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Introduction to Investments

by: Mr. April Weber

Introduction to Investments FIN 340

Mr. April Weber
Cal State Fullerton
GPA 3.97

Tsong Lai

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Tsong Lai
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This 32 page Class Notes was uploaded by Mr. April Weber on Wednesday September 30, 2015. The Class Notes belongs to FIN 340 at California State University - Fullerton taught by Tsong Lai in Fall. Since its upload, it has received 15 views. For similar materials see /class/217012/fin-340-california-state-university-fullerton in Finance at California State University - Fullerton.

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Date Created: 09/30/15
Chapter 06 Efficient Diversification Chapter 6 Ef cient Diversi cation 1 Erp 05 x 15 04 x 10 010 x 6 121 2 Fund D represents the single best addition to complement Stephenson s current portfolio given his selection criteria First Fund D s expected return 140 percent has the potential to increase the portfolio s return somewhat Second Fund D s relatively low correlation with his current portfolio 065 indicates that Fund D will provide greater diversi cation bene ts than any of the other alternatives except Fund B The result of adding Fund D should be a portfolio with approximately the same expected return and somewhat lower volatility compared to the original portfolio The other three funds have shortcomings in terms of either expected return enhancement or volatility reduction through diversi cation bene ts Fund A offers the potential for increasing the portfolio s return but is too highly correlated to provide substantial volatility reduction bene ts through diversi cation Fund B provides substantial volatility reduction through diversi cation bene ts but is expected to generate a return well below the current portfolio s return Fund C has the greatest potential to increase the portfolio s return but is too highly correlated to provide substantial volatility reduction bene ts through diversi cation a The mean return should be equal to the value computed in the spreadsheet The fund39s return is 3 lower in a recession but 3 higher in a boom However the variance of returns should be higher re ecting the greater dispersion of outcomes in the three scenarios b Calculation of mean return and variance for the stock fund A B C D E F 6 Col B Deviation Col B Rate of X from Expected Squared gtlt Scenario Probability Return C01 C Return Deviation C01 F Recession 03 14 42 24 576 1728 Normal 04 13 52 3 9 36 Boom 03 30 9 20 400 120 Expected Return 10 Variance 2964 Standard Deviation 17 22 Chapter 06 Efficient Diversification Stock Fund 03 24 04 3 03 20 Probabilit Bond X X Fund Col D Col E 10 240 0 0 10 200 Covariance Covariance has increased because the stock returns are more extreme in the recession and boom periods This makes the tendency for stock returns to be poor when bond returns are good and Vice versa even more dramatic a One would expect variance to increase because the probabilities of the extreme outcomes are now higher 3 Calculation of mean return and variance for the stock fund A B C D E F G C01 B Deviation from Col B Rate of X Expected Squared gtlt Scenario Probabilitx Return C01 C Return DBVM C01 F Recession 04 11 44 20 400 160 Normal 02 13 26 4 16 32 Boom 04 27 108 18 324 1296 Expected Return 9 Variance 2928 Standard Deviation 1711 Chapter 06 Efficient Diversification Stock Bond X X Col D Col E 0 0 10 180 Covariance Covariance has increased because the probabilities of the more extreme returns in the recession and boom periods are now higher This makes the tendency for stock returns to be poor when bond returns are good and Vice versa more dramatic E Subscript OP refers to the original portfolio ABC to the new stock and NP to the new portfolio i Eer we Er0p wABc ErABc 09 X 067 01 X 125 0728 ii Cov r X 601 X GABC 040 X 237 X 295 27966 5 280 iii GNP Wm2 6013 WABCZ GABCZ 2 Wm WABc COVOPABC12 092 X 2371 012 X 2952 2 X 09 X 01 X 280 2 22673 s 227 Fquot Subscript OP refers to the original portfolio GS to government securities and NP to the new portfolio i Eer Wop Erop wGs Eer 