This Document Contains Chapters 10 to 11 Brealey 5CE Solutions to Chapter 10 1. The extra 1 million burgers increase total costs by $.5 million. Therefore, variable cost = $.50 per burger. Fixed costs must then be $1.25 million, since the first 1 million burgers result in total cost of $1.75 million. 2. a. Average cost = $1.75 million / 1 million = $1.75/burger b. Average cost = $2.25 million / 2 million = $1.125/burger c. The fixed costs are spread across more burgers — thus the average cost falls. 3. a. (Revenue – expenses) changes by $1 million – $0.5 million = $0.5 million. After-tax profits increase by $0.5 million × (1 – .35) = $0.325 million. Because depreciation is unaffected, cash flow changes by an equal amount. b. Expenses increase from $5 million to $6 million. After-tax income and CF fall by $1 million × (1 – .35) = $0.65 million. 4. The 12%, 10-year annuity factor is 5.650. So the effect on NPV equals the change in CF × 5.650 a. $.325 million × 5.650 = $1.836 million increase $.65 million × 5.650 = $3.673 million decrease b. Fixed costs can increase until the point at which the higher costs (after taxes) reduce NPV by $2 million. Increase in fixed costs × (1 – T) × annuity factor(12%, 10 years) = $2 million Increase × (1 – .35) × 5.650 = $2 million Increase = $544,588 c. Accounting profits currently are $(10 – 5 – 2) million × (1 – .35) = $1.95 million. Pretax profits are currently $(10 – 5 –2) = $3 million. Fixed costs can increase by this amount ($ 3 million) before pretax profits are reduced to zero. 10-1 This Document Contains Chapters 10 to 11 5. Revenue = Price × quantity = $2 × 6 million = $12 million Expense = Variable cost + fixed cost = $1 × 6 million + $2 million = $8 million Depreciation = $5 million/5 years = $1 million per year CF = (1 − T) × (Revenue – expenses) + T × depreciation = .60 × ($12 million – $8 million) + .4 × $1 million = $2.8 million a. NPV = –$5 million + $2.8 million × annuity factor(5 years, 12%) = –$5 million + $2.8 million × 3.605 = $5.1 million b. If variable cost = $1.20, then expenses increase to $1.20 × 6 million + $2 million = $9.2 million. CF = .60 × ($12 million – $9.2 million) + .4 × $1 million = $2.08 million NPV = –$5 million + $2.08 million × 3.605 = $2.5 million c. If fixed costs = $1.5 million, expenses fall to ($1 × 6 million) + $1.5 million = $7.5 million CF = .60 × ($12 million – $7.5 million) + .4 × $1 million = $3.1 million NPV = –$5 million + $3.1 million × 3.605 = $6.2 million d. Call P the price per jar. Then Revenue = P × 6 million Expense = $1 × 6 million + $2 million = $8 million CF = (1 – .40) × (6P – 8) + .40 × 1 = 3.6P – 4.4 NPV = –5 + (3.6P – 4.4) × 3.605 = –20.862 + 12.978P NPV = 0 when P = $1.61 per jar 6. Base Case Best Case Worst Case Price $ 50 $ 55 $ 45 Variable Cost $ 30 $ 27 $ 33 Fixed Cost $300,000 $ 270,000 $ 330,000 Sales 30,000 33,000 27,000 10-2 CF = (1 – T) × [Revenue – Cash Expenses] + T × Depreciation Depreciation = $1 million/10 years = $100,000 per year Best-case CF = .65 [33,000 × (55 – 27) – 270,000] + . 35 × 100,000 = $460,100 Worst-case CF = .65 [27,000 × (45 – 33) – 330,000] + .35 × 100,000 = $ 31,100 10-Year Annuity factor at 14% discount rate = 5.2161 Best-case NPV = 5.2161 × $460,100 – $1,000,000 = $1,399,928 Worst-case NPV = 5.2161 × $ 31,100 – $1,000,000 = –$ 837,779 7. If price is higher, for example because of inflation, variable costs also may be higher. Similarly, if price is high because of strong demand for the product, then sales may be higher. It doesn’t make sense to formulate a scenario analysis in which uncertainty in each variable is treated independently. 8. At the break-even level of sales, which is 60,000 units, profit would be zero: Profit = 60,000 × (2 – variable cost per unit) – 20,000 – 10,000 = 0 Solve to find that variable cost per unit = $1.50 9. a. Each dollar of sales generates $0.70 of pretax profit. Depreciation is $100,000 and fixed costs are $200,000. Accounting break-even revenues are therefore: (200,000 + 100,000)/.70 = $428,571 The firm must sell 4,286 diamonds annually. b. Call Q the number of diamonds sold. Cash flow equals = (1 – .35)(Revenue – expenses) + .35 × depreciation = .65 (100Q – 30Q – 200,000) + .35 (100,000) = 45.5Q – 95,000 The 12%, 10-year annuity factor is 5.650. Therefore, for NPV to equal zero, (45.5Q – 95,000) × 5.650 = $1,000,000 10-3 257.075Q – 536,750 = 1,000,000 Q = 5,978 diamonds per year 10. a. Accounting break-even would increase because the depreciation charge will be higher. b. NPV break-even would decrease because the present value of the depreciation tax shield will be higher when all depreciation charges can be taken in the first five years. 11. Accounting break-even is unaffected since taxes paid are zero when pretax profit is zero, regardless of the tax rate. NPV break-even increases since the after-tax cash flow corresponding to any level of sales falls when the tax rate increases. 12. Cash flow = Net income + depreciation If depreciation is positive, then CF will be positive even when net income = 0. Therefore the level of sales necessary for CF break-even must be less than the level of sales necessary for zero-profit break-even. 13. If CF = 0 for the entire life of the project, then the PV of cash flows = 0, and project NPV will be negative in the amount of the required investment. 14. a. Variable cost = 75% of revenue. Additional profit per $1 of additional sales is therefore $0.25. Depreciation per year = $3000/5 = $600. Break-even sales level = Profit Fixed costs per $ of extra sales = 1000 + 600 .25 = $6400/year This sales level corresponds to a production level of $6400/$80 per unit = 80 units. To find NPV break-even sales, first calculate cash flow. With no taxes, CF = .25 × Sales – 1000. 10-4 The 10%, 5-year annuity factor is 3.7908. Therefore, if project NPV equals zero: PV(cash flows) – Investment = 0 3.7908 × (.25 × Sales – 1000) – 3000 = 0 .9477 × Sales – 3790.8 – 3000 = 0 Sales = $7166 This sales level corresponds to a production level of $7,166/$80, almost 90 units. b. Now taxes are 40% of profits. Accounting break-even is unchanged, since taxes are zero when profits = 0. To find NPV break-even, recalculate cash flow. CF = (1 – T) (Revenue – Cash Expenses) + T × Depreciation = .60 (.25 × Sales – 1000) + .40 × 600 = .15 × Sales – 360 The annuity factor is 3.7908, so we find NPV as follows: 3.7908 (.15 × Sales – 360) – 3000 = 0 Sales = $7,676 which corresponds to production of $7,676/$80, almost 96 units. 15. a. Accounting break-even increases: MACRS results in higher depreciation charges in the early years of the project, requiring a higher sales level for the firm to break even in terms of accounting profits. b. NPV break-even decreases. The accelerated depreciation increases the present value of the tax shield, and thus reduces the level of sales necessary to achieve zero NPV. c. MACRS makes the project more attractive. The PV of the tax shield is higher, so the NPV of the project at any given level of sales is higher. 