Assignment 6 - Business & Finance
Please check the attachment below
Instruction:
1. According to these documents named “Assignment 6 data”, “Chapter 6” and “Chapter 7”
2.
Complete the multiple questions in the “Assignment 6 questions” documents.
3. Please calculate the problem and write out the full steps and explanations.
Due on Tuesday, October 26th , Before 10 PM, US central time. (Chicago)
Assignment 6
1. Do Chapter 10 Problem 6 in BKM [Assume aA = 0 and aB = 0]
2. Do Chapter 10 Problem 7 in BKM
3. A mean-variance investor has relative risk aversion of 5. The investor chooses between a US stock market index and Treasury bills. Assume that the risk premium for the stock market is 6\% and the standard deviation for the stock market is 20\%.
a) What fraction of financial wealth should be allocated to the US stock market for a person at retirement with no human capital?
b) If the individual is an employee of the government (ph s = 0 and rrh = 0 05) and the present value of all future wages is 80\% of the persons total wealth portfolio at their current age. What fraction of financial wealth should be allocated to the US stock market currently?
c) If the individual is an investment banker (ph s = 0 6 and rrh = 0 15) and the present value of all future wages is 80\% of the persons total wealth portfolio at their current age. What fraction of financial wealth should be allocated to the US stock market currently?
d) Assume that there is no human capital remaining upon retirement. Should the fraction of financial wealth allocated to the stock market rise or fall as a government employee approaches retirement? Does this answer to this question change if the person is an investment banker instead? Briefly explain your answers.
4. Compare 10 test portfolios using the CAPM model and 3-factor model. Please do not include the raw data in your solutions.
a) Download assignment6data.xls from the compass2g website.
b) Create a variable for the excess return of each portfolio as the difference between the return on the portfolio and the risk free rate (labeled rf).
c) Regress the portfolio excess return on the excess return of the market (labeled mkt_rf). There should be regression results for ten separate regressions.
d) What is the interpretation of the estimated constant (intercept) in the ten regressions?
e) For which portfolios (portid) is the constant statistically different from zero at a 5\% level of significance and positive? For which of portfolios is the constant statistically different zero at a 5\% level of significance and negative?
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1
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f) According to the constants from the regressions of the excess return of the portfolio on the excess return of the market, does it appear that the constants are increasing or decreasing with the portfolio identifier (portid)? What trading strategy, if any, would you adopt to exploit this pattern?
g) Given your answer to part f), would you conclude that the market is efficient with regard to the information used to create these portfolios?
h) Regress the portfolio excess return on the excess return of the market, the SMB fac- tor, and the HML factor. Again, there should be regression results for another ten separate regressions.
i) For which portfolios (portid) is the constant statistically different from zero at a 5\% level of significance and positive? For which of these portfolios (portid) is the constant statistically different from zero at a 5\% level of significance and negative? What trading strategy, if any, would you adopt to exploit this pattern?
j) Assume that SMB and HML represent for systematic risk factors. Given your answer to part i) would you conclude that the market is efficient with regard to the information used to create these portfolios?
k) According to the other coefficients from the regressions of the excess return of the portfolio on the excess return of the market, SMB, and HML, which portfolios (portid) have a statistically significant loading (different from zero) on the value factor?
ret10
portid portid portid portid portid portid portid portid portid portid
1 2 3 4 5 6 7 8 9 10 rf mkt_rf smb hml
3.363749907 1.8373846339 2.4238167044 2.7717240712 4.4352622425 1.6371196715 2.1505142123 2.1967469137 2.6080115079 2.5342801257 0.3 2.28 -0.17 1.54
2.0758052629 1.2308992591 1.6981292047 0.2822848284 1.3624425159 2.3612695121 2.9527271646 4.365338316 4.9362599875 4.4953748297 0.26 1.46 0.08 2.84
1.1990847488 1.2340837871 1.5438168097 0.2256250126 3.2333360395 3.9939669953 5.188726817 4.0423422292 2.9938557741 3.2814619342 0.31 1.45 0.95 3.43
-2.610619048 2.2382514367 1.6519498895 1.6678177586 0.401413309 -0.2321228375 3.5179636814 0.3711317384 1.5880308795 0.1208978867 0.29 0.17 -1.37 -0.52
2.2956055922 2.7743039733 0.5810235998 0.0758713787 1.3826156983 0.965150015 2.3090682892 3.235157098 2.1399884312 3.3317420318 0.26 1.48 -0.89 1.91
0.158714865 1.8207980874 1.7262901293 3.7218746833 1.0195323934 0.9949191131 0.3764660192 0.9332000585 1.7551874712 4.1887772152 0.3 1.21 -0.3 0.75
0.7079028279 4.3912850391 1.0937957095 2.3670776447 1.8695772225 4.7455235658 1.0763464088 2.2292410688 1.5218268597 1.1383728943 0.3 1.71 0.23 0.69
-2.0727978668 -0.2554871669 -1.3951845037 -1.1516654216 0.4931486505 -0.7584641956 -1.318638448 0.0280594066 -0.1600288143 -2.0972417078 0.28 -1.41 0.09 0.12
3.2301227615 3.7347851162 0.7570008154 3.9651365025 3.4409623444 4.4826588028 3.9794999624 4.3166763238 6.3883782689 6.3087899202 0.28 2.77 -0.51 1.65
-0.8266447475 0.6117391775 1.4949963465 2.4994120461 0.2585157805 0.8243141406 1.5595514143 1.3601261391 2.3742339674 2.291238744 0.29 0.6 0.45 1.12
0.7185210119 0.3414678187 0.1918539064 0.5096054424 1.6481031334 0.6709949657 0.2755634655 0.3801270612 -2.9787023191 -2.5093343037 0.29 0.02 0.61 -1.96
1.155605175 1.1120656422 1.0529724797 0.9956563369 0.5361953748 -2.1912802815 -0.0919486747 -2.9243144229 -1.4189122492 -0.5931582098 0.31 0.06 -0.24 -2.59
6.9678601641 3.6338742974 0.3290717877 3.7839767124 2.622816492 4.8056668117 3.9239389615 4.8860815329 5.8111443677 9.1900987537 0.28 3.58 2.67 0.18
1.0766759261 0.4773691601 -0.7336790035 1.4259992083 2.0703234175 0.7472999064 -0.1206019392 1.8070740504 0.7046847082 2.1046808992 0.3 0.39 3.48 0.21
-1.7794716127 0.1129754785 -1.7149176897 -1.2182734627 -1.9633325899 -1.2881028434 -0.0491884336 0.3147141466 0.9835637657 -1.9444630517 0.36 -1.33 1.77 1.08
4.9001397768 5.0133721259 1.7870277238 2.5922218635 2.4903644749 3.7702782748 3.2623923227 2.4706953115 6.9033338478 6.2128781819 0.31 3.06 1.19 0.7
-0.0244661186 -1.8033601868 0.4989958143 0.4963161888 -0.5702900487 0.4863858524 -0.3290856363 0.5402657166 -2.7278053468 -2.0083506517 0.31 -0.75 0.03 -1.62
-4.6962962758 -5.8296264322 -2.889411685 -5.4034699092 -6.305463681 -5.9753175052 -6.0653311777 -4.4738955622 -7.9144884674 -6.718253691 0.35 -5.54 -4.35 0.57
2.3657878937 0.2687574253 -0.0401067542 0.7223576524 1.5617998641 2.4421712588 4.0345723686 2.0569937309 3.4766523255 4.1935306831 0.31 1.37 0.84 2.24
3.5279553838 3.1127248964 1.1271456221 2.3728279654 4.1906429797 3.4918756465 2.5361665318 2.5141858168 3.6608770769 3.4363419896 0.33 2.76 2.85 -1.06
3.7333163907 4.8520414816 3.7148568264 1.3788130322 2.937653778 1.6566047719 3.2184302869 4.6518857563 4.0843241309 6.3297377351 0.31 2.89 0.6 -0.09
3.9459203126 2.0517307989 1.9734502672 0.0241090858 3.1932074727 4.7639366095 3.616560877 5.0544066745 7.6689467385 5.5261670116 0.31 2.62 2.43 1.58