09 X 067 01 X 042 0645 ii Covrgtlt 601 X 6Gs0 gtlt 237gtlt 00 iii 6N1 Wop2 copz we 6552 2 Wm WGs COVOP GS12 092 X 2372 012 X 0 2 X 09 X 01 X 0 2 2133 s 213 0 Adding the riskfree government securities would result in a lower beta for the new portfolio The new portfolio beta will be a weighted average of the individual security betas in the portfolio the presence of the riskfree securities would lower that weighted average Chapter 06 Efficient Diversification d The comment is not correct Although the respective standard deviations and expected returns for the two securities under consideration are equal the covariances between each security and the original portfolio are unknown making it impossible to draw the conclusion stated For instance if the covariances are different selecting one security over the other may result in a lower standard deviation for the portfolio as a whole In such a case that security would be the preferred investment assuming all other factors are equal e Grace clearly expressed the sentiment that the risk of loss was more important to her than the opportunity for return Using variance or standard deviation as a measure of risk in her case has a serious limitation because standard deviation does not distinguish between positive and negative price movements 6 The parameters of the opportunity set are Ers 15 ErB 9 65 32 GB 23 p 015 rf 55 From the standard deviations and the correlation coef cient we generate the covariance matrix note that Covrs rB pcscB Bonds Stocks Bonds 5290 1104 Stocks 1104 10240 The minimumvariance portfolio proportions are 62 Covr r WMS B S B 65 SE 2CovrsrB 529 1104 1024 529 2 gtlt1104 wMinB 06858 03142 Chapter 06 Efficient Diversification The mean and standard deviation of the minimum variance portfolio are ErMin 03142 X 15 06858 X 9 1089 1 2 2 2 2 A cMin wscs WB6B 2WSWBC0V1 S1 B 031422 X 1024 068582 X 529 2 X 03142 X 06858 gtlt 110412 1994 in stocks in bonds Exp return Std dev 0000 10000 900 2300 2000 8000 1020 2037 3142 6858 1089 1994 Minimum variance 4000 6000 1140 2018 6000 4000 1260 2250 7075 2925 1325 2457 Tangency portfolio 8000 2000 1380 2668 10000 0000 1500 3200 7 Investment Opportunity Set 20 A 18 16 7 E 14 7 9 3 12 7 g 10 e 1 O a 8 e E e 7 2 0 1 1 1 0 1O 20 30 40 Standard Deviation Chapter 06 Efficient Diversification The graph approximates the points Er 6 Minimum Variance Portfolio 1089 1994 Tangency Portfolio 1325 2457 8 The rewardtovariability ratio of the optimal CAL is Erprf 21325 55 c 2457 P 03154 a The equation for the CAL is Erp rf 6P Ercrf 6C 55031546C Setting ErC equal to 12 yields a standard deviation of 2061 b The mean of the complete portfolio as a function of the proportion invested in the risky portfolio y is Erc 1 yrf yErp rf yErp rf 55 y1325 55 Setting Erc 12 3 y 08387 8387 in the risky portfolio 1 y 01613 1613 in Tbills From the composition of the optimal risky portfolio Proportion of stocks in complete portfolio 08387 gtlt 07075 05934 Proportion ofbonds in complete portfolio 08387 gtlt 02925 02453 Chapter 06 Efficient Diversification 10 Using only the stock and bond funds to achieve a mean of 12 we solve 12l5ws9l ws96W53WS05 Investing 50 in stocks and 50 in bonds yields a mean of 12 and standard deviation of 6p 0502 X 1024 0502 X 529 2 X 050 X 050 X 1104 2 2106 The ef cient portfolio with a mean of 12 has a standard deviation of only 2061 Using the CAL reduces the standard deviation by 45 basis points a Although it appears that gold is dominated by stocks gold can still be an attractive diversi cation asset If the correlation between gold and stocks is suf ciently low gold will be held as a component in the optimal portfolio b If gold had a perfectly positive correlation with stocks gold would not be a part of ef cient portfolios The set of risldretum combinations of stocks and gold would plot as a straight line with a negative slope See the following graph The graph shows that the stockonly portfolio dominates any portfolio containing gold This cannot be an equilibrium the price of gold must fall and its expected return must rise r Gold Expected Return 0 i i i 1 5 20 Standard Deviation Chapter 06 Efficient Diversification 12 Since Stock A and Stock B are