10-5 16. Figures in Thousands of Dollars Sales $16,000 − Variable cost 12,800 (80% of sales) − Fixed cost 2,000 − Depreciation 500 (includes depreciation on new checkout equipment) = Pretax profit 700 − Taxes (at 40%) 280 = Profit after tax $ 420 + Depreciation 500 = Cash flow $ 920 a. Cash flow increases by $140,000 from $780,000 (see Table 8.1) to $920,000. The cost of the investment is $600,000. Therefore, NPV = –600 + 140 × annuity factor(8%, 12 years) = –600 + 140 × 7.536 = $455.04 thousand = $455,040 b. The equipment reduces variable costs from 81.25% of sales to 80% of sales. Pretax savings are therefore 0.0125 × sales. On the other hand, depreciation charges increase by $600,000/12 = $50,000 per year. Therefore, accounting profits are unaffected if sales equal $50,000/.0125 = $4,000,000. c. The project reduces variable costs from 81.25% of sales to 80% of sales. Pretax savings are therefore .0125 × Sales. Depreciation increases by $50,000 per year. Therefore, after-tax cash flow increases by (1 – T) × (∆Revenue – ∆ Expenses) + T × (∆Depreciation) = (1 – .4) × (.0125 × Sales) + .4 × 50,000 = .0075 × sales + 20,000 For NPV to equal zero, the increment to cash flow times the 12-year annuity factor must equal the initial investment. ∆cash flow × 7.536 = 600,000 ∆cash flow = $79,618 Therefore, .0075 × Sales + 20,000 = 79,618 Sales = $7,949,067 NPV break-even is nearly double accounting break-even. 10-6 17. NPV will be negative. We’ve shown in the previous problem that the accounting break-even level of sales is less than NPV break-even. 18. Percentage change in profits equals percentage change in sales × DOL A sales decline of $0.5 million represents a change of $.5/$4 = 12.5 percent. Profits will fall by 7.5 × 12.5 = 93.75%, from $1 million to $.0625 million. Similarly, a sales increase will increase profits to $1.9375 million. 19. DOL = 1 + Fixed costs including depreciation Profit a. Profit = Revenues – variable cost – fixed cost – depreciation = $ 8,000 – $6,000 – $1,000 – $600 = $400 DOL = 1 + 1600 400 = 5.0 b. Profit = Revenues – variable cost – fixed cost – depreciation = $10,000 – $7,500 – $1,000 – $600 = $900 DOL = 1 + 1600 900 = 2.78 c. DOL is higher when profits are lower because a $1 change in sales leads to a greater percentage change in profits. 20. DOL = 1 + Fixed costs including depreciation Profit If profits are positive, DOL cannot be less than 1.0. At sales = $8000, profits for Modern Artifacts (if fixed costs and depreciation were zero) would be: $ 8000 × .25 = $2000 At sales of $10,000, profits would be $10,000 × .25 = $2500 Profit is one-quarter of sales regardless of the level of sales. If sales increase by 1%, so will profits. Thus DOL = 1. 10-7 21. a. Pretax profits currently equal Revenue – variable costs – fixed costs – depreciation = $6000 – $4000 – $1000 – $500 = $500 If sales increase by $300, expenses will increase by $200, and pretax profits will increase by $100, an increase of 20%. b. DOL = 1 + Fixed costs + depreciation Pretax profits = 1 + 1000 + 500 500 = 4 c. Percent change in profits = DOL × percent change in sales 20% = 4 × 5% 22. We compare expected NPV with and without testing. If the field is large, then: NPV = $8 million – $3 million = $5 million. If the field is small, then NPV = $2 million – $3 million = –$1 million. If the test is performed, and the field is found to be small, then the project is abandoned, and NPV = zero (minus the cost of the test, which is $.1 million). Therefore, without testing: NPV = .5 × $5 million + .5 × (−$1 million) = $2 million. With testing, expected NPV is higher: NPV = –$0.1 million + .5 × $5 million + .5 × 0 = $2.4 million. Therefore, it pays to perform the test. The decision tree is on the following page. 10-8 NPV = $5 million NPV = 0 (abandon) NPV = $5 million NPV = –$1 million Big oil field Small oil field Small oil field Big oil field Test (Cost = $100,000) Do not test 23. a. Expenses = (10,000 × $8) + $10,000 = $90,000 Revenue is either 10,000 × $12 = $120,000 or 10,000 × $6 = $60,000 Average CF = .5 × ($120,000 – $90,000) + .5 × ($60,000 – $90,000) = 0 b. If you can shut down the mine, CF in the low-price years will be zero. In that case: Average CF = .5 × ($120,000 – $90,000) + .5 × $0 = $15,000 (We assume fixed costs are incurred only if the mine is operating. The fixed costs do not rise with the amount of silver extracted, but are not incurred unless the mine is in production.) 10-9 24. a. Expected NPV = .5 × ($140 – $100) + .5 × ($50 – $100) = –$5 million Therefore, you should not build the plant. b. Now the worst-case value of the installed project is $90 million rather than $50 million. Expected NPV increases to a positive value: .5 × ($140 – $100) + .5 × ($90 – $100) = $15 million Therefore, you should build the plant. c. PV = $140 million Success Invest $100 million Failure Sell plant for $90 million 25. Options give you the ability to cut your losses or extend your gains. You benefit from good outcomes, but can limit damage from unsuccessful outcomes. The ability to change your actions (e.g. abandon or expand or change timing) is most important when the ultimate best course of action is most difficult to forecast. 26. Dell January 29, 2010- ($ millions) a. Variable cost - % of sales = 0.82 52,902 43,641 = 2009- Breakeven in revenue = $44,117 1 .82 624 6,465 852 = − + + b. % change in pretax profit = 0.3911 3,324 2,024 − 3,324 = − % change in sales = 0.1342 61,101 52,902 − 61,101 = − DOL = 2.914 0.1342 0.3911 = − − 10-10 DOL = 1 + profit Fixed cost = 1 + 4.923 2,024 624 + 6,465 + 852 = 27. a. Decision Tree (all figures in $000s) b. Joint Probability Calculations (for outcomes A through H): (A & B) 0.65 x 0.3 x 0.5 = 0.0975 (C & D) 0.65 x 0.5 x 0.5 = 0.1625 (E & F) 0.65 x 0.2 x 0.5 = 0.065 (G) 0.35 x 0.6 x 1.0 = 0.21 (H) 0.35 x 0.4 = 0.14 c. All dollar figures in 000s. Outcome Joint Probability NPV* ($) Product: Joint Prb. x NPV A 0.0975 2,924.52 $285.141 B 0.0975 2,856.90 278.548 C 0.1625 2,293.44 372.684 D 0.1625 2,225.82 361.696 E 0.0650 1,820.13 118.309 F 0.0650 1,752.51 113.913 G 0.2100 (1,296.72) (272.311) Initial investment ($1,300) t = 0 t = 1 t = 1 t = 2 t = 2 t = 2 t = 2 t = 2 t = 3 t = 3 t = 3 t = 3 t = 3 t = 3 t = 3 “success” 0.65 “failure” 0.35 $800 $1.00 0.3 0.5 0.2 0.6 0.4 0.5 0.5 0.5 0.5 0.5 0.5 $2,200 $1,800 $1,500 $1.50 $0 $2,235 $2,145 $1,835 $1,745 $1,535 $1,445 $1.50 STOP - abandon project A B C D E F G H OUTCOMES 10-11 H 0.1400 (1,299.09) (181.873) 1.000 E(NPV) = $1,076.11 *Sample NPV Calculation: By using your answer in part a, you can easily determine the project’s annual net cash flows for each outcome. Then, for each outcome, you can calculate the NPV for the project. This method can be applied individually to outcomes A through H. Below is a sample NPV calculation using outcome A. Year Net Cash Flows ($) PV Calculation Present Value ($) 0 (1,300) = − (1.10)0 1,300 (1,300) 1 800 (1.10)1 = 800 727.27 2 