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Sales growth Current assets/Sales Current liabilities/Sales Costs of goods sold/Sales Depreciation rate Interest rate on debt Interest paid on cash & marketable securities Tax rate Dividend payout ratio entered into the balance sheet and the interest is Year Income statement Sales Costs of goods sold Interest payments on debt Interest earned on cash & marketable securities Depreciation Profit before tax Taxes Profit after tax Dividends Retained earnings Balance sheet 1. PROJECT FINANCE In the project f 3.9 it is assumed that the firm pays off its initi installments of principal over five years. Change Debt repayment table (essentially a loan table from Cha the plug:
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FM-MTE 2 Preparation-S 2012 Solution

Oct 23, 2014

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Page 1: FM-MTE 2 Preparation-S 2012 Solution

Sales growth 15%Current assets/Sales 15%Current liabilities/Sales 8%Costs of goods sold/Sales 55%Depreciation rate 10%Interest rate on debt 10.00%Interest paid on cash & marketable securities 8.00%Tax rate 40%Dividend payout ratio 0%

entered into the balance sheet and the interest is put into the profit and loss statement.

Year12345

Year 0

Income statement

SalesCosts of goods soldInterest payments on debtInterest earned on cash & marketable securitiesDepreciationProfit before taxTaxesProfit after taxDividendsRetained earnings

Balance sheet

Cash and marketable securities - Current assets 200 Fixed assets At cost 2,000

1. PROJECT FINANCE In the project finance pro forma of section 3.9 it is assumed that the firm pays off its initial debt of 1,000 in equal installments of principal over five years. Change this assumption and assume instead that the firm pays off its debt in equal payments of interest and principal over five years.

Debt repayment table (essentially a loan table from Chapter 1): The principal amounts are

the plug: =C39-C28-C32

Page 2: FM-MTE 2 Preparation-S 2012 Solution

Depreciation - Net fixed assets 2,000 Total assets 2,200

Current liabilities 100 Debt Err:523Stock 1,100 Accumulated retained earnings - Total liabilities and equity Err:523

FREE CASH FLOW CALCULATION

Year 0

Profit after taxAdd back depreciationSubtract increase in current assetsAdd back increase in current liabilitiesSubtract increase in fixed assets at costAdd back after-tax interest on debt

Subtract after-tax interest on cash & mkt. securities

Free cash flow

RETURN ON EQUITY (ROE)

Year 0

Equity cash flow -1,100RETURN ON EQUITY (ROE) Err:523

Data table: ROE as a function of initial

equity investment 2,000

1,8001,6001,4001,2001,000

800

600400200

=B31-$B$7*(C30+B30)/2

Note that the cash flow generated by depreciation equals the increase in fixed assets at cost.

Page 3: FM-MTE 2 Preparation-S 2012 Solution

<-- No dividends until all the debt is paid off

Principal Debt Of which:

at begin. payment Repaidof year at end yr. Interest principal

Err:523 263.80 100.00 163.80 836.20 263.80 83.62 180.18 656.03 263.80 65.60 198.19 457.83 263.80 45.78 218.01 239.82 Err:523 23.98 Err:523

1 2 3 4 5

1,150 1,323 1,521 1,749 2,011 (633) (727) (836) (962) (1,106) (100) (84) (66) (46) (24) #VALUE!

Err:523 Err:523 Err:523 Err:523 Err:523 (211) (233) (257) (284) (314)

Err:523 Err:523 Err:523 Err:523 Err:523Err:523 Err:523 Err:523 Err:523 Err:523Err:523 Err:523 Err:523 Err:523 Err:523Err:523 Err:523 Err:523 Err:523 Err:523Err:523 Err:523 Err:523 Err:523 Err:523

Err:523 Err:523 Err:523 Err:522 Err:523 173 198 228 262 302

2,211 2,443 2,700 2,985 3,299

Page 4: FM-MTE 2 Preparation-S 2012 Solution

(211) (443) (700) (985) (1,299) 2,000 2,000 2,000 2,000 2,000

Err:523 Err:523 Err:523 Err:522 Err:523

92 106 122 140 161 Err:523 Err:523 Err:523 Err:523 Err:523 #VALUE!