perfectly negatively correlated a riskfree portfolio can be created and the rate of return for this portfolio in equilibrium will always be the risk free rate To nd the proportions of this portfolio with WA invested in Stock A and WB l 7 WA invested in Stock B set the standard deviation equal to zero With perfect negative correlation the portfolio standard deviation reduces to 6p AbSWA6A WBGB 0 40 WA 6017WA3 WA 060 The expected rate of return on this riskfree portfolio is Er 060 X 8 040 X 13 100 Therefore the riskfree rate must also be 100 Chapter 06 Efficient Diversification 13 Returns Large LT Year Stocks TBonds 1987 534 265 1988 1686 840 1989 3134 1949 1990 320 713 LT TBonds 1991 3066 1839 1992 771 779 1 1993 987 1548 1994 129 718 1995 3771 3167 1996 2307 081 1997 3317 1508 1998 2858 1352 1999 2104 874 2000 910 2027 2001 1189 421 2002 2210 1679 2003 2869 238 2004 1088 840 2005 491 729 2006 1550 151 Average 1302 8 92 Std deviation 1662 1010 Calculation of Investment Opportunity Set Portfolio ProEortions Portfolio Large LT Stocks TBonds Mean Std Dev 000 100 892 1078 010 090 933 1020 020 080 974 995 030 070 1015 1006 040 060 1056 1052 050 050 1097 1129 060 040 1138 1231 070 030 1179 1353 080 020 1220 1491 090 010 1261 1640 100 000 1302 1798 Min Var Por 01473 08527 952 1004 Chapter 06 Efficient Diversification 14 If the lending and borrowing rates are equal and there are no other constraints on portfolio choice then optimal risky portfolios of all investors will be identical However if the borrowing and lending rates are not equal then borrowers who are relatively risk averse and lenders who are relatively risk tolerant will have different optimal risky portfolios 15 No it is not possible to get such a diagram Even if the correlation between A and B were 10 the frontier would be a straight line connecting A and B 16 In the special case that all assets are perfectly positively correlated the portfolio standard deviation is equal to the weighted average of the componentasset standard deviations Otherwise as the formula for portfolio variance Equation 66 shows the portfolio standard deviation is less than the weighted average of the componentasset standard deviations The portfolio variance is a weighted sum of the elements in the covariance matrix with the products of the portfolio proportions as weights 17 The probability distribution is Probability Rate of Return 07 100 03 50 Expected return 07 X 100 03 X 50 55 Variance 07 X 100 552 03 X 50 552 4725 Standard deviation x 4725 6874 18 The expected rate of return on the stock will change by beta times the unanticipated change in the market return 12 X 8 7 10 7 24 Therefore the expected rate of return on the stock should be revised to 12 7 24 96 610 Chapter 06 Efficient Diversification 19 a Fquot The risk of the diversi ed portfolio consists primarily of systematic risk Beta measures systematic risk which is the slope of the security characteristic line SCL The two gures depict the stocks39 SCLs Stock B s SCL is steeper and hence Stock B s systematic risk is greater The slope of the SCL and hence the systematic risk of Stock A is lower Thus for this investor stock B is the riskiest The undiversi ed investor is exposed primarily to rmspeci c risk Stock A has higher rmspeci c risk because the deviations of the observations from the SCL are larger for Stock A than for Stock B Deviations are measured by the vertical distance of each observation from the SCL Stock A is therefore riskiest to this investor Chapter 06 Efficient Diversification 20 The answer will vary depending on the data set selected The following raw data is used to produce the subsequent results Chapter 06 Efficient Diversification Chapter 06 Efficient Diversification 21 A scatter plot results in the following diagram The slope of the regression line is 20 and intercept is 10 Generic 1 Return 3 y 10 20 x Percent 2 4 0 1 05 1 W Market Return Percent 22 a Restricting the portfolio to 20 stocks rather than 40 to 50 will very likely increase the risk of the portfolio due to the reduction in diversi cation Such an increase might be acceptable if the expected return is increased suf ciently b Hennessy could contain the increase in risk