice b. Which one of the four investments should Sonny choose? Returns Standard Investments Expected valu Deviatio 7000 Buy bonds 5.000 1.560 Buy commodity futures 12000 15.100 Buy options 8.000 8.850 Solution: Coefficient of variation(V)=standard deviation/expected value Buy stocks $4.000/$7,000=571 Buy bonds $l560/$5,000=312 BI dity futures $15,100/$12,000=1.258 Buy options $8.850/$8.000=1.106 a. Tom should buy the bonds because bonds have the lowest coefficient of variation b. Sonny should buy the commodity futures because they have the highest coefficient of variation by The McGraw-Hill Compa
Copyright © 2005 by The McGraw-Hill Companies, Inc. S-486 13-7. Tom Fears is highly risk-averse while Sonny Outlook actually enjoys taking a risk. a. Which one of the four investments should Tom choose? Compute the coefficients of variation to help you in your choice. b. Which one of the four investments should Sonny choose? Investments Returns – Expected Value Standard Deviation Buy stocks $ 7,000 $ 4,000 Buy bonds 5,000 1,560 Buy commodity futures 12,000 15,100 Buy options 8,000 8,850 Solution: Coefficient of variation (V) = standard deviation/expected value. Buy stocks $4,000/$7,000 = .571 Buy bonds $1,560/$5,000 = .312 Buy commodity futures $15,100/$12,000 = 1.258 Buy options $8,850/$8,000 = 1.106 a. Tom should buy the bonds because bonds have the lowest coefficient of variation. b. Sonny should buy the commodity futures because they have the highest coefficient of variation