2,200 (1.10)2 = 2,200 1,818.08 3 2,235 (1.10)3 = 2,235 1,679.16 NPVA = 2,924.52 OR using present value tables Year Net Cash Flow ($) Discount Factor (10%) Present Value ($) 0 (1,300) 1.000 (1,300) 1 800 0.909 727.28 2 2,200 0.826 1,818.08 3 2,235 0.751 1,679.16 NPVA = 2,924.52 28. a. Optimistic Pessimistic Price $ 60 $ 55 Sales units 50,000 30,000 Variable cost $30 $ 30 CF = (1 – T) × (Revenue – Cash Expenses) + T × Depreciation Optimistic CF = .65 × [(60 – 30) × 50,000] + .35 × 600,000 = $1,185,000 NPV = –6,000,000 + 1,185,000 × annuity factor(12%, 10 years) = $ 695,514 (using annuity tables, we will get $695,487) 10-12 Pessimistic CF = .65 × [ (55 – 30) × 30,000] + .35 × 600,000 = $ 697,500 NPV = –6,000,000 + 697,500 × annuity factor(12%, 10 years) = –$2,058,969 (using annuity tables, we will get -$2,058,985.5) Expected NPV = 21 × $695,514 + 21 × (−$2,058,969) = –$681,728 The firm will reject the project. b. If the project can be abandoned after 1 year, then it will be sold for $5.4 million. (There will be no taxes, since this also is the depreciated value of the equipment.) Cash flow at t = 1 equals CF from project plus sales price: $697,500 + $5,400,000 = $6,097,500 PV = 6,097,500 1.12 = $5,444,196 NPV in the abandonment scenario is: $5,444,196 – $6,000,000 = –$555,804 which is not as disastrous as the result in part (a). Expected NPV is now positive: 21 × $695,514 + 21 × (−$555,804) = $69,855 Because of the abandonment option, the project is now worth pursuing. 29. The additional after-tax cash flow from the expanded sales in the good outcome for the project is: .65 × [ 20,000 × (60 – 35)] = $325,000 As in the previous question, we assume that the firm decides whether to expand production after it learns the first-year sales results. At that point, the project will have a remaining life of 9 years. The present value as of the end of the first year is thus calculated using the 9-year annuity factor at an interest rate of 12%, which is 5.3282. The increase in NPV as of year 1 in this scenario is therefore 5.3282 × $325,000 = $1,731,665 and the increase in NPV as of time 0 is 10-13 $1,731,665/1.12 = $1,546,129 The probability of this outcome is 1/2, so the increase in expected NPV is $773,065. 10-14 Solution to Minicase for Chapter 10 The following spreadsheet presents the base-case analysis for the mining project. Inflation is assumed to be 3.5%, but most costs increase in line with inflation. Thus, we deal with real quantities in the spreadsheet, and keep all quantities except for depreciation at their constant real values. The real value of the depreciation expense thus falls by 3.5% per year. For example, real depreciation for the expensive design in year t is: real depreciation = $10 million/7 (1.035)t The real discount rate is 1.14/1.035 – 1 = .10 = 10%. Notice that the cheaper design seems to dominate the more expensive one. Even if the expensive design ends up costing $10 million, which appears to be the best-case outcome, the cheaper design saves $1.7 million up front, which results in higher net present value. If the cost overruns on the expensive design, the advantage of the cheap design will be even more dramatic. We are told that the two big uncertainties are construction costs and the price of the transcendental zirconium (TZ). The following table does a sensitivity analysis of the impact of these two variables on the NPV of the expensive design. The range of initial costs represents a pessimistic outcome of a 15% overrun (i.e., $1.5 million) combined with environmental regulation costs of an additional $1.5 million. The optimistic outcome, which we arbitrarily take to entail costs of only $8 million, is probably less relevant. It seems from the case description that there is little chance of costs coming in below $10 million. Range of input variables Resultant net present values Variable Pessimistic Expected Optimistic Pessimistic Expected Optimistic Initial cost $13 m $10 m $8 m –$0.72m $1.63m $3.20m TZ Price $7,500 $10,000 $14,000 –$1.06m $1.63m $5.94m The following table repeats this analysis for the cheaper design. Here, the uncertainty in initial cost is due solely to the environmental regulations. We are told that this design will not be subject to significant other cost overruns. Range of input variables Resultant net present values Variable Pessimistic Expected Optimistic Pessimistic Expected Optimistic Initial cost $9.8 m $8.3 m NA $0.68m $1.63m NA TZ Price $7,500 $10,000 $14,000 –$0.83m $1.86m $6.16m 10-15 Notice that the NPV of the cheaper design exceeds the NPV of the expensive one by about $0.22 million regardless of the price of TZ. In this case, there do not seem to be any inherent relationships among the chief uncertainties of this project. The price of TZ is likely to be unaffected by the cost of opening a new mine. Thus, scenario analysis does not add much information beyond that provided by sensitivity analysis. One can make a case for delaying construction. If the firm waits a year to see how the price of TZ evolves, the firm may avoid the negative NPV that would result from a low price. Whether it is worth waiting depends on the likelihood that the price will fall. There is less of a case to be made for delaying construction over the uncertainty of cost overruns. It is unlikely that much of the uncertainty regarding initial cost would be resolved by waiting –- the firm probably needs to go into production to learn if there will be overruns. If the firm goes ahead with the cheaper design, it does not seem necessary to wait to see how the environmental regulations turn out. NPV is positive regardless of the outcome for this variable, so it would not affect the decision of whether to go ahead with the project. The option to walk away from the project would be irrelevant, at least with regard to this variable. 10-16 Brealey 5CE Solutions to Chapter 11 1. Return = dividend + capital gain initial price = 2 + (44 − 40) 40 = .15 = 15% Dividend yield = dividend / initial price = 2/40 = .05 = 5% Capital gains yield = capital gains / initial price = 4/40 = .10 = 10% 2. Dividend yield = 2/40 = .05 = 5%. The dividend yield is unaffected; it is based on the initial price, not the final price. Capital gain = $36 – $40 = −$4 Capital gains yield = –4/40 = –.10 = – 10% 3. a. Rate of return = dividend + capital gain price = 2 + (38 − 40) 40 = 0 Real rate = 11 + nominal rate + inflation rate − 1 = 1 + 0 1 + .03 − 1 = –.0291 = –2.91% b. Rate of return = 2 + (40 − 40) 40 = .05 = 5% Real rate = 11 + nominal rate + inflation rate − 1 = 1.05 1.03 − 1 = .0194 = 1.94% c. Rate of return = 2 + (42 - 40) 40 = .10 = 10% Real rate = 11 + nominal rate + inflation rate − 1 = 1.10 1.03 − 1 = .0680 = 6.80% 10-1 4. Real return = 1 + nominal rate of return 1 + inflation rate − 1 Costaguana: Real return = 1.95 1.80 − 1 = .0833 = 8.33% Canada: Real return = 1.14 1.03 − 1 = .1067 = 10.67% Canada provides the higher real return despite the lower nominal return. Notice that the approximation real rate ≈ nominal rate – inflation rate would incorrectly suggest that the Costaguanan real rate was higher than the Canadian real rate. The approximation is valid only for low rates. 