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# Note 8 for Fin 526 at WSU

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This 10 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Washington State University taught by a professor in Fall. Since its upload, it has received 17 views.

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Date Created: 02/06/15

CHAPTER 8 Introduction to Risk Return and The Opportunity Cost of Capital Answers to Practice Questions 10 02 5 11 Recall from Chapter 4 that 1 rnominal 1 rreal X 1 inflation rate Therefore rreal 1 rnominaI1 inflation rate 1 a The real return on the stock market in each year was 1929 147 1930 237 1931 380 1932 05 1933 565 b From the results for Part a the average real return was 389 c The risk premium for each year was 1929 193 1930 307 1931 450 1932 109 1933 570 d From the results for Part c the average risk premium was 978 e The standard deviation 6 ofthe risk premium is calculated as follows 1 1 o193 o09782 o3o7 009782 o450 009782 0109 009782 o570 009782 0155739 0 xo155739 0394637 3946 Internet exercise answers will vary 81 12 13 14 15 a A longterm United States government bond is always absolutely safe in terms of the dollars received However the price ofthe bond fluctuates as interest rates change and the rate at which coupon payments received can be invested also changes as interest rates change And of course the payments are all in nominal dollars so in ation risk must also be considered b It is true that stocks offer higher longrun rates of return than do bonds but it is also true that stocks have a higher standard deviation of return 80 which investment is preferable depends on the amount of risk one is willing to tolerate This is a complicated issue and depends on numerous factors one of which is the investment time horizon lfthe investor has a short time horizon then stocks are generally not preferred 0 Unfortunately 10 years is not generally considered a suf cient amount of time for estimating average rates of return Thus using a 10year average is likely to be misleading The risk to Hippique shareholders depends on the market risk or beta ofthe investment in the black stallion The information given in the problem suggests that the horse has very high unique risk but we have no information regarding the horse s market risk 80 the best estimate is that this horse has a market risk about equal to that of other racehorses and thus this investment is not a particularly risky one for Hippique shareholders In the context ofa welldiversi ed portfolio the only risk characteristic of a single security that matters is the security s contribution to the overall portfolio risk This contribution is measured by beta Lonesome Gulch is the safer investment for a diversi ed investor because its beta 010 is lower than the beta of Amalgamated Copper 066 For a diversi ed investor the standard deviations are irrelevant XI 060 6 010 XJ 040 GJ 020 3 PIJ 1 2 2 2 2 2 0 Xl ol XJ OJ 2XIXJpIJoloJ 06020102 0402020220600401010020 00196 82 16 17 pU050 op2 XI20I2 XJ20J2 2XIXJpIJoloJ 06020102 040202022060040050010020 00148 pij0 2 2 2 2 2 0 Xl ol XJ OJ 2XIXJpIJoloJ 06020102 0402020220600400010020 00100 Refer to Figure 813 in the text V th 100 securities the box is 100 by 100 The variance terms are the diagonal terms and thus there are 100 variance terms The rest are the covariance terms Because the box has 100 times 100 terms altogether the number of covariance terms is 1002 100 9900 Half ofthese terms ie 4950 are different Once again it is easiest to think of this in terms of Figure 813 With 50 stocks all with the same standard deviation 030 the same weight in the portfolio 002 and all pairs having the same correlation coef cient 040 the portfolio variance is 02 5000220302 502 5000220400302 003708 o 0193 193 For a fully diversi ed portfolio portfolio variance equals the average covariance o2 030030040 0036 0 0190 190 Refer to Figure 813 in the text For each different portfolio the relative weight of each share is one divided by the number of shares n in the portfolio the standard deviation of each share is 040 and the correlation between pairs is 030 Thus for each portfolio the diagonal terms are the same and the offdiagonal terms are the same There are n diagonal terms and n2 n offdiagonal terms In general we have Variance n1n2042 n2 n1n2030404 8 3 For one share Variance 112042 0 0160000 For two shares Variance 2052042 2052030404 0104000 The results are summarized in the second and third columns ofthe table below Graphs are on the next page The underlying market risk that can not be diversi ed away is the second term in the formula for variance above Underlying market risk n2 n1n2030404 As n increases n2 n1n2 n1n becomes close to 1 so that the underlying market risk is 030404 0048 This is the same as Part a except that all of the offdiagonal terms are now equal to zero The results are summarized in the fourth and fifth columns ofthe table below Part a Part a Part c Part c No of Standard Standard Shares Variance Deviation Variance Deviation 1 160000 400 160000 400 2 104000 322 080000 283 3 085333 292 053333 231 4 076000 276 040000 200 5 070400 265 032000 179 6 066667 258 026667 163 7 064000 253 022857 151 8 062000 249 020000 141 9 060444 246 017778 133 10 059200 243 016000 126 84 Graphs for Part a Partfaiia Variance W 02 05 5 015 391 oquot E g 03 g 01 E g g 0 2 005 Fa a 01 0 0 0 2 4 6 a 10 12 0 2 4 6 a 10 12 Number ofSecurities Number of Securities Graphs for Part c Portfollia Variance Portfolio Standard Deviation 02 05 5 015 391 oquot E g 03 g 01 E g g 0 2 005 Fa a 01 0 0 0 2 4 6 a 10 12 0 2 4 6 a 10 12 Number ofSecurities Number