by making sure that he maintains reasonable diversi cation among the 20 stocks that remain in his portfolio This entails maintaining a low correlation among the remaining stocks As a practical matter this means that Hennessy would need to spread his portfolio among many industries rather than concentrating in just a few 23 Risk reduction bene ts from diversi cation are not a linear function of the number of issues in the portfolio See Figures 61 and 62 in the text Rather the incremental bene ts from additional diversi cation are most important when the portfolio is least diversi ed Restricting Hennessy to 10 issues instead of 20 issues would increase the risk of his portfolio by a greater amount than reducing the size of the portfolio from 30 to 20 stocks 24 The point is well taken because the committee should be concerned with the volatility of the entire portfolio Since Hennessy39s portfolio is only one of six welldiversi ed portfolios and is smaller than the average the concentration in fewer issues might have a minimal effect on the diversi cation of the total fund Hence unleashing Hennessy to do stock picking may be advantageous Chapter 06 Efficient Diversification 25 In the regression of the excess return of Stock ABC on the market the square of the correlation coef cient is 0296 which indicates that 296 of the variance of the excess return of ABC is explained by the market systematic risk 26 a Systematic risk refers to uctuations in asset prices caused by macroeconomic factors that are common to all risky assets hence systematic risk is often referred to as market risk Examples of systematic risk factors include the business cycle in ation monetary policy and technological changes Firmspeci c risk refers to uctuations in asset prices caused by factors that are independent of the market such as industry characteristics or rm characteristics Examples of rmspeci c risk factors include litigation patents management and nancial leverage Trudy should explain to the client that picking only the ve best ideas would most likely result in the client holding a much more risky portfolio The total risk of a portfolio or portfolio variance is the combination of systematic risk and rmspeci c risk The systematic component depends on the sensitivity of the individual assets to market movements as measured by beta Assuming the portfolio is well diversi ed the number of assets will not affect the systematic risk component of portfolio variance The portfolio beta depends on the individual security betas and the portfolio weights of those securities On the other hand the components of rmspeci c risk sometimes called nonsystematic risk are not perfectly positively correlated with each other and as more assets are added to the portfolio those additional assets tend to reduce portfolio risk Hence increasing the number of securities in a portfolio reduces rmspeci c risk For example a patent expiration for one company would not affect the other securities in the portfolio An increase in oil prices might hurt an airline stock but aid an energy stock As the number of randomly selected securities increases the total risk variance of the portfolio approaches its systematic variance Chapter 14 Financial Statement Analysis E 4 V39 0 Chapter 14 Financial Statement Analysis ROA EBITSales gtlt SalesTotal Assets ROS gtlt ATO The only way that Crusty Pie can have an ROS higher than the industry average and an ROA equal to the industry average is for its ATO to be lower than the industry average ABC s asset turnover must be above the industry average Since ROE is a function of net pro t and equity it is possible to maintain a stable ROE while net pro ts decline so long as equity also declines proportionally c Old plant and equipment is likely to have a low net book value making the ratio of net sales to average net xed assets higher This transaction would increase the current ratio The transaction reduces both current assets and current liabilities by the same amount but the reduction has a larger proportionate impact on current liabilities than on current assets Therefore the current ratio would increase This transaction would increase the asset turnover ratio