5. We use the relationship (with all rates expressed as decimals) that: Real rate = 1 + nominal rate 1 + inflation rate − 1 Asset class Nominal Return Inflation Real Rate Treasury bills 4.6% 3.2% 1.36% Gov’t bonds 6.5 3.2 3.20 Common stocks 11.5 3.2 8.04 6. The nominal interest rate cannot be negative. If it were, investors would choose to hold cash (which pays a return of zero) rather than buy a bill providing a negative return. On the other hand, the real expected rate of return is negative if the inflation rate exceeds the nominal return. 7. Average price of the stocks in the market of the year 2005 is taken as the base value to calculate the equal weighted index (like the DJIA). The equal weight index value in 2005 is 1000 times the ratio of the average stock prices for that year. So, the 2005 Index using DJIA method is 74.81/74.81 x 1000 = 1000. To calculate the index value in the next year divide that year’s stock prices by the base year (2005) value and multiply by 1000. So, the 2006 Index using DJIA method is 98.71/74.81 x 1000 = 1319.48. This index value in 2007 is 105.36/74.81 x 1000 = 1408.37. For the market-value weighted index the total market value of stock is calculated for the base year, 2005, and the index value for the base year is 20,166)/20,166 x 1000 = set as 1000 for that year. So the market-value index value in 2006 is 25,121/20,166 x 1000 = 1245.7. All of the values are in the table: 10-2 Year Average price of stocks in market Index (using DJIA method) Total market value of stocks Index (using S&P method) 2005 74.81 1000.00 20,166 1000.00 2006 98.71 1319.48 25,121 1245.7 2007 105.36 1408.37 26,112 1294.8 2008 102.40 1368.80 24,734 1226.5 2009 102.37 1368.40 24,360 1208.0 8. Annual Rates of Return for stocks and portfolio Year San Tome Mining Sulaco Markets National Central Railway Minerva Shipping Azuera Inc. Portfolio Value (1/5 of each stock price) Portfolio Rate of Return 2005 74.810 2006 .0554 .8078 .1207 .4002 .1201 98.710 .3195 2007 .0052 -.0601 .1036 .2024 .1043 105.360 .0674 2008 -.1030 -.4511 -.1131 .5264 -.1052 102.402 -.0281 2009 .0013 .0054 .3795 -.1764. .1864 102.372 -.0003 Average rate of return -.0103 .0755 .1227 .2382 .0764 .0896 Standard deviation of return .0665 .5281 .2016 .3069 .1262 .1584 The simple average of the individual stocks’ standard deviation is 0.2460 or 24.6%. The standard deviation of the equal-weighted portfolio, shown in the table, is 15.84%. This is striking evidence of the benefits of diversification. Note: Since the question works with observed data, the sample standard deviations are calculated. Thus for each stock the average rate of return is calculated. Then, for each year, the squared difference between the year’s return and the average rate of return for all years is calculated. The squared deviations are summed and divided by 4 (the number of annual returns minus 1). This gives the sample variance. The sample standard deviation is the square root of the sample variance. 10-3 9. a. Subtract the T-bill rate of return to calculate the risk premium: TSX T-Bill Long Bond TSX risk premium Long bond risk premium 2006 0.1701 0.0397 0.1505 0.1304 0.1108 2007 0.0983 0.0428 0.0322 0.0555 -0.0106 2008 -0.33 0.0264 0.033 -0.3564 0.0066 2009 0.3505 0.0046 0.136 0.3459 0.1314 2010 0.1761 0.0049 -0.043 0.1712 -0.0479 average 0.0930 0.0237 0.0617 0.0693 0.0381 Std. Dev. 0.2540 0.0183 0.0807 0.2608 0.0787 b. The average TSX risk premium was 6.93 %. The average long bond risk premium was 3.81% for these five years. These results are largely due to the volatile performance from 2006-2010. c. A fast way to calculate standard deviation of a sample of data is using a spreadsheet, such as Excel. In Excel, use the STDEV function. Alternatively, the standard deviation can be calculated by hand. First, calculate the sample variance, then take the square root. The sample variance is the sum of the squared deviations from the mean, divided by the number of observations minus 1. We illustrate with the TSX risk premium: Variance of TSX risk premium = [1/(5-1)] × [(.1304 – .0693)2 + (.0555 – .0693)2 + (-.3564 – .0693)2 + (.3459 – .0693)2 + (.1712 – .0693)2 = .0680 Standard deviation of TSX risk premium = .0680 = 26.08% We would expect that the risk premium standard deviation would be higher for the TSX than for the Long Bond portfolio. This is what we find: the TSX risk premium has a 26.08% standard deviation and the Long Bond risk premium has a 7.87% standard deviation. There is more variation in the TSX risk premium because there is more variation in the TSX return than for the Long Bond portfolio. 10. In 2010, the S&P/TSX was more than four times its 1990 level. Therefore a 40- point movement was far less significant in percentage terms than in 1990. We would expect to see more 40-point days even if market risk as measured by percentage returns is no higher than in 1990. 11. Investors would not have invested in bonds if they had expected to earn negative average returns. Unanticipated events must have led to these results. For example, 10-4 inflation and nominal interest rates during this period rose to levels not seen for decades. These increases, which resulted in large capital losses on long-term bonds, were almost surely unanticipated by investors who bought those bonds in prior years. The results from this period demonstrate the perils of attempting to measure “normal” maturity (or risk) premiums from historical data. While experience over long periods may be a reasonable guide to normal premiums, the realized premium over short periods may contain little information about expectations of future premiums. 12. If investors become less willing to bear investment risk, they will require a higher risk premium for holding risky assets. Security prices will fall until the expected rates of return on those securities rise to the now-higher required rates of return. 13. Based on the historical risk premium of the TSX (7.0 percent), and the current level of the risk-free rate (about 0.5 percent), one would predict an expected rate of return of 7.5 percent. If the stock has the same systematic risk, it also should provide this expected return. Therefore, the stock price equals the present value of cash flows for a one-year horizon. P0 = 1.075 2 + 50 = $48.37 14. Boom 5 + (195−90) 90 = 122.22% Normal 2 + (100−90) 90 = 13.33% Recession 0 + (0−90) 90 = –100% Expected return = .3 × 122.22 + .5 × 13.33 + .2 × (−100)= 23.33% Variance = 0.3 × (122.22 − 23.33)2 + .5 × (13.33−23.33)2 + .2 × (−100−23.33)2 = 6025.8 Standard deviation = variance = 77.63% 15. The bankruptcy lawyer does well when the rest of the economy is floundering, but does poorly when the rest of the economy is flourishing and the number of bankruptcies is down. Therefore, the Tower of Pita is a good hedge. When the 10-5 economy does well and the lawyer’s bankruptcy business suffers, the stock return is excellent, thereby stabilizing total income. The owner of the gambling casino probably does well when the economy is flourishing and less well when it is doing poorly. For the casino owner, holding Tower of Pita stock will not stabilize total income as much as it does for the bankruptcy lawyer. 