of Securities 85 18 Internet exercise answers will vary depending on time period 19 The table below uses the format of Figure 813 in the text in order to calculate the portfolio variance The portfolio variance is the sum ofall the entries in the matrix Portfolio variance equals 00292516 Alcan BP Deutsche Fiat Heineken LVMH Nestle Alcan 00018002 00003792 00009669 00006528 00002085 00009959 00000669 BP 00003792 00006909 00004973 00003505 00001292 00003143 00001503 Deutsche 00009669 00004973 00018490 00011467 00002324 00010664 00002035 Fiat 00006528 00003505 00011467 00026302 00002142 00009539 00002629 Heineken 00002085 00001292 00002324 00002142 00006038 00003591 00002422 LVMH 00009959 00003143 00010664 00009539 00003591 00019612 00002706 Nestle 00000669 00001503 00002035 00002629 00002422 00002706 00003887 20 Internet exercise answers will vary depending on time period 21 Safest means lowest risk in a portfolio context this means lowest variance of return Half ofthe portfolio is invested in Alcan stock and half of the portfolio must be invested in one ofthe other securities listed Thus we calculate the portfolio variance for six different portfolios to see which is the lowest The safest attainable portfolio is comprised of Alcan and Nestle Stocks Portfolio Variance BP 0039806 Deutsche 0068393 Fiat 0070266 Heineken 0034557 LVMH 0070476 Nestle 0028453 22 a In general we expect a stock s price to change by an amount equal to beta gtlt change in the market Beta equal to 025 implies that ifthe market rises by an extra 5 the expected change in the stock s rate of return is 125 lfthe market declines an extra 5 then the expected change is 125 Safest implies lowest risk Assuming the welldiversified portfolio is invested in typical securities the portfolio beta is approximately one The largest reduction in beta is achieved by investing the 20000 in a stock with a negative beta Answer iii is correct 86 23 24 Expected portfolio return XA ERA XB ER B 12 012 LetXB 1 XA XA 010 1 XA 015 012 2 XA 060 and X3 1 XA 040 Portfolio variance XA2 0A2 X32 032 2XAXB pAB voB 0602 202 0402 402 20600400502040 592 Standard deviation o V592 2433 Internet exercise answers will vary depending on time period 87 Challenge Questions 25 a In general Portfolio variance 6P2 X12612 X22622 2X1X2p126162 Thus cp2 o5229322o52292722o5o505929322927 cp2 682267 Standard deviation 61 2612 We can think of this in terms of Figure 813 in the text with three securities One of these securities Tbills has zero risk and hence zero standard deviation Thus cp2 13229322132292722131305929322927 cp2 303230 Standard deviation 61 1741 Another way to think ofthis portfolio is that it is comprised of onethird TBills and twothirds a portfolio which is half Dell and half Home Depot Because the risk of Tbills is zero the portfolio standard deviation is two thirds of the standard deviation computed in Part a above Standard deviation 232612 1741 V th 50 margin the investor invests twice as much money in the portfolio as he had to begin with Thus the risk is twice that found in Part a when the investor is investing only his own money Standard deviation 2 X 2612 5224 V th 100 stocks the portfolio is well diversified and hence the portfolio standard deviation depends almost entirely on the average covariance of the securities in the portfolio measured by beta and on the standard deviation of the market portfolio Thus for a portfolio made up of 100 stocks each with beta 125 the portfolio standard deviation is approximately 125 X 15 1875 For stocks like Home Depot it is 153 X 15 2295 8 8 26 For a twosecurity portfolio the formula for portfolio risk is If security one is Treasury bills and security two is the market portfolio then 61 is Portfolio variance X12612 X22622 2X1X2pp126162 zero 62 is 20 Therefore Portfolio variance X22622 x220202 Standard deviation 020x2 Portfolio expected return x1006 x2006 085 Portfolio expected return 006x1 0145x2 Expected Standard PonfOIIO X1 X2 Return Deviation 1 10 00 0060 0000 2 08 02 0077 0040 3 06 04 0094 0080 4 04 06 0111 0120 5 02 08 0128 0160 6 00 10 0145 0200 Expected Return Portfolio Return amp Risk N 01 l P A l 005 0 005 01 015 Standard Deviation 02 025 89 27 28 Internet exercise answers will vary The matrix below displays the variance for each ofthe seven stocks along the diagonal and each of the covariances in the offdiagonal cells Alcan BP Deutsche Fiat Heineken LVMH Nestle Alcan 00882090 00185803 00473804 00319869 00102168 00487971 00032789 BP 00185803 00338560 00243690 00171746 00063296 00154008 00073637 Deutsche 00473804 00243690 00906010 00561907 00113898 00522536 00099691 Fiat 00319869 00171746 00561907 01288810 00104972 00467418 00128809 Heineken 00102168 00063296 00113898 00104972 00295840 00175956 00118680 LVMH 00487971 00154008 00522536 00467418 00175956 00961000 00132618 Nestle 00032789 00073637 00099691 00128809 00118680 00132618 00190440 The covariance ofAlcan with the market portfolio oAican Market is the mean of the seven respective covariances between Alcan and each of the seven stocks in the portfolio The covariance of Alcan with itself is the variance of Alcan Therefore oAican Market is equal to the average of the seven covariances in the rst row or equivalently the average of the seven covariances in the rst column Beta for Alcan is equal to the covariance divided by the market variance see Practice Question 10 The covariances and betas are displayed in the table below Covariance Beta Alcan 00354928 12134 BP 00175820 06011 Deutsche 00417362 14268 Fiat 00434790 14864 Heineken 00139259 04761 LVMH 00414501 14170 Nestle 00110952 03793 810

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