-0.163273415 0.082342122 0.913508787 -2.4417756704 2.1190943165 2.0365578059 0.8946961423 1.2856722281 0.2205976632 2.3442069671 0.35 -0.04 4.66 0.31
1.2438000716 1.3080182978 0.1810447689 0.0711380777 0.1357995657 2.2543638678 2.387412478 4.3642994096 4.1297761638 2.7882839598 0.33 1.02 2.08 1.98
-0.1945769192 0.58838481 0.4590185613 -0.85156031 0.8990994093 4.4335837142 4.377274937 3.8871406267 3.1467111661 8.8524212638 0.38 0.83 3.86 3.53
-0.8605966082 -1.0958533488 -3.0473065042 -0.9431768319 -0.9056240394 -0.4026773221 -0.0608190444 0.1761360813 0.9078424101 -0.6686566697 0.35 -1.21 4.44 0.43
-0.9345150782 -2.1545265538 -2.1003962659 -4.1423763008 -2.2307946789 -0.5462947088 -2.419764606 -2.2292725888 -2.142372886 -4.4497968263 0.38 -2.47 0.97 -2.06
2.0874606551 2.4840259705 3.8156681321 0.9202022163 2.0270223967 2.5547385389 3.5413089411 4.0142040051 1.406154281 2.9099469729 0.34 2.14 3.44 -0.42
-4.3418067893 -5.2089072074 -4.2625949099 -4.0104324458 -6.9069672182 -5.5002261952 -6.7256280377 -8.0651840193 -7.4256315558 -8.7385460367 0.41 -5.66 -4.66 -1.63
-1.4166881897 -2.2074817217 -2.5128658662 -0.6077863013 -0.3808849778 0.1036855681 1.0858076598 1.9494587111 -0.3999476298 -1.4989232868 0.38 -1.41 1.05 0.59
-1.9993081571 -1.2177005303 -1.1482636469 -1.9883962838 -0.1685256752 -1.1125190337 -1.1807545491 0.4652783419 -0.7984480344 -2.0542891627 0.35 -1.64 -0.33 0.84
-7.297870547 -7.5743252404 -8.8458073878 -7.6925389164 -7.9614533387 -5.3501422317 -7.3090343792 -7.0001628863 -8.0248719238 -9.1317182233 0.41 -7.95 -3.31 0.56
-2.956993477 0.1060428601 -0.2175630268 1.10486619 -0.2059253429 0.1666779874 -0.1120987257 0.786697188 -1.8770817524 -1.1523452085 0.4 -1.1 -1.1 0.53
2.3882341452 1.7915494399 6.1722986343 3.6922139831 7.7166734458 7.5220361655 5.6852725263 4.8775641671 3.3886132171 2.0207277459 0.45 3.78 -6.62 2.91
8.3242692244 0.0686189443 0.5629305372 1.8236706622 -1.7417851622 -1.6541704675 0.4991346972 -0.8865240237 0.1332369387 -0.3922880112 0.4 1.35 4.37 -4.65
-1.0104307997 -0.3282291464 1.4356554759 0.6366543128 1.4955754289 1.7519629576 1.2319931691 0.4074681182 1.4730957357 0.2781129341 0.4 0.22 1.87 -1.33
7.9433473865 9.6561294685 6.4887629736 8.5670475603 3.4708552358 7.4616059724 9.9804134643 9.9586095349 13.7289998003 16.4972064862 0.43 8.12 8.47 1.89
4.331470123 -0.1408172739 1.1685285321 0.5392485321 -0.4999608658 -0.0663425826 -1.4350209071 -0.4884644569 -0.2570454378 1.3338695961 0.36 0.73 3.38 -2.27
4.4659846348 4.9667963794 3.9791246874 3.2327382799 3.7777516824 3.4379207555 5.7168393747 3.99961925 6.3055697632 5.6884224104 0.39 3.95 1.76 0.15
6.9714472794 7.519084575 2.8026653575 3.6680920977 1.2858752492 0.8807119607 3.192683738 3.9669385613 2.8614330399 4.6963518696 0.32 3.84 0.6 -2.61
-5.5051433193 -5.8024415585 -1.9713132114 -3.6459712225 -3.5670061556 -4.3565966747 -3.021242192 -2.118710743 -3.4574135675 -3.1884893226 0.33 -4.26 2 0.81
3.4418057868 0.0899680904 0.3970577438 2.5152184994 1.722690791 3.1126725734 2.0954524511 6.3616149946 4.8226902297 4.2116549434 0.27 2.42 6.01 0.88
2.8143980866 7.1730457636 6.0095285443 5.1744710502 6.0908713161 -0.248779625 7.5003291759 6.0467455995 10.0001209944 7.5626086958 0.32 4.6 3.09 2.69
-1.0382202227 -0.2967876853 -0.793042716 -0.802624091 -1.0054685966 -1.1114722606 -1.2167566812 -0.9798504019 2.5123112217 0.2799831509 0.31 -0.94 0.48 1.48
5.007479422 3.6836705998 3.743410208 1.4163896806 3.429577047 3.3519005432 1.5312905458 4.2432598434 2.5025977982 2.6263025791 0.32 3.11 3.08 -2.46
2.2379139154 -4.0545657041 -3.1956319081 -4.8343450665 -2.2997611279 -3.9448981398 -5.4546659751 -4.6894529659 -4.8465168462 -3.0527355767 0.39 -3.13 1.44 -3.39
2.4079295949 0.5077334728 0.0604988443 2.0316606015 0.1193677231 0.3122879201 -0.2314544871 -0.8515332244 -1.5681212657 0.741445126 0.36 0.43 0.19 -1.72
2.3079286857 1.9738346194 3.2771440989 2.9023859387 4.0311366085 1.7536681569 6.2693528157 5.5374211648 4.6519601347 5.1902359734 0.33 3.04 5.73 -0.44
-8.7533305392 -4.5188013379 -4.254286152 -3.8186791091 -1.9211944433 1.2500614391 -2.4810737345 0.8934977848 -0.9158832512 -0.2198223782 0.4 -4.03 3.9 4.79
-3.7090357768 -2.9523816665 -3.744919104 -2.3269224781 -2.1180683609 -2.339317683 -3.7774872824 -4.5255990316 -3.5724471316 -4.1460985508 0.39 -3.75 -2.93 1.21
4.0597321068 1.0937068224 -0.6159143731 -2.1854047549 0.6813497556 -1.1942054012 -1.3949027813 -1.2389073228 0.8026906129 -0.7095478979 0.38 0.13 -1.27 -0.52
12.0870864618 10.9229961025 8.1251513841 8.3282894483 4.9466964631 4.2983072811 9.867134007 11.1064380746 11.7157900736 14.8306934067 0.43 8.98 5.72 -1.06
4.8023541663 0.1596813278 2.1895430779 1.1385632928 0.5130120393 0.9498102856 0.8540532112 4.9986233928 5.3972405868 5.7603798049 0.45 2.25 6.42 0.85
-1.3959761824 0.6839233099 0.7357729907 2.8326518704 2.7412423919 4.3641627291 2.2443120906 2.7633585602 -1.1512885783 -1.5168636558 0.43 0.72 -0.17 0.73
-5.9345531971 -5.8003225339 -3.9394911001 -1.9475505769 -1.2292584862 -0.9178785844 2.6971952547 0.4443255796 1.5803573528 -1.3715698703 0.48 -2.68 -1.32 5.45
1.0526476467 2.2291764574 1.2257197721 1.3075132772 0.8828115335 2.4088151706 2.2881119858 1.9885680903 2.6887538301 3.2665024553 0.42 1.38 2.35 0.97
0.9971830454 5.4823977317 5.0544633098 6.1708881483 4.4015906691 4.0598158255 3.7667222425 4.9738346869 4.5978128435 8.1929892825 0.43 4.02 2.81 0.24
-3.3920389194 -1.4712046801 2.15428035 2.0528917463 0.928846352 1.7790837417 1.3785167277 3.9867943241 1.9670304708 3.1458625528 0.44 0.46 -0.48 2.96
7.4407471385 8.004213935 4.1072147838 2.6581021797 5.6675826186 6.7156446076 6.858426657 4.7422754621 7.1495696119 2.9074150011 0.42 5.43 2.38 -0.91
-5.0036117576 -2.5036410048 -4.5974994872 -3.7464000857 -3.8661650859 -1.7658520922 -3.2724760923 -1.2863245097 -4.9264131752 -1.5460530624 0.43 -3.82 3.44 0.06
-2.9888419133 -1.9008256293 -0.2973634205 -0.8218832283 0.1000518416 0.2689278979 0.7304254871 -1.6264160815 0.162768977 1.0374364388 0.53 -1.2 -0.79 1.57
-4.3279590905 -6.3144490436 -5.028733157 -4.2047852698 -4.9264989297 -6.1039753694 -5.2215175206 -5.8244756409 -4.4307534153 -6.7119527555 0.46 -5.82 -3.9 0.93
3.903843982 3.8625071698 4.1850758558 2.9965712346 1.4386470562 3.716782583 3.6500253349 2.811487604 2.7855168776 3.534490845 0.46 2.59 -0.26 -0.45
3.7980156321 3.3719728884 3.1315685216 1.401803119 -0.028398972 1.8751811683 -0.2677045761 -0.1758681697 5.8208858145 1.4702315956 0.53 1.52 -0.84 0.04
-0.2472076767 -0.2858934714 -0.7371757736 1.2101938562 0.5514674472 0.1821709676 1.967956958 2.1134289223 0.6061375824 0.5722025052 0.48 0.02 -0.28 0.72
-0.9164113507 -7.1022692393 -6.1163046779 -7.1090066902 -8.0629632942 -8.6302430831 -8.5684547594 -8.4694201181 -6.8034886565 -8.9367966885 0.51 -7.25 -5.36 -1.13
-4.5045666781 -5.0545308553 -6.991139095 -5.5917394496 -8.6046579558 -6.5710754345 -7.5475876803 -7.4039070819 -6.0997766426 -6.5139386489 0.53 -7.05 -3.21 1.38
7.1981311236 7.4099307269 8.0934624033 4.9447879233 5.050680207 3.0496646318 5.1558268153 3.9098364404 3.1779274683 2.0599955975 0.5 4.65 0.9 -3.8
-0.0904844222 -1.0273486465 -0.9251141315 -1.1641620745 -2.1169796133 -1.6555635561 -4.9603492345 -4.072988883 -5.2109683293 -3.1342062034 0.62 -2.88 1.18 -3.2
6.6279862851 6.7380373012 7.0742068293 6.1112301926 5.2204068005 4.9558784425 4.8551472887 2.7725094259 4.1337164422 5.9380787537 0.6 4.96 3.76 -3.11
-1.0455897161 -2.7884315486 -3.7038814035 -3.4909881649 -2.9543598072 -3.4661409078 -3.6208878329 -4.6558947405 -2.7718878391 -5.9343818123 0.52 -3.74 -2.54 -1.11