1,100 1,100 1,100 1,100 1,100 Err:523 Err:523 Err:523 Err:523 Err:523Err:523 Err:523 Err:523 Err:522 Err:523

1 2 3 4 5

Err:523 Err:523 Err:523 Err:523 Err:523211 233 257 284 314

28 (26) (30) (34) (39)(8) 14 16 18 21

(211) (233) (257) (284) (314)100 84 66 46 24

Err:523 Err:523 Err:523 Err:523 Err:523Err:523 Err:523 Err:523 Err:523 Err:523

1 2 3 4 5

Err:523 Err:523 Err:523 Err:523 Err:523 #VALUE!#VALUE!

Err:523 #VALUE!11.40%

11.98%12.70%13.59%14.73%16.27%18.42%

21.70%27.37%

40.10%

0 200 400 600 800 1000 1200 1400 1600 1800 20000%

5%

10%

15%

20%

25%

30%

35%

40%

45% ROE as a Function of Initial Equity Investment

Equity investment

RO

E

Page 5: FM-MTE 2 Preparation-S 2012 Solution

0 200 400 600 800 1000 1200 1400 1600 1800 20000%

5%

10%

15%

20%

25%

30%

35%

40%

45% ROE as a Function of Initial Equity Investment

Equity investment

RO

E

Page 6: FM-MTE 2 Preparation-S 2012 Solution

0 200 400 600 800 1000 1200 1400 1600 1800 20000%

5%

10%

15%

20%

25%

30%

35%

40%

45% ROE as a Function of Initial Equity Investment

Equity investment

RO

E

Page 7: FM-MTE 2 Preparation-S 2012 Solution

0 200 400 600 800 1000 1200 1400 1600 1800 20000%

5%

10%

15%

20%

25%

30%

35%

40%

45% ROE as a Function of Initial Equity Investment

Equity investment

RO

E

Page 8: FM-MTE 2 Preparation-S 2012 Solution

Lessee's problem below, Lessor's problem in row 30

Lessee's problem

Asset cost 250,000Interest rate 9%Lease rental payment 50,000Annual depreciation 30%Tax rate 0%

Year 0 1

After-tax cash flows from leasing

After-tax lease rental -50,000 -50,000

After-tax cash flows from buying the asset

Asset cost -250,000depreciation tax shield 0net cash flow from buying -250,000 0

differential cash flow

lease minus buy 200,000 -50,000

IRR of differential cash flow 7.93%

decision?? lease <-- =IF(B23<(1-B8)*B5,"lease","buy")

Lessor's problem

asset cost 250,000interest rate 8%lease rental payment 50,000annual depreciation 20%tax rate 20%

year 0 1

after-tax cash flows from leasing

after-tax lease rental -40,000 -40,000

2. MEGA Corporation is considering leasing an asset from BASIS A Corporation. Here are the relevant facts: Asset cost $250,000; Depreciation schedule:Year1: 30%; Year 2: 22%; Year 3: 19.20%; Year 4: 11.52%; Year 5: 11.52%; Year 6: 5.76%. Lease term 6 years. Lease payment $50,000 per year, at the beginning of years 0, 1, …, 5 Asset residual value =Zero; Tax rates MEGA:TC = 0% (MEGA has tax-loss carryforwards that prevent it from utilizing any additional tax shields) ;BASIS A: TC = 20%.Show that it will be advantageous both for MEGA to lease the asset and for BI to purchase the asset in order to lease it out to MEGA.

Page 9: FM-MTE 2 Preparation-S 2012 Solution

after-tax cash flows from buying the asset

asset cost -250,000depreciation tax shield 10,000net cash flow from buying -250,000 10,000

differential cash flow

lease minus buy 210,000 -50,000

IRR of differential cash flow 6.20%

decision?? don't lease <-- =IF(B50>(1-B35)*B32,"lease","don't lease")

Page 10: FM-MTE 2 Preparation-S 2012 Solution

22% 19.20% 11.52% 11.52% 5.76%

2 3 4 5 6

-50,000 -50,000 -50,000 -50,000

0 0 0 0 00 0 0 0 0

-50,000 -50,000 -50,000 -50,000 0

<-- =IF(B23<(1-B8)*B5,"lease","buy")