Sales should remain unaffected but assets are reduced SmileWhite has the higher quality of earnings for several reasons SmileWhite amortizes its goodwill over a shorter period than does QuickBrush SmileWhite therefore presents more conservative earnings because it has greater goodwill amortization expense SmileWhite depreciates its property plant and equipment using an accelerated method This results in earlier recognition of depreciation expense so that income is more conservatively stated SmileWhite s bad debt allowance as a percent of receivables is greater SmileWhite therefore recognizes higher baddebt expense than does QuickBrush If the actual collection experience for the two rms is comparable then SmileWhite has the more conservative recognition policy Chapter 14 Financial Statement Analysis 7 ROE M 55 X 20 X 22 242 equ1ty 8 Par value 20000 X 20 400000 Retained earnings 5000000 Addition to Retained earnings 70000 Book value of equity 5470000 Book value per share 547000020000 27350 Palomba Pizza Stores Statement of Cash Flows For Year Ended December 31 2007 Cash ows from operating activities Cash collections from customers 250000 Cash payments to suppliers 85000 Cash payments for salaries 45000 Cash payments for interest M Net cash provided by operating activities 110000 Cash ows from investing activities Sale of equipment 38000 Purchase of equipment 30000 Purchase ofland 1 14 000 Net cash provided by used in investing activities 6000 Cash ows from nancing activities Retirement of commons stock 25000 Payment of dividends 35 000 Net cash provided by used in nancing activities 160000 Net increase in cash 44000 Cash at beginning of year 50 000 Cash at end ofyear 94000 Chapter 14 Financial Statement Analysis b Cash ow from operations CFO focuses on measuring the cash ow generated by operations not on measuring pro tability If used as a measure of performance CFO is less subject to distortion than the net income gure Analysts use the CFO as a check on the quality of earnings The CFO then becomes a check on the reported net earnings gure although not as a substitute for net earnings Companies with high net income but low CFO may be using income recognition techniques that are suspect The ability of a rm to generate cash from operations on a consistent basis is one indication of the nancial health of the rm For most rms CFO is the life blood of the rm Analysts search for trends in CFO to indicate future cash conditions and the potential for cash ow problems Cash ow from investing activities CFI is an indication of how the rm is investing its excess cash The analyst must consider the ability of the rm to continue to grow and expand activities CFI is a good indication of the attitude of management in this area Analysis of this component of total cash ow indicates the type of capital expenditures being made by management to either expand or maintain productive capability CFI is also an indicator of the rm s nancial exibility and its ability to generate suf cient cash to respond to unanticipated needs and opportunities Decreasing CFI may be a sign of a slowdown in growth of the rm Cash ow from nancing activities CFF presents the feasibility of nancing the sources of nancing and an indication of the types of sources management supports Continued debt nancing may signal a future cash ow problem The dependency of a rm on external sources of nancing either debt or equity nancing may present troubles in the future with regard to debt servicing and maintaining dividend policy Analysts also use CFF as an indication of the quality of earnings It offers insights into the nancial habits of management and potential future policies 10 Chicago Refrigerator Co Cash receivables 7 325 3599 a Quick Ratio 099 Current liabilities 3945 b ROA EBIT Net income before tax interest expense Assets Average assets w 0364 364 05 X 8058 4792 143 Chapter 14 Financial Statement Analysis c ROE 7 Net 1ncome preferred d1vidends Average common equ1ty Preferred dividends 01 X 25 X 18000 45000 Common equity in 2005 829 575 1949 3353 million Common equity in 2004 550 450 1368 2368 million ROE 7 Net 1ncome preferred d1vidends 7 1265 45 7 0426 7 426 Average common equ1ty 0 