0.1587771615 2.5483184546 -3.1861599903 -2.136854308 -1.8550197189 -2.0271427413 -1.7066281702 -2.170599304 -4.6026761848 -3.511979525 0.64 -2.61 -3.67 -3.05
-9.0829515602 -7.4198082797 -7.402327532 -6.7063764428 -5.7085482771 -6.8770954382 -9.4354738861 -7.7130327266 -6.0306591638 -4.3370968914 0.6 -7.93 2.9 3.01
2.2612970313 5.5254912129 3.5173511307 4.5363409162 7.7854324408 8.3735787804 6.9308167409 6.9420573205 7.7044813507 7.7255157084 0.62 5.05 -2.37 4.03
-2.5524344118 -4.1735871945 -2.5579708587 0.409144036 -0.8966607085 3.3500556563 0.5710749722 1.4404683393 1.0987265877 1.2044372557 0.57 -1.04 -2.35 4.27
-12.3193091361 -11.287725182 -13.7388094934 -8.7103926165 -11.9697550674 -9.6222892137 -8.7578916715 -8.5338392757 -7.9231946725 -8.6881322309 0.5 -11.03 -6.11 6.34
-7.3367907811 -10.1772890125 -10.7931660713 -5.7861509967 -8.1076538114 -7.0650840666 -4.2723502937 -1.98236846 -3.2177928876 -6.4179234293 0.53 -6.96 -4.46 3.57
-8.1324792772 -4.0977047039 -5.3367414163 -3.7893858804 -4.1675549412 -4.4850370223 -2.4422766244 -2.2080923484 -6.8703759977 -6.9347954148 0.58 -5.69 -2.16 0.79
2.3567688065 9.1206873245 7.4107312593 9.1481901088 7.3605829764 8.8854843888 10.708551952 9.9023648833 8.9165481941 5.3226116929 0.52 6.9 -0.56 1.01
4.7429262008 3.2598128658 4.6882895617 6.7354503229 4.3904680315 5.5844655714 5.6898151582 4.0362605422 6.3026216905 6.3478907805 0.53 4.47 1.52 1.04
8.6676021254 7.4484904393 6.4731097785 1.296311314 1.8729331404 1.9014147061 1.8209994384 1.276315736 4.9969872083 6.9694512375 0.54 4.21 8.61 -5.54
-0.4285395272 -1.6418539167 -1.8341876186 -1.9418357759 -2.7676587442 -4.64348179 1.5103325974 -2.8613186933 -2.505045696 -4.8219410385 0.46 -2.28 -4.2 0.3
4.4036224256 6.0970981187 4.5463375266 6.3770871884 3.9951631246 5.1712227452 5.1844524832 6.4383590648 5.352233657 4.0848643261 0.46 4.58 -4.09 1.68
4.6424028782 4.5478952248 7.3197207607 5.7224469324 7.5857138478 8.3523736995 5.4562151306 6.5224049149 6.8164056088 9.1757943266 0.42 5.65 2.97 1
4.1824953564 4.9152780266 5.9594468947 4.4656393549 5.7978214743 4.571732865 0.9640178782 6.5140675632 6.443940323 9.3777629624 0.38 4.82 7.44 1.34
2.4243795188 2.4938429669 2.2262615947 1.2059364198 1.1869394306 -0.0110053443 3.8026980265 -1.3329194247 1.8234593411 1.2051514871 0.33 1.36 1.9 -1.34
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1.7414106926 4.9549976736 4.2660633626 5.7719323669 6.5851735737 4.4878687845 6.0302518614 4.6948154042 6.3766995742 5.2899835248 0.81 3.88 1.17 1.42
-1.3310024176 1.0744725268 2.0450242192 2.245247066 1.750170315 1.8993634384 1.4508400986 1.8424695985 3.6896780496 3.4311577672 0.77 0.73 1.31 1.75
6.3400271059 7.9845003609 7.6563662051 6.9953438529 7.1482853476 4.1114216397 6.3757730402 5.6423505635 7.5965916564 6.6419896948 0.77 5.7 2.07 -1.53
-1.7153569087 -0.4349291563 -0.0691347189 1.1485992479 1.6205537878 0.6108302162 0.6423706985 -1.1179504914 0.0732663458 -0.5497522873 0.83 -0.69 -0.29 -0.91
-6.5217401516 -6.9607475617 -7.0403674082 -5.8930836486 -7.0783569538 -6.8242214487 -8.2463886637 -8.8603558937 -8.2859625821 …
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that the SML of the CAPM must also apply to well-diversified portfolios simply by virtue
of the “no-arbitrage” requirement of the APT.
Individual Assets and the APT We have demonstrated that if arbitrage opportuni-
ties are to be ruled out, each well-diversified portfolio’s expected excess return must be
proportional to its beta. The question is whether this relationship tells us anything about
the expected returns on the component stocks. The answer is that if this relationship is to
be satisfied by all well-diversified portfolios, it must be satisfied by almost all individual
securities, although a full proof of this proposition is somewhat difficult. We can illustrate
the argument less formally.
Suppose that the expected return–beta relationship is violated for all single assets. Now
create a pair of well-diversified portfolios from these assets. What are the chances that in
spite of the fact that for any pair of assets the relationship does not hold, the relationship
will hold for both well-diversified portfolios? The chances are small, but it is perhaps
possible that the relationships among the single securities are violated in offsetting ways
so that somehow it holds for the pair of well-diversified portfolios.
Now construct yet a third well-diversified portfolio. What are the chances that the
violations of the relationships for single securities are such that the third portfolio also
will fulfill the no-arbitrage expected return–beta relationship? Obviously, the chances
are smaller still, but the relationship is possible. Continue with a fourth well-diversified
portfolio, and so on. If the no-arbitrage expected return–beta relationship has to hold for
each of these different, well-diversified portfolios, it must be virtually certain that the
relationship holds for all but a small number of individual securities.
We use the term virtually certain advisedly because we must distinguish this conclusion
from the statement that all securities surely fulfill this relationship. The reason we cannot
make the latter statement has to do with a property of well-diversified portfolios.
Recall that to qualify as well diversified, a portfolio must have very small positions
in all securities. If, for example, only one security violates the expected return–beta rela-
tionship, then the effect of this violation on a well-diversified portfolio will be too small
to be of importance for any practical purpose, and meaningful arbitrage opportunities
will not arise. But if many securities violate the expected return–beta relationship, the
relationship will no longer hold for well-diversified portfolios, and arbitrage opportuni-
ties will be available. Consequently, we conclude that imposing the no-arbitrage condi-
tion on a single-factor security market implies maintenance of the expected return–beta
relationship for all well-diversified portfolios and for all but possibly a small number of
individual securities.
Well-Diversified Portfolios in Practice
What is the effect of diversification on portfolio standard deviation in practice, where
portfolio size is not unlimited? To illustrate, we work out the residual standard deviation
(SD) of portfolios of different size under ideal conditions, with equal weights on each com-
ponent stock. These calculations appear in Table 10.1. The table shows portfolio residual
SD as a function of the number of stocks. Equally weighted, 1,000-stock portfolios achieve
small but not negligible standard deviations of 1.58\% when residual risk is 50\% and 3.16\%
when residual risk is 100\%.!For 10,000-stock portfolios, the SDs are close to negligible,
verifying that diversification can eliminate residual risk, at least in principle, if the invest-
ment universe is large enough.
What is a “large” portfolio? Many widely held ETFs or mutual funds hold hundreds of
shares, and some funds and indexes such as the Wilshire 5000 contain thousands. Thus, a
portfolio of 1,000 stocks is not unheard of, but a portfolio of 10,000 shares is. Therefore,
for plausible portfolios, the standard deviations in Table 10.1 make it clear that even broad
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a small number of securities. Because it focuses on the no-arbitrage condition, without the
further assumptions of the market or index model, the APT cannot rule out a violation of
the expected return–beta relationship for any particular asset. For this, we need the CAPM
assumptions and its mean-variance dominance arguments.