32% 19.20% 11.52% 11.52% 5.76%

2 3 4 5 6

-40,000 -40,000 -40,000 -40,000

MEGA Corporation is considering leasing an asset from BASIS A Corporation. Here are the relevant facts:Asset cost $250,000; Depreciation schedule:Year1: 30%; Year 2: 22%; Year 3: 19.20%; Year 4: 11.52%; Year 5: 11.52%; Year 6: 5.76%. Lease

term 6 years. Lease payment $50,000 per year, at the beginning of years 0, 1, …, 5 Asset residual value =Zero; Tax rates MEGA:TC = 0% (MEGA has tax-loss carryforwards that prevent it from utilizing any additional tax shields) ;BASIS A: TC = 20%.Show that it will be advantageous both for MEGA to lease the asset and for BI to purchase the asset in order to lease it out to MEGA.

Page 11: FM-MTE 2 Preparation-S 2012 Solution

16,000 9,600 5,760 5,760 2,88016,000 9,600 5,760 5,760 2,880

-56,000 -49,600 -45,760 -45,760 -2,880

<-- =IF(B50>(1-B35)*B32,"lease","don't lease")

Page 12: FM-MTE 2 Preparation-S 2012 Solution

BASIC LEVERAGED LEASE EXAMPLE

cost of asset 1,000,000lease term 15residual value 300,000equity 200,000debt 800,000 15-year term loan, equal payments of interest & principalinterest 10%annual debt payment 105,179 annual rent received 110,000 tax rate 40%

rental or

year equity invested salvage

0 -200,0001 110,0002 110,0003 110,0004 110,0005 110,0006 110,0007 110,0008 110,0009 110,000

10 110,00011 110,00012 110,00013 110,00014 110,00015 110,00016 300,000

MPM 4.18% <-- Solved by SolverIRR 4.18%

CALCULATING THE MULTIPLE-PHASES METHOD (MPM) RETURN--REQUIRES SOLVERinvestment at

year beginning of year cash flow

1 200,000 19,488

3. Your company is considering either purchasing or leasing an asset that costs $1,000,000. The asset, if purchased, will be depreciated on a straight-line basis over 15 years to a zero residual value. A leasing company is willing to lease the asset for $300,000 per year; the first payment on the lease is due at the time the lease is undertaken (i.e., year 0), and the remaining five payments are due at the beginning of years 1-15. Your company has a tax rate tc = 40 percent and can borrow at 10 percent from its bank. Reconsider the leveraged-leasing example. Show that if depreciation is straight line over 15 years, then the MPM rate of return is equal to the IRR. Explain.

Page 13: FM-MTE 2 Preparation-S 2012 Solution

2 188,869 18,4803 178,281 17,3734 168,358 16,1545 159,239 14,8136 151,079 13,3397 144,054 11,7178 138,356 9,9339 134,205 7,970

10 131,843 5,81111 131,541 3,43612 133,601 82413 138,360 -2,05014 146,192 -5,21115 157,511 -8,68816 172,780 180,00017 0

You score is:

Page 14: FM-MTE 2 Preparation-S 2012 Solution

BASIC LEVERAGED LEASE EXAMPLE

15-year term loan, equal payments of interest & principal

principal repayment

at start loan of CASH

depreciation of year payment interest principal FLOW

-200,00066,667 800,000 105,179 80,000 25,179 19,488 #VALUE!66,667 774,821 105,179 77,482 27,697 18,48066,667 747,124 105,179 74,712 30,467 17,37366,667 716,657 105,179 71,666 33,513 16,15466,667 683,144 105,179 68,314 36,865 14,81366,667 646,280 105,179 64,628 40,551 13,33966,667 605,728 105,179 60,573 44,606 11,71766,667 561,122 105,179 56,112 49,067 9,93366,667 512,056 105,179 51,206 53,973 7,97066,667 458,082 105,179 45,808 59,371 5,81166,667 398,711 105,179 39,871 65,308 3,43666,667 333,403 105,179 33,340 71,839 82466,667 261,565 105,179 26,156 79,023 -2,05066,667 182,542 105,179 18,254 86,925 -5,21166,667 95,617 105,179 9,562 95,617 -8,688