16. Rate of Return Boom 0 + (18−25) 25 = –28% Normal 1 + (26−25) 25 = 8% Recession 3 + (34−25) 25 = 48% Expected return = .3 × (−28%) + .5 × 8% + .2 × 48% = 5.2% Variance = .3 × (−28 – 5.2)2 + .5 × (8 – 5.2)2 + .2 × (48 – 5.2)2 = 700.96 Standard deviation = 700.96 = 26.5% Portfolio Rate of Return Boom (−28 + 122.22)/2 = 47.11% Normal (8 + 13.33)/2 = 10.665% Recession (48 –100)/2 = –26.0% Expected return = .3 × 47.11% + .5 × 10.665% + .2 × (-26.0%) = 14.27% Variance = .3 × (47.11 – 14.27)2 + .5 × (10.665 – 14.27)2 + .2 × (-26.0 – 14.27)2 = 654.4 Standard deviation = 654.4 = 25.6% Standard deviation is lower than for either firm individually because the variations in the returns of the two firms serve to offset each other. When one firm does poorly, the other does well, which reduces the risk of the combination of the two. 17. a. Interest rates tend to fall at the outset of a recession and rise during boom periods. Because bond prices move inversely with interest rates, bonds will provide higher returns during recessions when interest rates fall. 10-6 b. rstock = .2 × (−5%) + .6 × 15% + .2 × 25% = 13% rbonds = .2 × 14% + .6 × 8% + .2 × 4% = 8.4% Variance(stocks) = .2 × (−5−13)2 + .6 × (15−13)2 + .2 × (25 – 13)2 = 96 Standard deviation(stocks) = 96 = 9.80% Variance(bonds) = .2 × (14−8.4)2 + .6 × (8−8.4)2 + .2 × (4−8.4)2 = 10.24 Standard deviation(bonds) = 10.24 = 3.20% c. Stocks have higher expected return and higher volatility. More risk averse investors will choose bonds, while others will choose stocks. 18. a. Recession (−5% × .6) + (14% × .4) = 2.6% Normal (15% × .6) + ( 8% × .4) = 12.2% Boom (25% × .6) + ( 4% × .4) = 16.6% b. Expected return = .2 × 2.6% + .6 × 12.2% + .2 × 16.6% = 11.16% Variance = .2 × (2.6 – 11.16)2 + .6 × (12.2 – 11.16)2 + .2 × (16.6 – 11.16)2 = 21.22 Standard deviation = 21.22 = 4.61% c. The investment opportunities have these characteristics: Mean Return Standard Deviation Stocks 13.0% 9.80% Bonds 8.4 3.20 Portfolio 11.16 4.61 The best choice depends on the degree of your aversion to risk. Nevertheless, we suspect most people would choose the portfolio over stocks since it gives almost the same return with much lower volatility. This is the advantage of diversification. d. To calculate the correlation coefficient, rearrange the formula for the portfolio standard deviation as we did in Check Point 10.7. Correlation between bond and stock returns = (σp2 – xs2 σs2 – xb2 σb2) / ( 2 xs xb σs σb) = (.04612 – .62× .0982 – .42 × .0322) / ( 2 × .6 × .4 × .098 × .032) = -.995 10-7 The stocks and bonds are almost perfectly negatively correlated. 19. If we use historical averages to compute the “normal” risk premium, then our estimate of “normal” returns and “normal” risk premiums will fall when we include a year with a negative market return. This makes sense if we believe that each additional year of data reveals new information about the “normal” behavior of the market portfolio. We should update our beliefs as additional observations about the market become available. 20. Risk reduction is most pronounced when the stock returns vary against each other. When one firm does poorly, the other will tend to do well, thereby stabilizing the return of the overall portfolio. By contrast stock returns that move together provide no risk reduction. If stock returns are independent, some risk reduction (variability reduction) occurs but it is less than if the stock returns vary against each other. 21. a. General Steel ought to have more sensitivity to broad market movements. Steel production is more sensitive to changes in the economy than is food consumption. b. Exotic World Tours Agency sells a luxury good (expensive vacations) while General Cinema sells movies, which are less sensitive to changes in the economy. Exotic World Tours Agency will have greater market risk. 22. a. Expected return = .5 × (-20%) + .5 × 30% = 5% Standard deviation = [ .5 × (-20% - 5%)2 + .5 × (30% - .5%)2]1/2 = 25% The expected rate of return on the stock is 5 percent. The standard deviation is 25 percent. b. Because the stock offers a risk premium of zero (its expected return is the same as for Treasury bills), it must have no market risk. All the risk must be diversifiable, and therefore of no concern to investors. 23. Sassafras is not a risky investment to a diversified investor. Its return is better when the economy enters a recession. Therefore, the company risk offsets the risk of the rest of the portfolio. It is a portfolio stabilizer despite the fact that there is a 90 percent chance of loss. (Compare Sassafras to purchasing an insurance policy. Most of the time, you will lose money on your insurance policy. But the policy will pay off big if you suffer losses elsewhere — for example, if your house burns down. For this reason, we view insurance as a risk-reducing hedge, not as speculation. Similarly, Sassafras 10-8 may be viewed as analogous to an insurance policy on the rest of your portfolio since it tends to yield higher returns when the rest of the economy is faring poorly.) In contrast, the Leaning Tower of Pita has returns that are positively correlated with the rest of the economy. It does best in a boom and goes out of business in a recession. For this reason, Leaning Tower would be a risky investment to a diversified investor since it increases exposure to the macroeconomic or market risk to which the investor is already exposed. 24. a. Portfolio expected return = .3 × 9% + .7 × 8% = 8.3% Portfolio standard deviation = [.32 × .22 +.72 × .252 + 2 × .3 × .7 × .2 × .2 × .25]1/2 = .196 = 19.6% b. With correlation of .7, the portfolio standard deviation is = [.32 × .22 +.72 × .252 + 2 × .3 × .7 × .7 × .2 × .25]1/2 = .221 = 22.1% c. The higher is the correlation between two variables, the less potential for diversification. In (a), with correlation of only .2, the portfolio standard deviation is less than the standard deviation of return of either of the two stocks in the portfolio. However, with the higher correlation of .7, the stocks’ return move more closely together and forming a portfolio only somewhat reduces total variability. 