Moreover, while the APT is built on the foundation of well-diversified portfolios,
we’ve seen, for example in Table 10.1, that even large portfolios may have non-negligible
residual risk. Some indexed portfolios may have hundreds or thousands of stocks, but
active portfolios generally cannot, as there is a limit to how many stocks can be actively
analyzed in search of alpha.!
Despite these shortcomings, the APT is valuable. First, recall that the CAPM requires
that almost all investors be mean-variance optimizers. The APT frees us of this assump-
tion. It is sufficient that a small number of sophisticated arbitrageurs scour the market for
arbitrage opportunities.!
Moreover, when we replace the unobserved market portfolio of the CAPM with an
observed, broad index portfolio that may not be efficient, we no longer can be sure that the
CAPM predicts risk premiums of all assets with no bias. Therefore, neither model is free
of limitations.!
In the end, however, it is noteworthy and comforting that despite the very different
paths they take to get there, both models arrive at the same security market line. Most
important, they both highlight the distinction between firm-specific and systematic risk,
which is at the heart of all modern models of risk and return.
The APT and Portfolio Optimization in a Single-Index Market
The APT is couched in a single-factor market3 and applies with perfect accuracy to
well-diversified portfolios. It shows arbitrageurs how to generate infinite profits if the risk
premium of a well-diversified portfolio deviates from Equation 10.6. The trades executed
by these arbitrageurs are the enforcers of the accuracy of this equation.
In effect, the APT shows how to take advantage of security mispricing when diversi-
fication opportunities are abundant. When you lock in and scale up an arbitrage oppor-
tunity you’re sure to be rich as Croesus regardless of the composition of the rest of your
portfolio—but only if the arbitrage portfolio is truly risk-free! However, if the arbitrage
position is not perfectly well diversified, an increase in its scale (borrowing cash, or
borrowing shares to go short) will increase the risk of the arbitrage position, potentially
without bound. The APT ignores this complication.
Now consider an investor who confronts this single-factor market, and whose
security analysis reveals an underpriced asset (or portfolio), that is, one whose risk
premium implies a positive alpha. This investor can follow the advice woven throughout
Chapters 6, 7, and 8 to construct an optimal risky portfolio. The optimization process will
consider both the potential profit from a position in the mispriced asset, as well as the risk
of the overall portfolio and efficient diversification. As we saw in Chapter 8, the Treynor-
Black (T-B) procedure can be summarized as follows.4
1. Estimate the risk premium and standard deviation of the benchmark (index)
portfolio, RPM and M.
3The APT is easily extended to a multifactor market, as we show later.
4The tediousness of some of the expressions involved in the T-B method should not deter anyone. The calcula-
tions are straightforward using a spreadsheet. The estimation of the risk parameters also is a relatively straight-
forward statistical task. The real difficulty is to uncover security alphas and the macro-factor risk premium, RPM.
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2. Place all the assets that are mispriced into an active portfolio. Call the alpha of the
active portfolio !A, its systematic-risk coefficient A, and its residual risk #(eA).
Your optimal risky portfolio will allocate to the active portfolio a weight, w A *:
w A 0 =
! A $ # 2 ( e A ) __________
E( R M )$ # M 2
; w A * \%=\%
w A 0 _____________
1\%+\% w A 0 (1\%&\%A)
The allocation to the passive portfolio is then\% w A * \%=\%1\%&\% w A * . With this allocation,
the increase in the Sharpe ratio of the optimal portfolio, SP, over that of the
passive portfolio, SM, depends on the information ratio of the active portfolio,
IRA = !A/#(eA).
3. To maximize the Sharpe ratio of the risky portfolio, you maximize the information
ratio of the active portfolio. This is achieved by allocating to each asset in the active
portfolio a portfolio weight proportional to: wAi = !i /#2(ei ).
Now see what happens in the T-B model when the residual risk of the active portfolio
is zero. This is essentially the assumption of the APT, that a well-diversified portfolio
(with zero residual risk) can be formed. When the residual risk of the active portfolio goes
to zero, the position in it goes to infinity. This is precisely the same implication as the
APT: When portfolios are well-diversified, you will scale up an arbitrage position without
bound. Similarly, when the residual risk of an asset in the active T-B portfolio is zero, it
will displace all other assets from that portfolio, and thus the residual risk of the active
portfolio will be zero and will elicit the same extreme portfolio response.
However, we have seen that, in practice, it is unlikely that residual risk can be driven
all the way to zero. When residual risk is not zero, the T-B procedure produces the optimal
risky portfolio, which is a compromise between seeking alpha and shunning potentially
diversifiable risk. In contrast, by assuming residual risk can be diversified away, the APT
ignores it altogether. When residual risk can be made small through diversification, the
T-B model prescribes very aggressive (large) positions in mispriced securities that exert
great pressure on equilibrium risk premiums to eliminate nonzero alpha values. The T-B
model does what the APT is meant to do, but with more flexibility in terms of accommo-
dating the practical limits to diversification. In this sense, Treynor and Black anticipated
the development of the APT.
We have assumed so far that only one systematic factor affects stock returns. This simpli-
fying assumption is in fact too simplistic. We’ve noted that it is easy to think of several
factors driven by the business cycle that might affect stock returns: interest rate fluctua-
tions, inflation rates, and so on. Presumably, exposure to any of these factors will affect a
stock’s risk and hence its expected return. We can derive a multifactor version of the APT
to accommodate these multiple sources of risk.
Suppose that we generalize the single-factor model expressed in Equation 10.1 to a two-
factor model:
Ri\%=\%E(Ri )\%+\%i1 F1\%+\%i 2 F2\%+\%ei (10.7)
In Example 10.2, factor 1 was the departure of GDP growth from expectations, and factor 2
was the unanticipated change in interest rates. Each factor has zero expected value because
each measures the surprise in the systematic variable rather than the level of the variable.
10.4 A Multifactor APT
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Similarly, the firm-specific component of unexpected return, ei, also has zero expected
value. Extending such a two-factor model to any number of factors is straightforward.
The benchmark portfolios in the APT are!factor portfolios, which are well-diversified
portfolios constructed to have a beta of 1 on one of the factors and a beta of zero on
any other factor. We can think of each factor portfolio as a tracking portfolio. That is,
the returns on such a portfolio track the evolution of one particular source of macroeco-
nomic risk but are uncorrelated with other sources of risk. It is possible to form such factor
portfolios because we have a large number of securities to choose from, and a relatively
small number of factors. The multidimensional SML predicts that the contribution of each
source of risk to the security’s total risk premium equals the factor beta times the risk
premium of the factor portfolio tracking that source of risk. We illustrate with an example.
Suppose that the two factor portfolios, portfolios ! and , have expected returns E(r!) = !#\%
and E(r) = !\% and that the risk-free rate is $\%. The risk premium on the first factor portfolio
is !#\% $\% = \%\%, and that on the second factor portfolio is !\% $\% = &\%.
Now consider a well-diversified portfolio, portfolio A, with beta on the first factor portfolio,
#A! = ., and beta on the second factor portfolio, #A = .(. The multifactor APT states that
the overall risk premium on this portfolio must equal the sum of the risk premiums required as
compensation for each source of systematic risk. The risk premium attributable to risk factor
! is the portfolio’s exposure to factor !, #A!, multiplied by the risk premium earned on the
first factor portfolio, E(r!) rf. Therefore, the portion of portfolio A’s risk premium that is com-
pensation for its exposure to the first factor is #A!)[E(r!) rf!!] = .(!#\% $\%) = *\%. Similarly,
the risk premium attributable to risk factor is #A)[E(r) rf!!] = .((!\% $\%) = \%\%. The total
risk premium on the portfolio is *\% + \%\% = +\% and the total expected return on the portfolio
should be $\% + +\% = !*\%.
Example 10.3 Multifactor SML
Using the numbers in Example !#.*:
E(rQ))!=!$,+,.!$!(!#!!$),+,.(!$!(!!!$)!=!!*\%
Suppose the expected return on portfolio A from Example !#.* were !\% rather than !*\%.
This return would give rise to an arbitrage opportunity. Form a portfolio from the factor
Example 10.4 Mispricing and Arbitrage
To generalize Example 10.3, note that the factor exposures of any portfolio, P, are given
by its betas, #P1 and #P2. A competing portfolio, Q, can be formed by investing in factor
portfolios with the following weights: #P1 in the first factor portfolio, #P2 in the second
factor portfolio, and 1 #P1 #P2 in T-bills. By construction, portfolio Q will have betas
equal to those of portfolio P and expected return of
E(rQ )!=!#P1 E(r1)!+!#P2 E(r2)!+!(1!!#P1!!#P2)rf
=!rf! +!#P1 [ E(r1)!!rf ] +!#P2 [ E(r2)!!rf ]
(10.8)
This is a two-factor SML, and, as Example 10.4 shows, any well-diversified portfolio with
the same betas must have the same expected return as long as capital markets do not allow
for easy arbitrage opportunities.