5 X 3353 2368 d Earnings per share m 177 05 X 829 550 e Pro t margin EBIT W 0194 194 Sales 12065 f Times interest earned i W 300 Interest expense 78 Cost of goods sold 7 8048 g Inventory turnover 19 Average inventory 05 X 1415 2423 Average assets 7 05 X 4792 8058 7 Average common equity 0 5 X 2368 3353 h Leverage ratio 225 a The use of FIFO during a period of de ation means that higherhistoricalcost goods are taken out of inventory So accounting income is lower and assets are lower Chapter 14 Financial Statement Analysis 15 a R OE Net pro t Net pro t X PretaX pro t X EBIT X Sales Assets Equity PretaX pro t EBIT gtlt Sales Assets Equity Taxburden gtlt Interest burden gtlt Pro t margin gtlt Asset turnover gtlt Leverage Tax burden m 06335 PretaX Pro t 805 PretaX pro t 805 7 09699 EBIT 830 EBIT 830 201615 Sales 5140 Interest burden Pro t margin Sales 5140 16581 Assets 3100 Asset turnover Leverage Assets m 14091 Equity 2200 b ROE 06335 gtlt 09699 gtlt 01615 gtlt 16581 gtlt 14091 02318 2318 c g ROE gtlt plowback 02318 gtlt 01608 1608 Chapter 14 Financial Statement Analysis 18 a QuickBrush has had higher sales and earnings growth per share than SmileWhite Margins are also higher But this does not necessarily mean that QuickBrush is a better investment SmileWhite has a higher ROE which has been stable while QuickBrush s ROE has been declining We can use Du Pont analysis to identify the source of the difference in ROE Component De nition QuickBrush SmileWhite Tax burden 1 7 t Net pro tPretaX pro t 674 660 Interest burden PretaX pro tEBIT 100 0955 Pro t margin EBIT Sales 8 5 65 Asset turnover Sales Assets 142 35 5 Leverage Assets Equity 147 148 ROE Net pro tEquity 120 214 While tax burden interest burden and leverage are similar pro t margin and asset turnover differ Although SmileWhite has a lower pro t margin it has far higher asset turnover Sustainable growth ROE gtlt plowback ratio Plowback Sustainable Ludlow s ROE ratio growth rate estimate QuickBrush 120 100 120 300 SmileWhite 214 034 73 100 Ludlow has overestimated the sustainable growth rate for each company QuickBrush has little ability to increase its sustainable growth because plowback already equals 100 SmileWhite could increase its sustainable growth by increasing its plowback ratio Fquot QuickBrush s recent EPS growth has been achieved by increasing book value per share not by achieving greater pro ts per dollar of equity Since EPS is equal to Book value per share gtlt ROE a rm can increase EPS even if ROE is declining this is the case for QuickBrush QuickBrush s book value per share has more than doubled in the last two years Book value per share can increase either by retaining earnings or by issuing new stock at a market price greater than book value QuickBrush has been retaining all earnings but the increase in the number of outstanding shares indicates that it has also issued a substantial amount of stock 146 Chapter 14 Financial Statement Analysis 19 a 2001 2005 Operating margin Operat1ng1ncome deprec1aton 38 3 6 5 76 9 6 8 Sales 542 979 Asset turnover sa les 221 E 336 Total Assets 245 291 Interest Burden w i 0914 i 100 Operat1ng1ncome Deprec1aton 38 3 76 9 Financial Leverage M E 154 E 132 Shareholders Equ1ty 159 220 Income tax rate E 4063 a 5522 PretaX 1ncome 32 67 Using the Du Pont formula ROE200117 04063 gtlt 0914 X 0065 X 221 X 154 0120 120 ROE2005 1 05522 X 10 X 0068 X 336 X 132 0135 135 b i Asset turnover measures the ability of a company to minimize the level of assets current or xed to support its level of sales The asset turnover increased substantially over the period thus contributing to an increase in the ROE ii Financial leverage measures the amount of nancing not including equity but including short and longterm debt that the rm uses Financial leverage declined over the period thus adversely affecting the ROE Since asset turnover increased substantially more than nancial leverage declined the net effect was an increase in ROE Chapter 05 Risk and Return Past and Prologue Chapter 5 Risk and Return Past and Prologue V12312007 7 V111991 X 1 g7 7 100000 X 1057 7 14071004 2 i and ii The standard deviation is nonnegative 3 c Determines most of the portfolio s return