25. a. The following table contains the annual rates of return, the five-year average rate of return and the standard deviation of the rates of return for each index and the portfolio with one-third in each of the indexes: TSX T-Bill Long Bond Portfolio 2006 0.1701 0.0397 0.1505 0.1201 2007 0.0983 0.0428 0.0322 0.0578 2008 -0.3300 0.0264 0.033 -0.0902 2009 0.3505 0.0046 0.136 0.1637 2010 0.1761 0.0049 -0.043 0.0460 average 0.0930 0.0237 0.0617 0.0595 std dev 0.2540 0.0183 0.0807 0.0963 b. The table summarizes the calculations from (a): Average return (%) Standard deviation (%) TSX 300 9.30 25.4 Long Bond 6.17 8.07 Treasury Bill 2.37 1.83 10-9 Portfolio 5.95 9.63 The average standard deviation of the three securities is 11.8% = (25.5+8.07+1.83)/3, higher than the portfolio standard deviation of 9.63%, showing the benefit of diversification. If there were no benefits from diversification, the portfolio standard deviation would simply be the average of the standard deviations of each of the securities in the portfolio, weighted by their portfolio weights (here the weights are each 1/3). 26. The correlation coefficients between the 4 annual rates of return on San Tome Mining (STM) and each of the stocks are as follows: San Tome Sulaco National Central Minerva Azuera Mining Markets Railway Shipping INC Correlation with STM 1.000 .8916 .6070 -.3503 .8459 As expected, the correlation of San Tome Mining with itself is 1. The stock offering the best diversification benefit is Minerva Shipping. Its return is most negatively correlated with San Tome’ rate of return. 27. Internet: Arithmetic Average Stocks T.Bills T.Bonds 1928-2010 11.31% 3.70% 5.28% 1961-2010 11.10% 5.27% 6.96% 2001-2010 3.54% 2.18% 5.80% a. From the above tables, the overall risk premium is bigger when using the Treasury Bill as the risk free security than using Treasury Bonds as the risk free security. This makes sense: Treasury Bills are less risky than Treasury Bonds, making the difference in risk between Treasury Bills and the market index bigger than the difference in risk Risk Premium Stocks - T.Bills Stocks - T.Bonds 1928-2010 7.62% 6.03% 1961-2010 5.83% 4.13% 2001-2010 1.37% -2.26% 10-10 between Treasury Bonds and the market index. b. The risk premium becomes smaller over time. 28. Internet: AGF Canada Class http://www.theglobeandmail.com/globe-investor/funds-and- etfs/funds/summary/?id=28560 3yr risk: 18.04 TD Precious Metals http://www.globefund.com/servlet/Page/document/v5/data/fund?style=na_eq&id=183 50&gf_uid=globeandmail.gf.03428539934 3yr risk: 44.81 TD Energy http://www.globefund.com/servlet/Page/document/v5/data/fund?style=na_eq&id=183 45&gf_uid=globeandmail.gf.03428539934 3yr risk: 27.35 TD Entert. & Communications GIF II http://www.globefund.com/servlet/Page/document/v5/data/fund?style=globe_eq&id=5 2905&gf_uid=globeandmail.gf.03428539934 3yr risk: 19.73 TD Health Sciences http://www.globefund.com/servlet/Page/document/v5/data/fund?style=na_eq&id=259 95&gf_uid=globeandmail.gf.03428539934 3 yr risk: 15.30 Except for TD Health Sciences, the other sectors all have higher risk than the index fund. It indicates some sectors have risk above Index, some below Index. 10-11 29. Expected results: Students have experience calculating rates of return for companies. They will see differences in dividend and capital gains yields. One thing to note: The S&P database provides a rolling 5 years worth of stock prices. At the time this data was retrieved, there were not 5 December closing prices. If another month had been selected, five years of data would have been available. Company Year December Closing Price Dividend Dividend Yield Capital Gains Yield Rate of return Magna 2007 80.43 0.575 (MGA) 2008 29.93 0.63 0.0078 -0.6279 -0.6200 2009 50.58 0.09 0.0030 0.6899 0.6930 2010 52.00 0.42 0.0083 0.0281 0.0364 Ford 2007 6.65 0 (F) 2008 2.29 0 0.0000 -0.6556 -0.6556 2009 10.00 0 0.0000 3.3668 3.3668 2010 16.79 0 0.0000 0.6790 0.6790 Microsoft 2007 35.60 0.41 (MSFT) 2008 19.44 0.46 0.0129 -0.4539 -0.4410 2009 30.48 0.52 0.0267 0.5679 0.5947 2010 27.91 0.55 0.0180 -0.0843 -0.0663 10-12 30. Expected results: Students will see diversification in action. MG Return RIM Return RY Return RCI.B Return HSE Return 4-Jul-11 46.60 -0.106 23.93 -0.142 51.40 -0.058 36.48 -0.045 26.75 0.017 1-Jun- 11 52.15 0.112 27.88 -0.326 54.58 -0.026 38.19 0.044 26.3 -0.113 2-May- 11 46.90 -0.031 41.35 -0.103 56.06 -0.050 36.58 0.032 29.65 0.013 1-Apr- 11 48.38 0.046 46.09 -0.159 59.01 0.002 35.46 0.016 29.28 0.005 1-Mar- 11 46.25 -0.024 54.83 -0.145 58.90 0.056 34.9 0.038 29.14 -0.007 1-Feb- 11 47.37 -0.182 64.15 0.087 55.76 0.058 33.61 -0.019 29.35 0.110 4-Jan- 11 57.88 0.126 58.99 0.016 52.70 0.036 34.27 0.011 26.44 0.017 1-Dec- 10 51.39 0.060 58.07 -0.082 50.89 -0.048 33.91 -0.046 26 0.085 1-Nov- 10 48.50 0.070 63.25 0.090 53.45 0.010 35.56 -0.015 23.96 -0.022 1-Oct- 10 45.33 0.093 58.01 0.158 52.90 0.024 36.1 -0.035 24.5 0.012 1-Sep- 10 41.49 0.017 50.10 0.096 51.67 0.051 37.41 0.049 24.21 0.007 3-Aug- 10 40.81 0.089 45.70 -0.227 49.17 -0.051 35.65 0.035 24.04 -0.005 2-Jul-10 37.46 0.091 59.15 0.130 51.79 0.070 34.44 0.028 24.16 0.001 1-Jun- 10 34.33 -0.029 52.33 -0.177 48.41 -0.080 33.5 -0.036 24.14 -0.053 3-May- 10 35.37 0.084 63.55 -0.122 52.63 -0.105 34.75 0.005 25.5 -0.061 1-Apr- 10 32.62 0.063 72.40 -0.038 58.81 0.045 34.57 0.044 27.16 -0.014 1-Mar- 10 30.69 0.047 75.25 0.009 56.30 0.046 33.12 0.010 27.55 0.082 1-Feb- 10 29.30 0.020 74.55 0.105 53.81 0.087 32.8 0.039 25.46 0.023 4-Jan- 10 28.72 0.105 67.47 -0.050 49.52 -0.064 31.56 0.021 24.88 -0.116 1-Dec- 09 26.00 0.059 71.03 0.165 52.93 -0.007 30.92 0.029 28.14 0.086 2-Nov- 09 24.56 0.178 60.97 -0.045 53.31 0.037 30.04 0.009 25.91 -0.018 1-Oct- 09 20.85 -0.064 63.82 -0.118 51.43 -0.039 29.77 0.050 26.39 -0.054 1-Sep- 09 22.28 -0.080 72.38 -0.099 53.53 0.020 28.35 0.013 27.91 0.023 10-13 4-Aug- 09 24.22 -0.090 80.35 -0.018 52.50 0.101 27.99 0.007 27.29 -0.061 2-Jul-09 26.61 0.104 81.82 -0.010 47.70 0.089 27.8 0.001 29.07 -0.025 1-Jun- 09 24.10 0.364 82.68 -0.033 43.80 0.088 27.77 -0.078 29.83 -0.029 1-May- 09 17.67 -0.117 85.53 0.038 40.24 0.033 30.11 0.116 30.73 0.170 1-Apr- 09 20.00 0.221 82.43 0.513 38.95 0.164 26.97 0.009 26.27 0.081 2-Mar- 09 16.38 0.037 54.49 0.072 33.46 0.189 26.72 -0.019 24.31 -0.017 2-Feb- 09 15.80 -0.049 50.84 -0.250 28.13 0.017 27.23 -0.135 24.72 -0.096 2-Jan- 09 16.62 -0.066 67.78 0.369 27.66 -0.144 31.47 -0.055 27.36 -0.014 1-Dec- 08 17.79 0.028 49.50 -0.092 32.33 -0.164 33.3 0.019 27.75 -0.035 3-Nov- 08 17.30 -0.110 54.50 -0.107 38.67 -0.078 32.68 0.034 28.76 -0.103 1-Oct- 08 19.44 -0.261 61.02 -0.149 41.94 -0.063 31.6 0.014 32.07 -0.181 2-Sep- 08 26.29 -0.103 71.71 -0.446 44.74 0.036 31.17 -0.103 39.16 -0.058 1-Aug- 08 29.32 0.017 129.37 0.028 43.19 0.032 34.75 0.120 41.59 0.047 2-Jul-08 28.83 -0.006 125.84 0.051 41.86 0.043 31.04 -0.126 39.72 -0.071 2-Jun- 08 29.00 -0.133 119.69 -0.129 40.14 -0.098 35.5 -0.095 42.77 -0.021 1-May- 08 33.46 -0.052 137.41 0.121 44.50 0.058 39.21 -0.020 43.68 0.105 1-Apr- 08 35.30 -0.005 122.59 0.061 42.06 0.012 39.99 0.214 39.54 0.132 3-Mar- 08 35.49 0.039 115.49 0.126 41.57 -0.029 32.95 -0.037 34.94 -0.036 1-Feb- 08 34.15 -0.084 102.53 0.090 42.82 -0.025 34.23 0.012 36.26 0.011 2-Jan- 08 37.30 -0.017 94.06 -0.164 43.91 0.009 33.82 -0.147 35.88 -0.067 3-Dec- 07 37.93 0.002 112.56 -0.011 43.51 -0.043 39.64 0.086 