180,000

<-- Solved by Solver

CALCULATING THE MULTIPLE-PHASES METHOD (MPM) RETURN--REQUIRES SOLVERattribution of cash flow

income investment Explanation: When all of the cash flows8,357 11,131 from the lease are positive, the MPM return

Your company is considering either purchasing or leasing an asset that costs $1,000,000. The asset, if purchased, will be depreciated on a straight-line basis over 15 years to a zero residual value. A leasing company is willing to lease the asset for $300,000 per year; the first payment on the lease is due at the time the lease is undertaken (i.e., year 0), and the remaining five payments are due at the beginning of years 1-15. Your company has a tax rate tc = 40 percent and can borrow at 10 percent from its bank. Reconsider the leveraged-leasing example. Show that if depreciation is straight line

Page 15: FM-MTE 2 Preparation-S 2012 Solution

7,892 10,588 will be the same as the IRR. When the depreciation7,450 9,923 is straight-line over the life of the asset, this7,035 9,119 will be the case.6,654 8,1606,313 7,0266,019 5,6975,781 4,1515,608 2,3625,509 3025,497 -2,0605,583 -4,7595,781 -7,8316,109 -11,3196,582 -15,2697,220 172,780

Page 16: FM-MTE 2 Preparation-S 2012 Solution

will be the same as the IRR. When the depreciation

Page 17: FM-MTE 2 Preparation-S 2012 Solution

BASIC LEVERAGED LEASE EXAMPLE

To answer this question, you have to run the Solver macro found on the This program can also be run by [ctr]+a.

cost of asset 1,000,000lease term 15residual value 300,000equity 200,000debt 800,000 15-year term loan, equal payments of interest & principalinterest 10%annual debt payment 105,179 annual rent received 150,000 tax rate 40%

rental or

year equity invested salvage

0 -200,0001 150,0002 150,0003 150,0004 150,0005 150,0006 150,0007 150,0008 150,0009 150,000

10 150,00011 150,00012 150,00013 150,00014 150,00015 150,00016 300,000

Start 120,000

4.Your company is considering either purchasing or leasing an asset that costs $1,000,000. The asset, if purchased, will be depreciated on a straight-line basis over 15 years to a zero residual value. A leasing company is willing to lease the asset for $300,000 per year; the first payment on the lease is due at the time the lease is undertaken (i.e., year 0), and the remaining five payments are due at the beginning of years 1-15. Your company has a tax rate tc = 40 percent and can borrow at 10 percent from its bank. In the leveraged-lease example , find the lowest lease rental so that the MPM is equal to the IRR (assume the original depreciation schedule).

Page 18: FM-MTE 2 Preparation-S 2012 Solution

Increase 2,000 MPM 31.34%IRR 34.61% #VALUE!

0

Year Cash flow

1 200,000 73,9412 188,745 113,7743 134,129 84,6664 91,502 63,4875 56,694 47,8276 26,637 46,3527 -11,366 44,7308 -56,096 25,1869 -81,282 5,303

10 -86,585 3,14411 -89,729 76912 -90,499 -1,84313 -88,656 -4,71614 -83,939 -7,87715 -76,062 -11,35416 -64,708 180,00017 -244,708

Results

rent MPM IRR( see Data Table in right)

120,000 31.34% 19.68%122,000 31.34% 20.90%124,000 31.34% 22.07%126,000 31.34% 23.19%128,000 31.34% 24.28%130,000 31.34% 25.34%132,000 31.34% 26.36%134,000 31.34% 27.35%136,000 31.34% 28.33%138,000 31.34% 29.28%140,000 31.34% 30.21%142,000 31.34% 31.12%144,000 31.34% 32.01%146,000 31.34% 32.89%148,000 31.34% 33.76%150,000 31.34% 34.61%

Investment at beginning of year

Page 19: FM-MTE 2 Preparation-S 2012 Solution

AN EXPLANATION OF EXERCISE 2

The VBA program on this workbook runs the Solver for a number of different values of the lease rent. The startingvalue is given in the cell "Start" (B40), and the increases are given by the cell "Increase" (B41).Thus, for the values given in this particular case, [ctr]+a will find the MPM for rents of 120000,122000, 124000 ...

Note that the MPM <= IRR.Here you can see that the smallest rent for which the MPM = IRR is 142,000 (this number is, of course, approximate).