and volatility over time 4 Er 7 03 X 44 04 X 14 03 X 716 7 14 62 7 03 X 44 7 142 04 X 14 7142 03 X 716 71427 540 c 7 2324 The mean is unchanged but the standard deviation has increased a The holding period returns for the three scenarios are Boom 50 7 40 240 030 3000 Normal 43 7 40 140 010 1000 Recession 34 7 40 05040 701375 71375 EHPR 13 X 30 13 X 10 13 X 71375 875 62HPR 13 X 30 7 8752 13 X 10 7 8752 13 X 71375 7 8752 31979 6 V31979 1788 b Er 7 05 X 875 05 X 4 7 6375 c 7 05 gtlt1788 7 894 Chapter 05 Risk and Return Past and Prologue 6 gt1 gt0 Investment 3 For each portfolio Utility Er 7 05 X 4 X 62 Investment Er c U 1 012 030 00600 2 015 050 03500 3 021 016 01588 4 024 021 01518 We choose the portfolio with the highest utility value Investment 4 When an investor is risk neutral A 0 so that the portfolio with the highest utility is the portfolio with the highest expected return Erx 02 X 720 05 X 18 03 X 50 20 Ery 02 X 715 05 X 20 03 X 10 10 6X2 02 X 720 7 202 05 X 18 7 202 03 X 50 7 202 592 ox 2433 6y 02 X 715 7 102 05 X 20 7102 03 X 10 7102 175 m 1323 Er 09 X 20 01 X 10 19 The probability is 050 that the state of the economy is neutral Given a neutral economy the probability that the performance of the stock will be poor is 030 and the probability of both a neutral economy and poor stock performance is 030 X 050 015 Er 01 X 15 06 X 13 03 X 7 114 Chapter 05 Risk and Return Past and Prologue 14 a Timeweighted average returns are based on yearbyyear rates of return Year Return capital gains dividendprice 20052006 110 7100 4100 1400 20062007 90 7110 4110 71455 20072008 95 7 90 490 1000 Arithmetic mean 315 Geometric mean 233 b Time Cash ow 1 39 quot 0 300 Purchase of three shares at 100 per share 1 208 Purchase oftwo shares at 110 plus dividend income on three shares held 2 110 Dividends on ve shares plus sale of one share at 90 3 396 Dividends on four shares plus sale of four shares at 95 per share 396 l l l l l l 1 Date 1105 1106 1107 1108 1 l l l l l l l l 208 300 Dollarweighted return Internal rate of return 701661 Chapter 05 Risk and Return Past and Prologue 15 a Erp irf 12A6p2 12 X 4 X 0202 008 80 b 009 12A6p2 12 X A X 0202 2 A 00912 X 004 45 c Increased risk tolerance means decreased risk aversion A which results in a decline in risk premiums 16 For the period 1926 7 2006 the mean annual risk premium for large stocks over T bills is 842 Er Riskfree rate Risk premium 5 842 1342 17 In the table below we use data from Table 53 and the approximation r E R 7 i Large Stocks r E 1219 313 906 Small Stocks r 1814 313 1501 LongTerm TBonds r E 564 313 251 TBills r E 377 313 064 Next we compute real rates using the exact relationship 1 R R i r 1 11 11 Large Stocks r 0090610313 879 Small Stocks r 0150110313 1455 LongTerm TBonds r 0025110313 243 TBills r 0006410313 062 a The expected cash ow is 05 X 50000 05 X 150000 100000 With a risk premium of 10 the required rate of return is 15 Therefore if the value of the portfolio is X then in order to earn a 15 expected return X115 100000 3 X 86957 Chapter 05 Risk and Return Past and Prologue b 0 3 1 E Ifthe portfolio is purchased at 86957 and the expected payoffis 100000 then the expected rate of return Er is 100000 86957 015 150 86957 The portfolio price is set to equate the expected return with the required rate of return If the risk premium over Tbills is now 15 then the required return is 5 15 20 The value of the portfolio X must satisfy X120 100 000 3 X 83333 For a given expected cash ow portfolios that command greater risk premia must sell at lower prices The extra discount from expected value is a penalty for risk Erp 03 X 7 07 X 17 14 per year 6p 07 X 27 189 per year Investment Security Proportions T Bills 300 Stock A 07 X 27 189 Stock B 07 X 33 231 Stock C 07 X 40 280 17 7 Your Rewardtovar1ab111ty rat1o 03704 Client s Rewardtovariability ratio 1487 7 03704 Chapter 05 Risk and Return Past and Prologue d 130 0 A P CALslope3704 17 client 7 C 1 89 27 20 a Mean ofportfolio 1 7yrf y rP rf rP 7rfy 7 10y If the expected rate of return for the portfolio is 15 then solving for y 15710y3y 08 Therefore in order to achieve an expected rate of return of 15 the client must invest 80 of total funds in the