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Using the APT to Find Cost of Capital
W
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Elton, Gruber, and Mei* use the APT to derive the cost of
capital for electric utilities. They consider five potential
systematic risk factors: unanticipated developments in the
term structure of interest rates, the level of interest rates,
inflation rates, the business cycle (measured by GDP),
foreign exchange rates, and a summary measure they devise
to measure other macro factors.
Their first step is to estimate the risk premium associated
with exposure to each risk source. They accomplish this in a
two-step strategy (which we will describe in considerable detail
in Chapter $!):
$. Estimate “factor loadings” (i.e., betas) of a large sample
of firms. Regress returns of $\%\% randomly selected stocks
against the five systematic factors. They use a time-series
regression for each stock (e.g., &\% months of data), there-
fore estimating $\%\% regressions, one for each stock.
. Estimate the reward earned per unit of exposure to each
risk factor. For each month, regress the return of each stock
against the five betas estimated. The coefficient on each
beta is the extra average return earned as beta increases
(i.e., it is an estimate of the risk premium for that risk factor
from that month’s data). These estimates are of course sub-
ject to sampling error. Therefore, average the risk premium
estimates across the $ months in each year. The average
response of return to risk is less subject to sampling error.
The risk premiums are in the middle column of the table in the
next column.
Notice that some risk premiums are negative. The interpre-
tation of this result is that risk premium should be positive for
risk factors you don’t want exposure to, but negative for factors
you do want exposure to. For example, you should desire secu-
rities that have higher returns when inflation increases and be
willing to accept lower expected returns on such securities; this
shows up as a negative risk premium.
The study finds that average returns are related to factor
betas as follows:
rf+.#( !term struc#.\%($ !int rate
# .\%#) !ex rate+.\%#$ !bus cycle#.\%&) !inflation+.(!\% !other
Finally, to obtain the cost of capital for a particular firm, the
authors estimate the firm’s betas against each source of risk,
multiply each factor beta by the “cost of factor risk” from the
table above, sum over all risk sources to obtain the total risk
premium, and add the risk-free rate.
For example, the beta estimates for Niagara Mohawk appear
in the last column of the table above. Therefore, its cost of
capital is
Costofcapital=rf+.#($$.\%&$(#\%.($(# .#$&*)
# .\%#)($.!!()+.\%#$(.$))
# .\%&)(# .(\%)+.(!\%(.!\%#&)
=rf+.*
In other words, the monthly cost of capital for Niagara Mohawk
is .*\% above the monthly risk-free rate. Its annualized risk
premium is therefore .*\% $ $ = +.&#\%.
*Edwin J. Elton, Martin J. Gruber, and Jianping Mei, “Cost of Capital Using
Arbitrage Pricing Theory: A Case Study of Nine New York Utilities,” Finan-
cial Markets, Institutions, and Instruments ! (August $))#), pp. #&–&+.
Factor
Factor Risk
Premium
Factor Betas for
Niagara Mohawk
Term structure !.#$ \%.!&\%$
Interest rates #!.!$\% ##.\%&
Exchange rates #!.!( \%.)#)$
Business cycle !.!\% !.\%#(#
Inflation #!.!&( #!.$##!
Other macro factors !.$)!* !.)!&
The APT shows us how multiple risk factors can result in a multifactor SML. But how
can we identify the most likely sources of systematic risk? One approach comes from
Merton’s multifactor CAPM, discussed in Chapter 9, in which the extra-market risk factors
are due to hedging demands against a range of risks associated with either consumption
or investment opportunities. Another approach, which is more pervasive today, uses firm
characteristics that seem on empirical grounds to proxy for exposure to systematic risk.
The factors chosen are variables that on past evidence have predicted average returns well
and therefore may be capturing risk premiums. One example of this approach is the Fama
10.5 The Fama-French (FF) Three-Factor Model
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and French three-factor model and its variants, which have come to dominate empirical
research in security returns:6
Rit!=!i!+!#iM RMt!+!#iSMB SMBt!+!#iHML HMLt!+!eit (10.9)
where
SMB = Small Minus Big (i.e., the return of a portfolio of small stocks in excess of
the return on a portfolio of large stocks).
HML = High Minus Low (i.e., the return of a portfolio of stocks with a high
book-to-market ratio in excess of the return on a portfolio of stocks with
a low book-to-market ratio).
Note that in this model the market index does play a role and is expected to capture
systematic risk originating from macroeconomic factors.
These two extra-market factors are chosen because of long-standing observations that
firm size, measured by market capitalization (the market value of outstanding equity),
and the book-to-market ratio (book value per share divided by stock price) predict devia-
tions of average stock returns from levels consistent with the CAPM. Fama and French
justify this model on empirical grounds: While SMB and HML are not themselves
obvious candidates for relevant risk factors, the argument is that these variables may proxy
for hard-to-measure more-fundamental variables. For example, Fama and French point
out that firms with high book-to-market ratios are more likely to be in financial distress
and that small stocks may be more sensitive to changes in business conditions. Thus, these
variables may capture sensitivity to risk factors in the macroeconomy. More evidence on
the Fama-French model appears in Chapter 13.
The problem with empirical approaches such as the Fama-French model is that the
extra-market factors in these models cannot be clearly identified with a source of risk
that is of obvious concern to a significant group of investors. Black7 points out that when
researchers scan and rescan the database of security returns in search of explanatory fac-
tors (an activity often called data-snooping), they may eventually uncover past “patterns”
that are due purely to chance. However, Fama and French have shown that size and book-
to-market ratios have predicted average returns in different time periods and in markets all
over the world, thus mitigating potential effects of data-snooping.
The risk premiums associated with Fama-French factors raise the question of whether
they reflect a multi-index ICAPM based on extra-market hedging demands or just
represent yet-unexplained anomalies, where firm characteristics are correlated with
alpha values. This is an important distinction for the debate over the proper interpreta-
tion of the model, because the validity of FF-style models may signify either a devia-
tion from rational equilibrium (as there is no rational reason to prefer one or another
of these firm characteristics per se) or indicate that firm characteristics identified as
empirically associated with average returns are correlated with other (harder to specify)
risk factors.
The issue is still unresolved and is revisited in Chapter 13.
6Eugene F. Fama and Kenneth R. French, “Multifactor Explanations of Asset Pricing Anomalies,” Journal of
Finance 51 (1996), pp. 55–84.
7Fischer Black, “Beta and Return,” Journal of Portfolio Management 20 (1993), pp. 8–18.
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!# P A R T I I I Equilibrium in Capital Markets
single-factor model
multifactor model
factor loading
factor beta
KEY TERMS arbitrage pricing theory (APT)
arbitrage
Law of One Price
risk arbitrage
well-diversified portfolio
factor portfolio
Single-factor model: Ri = E(Ri ) + !i F + ei
Multifactor model (here, 2 factors, F1 and F2): Ri = E(Ri ) + !i1F1 + !i2 F2 + ei
Single-index model: Ri = i + !i RM + ei
KEY EQUATIONS
1. Multifactor models seek to improve the explanatory power of single-factor models by explicitly
accounting for the various components of systematic risk. These models use indicators intended
to capture a wide range of macroeconomic risk factors.
2. Once we allow for multiple risk factors, we conclude that the security market line also ought to be
multidimensional, with exposure to each risk factor contributing to the total risk premium of the
security.
3. A (risk-free) arbitrage opportunity arises when two or more security prices enable investors
to construct a zero-net-investment portfolio that will yield a sure profit. The presence of arbi-
trage opportunities will generate a large volume of trades that puts pressure on security prices.
This pressure will continue until prices reach levels that preclude such arbitrage.
4. When securities are priced so that there are no risk-free arbitrage opportunities, we say that they
satisfy the no-arbitrage condition. Price relationships that satisfy the no-arbitrage condition are
important because we expect them to hold in real-world markets.
5. Portfolios are called “well diversified” if they include a large number of securities and the invest-
ment proportion in each is sufficiently small. The proportion of a security in a well-diversified
portfolio is small enough so that for all practical purposes a reasonable change in that security’s
rate of return will have a negligible effect on the portfolio’s rate of return.
6. In a single-factor security market, all well-diversified portfolios have to satisfy the expected
return–beta relationship of the CAPM to satisfy the no-arbitrage condition. If all well-diversified
portfolios satisfy the expected return–beta relationship, then individual securities also must
satisfy this relationship, at least approximately.
7. The APT does not require the restrictive assumptions of the CAPM and its (unobservable) market
portfolio. The price of this generality is that the APT does not guarantee this relationship for all
securities at all times.
8. A multifactor APT generalizes the single-factor model to accommodate several sources of
systematic risk. The multidimensional security market line predicts that exposure to each risk
factor contributes to the security’s total risk premium by an amount equal to the factor beta times
the risk premium of the factor portfolio that tracks that source of risk.