38.47 0.124 1-Nov- 07 37.85 -0.103 113.83 -0.033 45.45 -0.054 36.51 -0.137 34.22 -0.092 1-Oct- 07 42.19 -0.066 117.75 0.205 48.05 0.027 42.29 0.063 37.68 0.062 4-Sep- 07 45.17 0.012 97.75 0.084 46.77 0.013 39.78 -0.054 35.48 0.070 10-14 1-Aug- 07 44.64 0.020 90.21 0.182 46.17 0.005 42.04 -0.005 33.16 -0.076 3-Jul-07 43.78 -0.042 76.33 0.068 45.95 -0.037 42.25 0.062 35.9 -0.037 1-Jun- 07 45.70 0.026 71.47 0.209 47.72 -0.029 39.79 0.027 37.28 -0.016 1-May- 07 44.54 0.085 59.10 0.216 49.12 0.008 38.75 0.043 37.9 0.059 2-Apr- 07 41.06 0.014 48.60 -0.074 48.73 0.013 37.17 0.126 35.78 0.049 1-Mar- 07 40.49 0.012 52.50 -0.040 48.09 0.063 33.01 -0.009 34.1 0.063 1-Feb- 07 40.02 -0.065 54.69 0.093 45.24 -0.009 33.32 0.049 32.08 0.019 2-Jan- 07 42.82 -0.021 50.02 0.007 45.66 -0.009 31.77 0.049 31.49 -0.033 1-Dec- 06 43.76 0.069 49.67 -0.061 46.08 0.042 30.28 -0.007 32.57 -0.004 1-Nov- 06 40.92 0.049 52.92 0.204 44.23 0.070 30.5 0.042 32.7 0.103 2-Oct- 06 38.99 0.036 43.97 0.151 41.34 0.014 29.28 0.098 29.65 -0.006 1-Sep- 06 37.64 0.023 38.20 0.256 40.77 0.011 26.67 0.074 29.83 -0.059 1-Aug- 06 36.79 -0.036 30.42 0.230 40.34 0.064 24.84 0.179 31.71 0.002 31-Jul- 06 38.15 24.73 37.91 21.07 31.66 a. MG RIM RY RCI.B HSE mean 0.79% 1.30% 0.72% 1.16% -0.05% stdev 9.72% 16.45% 6.52% 6.99% 6.92% b. Companies Correlation MG RIM 0.1941 MG RY 0.3129 MG RCI.B 0.0282 MG HSE 0.0867 RIM RY 0.2910 RIM RCI.B 0.2724 RIM HSE 0.3929 RY RCI.B 0.0877 RY HSE 0.3075 10-15 Highest Correlation: RIM (Research in Motion) and HSE (Husky Energy) Lowest Correlation: MG (Magna International) and RCI.B (Rogers Communications) c. Companies Correlation RY CM 0.6263 Correlation between Royal Bank of Canada and Canadian Imperial Bank of Commerce is 0.6263, which is higher than those found in part (b). It is expected that their correlation is higher because both companies share similar unique risks while operating in the same industry; thus, stock returns is expected to be more correlated. 31. MG Return RIM Return RY Return RCI.B Return HSE Return Portfolio Return 4-Jul-11 46.60 -0.106 23.93 -0.142 51.40 -0.058 36.48 -0.045 26.75 0.017 -0.0668 1-Jun- 11 52.15 0.112 27.88 -0.326 54.58 -0.026 38.19 0.044 26.3 -0.113 -0.0618 2-May- 11 46.90 -0.031 41.35 -0.103 56.06 -0.050 36.58 0.032 29.65 0.013 -0.0278 1-Apr- 11 48.38 0.046 46.09 -0.159 59.01 0.002 35.46 0.016 29.28 0.005 -0.0181 1-Mar- 11 46.25 -0.024 54.83 -0.145 58.90 0.056 34.9 0.038 29.14 -0.007 -0.0163 1-Feb- 11 47.37 -0.182 64.15 0.087 55.76 0.058 33.61 -0.019 29.35 0.110 0.0110 4-Jan- 11 57.88 0.126 58.99 0.016 52.70 0.036 34.27 0.011 26.44 0.017 0.0410 1-Dec- 10 51.39 0.060 58.07 -0.082 50.89 -0.048 33.91 -0.046 26 0.085 -0.0063 1-Nov- 10 48.50 0.070 63.25 0.090 53.45 0.010 35.56 -0.015 23.96 -0.022 0.0267 1-Oct- 10 45.33 0.093 58.01 0.158 52.90 0.024 36.1 -0.035 24.5 0.012 0.0502 1-Sep- 10 41.49 0.017 50.10 0.096 51.67 0.051 37.41 0.049 24.21 0.007 0.0440 3-Aug- 10 40.81 0.089 45.70 -0.227 49.17 -0.051 35.65 0.035 24.04 -0.005 -0.0317 2-Jul-10 37.46 0.091 59.15 0.130 51.79 0.070 34.44 0.028 24.16 0.001 0.0640 1-Jun- 10 34.33 -0.029 52.33 -0.177 48.41 -0.080 33.5 -0.036 24.14 -0.053 -0.0751 3-May- 10 35.37 0.084 63.55 -0.122 52.63 -0.105 34.75 0.005 25.5 -0.061 -0.0398 1-Apr- 32.62 0.063 72.40 -0.038 58.81 0.045 34.57 0.044 27.16 -0.014 0.0198 10-16 10 1-Mar- 10 30.69 0.047 75.25 0.009 56.30 0.046 33.12 0.010 27.55 0.082 0.0390 1-Feb- 10 29.30 0.020 74.55 0.105 53.81 0.087 32.8 0.039 25.46 0.023 0.0549 4-Jan- 10 28.72 0.105 67.47 -0.050 49.52 -0.064 31.56 0.021 24.88 -0.116 -0.0210 1-Dec- 09 26.00 0.059 71.03 0.165 52.93 -0.007 30.92 0.029 28.14 0.086 0.0664 2-Nov- 09 24.56 0.178 60.97 -0.045 53.31 0.037 30.04 0.009 25.91 -0.018 0.0321 1-Oct- 09 20.85 -0.064 63.82 -0.118 51.43 -0.039 29.77 0.050 26.39 -0.054 -0.0452 1-Sep- 09 22.28 -0.080 72.38 -0.099 53.53 0.020 28.35 0.013 27.91 0.023 -0.0248 4-Aug- 09 24.22 -0.090 80.35 -0.018 52.50 0.101 27.99 0.007 27.29 -0.061 -0.0123 2-Jul-09 26.61 0.104 81.82 -0.010 47.70 0.089 27.8 0.001 29.07 -0.025 0.0317 1-Jun- 09 24.10 0.364 82.68 -0.033 43.80 0.088 27.77 -0.078 29.83 -0.029 0.0624 1-May- 09 17.67 -0.117 85.53 0.038 40.24 0.033 30.11 0.116 30.73 0.170 0.0481 1-Apr- 09 20.00 0.221 82.43 0.513 38.95 0.164 26.97 0.009 26.27 0.081 0.1976 2-Mar- 09 16.38 0.037 54.49 0.072 33.46 0.189 26.72 -0.019 24.31 -0.017 0.0525 2-Feb- 09 15.80 -0.049 50.84 -0.250 28.13 0.017 27.23 -0.135 24.72 -0.096 -0.1027 2-Jan- 09 16.62 -0.066 67.78 0.369 27.66 -0.144 31.47 -0.055 27.36 -0.014 0.0180 1-Dec- 08 17.79 0.028 49.50 -0.092 32.33 -0.164 33.3 0.019 27.75 -0.035 -0.0487 3-Nov- 08 17.30 -0.110 54.50 -0.107 38.67 -0.078 32.68 0.034 28.76 -0.103 -0.0728 1-Oct- 08 19.44 -0.261 61.02 -0.149 41.94 -0.063 31.6 0.014 32.07 -0.181 -0.1279 2-Sep- 08 26.29 -0.103 71.71 -0.446 44.74 0.036 31.17 -0.103 39.16 -0.058 -0.1349 1-Aug- 08 29.32 0.017 129.37 0.028 43.19 0.032 34.75 0.120 41.59 0.047 0.0487 2-Jul-08 28.83 -0.006 125.84 0.051 41.86 0.043 31.04 -0.126 39.72 -0.071 -0.0217 2-Jun- 08 29.00 -0.133 119.69 -0.129 40.14 -0.098 35.5 -0.095 42.77 -0.021 -0.0951 1-May- 08 33.46 -0.052 137.41 0.121 44.50 0.058 39.21 -0.020 43.68 0.105 0.0424 1-Apr- 35.30 -0.005 122.59 0.061 42.06 0.012 39.99 0.214 39.54 0.132 0.0826 10-17 08 3-Mar- 08 35.49 0.039 115.49 0.126 41.57 -0.029 32.95 -0.037 34.94 -0.036 0.0125 1-Feb- 08 34.15 -0.084 102.53 0.090 42.82 -0.025 34.23 0.012 36.26 0.011 0.0007 2-Jan- 08 37.30 -0.017 94.06 -0.164 43.91 0.009 33.82 -0.147 35.88 -0.067 -0.0772 3-Dec- 07 37.93 0.002 112.56 -0.011 43.51 -0.043 39.64 0.086 38.47 0.124 0.0316 1-Nov- 07 37.85 -0.103 113.83 -0.033 45.45 -0.054 36.51 -0.137 34.22 -0.092 -0.0838 1-Oct- 07 42.19 -0.066 117.75 0.205 48.05 0.027 42.29 0.063 37.68 0.062 0.0582 4-Sep- 07 45.17 0.012 97.75 0.084 46.77 0.013 39.78 -0.054 35.48 0.070 0.0249 1-Aug- 07 44.64 0.020 90.21 0.182 46.17 0.005 42.04 -0.005 33.16 -0.076 0.0250 3-Jul-07 43.78 -0.042 76.33 0.068 45.95 -0.037 42.25 0.062 35.9 -0.037 0.0027 1-Jun- 07 45.70 0.026 71.47 0.209 47.72 -0.029 39.79 0.027 37.28 -0.016 0.0435 1-May- 07 44.54 0.085 59.10 0.216 49.12 0.008 38.75 0.043 37.9 0.059 0.0821 2-Apr- 07 41.06 0.014 48.60 -0.074 48.73 0.013 37.17 0.126 35.78 0.049 0.0257 1-Mar- 07 40.49 0.012 52.50 -0.040 48.09 0.063 33.01 -0.009 34.1 0.063 0.0177 1-Feb- 07 40.02 -0.065 54.69 0.093 45.24 -0.009 33.32 0.049 32.08 0.019 0.0173 2-Jan- 07 42.82 -0.021 50.02 0.007 45.66 -0.009 31.77 0.049 31.49 -0.033 -0.0015 1-Dec- 06 43.76 0.069 49.67 -0.061 46.08 0.042 30.28 -0.007 32.57 -0.004 0.0077 1-Nov- 06 40.92 0.049 52.92 0.204 44.23 0.070 30.5 0.042 32.7 0.103 0.0935 2-Oct- 06 38.99 0.036 43.97 0.151 41.34 0.014 29.28 0.098 29.65 -0.006 0.0585 1-Sep- 06 37.64 0.023 38.20 0.256 40.77 0.011 26.67 0.074 29.83 -0.059 0.0608 1-Aug- 06 36.79 -0.036 30.42 0.230 40.34 0.064 24.84 0.179 31.71 0.002 0.0878 31-Jul- 06 38.15 24.73 37.91 21.07 31.66 MG RIM RY RCI.B HSE Portfolio mean 0.79% 1.30% 0.72% 1.16% -0.05% 0.78% stdev 9.72% 16.45% 6.52% 6.99% 6.92% 6.02% 10-18 Mean standard deviation of the standard deviations of all five stocks: 0.0932 - calculated as average of the individual stocks’ standard deviations Portfolio standard deviation: 0.0602 - calculated using the monthly rates of return of the portfolio Notice that the portfolio has lower monthly standard deviation than the simple average of the individual stocks’ standard deviations. This is due to the fact that the stock returns are less than perfectly correlated. Diversification reduces portfolio standard deviation. 