risky portfolio and 20 in Tbills b Investment Security Proportions TBills 200 Stock A 08 X 27 216 Stock B 08 X 33 264 Stock C 08 X 40 320 c 6p 08 X 27 216 per year Chapter 05 Risk and Return Past and Prologue 21 a Portfolio standard deviation 6p y X 27 If the client wants a standard deviation of 20 then y 2027 07407 7407 in the risky portfolio b Expected rate of return 7 10y 7 07407 X 10 14407 22 13 7 a Slope ofthe CML 25 024 See the diagram on the next page b My fund allows an investor to achieve a higher expected rate of return for any given standard deviation than would a passive strategy ie a higher expected return for any given level of risk 20 18 CAL slope3704 16 14 CIVIL slope24 12 Be 10 c 8 L 6 4 2 0 1 l 1 0 1 0 20 30 039 Chapter 05 Risk and Return Past and Prologue 23 a With 70 of his money in my funds portfolio the client has an expected rate of return of 14 per year and a standard deviation of 189 per year If he shifts that money to the passive portfolio which has an expected rate of return of 13 and standard deviation of 25 his overall expected return and standard deviation would become ErC If 4 07I M rf In this case rf 7 and rM 13 Therefore Erc 7 07 X 6 112 The standard deviation of the complete portfolio using the passive portfolio would be cc 07 X 6M 07 X 25 175 Therefore the shift entails a decline in the mean from 14 to 112 and a decline in the standard deviation from 189 to 175 Since both mean return and standard deviation fall it is not yet clear whether the move is beneficial The disadvantage of the shift is apparent from the fact that if my client is willing to accept an expected return on his total portfolio of 112 he can achieve that return with a lower standard deviation using my fund portfolio rather than the passive portfolio To achieve a target mean of 112 we rst write the mean of the complete portfolio as a function of the proportions invested in my fund portfolio y Erc 7 y17 7 7 7 10y Because our target is Erc 112 the proportion that must be invested in my fund is determined as follows 112 7 112710y3y 042 The standard deviation of the portfolio would be 6c y X 27 042 X 27 1134 Thus by using my portfolio the same 112 expected rate ofretum can be achieved with a standard deviation of only 1134 as opposed to the standard deviation of 175 using the passive portfolio Chapter 05 Risk and Return Past and Prologue b The fee would reduce the rewardtovariability ratio ie the slope of the CAL Clients will be indifferent between my fund and the passive portfolio if the slope of the afterfee CAL and the CML are equal Let f denote the fee 17 7 f7 10 f 27 77 13 7 T Slope of CAL with fee Slope of CML which requires no fee 024 Setting these slopes equal and solving for f 10 f 27 X 024 648 f 10 648 352 per year 24 Assuming no change in tastes that is an unchanged risk aversion investors perceiving higher risk will demand a higher risk premium to hold the same portfolio they held before If we assume that the riskfree rate is unaffected the increase in the risk premium would require a higher expected rate of return in the equity market 25 Expected return for your fund Tbill rate risk premium 6 10 16 Expected return of client s overall portfolio 06 X 16 04 X 6 12 Standard deviation of client s overall portfolio 06 X 14 84 R1sk prem1um E 071 26 Reward to variability ratio Standard dev1at1on 14 Chapter 05 Risk and Return Past and Prologue 27 Risk Premium Sharpe Mean SD 39 19261946 2169 5677 19471966 1350 2407 056 19671986 1163 3560 033 19872006 1031 2460 042 19262006 1437 3753 038 source Data in Table 53 a In two of the ve time frames presented small stocks provide better ratios than large stocks b Small stocks show a declining trend in risk but the decline is not stable 28 Large Cap Real Risk Premium Sharpe Returns Mean SD Ratio 19261946 834 2773 0 30 19471966 1274 1816 070 19671986 526 18 42 0 29 19872006 993 16 83 0 59 19262006 906 2064 0 44 source Data in Table 53 29 Small Cap Real Risk Premium Sharpe Returns Mean SD Ratio 19261946 2064 56 20 0 37 19471966 1132 24 74 0 46 19671986 533 36 42 015 19872006 723 25 22 0 29 19262006 1125 37 91 0 30 source Data in Table 53


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