9. The multifactor extension of the single-factor CAPM, the ICAPM, predicts the same multidimen-
sional security market line as the APT. The ICAPM suggests that priced extra-market risk factors
will be the ones that lead to significant hedging demand by a substantial fraction of investors.
Other approaches to the multifactor APT are more empirically based, where the extra-market
factors are selected based on past ability to predict risk premiums.
SUMMARY
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1. Suppose that two factors have been identified for the U.S. economy: the growth rate of industrial
production, IP, and the inflation rate, IR. IP is expected to be 3\%, and IR 5\%. A stock with a
beta of 1 on IP and .5 on IR currently is expected to provide a rate of return of 12\%. If industrial
production actually grows by 5\%, while the inflation rate turns out to be 8\%, what is your revised
estimate of the expected rate of return on the stock?
2. The APT itself does not provide guidance concerning the factors that one might expect to
determine risk premiums. How should researchers decide which factors to investigate? Why,
for example, is industrial production a reasonable factor to test for a risk premium?
3. If the APT is to be a useful theory, the number of systematic factors in the economy must be
small. Why?
4. Suppose that there are two independent economic factors, F1 and F2. The risk-free rate is 6\%,
and all stocks have independent firm-specific components with a standard deviation of 45\%.
Portfolios A and B are both well-diversified with the following properties:
Portfolio Beta on F! Beta on F$ Expected Return
A !. #.$ \%!\%
B #.# !$.# #&\%
What is the expected return–beta relationship in this economy?
5. Consider the following data for a one-factor economy. Both portfolios are well diversified.
Portfolio E(r!) Beta
A !#\% !.#
F \% $.$
Suppose that another portfolio, portfolio E, is well diversified with a beta of .6 and expected
return of 8\%. Would an arbitrage opportunity exist? If so, what would be the arbitrage strategy?
6. Assume that both portfolios A and B are well diversified, that E(rA) = 12\%, and E(rB) = 9\%. If the
economy has only one factor, and A = 1.2, whereas B = .8, what must be the risk-free rate?
7. Assume that stock market returns have the market index as a common factor, and that all stocks
in the economy have a beta of 1 on the market index. Firm-specific returns all have a standard
deviation of 30\%.
Suppose that an analyst studies 20 stocks and finds that one-half of them have an alpha of
+2\%, and the other half have an alpha of !2\%. Suppose the analyst invests $1 million in an
equally weighted portfolio of the positive alpha stocks, and shorts $1 million of an equally
weighted portfolio of the negative alpha stocks.
a. What is the expected profit (in dollars) and standard deviation of the analyst’s profit?
b. How does your answer change if the analyst examines 50 stocks instead of 20 stocks?
100 stocks?
PROBLEM SETS
Multifactor SML (here, 2 factors, labeled 1 and 2):
E(ri )#=#rf#+#i1 [ E(r1)#!#rf ] +#i2 [ E(r2)#!#rf ]
=#rf#+#i1 E(R1)#+#i2 E(R2)
where i1 and i2#measure the stock’s typical response to returns on each factor portfolio and the risk
premiums on the two factor portfolios are E(R1) and E(R2).
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b. Suppose that the market expects the values for the three macro factors given in column 1
below, but that the actual values turn out as given in column 2. Calculate the revised expec-
tations for the rate of return on the stock once the “surprises” become known.
Factor Expected Value Actual Value
Inflation !\% \%
Industrial production # $
Oil prices \% &
11. Suppose that the market can be described by the following three sources of systematic risk with
associated risk premiums.
Factor Risk Premium
Industrial production (I!) $\%
Interest rates (R) \%
Consumer confidence (C!)
The return on a particular stock is generated according to the following equation:
r!=!15\%!+!1.0I!+!.5R!+!.75C!+!e
Find the equilibrium rate of return on this stock using the APT. The T-bill rate is 6\%. Is the
stock over- or underpriced? Explain.
12. As a finance intern at Pork Products, Jennifer Wainwright’s assignment is to come up with fresh
insights concerning the firm’s cost of capital. She decides that this would be a good opportunity
to try out the new material on the APT that she learned last semester. She decides that three
promising factors would be (a) the return on a broad-based index such as the S&P 500; (b) the
level of interest rates, as represented by the yield to maturity on 10-year Treasury bonds; and
(c) the price of hogs, which is particularly important to her firm. Her plan is to find the beta of
Pork Products against each of these factors by using a multiple regression and to estimate the
risk premium associated with each exposure factor. Comment on Jennifer’s choice of factors.
Which are most promising with respect to the likely impact on her firm’s cost of capital?
Can you suggest improvements to her specification?
Use the following information to answer Problems 13 through 16:
Orb Trust (Orb) has historically leaned toward a passive management style of its portfolios.
The only model that Orb’s senior management has promoted in the past is the capital asset pricing
model (CAPM). Now Orb’s management has asked one of its analysts, Kevin McCracken, CFA,
to investigate the use of the arbitrage pricing theory (APT) model.
McCracken believes that a two-factor APT model is adequate, where the factors are the
sensitivity to changes in real GDP and changes in inflation. McCracken has concluded that
the factor risk premium for real GDP is 8\% while the factor risk premium for inflation is 2\%.
He estimates for Orb’s High Growth Fund that the sensitivities to these two factors are 1.25 and
1.5, respectively. Using his APT results, he computes the equilibrium expected return of the fund.
For comparison purposes, he then uses fundamental analysis to compute the actually expected
return of Orb’s High Growth Fund. McCracken finds that the two estimates of the Orb High
Growth Fund’s expected return are equal.
McCracken asks a fellow analyst, Sue Kwon, to provide an estimate of the expected return of
Orb’s Large Cap Fund based on fundamental analysis. Kwon, who manages the fund, says that
the expected return is 8.5\% above the risk-free rate. McCracken then applies the APT model to
the Large Cap Fund. He finds that the sensitivities to real GDP and inflation are .75 and 1.25,
respectively.
McCracken’s manager at Orb, Jay Stiles, asks McCracken to construct a portfolio that has a
unit sensitivity to real GDP growth but is not affected by inflation. McCracken is confident in
his APT estimates for the High Growth Fund and the Large Cap Fund. He then computes the
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!! P A R T I I I Equilibrium in Capital Markets
sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0,
respectively. McCracken will use his APT results for these three funds to accomplish the task of
creating a portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the
fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees
who live off the steady income of their investments. McCracken does not agree with Stiles, but
says that the fund would be a good choice if upcoming supply side macroeconomic policies of the
government are successful.
13. According to the APT, if the risk-free rate is 4\%, what should be McCracken’s estimate of the
expected return of Orb’s High Growth Fund?
14. With respect to McCracken’s APT model estimate of Orb’s Large Cap Fund and the information
Kwon provides, is an arbitrage opportunity available?
15. If the GDP Fund is constructed from the other three funds, which of the following would be its
weight in the Utility Fund? (a) !2.2; (b) !3.2; or (c) .3.
16. With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund
would be appropriate:
a. McCracken is correct and Stiles is wrong.
b. Both are correct.
c. Stiles is correct and McCracken is wrong.
17. Assume a universe of n (large) securities for which the largest residual variance is not larger
than n M 2 . Construct as many different weighting schemes as you can that generate well-
diversified portfolios.
18. Derive a more general (than the numerical example in the chapter) demonstration of the APT
security market line:
a. For a single-factor market.
b. For a multifactor market.
19. Small firms generally have relatively high loadings (high betas) on the SMB (small minus big)
factor.
a. Explain why this is not surprising.
b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit
of the merged company. Would you expect the stock market behavior of the merged firm to
differ from that of a portfolio of the two previously independent firms?
c. How does the merger affect market capitalization?
d. What is the prediction of the Fama-French model for the risk premium on the merged firm
compared to the weighted average of the two component companies?#
e. Do we see here a problem in applying the FF model?
1. Jeffrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced
securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In compar-
ing CAPM and APT, the consultant makes the following arguments:
a. Both the CAPM and APT require a mean-variance efficient market portfolio.
b. Neither the CAPM nor the APT assumes normally distributed security returns.
c. The CAPM assumes that one specific factor explains security returns but APT does not.
State whether each of the consultant’s arguments is correct or incorrect. Indicate, for each
incorrect argument, why the argument is incorrect.
2. Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8\%.
Portfolio Expected Return Beta
X !\% !.##
Y !$ #.$\%
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330 PART III Equilibrium in Capital Markets
sensitivities for a third fund, Orb’s Utility Fund, which has sensitivities equal to 1.0 and 2.0,
respectively. McCracken will use his APT results for these three funds to accomplish the task of
creating a portfolio with a unit exposure to real GDP and no exposure to inflation. He calls the
fund the “GDP Fund.” Stiles says such a GDP Fund would be good for clients who are retirees
who live off the steady income of their investments. McCracken does not agree with Stiles, but
says that the fund would be a good choice if upcoming supply side macroeconomic policies of the
government are successful.