32. MG Return RIM Return RY Return RCI.B Return HSE Return Portfolio Value Portfolio Return 4-Jul-11 122.15 -0.106 96.77 -0.142 135.58 -0.058 173.14 -0.045 84.49 0.017 612.13 -0.0693 1-Jun- 11 136.70 0.112 112.74 -0.326 143.97 -0.026 181.25 0.044 83.07 -0.113 657.73 -0.0674 2-May- 11 122.94 -0.031 167.21 -0.103 147.88 -0.050 173.61 0.032 93.65 0.013 705.28 -0.0334 1-Apr- 11 126.82 0.046 186.37 -0.159 155.66 0.002 168.30 0.016 92.48 0.005 729.62 -0.0349 1-Mar- 11 121.23 -0.024 221.71 -0.145 155.37 0.056 165.64 0.038 92.04 -0.007 755.99 -0.0343 1-Feb- 11 124.17 -0.182 259.40 0.087 147.09 0.058 159.52 -0.019 92.70 0.110 782.87 0.0096 4-Jan- 11 151.72 0.126 238.54 0.016 139.01 0.036 162.65 0.011 83.51 0.017 775.43 0.0383 1-Dec- 10 134.71 0.060 234.82 -0.082 134.24 -0.048 160.94 -0.046 82.12 0.085 746.82 -0.0280 1-Nov- 10 127.13 0.070 255.76 0.090 140.99 0.010 168.77 -0.015 75.68 -0.022 768.33 0.0360 1-Oct- 10 118.82 0.093 234.57 0.158 139.54 0.024 171.33 -0.035 77.38 0.012 741.65 0.0570 1-Sep- 10 108.75 0.017 202.59 0.096 136.30 0.051 177.55 0.049 76.47 0.007 701.66 0.0526 3-Aug- 10 106.97 0.089 184.80 -0.227 129.70 -0.051 169.20 0.035 75.93 -0.005 666.60 -0.0661 2-Jul-10 98.19 0.091 239.18 0.130 136.61 0.070 163.46 0.028 76.31 0.001 713.75 0.0741 1-Jun- 10 89.99 -0.029 211.61 -0.177 127.70 -0.080 158.99 -0.036 76.25 -0.053 664.53 -0.0946 3-May- 10 92.71 0.084 256.98 -0.122 138.83 -0.105 164.93 0.005 80.54 -0.061 733.99 -0.0629 1-Apr- 10 85.50 0.063 292.76 -0.038 155.13 0.045 164.07 0.044 85.79 -0.014 783.26 0.0075 1-Mar- 10 80.45 0.047 304.29 0.009 148.51 0.046 157.19 0.010 87.02 0.082 777.45 0.0280 1-Feb- 10 76.80 0.020 301.46 0.105 141.94 0.087 155.67 0.039 80.42 0.023 756.29 0.0696 4-Jan- 10 75.28 0.105 272.83 -0.050 130.63 -0.064 149.79 0.021 78.58 -0.116 707.10 -0.0322 1-Dec- 09 68.15 0.059 287.22 0.165 139.62 -0.007 146.75 0.029 88.88 0.086 730.62 0.0809 10-19 2-Nov- 09 64.38 0.178 246.54 -0.045 140.62 0.037 142.57 0.009 81.84 -0.018 675.95 0.0043 1-Oct- 09 54.65 -0.064 258.07 -0.118 135.66 -0.039 141.29 0.050 83.35 -0.054 673.03 -0.0587 1-Sep- 09 58.40 -0.080 292.68 -0.099 141.20 0.020 134.55 0.013 88.16 0.023 714.99 -0.0415 4-Aug- 09 63.49 -0.090 324.91 -0.018 138.49 0.101 132.84 0.007 86.20 -0.061 745.92 -0.0057 2-Jul-09 69.75 0.104 330.85 -0.010 125.82 0.089 131.94 0.001 91.82 -0.025 750.19 0.0151 1-Jun- 09 63.17 0.364 334.33 -0.033 115.54 0.088 131.80 -0.078 94.22 -0.029 739.06 0.0010 1-May- 09 46.32 -0.117 345.86 0.038 106.15 0.033 142.90 0.116 97.06 0.170 738.29 0.0555 1-Apr- 09 52.42 0.221 333.32 0.513 102.74 0.164 128.00 0.009 82.98 0.081 699.47 0.2600 2-Mar- 09 42.94 0.037 220.34 0.072 88.26 0.189 126.82 -0.019 76.78 -0.017 555.14 0.0504 2-Feb- 09 41.42 -0.049 205.58 -0.250 74.20 0.017 129.24 -0.135 78.08 -0.096 528.51 -0.1562 2-Jan- 09 43.56 -0.066 274.08 0.369 72.96 -0.144 149.36 -0.055 86.42 -0.014 626.38 0.0841 1-Dec- 08 46.63 0.028 200.16 -0.092 85.28 -0.164 158.04 0.019 87.65 -0.035 577.77 -0.0585 3-Nov- 08 45.35 -0.110 220.38 -0.107 102.00 -0.078 155.10 0.034 90.84 -0.103 613.67 -0.0696 1-Oct- 08 50.96 -0.261 246.74 -0.149 110.63 -0.063 149.98 0.014 101.30 -0.181 659.60 -0.1188 2-Sep- 08 68.91 -0.103 289.97 -0.446 118.02 0.036 147.94 -0.103 123.69 -0.058 748.52 -0.2590 1-Aug- 08 76.85 0.017 523.13 0.028 113.93 0.032 164.93 0.120 131.36 0.047 1010.20 0.0440 2-Jul-08 75.57 -0.006 508.86 0.051 110.42 0.043 147.32 -0.126 125.46 -0.071 967.62 -0.0019 2-Jun- 08 76.02 -0.133 483.99 -0.129 105.88 -0.098 168.49 -0.095 135.09 -0.021 969.46 -0.1063 1-May- 08 87.71 -0.052 555.64 0.121 117.38 0.058 186.09 -0.020 137.97 0.105 1084.79 0.0699 1-Apr- 08 92.53 -0.005 495.71 0.061 110.95 0.012 189.80 0.214 124.89 0.132 1013.88 0.0827 3-Mar- 08 93.03 0.039 467.00 0.126 109.65 -0.029 156.38 -0.037 110.36 -0.036 936.43 0.0474 1-Feb- 08 89.52 -0.084 414.60 0.090 112.95 -0.025 162.46 0.012 114.53 0.011 894.05 0.0303 2-Jan- 08 97.77 -0.017 380.35 -0.164 115.83 0.009 160.51 -0.147 113.33 -0.067 867.79 -0.1136 3-Dec- 07 99.42 0.002 455.16 -0.011 114.77 -0.043 188.13 0.086 121.51 0.124 979.00 0.0190 1-Nov- 07 99.21 -0.103 460.29 -0.033 119.89 -0.054 173.28 -0.137 108.09 -0.092 960.76 -0.0701 1-Oct- 07 110.59 -0.066 476.14 0.205 126.75 0.027 200.71 0.063 119.01 0.062 1033.21 0.1016 4-Sep- 118.40 0.012 395.27 0.084 123.37 0.013 188.80 -0.054 112.07 0.070 937.91 0.0331 10-20 07 1-Aug- 07 117.01 0.020 364.78 0.182 121.79 0.005 199.53 -0.005 104.74 -0.076 907.84 0.0574 3-Jul-07 114.76 -0.042 308.65 0.068 121.21 -0.037 200.52 0.062 113.39 -0.037 858.53 0.0205 1-Jun- 07 119.79 0.026 289.00 0.209 125.88 -0.029 188.85 0.027 117.75 -0.016 841.27 0.0664 1-May- 07 116.75 0.085 238.98 0.216 129.57 0.008 183.91 0.043 119.71 0.059 788.92 0.0925 2-Apr- 07 107.63 0.014 196.52 -0.074 128.54 0.013 176.41 0.126 113.01 0.049 722.12 0.0176 1-Mar- 07 106.13 0.012 212.29 -0.040 126.85 0.063 156.67 -0.009 107.71 0.063 709.65 0.0068 1-Feb- 07 104.90 -0.065 221.15 0.093 119.34 -0.009 158.14 0.049 101.33 0.019 704.85 0.0287 2-Jan- 07 112.24 -0.021 202.26 0.007 120.44 -0.009 150.78 0.049 99.46 -0.033 685.19 0.0022 1-Dec- 06 114.71 0.069 200.85 -0.061 121.55 0.042 143.71 -0.007 102.87 -0.004 683.69 -0.0033 1-Nov- 06 107.26 0.049 213.99 0.204 116.67 0.070 144.76 0.042 103.28 0.103 685.96 0.1034 2-Oct- 06 102.20 0.036 177.80 0.151 109.05 0.014 138.97 0.098 93.65 -0.006 621.67 0.0691 1-Sep- 06 98.66 0.023 154.47 0.256 107.54 0.011 126.58 0.074 94.22 -0.059 581.47 0.0691 1-Aug- 06 96.44 -0.036 123.01 0.230 106.41 0.064 117.89 0.179 100.16 0.002 543.90 0.0878 31-Jul- 06 100.00 100.00 100.00 100.00 100.00 500.00 MG RIM RY RCI.B HSE Equal Portfolio Value Portfolio mean 0.79% 1.30% 0.72% 1.16% -0.05% 0.78% 0.65% stdev 9.72% 16.45% 6.52% 6.99% 6.92% 6.02% 7.80% Mean standard deviation of the standard deviations of all five stocks: 0.0932 Equal-Weighted Portfolio standard deviation: 0.0602 Value-Weighted Portfolio standard deviation: .0780 The value-weighted portfolio has lower monthly standard deviation than the simple average of the individual stocks’ standard deviation due to diversification. However, standard deviation in a value-weighted portfolio is higher than an equal-weighted portfolio as varying weightings to each stock will affect the standard deviation accordingly. 10-21 Solution Manual for Fundamentals of Corporate Finance Richard A. Brealey, Stewart C. Myers, Alan J. Marcus, Elizabeth Maynes, Devashis Mitra 9780071320573, 9781259272011
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