13. According to the APT, if the risk-free rate is 4\%, what should be McCracken’s estimate of the
expected return of Orb’s High Growth Fund?
14. With respect to McCracken’s APT model estimate of Orb’s Large Cap Fund and the information
Kwon provides, is an arbitrage opportunity available?
15. If the GDP Fund is constructed from the other three funds, which of the following would be its
weight in the Utility Fund? (a) −2.2; (b) −3.2; or (c) .3.
16. With respect to the comments of Stiles and McCracken concerning for whom the GDP Fund
would be appropriate:
a. McCracken is correct and Stiles is wrong.
b. Both are correct.
c. Stiles is correct and McCracken is wrong.
17. Assume a universe of n (large) securities for which the largest residual variance is not larger
than n
M
2
. Construct as many different weighting schemes as you can that generate well-
diversified portfolios.
18. Derive a more general (than the numerical example in the chapter) demonstration of the APT
security market line:
a. For a single-factor market.
b. For a multifactor market.
19. Small firms generally have relatively high loadings (high betas) on the SMB (small minus big)
factor.
a. Explain why this is not surprising.
b. Now suppose two unrelated small firms merge. Each will be operated as an independent unit
of the merged company. Would you expect the stock market behavior of the merged firm to
differ from that of a portfolio of the two previously independent firms?
c. How does the merger affect market capitalization?
d. What is the prediction of the Fama-French model for the risk premium on the merged firm
compared to the weighted average of the two component companies?
e. Do we see here a problem in applying the FF model?
1. Jeffrey Bruner, CFA, uses the capital asset pricing model (CAPM) to help identify mispriced
securities. A consultant suggests Bruner use arbitrage pricing theory (APT) instead. In compar-
ing CAPM and APT, the consultant makes the following arguments:
a. Both the CAPM and APT require a mean-variance efficient market portfolio.
b. Neither the CAPM nor the APT assumes normally distributed security returns.
c. The CAPM assumes that one specific factor explains security returns but APT does not.
State whether each of the consultant’s arguments is correct or incorrect. Indicate, for each
incorrect argument, why the argument is incorrect.
2. Assume that both X and Y are well-diversified portfolios and the risk-free rate is 8\%.
PortfolioExpected ReturnBeta
X16\%1.00
Y120.25
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In this situation you would conclude that portfolios X and Y:
a. Are in equilibrium.
b. Offer an arbitrage opportunity.
c. Are both underpriced.
d. Are both fairly priced.
3. A zero-investment portfolio with a positive alpha could arise if:
a. The expected return of the portfolio equals zero.
b. The capital market line is tangent to the opportunity set.
c. The Law of One Price remains unviolated.
d. A risk-free arbitrage opportunity exists.
4. According to the theory of arbitrage:
a. High-beta stocks are consistently overpriced.
b. Low-beta stocks are consistently overpriced.
c. Positive alpha investment opportunities will quickly disappear.
d. Rational investors will pursue arbitrage opportunities consistent with their risk tolerance.
5. The general arbitrage pricing theory (APT) differs from the single-factor capital asset pricing
model (CAPM) because the APT:
a. Places more emphasis on market risk.
b. Minimizes the importance of diversification.
c. Recognizes multiple unsystematic risk factors.
d. Recognizes multiple systematic risk factors.
6. An investor takes as large a position as possible when an equilibrium price relationship is vio-
lated. This is an example of:
a. A dominance argument.
b. The mean-variance efficient frontier.
c. Arbitrage activity.
d. The capital asset pricing model.
7. The feature of the general version of the arbitrage pricing theory (APT) that offers the greatest
potential advantage over the simple CAPM is the:
a. Identification of anticipated changes in production, inflation, and term structure of interest
rates as key factors explaining the risk–return relationship.
b. Superior measurement of the risk-free rate of return over historical time periods.
c. Variability of coefficients of sensitivity to the APT factors for a given asset over time.
d. Use of several factors instead of a single market index to explain the risk–return relationship.
8. In contrast to the capital asset pricing model, arbitrage pricing theory:
a. Requires that markets be in equilibrium.
b. Uses risk premiums based on micro variables.
c. Specifies the number and identifies specific factors that determine expected returns.
d. Does not require the restrictive assumptions concerning the market portfolio.
E$INVESTMENTS EXERCISES
One of the factors in the APT model specified in an influential paper by Chen, Roll, and Ross*\%is
the percent change in unanticipated inflation. Who gains and who loses when inflation changes?
Go to http://hussmanfunds.com/rsi/infsurprises.htm to see a graph of the Inflation Surprise
Index and Economists’ Inflation Forecasts.
*See Nai-Fu Chen, Richard Roll, and Stephen Ross, “Economic Forces and the Stock Market,” Journal of
Business\%59 (1986).
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!! P A R T I I I Equilibrium in Capital Markets
SOLUTIONS TO CONCEPT CHECKS
1. The GDP beta is 1.2 and GDP growth is 1\% better than previously expected. So you will increase
your forecast for the stock return by 1.2 ! 1\% = 1.2\%. The revised forecast is for an 11.2\% return.
2. a. This portfolio is not well diversified. The weight on the first security does not decline as n
increases. Regardless of how much diversification there is in the rest of the portfolio, you will
not shed the firm-specific risk of this security.
b. This portfolio is well diversified. Even though some stocks have three times the weight
of other stocks (1.5/n versus .5/n), the weight on all stocks approaches zero as n increases.
The impact of any individual stock’s firm-specific risk will approach zero as n becomes
ever larger.
3. The equilibrium return is E(r) = rf + P1 [ E(r1) # rf ] + P2 [ E(r2) # rf ] . Using the data in
Example 10.4:
E(r)$=$4$+$.2$ !$(10$#$4)$+$1.4$!$(12$#$4)$=$16.4\%
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!# P A R T I I I Equilibrium in Capital Markets
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The index model introduced in Chapter 8 gave us a way of decomposing stock variability
into market or systematic risk, due largely to macroeconomic events, versus firm-specific
or idiosyncratic effects that can be diversified in large portfolios. In the single-index model,
the return on a broad market-index portfolio summarized the impact of the macro factor.
In Chapter 9 we introduced the possibility that risk premiums may also depend on cor-
relations with extra-market risk factors, such as inflation, or changes in the parameters
describing future investment opportunities: interest rates, volatility, market-risk premi-
ums, and betas. For example, returns on an asset whose return increases when inflation
increases can be used to hedge uncertainty in the future inflation rate. Its price may rise
and its risk premium may fall as a result of investors’ extra demand for this asset.
Risk premiums of individual securities should reflect their sensitivities to changes
in extra-market risk factors just as their betas on the market index determine their risk
premiums in the simple CAPM. When securities can be used to hedge these factors,
the resulting hedging demands will turn the SML into a multifactor model, with each
significant risk source generating an additional factor. Risk factors can be represented
either by returns on these hedge portfolios (just as the index portfolio represents the
market factor), or more directly by changes in the risk factors themselves, for example,
changes in interest rates or inflation.
Factor Models of Security Returns
We begin with a familiar single-factor model like the one introduced in Chapter 8. Uncer-
tainty in asset returns has two sources: a common or macroeconomic factor and firm-
specific events. By construction, the common factor has zero expected value because it
measures!new information concerning the macroeconomy; new information implies a revi-
sion to current expectations, and if initial expectations are rational, then such revisions
should average out to zero.
If we call F the deviation of the common factor from its expected value, i the sensi-
tivity of firm i to that factor, and ei the firm-specific disturbance, the factor model states
that the actual excess return on firm i will equal its initially expected value plus a (zero
expected value) random amount attributable to unanticipated economywide events, plus
another (zero expected value) random amount attributable to firm-specific events.
Formally, the single-factor model of excess returns is described by Equation 10.1:
Ri!=!E(Ri )!+!i F!+!ei (10.1)
where E(Ri ) is the expected excess return on stock i. Notice that if the macro factor has a
value of 0 in any particular period (i.e., no macro surprises), the excess return on the secu-
rity will equal its previously expected value, E(Ri ), plus the effect of firm-specific events
only. The nonsystematic components of returns, the ei s, are assumed to be uncorrelated
across stocks and with the factor F.
10.1 Multifactor Models: A Preview
To illustrate the factor model, suppose that the macro factor, F,!represents!news about the
state of the business cycle, which we will measure by the unexpected percentage change
in gross domestic product (GDP). The consensus is that GDP will increase by \% this year.
Suppose also that a stock’s value is #.$. If GDP increases by only \%\%, then the value
Example 10.1 Factor Models
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!# P A R T I I I Equilibrium in Capital Markets
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beta on GDP. But the utility’s stock price may have a relatively high sensitivity to interest
rates. Because the cash flow generated by the utility is relatively stable, its present value
behaves much like that of a bond, varying inversely with interest rates. Conversely, the per-
formance of the airline is very sensitive to economic activity but is less sensitive to interest
rates. It will have a high GDP beta and a lower interest rate beta. Suppose that on a particu-
lar day, a news item suggests that the economy will expand. GDP is expected to increase,
but so are interest rates. Is the “macro news” on this day good or bad? For the utility, this
is bad news: Its dominant sensitivity is to rates. But for the airline, which responds more
to GDP, this is good news. Clearly a one-factor or single-index model cannot capture such
differential responses to varying sources of macroeconomic uncertainty.
Factor betas can provide a framework for a hedging strategy. The idea for an investor
who wishes to hedge a source of risk is to establish an opposite factor exposure to offset
that particular source of risk. Often, futures contracts can be used to hedge particular factor
exposures. We explore this application in Chapter 22.
As it stands, however, the multifactor model is no more than a description of the fac-
tors that affect security returns. There is no “theory” in the equation. The obvious question
left unanswered by a factor model like Equation 10.2 is where E(R) comes from, in other
words, what determines a security’s expected excess rate of return. This is where we need
a theoretical model of equilibrium security returns. We therefore now turn to arbitrage
pricing theory to help determine the expected value, E(R), in Equations 10.1 and 10.2.
Stephen Ross developed the arbitrage pricing theory (APT) in 1976.1 Like the CAPM,
the APT predicts a security market line linking expected returns to risk, but the path it
takes to the SML is quite different. Ross’s APT relies on three key propositions: (1) Secu-
rity returns can be described by a factor model; (2) there are sufficient securities to diver-
sify away idiosyncratic risk; and (3) well-functioning security markets do not allow for the
persistence of arbitrage opportunities. We begin with a simple version of Ross’s model,
which assumes that only one systematic factor affects security returns. Once we under-
stand how the model works, it will be much easier to see how it can be generalized to
accommodate more than one factor.
10.2 Arbitrage Pricing Theory
1Stephen A. Ross, “Return, Risk and Arbitrage,” in I. Friend and J. Bicksler, eds., Risk and Return in Finance
(Cambridge, MA: Ballinger, 1976).
Suppose we estimate the two-factor model in Equation !.# for Northeast Airlines and find
the following result:
R!=!.!$$\%+\%!.#(GDP)!!.$(IR)\%+\%e
This tells us that, based on currently available information, the expected excess rate of
return for Northeast is !$.$\%, but that for every percentage point increase in GDP beyond
current expectations, the return on Northeast’s shares increases on average by !.#\%, while for
every unanticipated percentage point that interest rates increase, Northeast’s shares fall on
average by .$\%.
Example 10.2 Risk Assessment Using Multifactor Models
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Arbitrage, Risk Arbitrage, and Equilibrium
An arbitrage opportunity arises when an investor can earn riskless profits without
making a net investment. A trivial example of an arbitrage opportunity would arise if shares
of a stock sold for different prices on two different exchanges. For example, suppose IBM
sold for $165 on the NYSE but only $163 on NASDAQ. Then you could buy the shares on
NASDAQ and simultaneously sell them on the NYSE, clearing a riskless profit of $2 per
share without tying up any of your own capital. The Law of One Price states that if two
assets are equivalent in all economically relevant respects, then they should have the same
market price. The Law of One Price is enforced by arbitrageurs: If they observe a violation
of the law, they will engage in arbitrage activity—simultaneously buying the asset where it
is cheap and selling where it is expensive. In the process, they will bid up the price where
it is low and force it down where it is high until the arbitrage opportunity is eliminated.
Strategies that exploit violations of the Law of One Price all involve long–short
positions. You buy the relatively cheap asset and sell the relatively overpriced one. The net
investment, therefore, is zero. Moreover, the position is riskless. Therefore, any investor,
regardless of risk aversion or wealth, will want to take an infinite position in it. Because
those large positions will quickly force prices up or down until the opportunity vanishes,
security prices should satisfy a “no-arbitrage condition,” that is, a condition that rules out
the existence of arbitrage opportunities.
The idea that market prices will move to rule out arbitrage opportunities is perhaps the
most fundamental concept in capital market theory. Violation of this restriction would
indicate the grossest form of market irrationality.
There is an important difference between arbitrage and risk–return dominance argu-
ments in support of equilibrium price relationships. A dominance argument holds that
when an equilibrium price relationship is violated, many investors will make limited
portfolio changes, depending on their degree of risk aversion. Aggregation of these limited
portfolio changes is required to create a large volume of buying and selling, which in turn
restores equilibrium prices. By contrast, when arbitrage opportunities exist, each inves-
tor wants to take as large a position as possible; hence it will not take many investors to
bring about the price pressures necessary to restore equilibrium. Therefore, implications
for prices derived from no-arbitrage arguments are stronger than implications derived from
a risk–return dominance argument.
The CAPM is an example of a dominance argument, implying that all investors hold
mean-variance efficient portfolios. If a security is mispriced, then investors will tilt their
portfolios toward the underpriced and away from the overpriced securities. Pressure on
equilibrium prices results from many investors shifting their portfolios, each by a relatively
small dollar amount. The assumption that a large number of investors are mean-variance
optimizers is critical. In contrast, the implication of a no-arbitrage condition is that a few
investors who identify an arbitrage opportunity will mobilize large dollar amounts and
quickly restore equilibrium.
Practitioners often use the terms arbitrage and arbitrageurs more loosely than our strict
definition. Arbitrageur often refers to a professional searching for mispriced securities in
specific areas such as merger-target stocks, rather than to one who seeks strict (risk-free)
arbitrage opportunities. Such activity is sometimes called risk arbitrage to distinguish it
from pure arbitrage.
Well-Diversified Portfolios
We begin by considering the risk of a portfolio of stocks in a single-factor market. We
first show that if a portfolio is well diversified, its firm-specific or nonfactor risk becomes
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For example
The inbound logistics for William Instrument refer to purchase components from various electronic firms. During the purchase process William need to consider the quality and price of the components. In this case
4. A U.S. Supreme Court case known as Furman v. Georgia (1972) is a landmark case that involved Eighth Amendment’s ban of unusual and cruel punishment in death penalty cases (Furman v. Georgia (1972)
With covid coming into place
In my opinion
with
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The ability to view ourselves from an unbiased perspective allows us to critically assess our personal strengths and weaknesses. This is an important step in the process of finding the right resources for our personal learning style. Ego and pride can be
· By Day 1 of this week
While you must form your answers to the questions below from our assigned reading material
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5 The family dynamic is awkward at first since the most outgoing and straight forward person in the family in Linda
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The most important benefit of my statistical analysis would be the accuracy with which I interpret the data. The greatest obstacle
From a similar but larger point of view
4 In order to get the entire family to come back for another session I would suggest coming in on a day the restaurant is not open
When seeking to identify a patient’s health condition
After viewing the you tube videos on prayer
Your paper must be at least two pages in length (not counting the title and reference pages)
The word assimilate is negative to me. I believe everyone should learn about a country that they are going to live in. It doesnt mean that they have to believe that everything in America is better than where they came from. It means that they care enough
Data collection
Single Subject Chris is a social worker in a geriatric case management program located in a midsize Northeastern town. She has an MSW and is part of a team of case managers that likes to continuously improve on its practice. The team is currently using an
I would start off with Linda on repeating her options for the child and going over what she is feeling with each option. I would want to find out what she is afraid of. I would avoid asking her any “why” questions because I want her to be in the here an
Summarize the advantages and disadvantages of using an Internet site as means of collecting data for psychological research (Comp 2.1) 25.0\% Summarization of the advantages and disadvantages of using an Internet site as means of collecting data for psych
Identify the type of research used in a chosen study
Compose a 1
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effect relationship becomes more difficult—as the researcher cannot enact total control of another person even in an experimental environment. Social workers serve clients in highly complex real-world environments. Clients often implement recommended inte
I think knowing more about you will allow you to be able to choose the right resources
Be 4 pages in length
soft MB-920 dumps review and documentation and high-quality listing pdf MB-920 braindumps also recommended and approved by Microsoft experts. The practical test
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One thing you will need to do in college is learn how to find and use references. References support your ideas. College-level work must be supported by research. You are expected to do that for this paper. You will research
Elaborate on any potential confounds or ethical concerns while participating in the psychological study 20.0\% Elaboration on any potential confounds or ethical concerns while participating in the psychological study is missing. Elaboration on any potenti
3 The first thing I would do in the family’s first session is develop a genogram of the family to get an idea of all the individuals who play a major role in Linda’s life. After establishing where each member is in relation to the family
A Health in All Policies approach
Note: The requirements outlined below correspond to the grading criteria in the scoring guide. At a minimum
Chen
Read Connecting Communities and Complexity: A Case Study in Creating the Conditions for Transformational Change
Read Reflections on Cultural Humility
Read A Basic Guide to ABCD Community Organizing
Use the bolded black section and sub-section titles below to organize your paper. For each section
Losinski forwarded the article on a priority basis to Mary Scott
Losinksi wanted details on use of the ED at CGH. He asked the administrative resident