Your score is:

Page 20: FM-MTE 2 Preparation-S 2012 Solution

15-year term loan, equal payments of interest & principal

principal repayment

at start loan of CASH

depreciation of year payment interest principal FLOW

-200,000142,800 800,000 105,179 80,000 25,179 73,941244,900 774,821 105,179 77,482 27,697 113,774174,900 747,124 105,179 74,712 30,467 84,666125,000 716,657 105,179 71,666 33,513 63,487

89,200 683,144 105,179 68,314 36,865 47,82789,200 646,280 105,179 64,628 40,551 46,35289,200 605,728 105,179 60,573 44,606 44,73044,800 561,122 105,179 56,112 49,067 25,186

512,056 105,179 51,206 53,973 5,303458,082 105,179 45,808 59,371 3,144398,711 105,179 39,871 65,308 769333,403 105,179 33,340 71,839 -1,843261,565 105,179 26,156 79,023 -4,716182,542 105,179 18,254 86,925 -7,877

95,617 105,179 9,562 95,617 -11,354180,000

34.610324%

Your company is considering either purchasing or leasing an asset that costs $1,000,000. The asset, if purchased, will be depreciated on a straight-line basis over 15 years to a zero residual value. A leasing company is willing to lease the asset for $300,000 per year; the first payment on the lease is due at the time the lease is undertaken (i.e., year 0), and the remaining five payments are due at the beginning of years 1-15. Your company has a tax rate tc = 40 percent and can borrow at 10 percent from its bank. In the leveraged-lease example , find the lowest lease rental so that the MPM is equal to the IRR (assume the original depreciation schedule).

Page 21: FM-MTE 2 Preparation-S 2012 Solution

Income Investment

62,686 11,25559,158 54,61642,040 42,62628,679 34,80817,770 30,057

8,349 38,0030 44,7300 25,1860 5,3030 3,1440 7690 -1,8430 -4,7160 -7,8770 -11,3540 180,000

Rent IRRIRR - MPM 34.610324%

11.66% 120,000 0.1967789110.44% 122,000 0.20898373

9.27% 124,000 0.220683428.15% 126,000 0.231943657.06% 128,000 0.24281856.01% 130,000 0.253352974.98% 132,000 0.26358483.99% 134,000 0.273545943.02% 136,000 0.283263672.07% 138,000 0.292761421.14% 140,000 0.302059480.23% 142,000 0.31117553

-0.67% 144,000 0.32012504-1.55% 146,000 0.32892167-2.41% 148,000 0.33757749-3.27% 150,000 0.34610324

Page 22: FM-MTE 2 Preparation-S 2012 Solution

The VBA program on this workbook runs the Solver for a number of different values of the lease rent. The starting

Thus, for the values given in this particular case, [ctr]+a will find the MPM for rents of 120000,122000, 124000 ...

Here you can see that the smallest rent for which the MPM = IRR is 142,000 (this number is, of course, approximate).

Page 23: FM-MTE 2 Preparation-S 2012 Solution

5. Innovation Fund and Index 500 Fund ReturnsThe Index 500 fund mimics the SP500 index; the Innovation Fund is an "aggressive" growth fund

Year Ended SP 500 Innivation31 December Index Fund

2007 45.45% 21.00%2006 23.20% 25.55%2005 39.80% 33.40%2004 2.30% -0.45%2003 14.10% 23.00%2002 8.60% 17.50%2001 30.50% 28.60%2000 -3.10% -15.50%1999 31.70% 15.00%1998 16.60% 28.70%1997 5.30% 7.40%1996 16.00% 19.50%1995 31.80% 18.40%1994 8.70% 5.70%

Average 19.35% 16.27%Standard deviation 14.18% 12.68%Covariance of returns 0.0129

Portfolio returnsPercentage of SP500 15%Percentage of Innivation F 85%

Expected portfolio return 16.73%Variance of portfolio return 1.53%Standard dev. of port. retur 12.39%

Sigma Exp. return12.39% 16.73%

0 12.68% 16.27%0.1 12.47% 16.58%0.2 12.33% 16.89%0.3 12.28% 17.20%0.4 12.32% 17.50%0.5 12.44% 17.81%0.6 12.64% 18.12%0.7 12.92% 18.43%

Following are annual return statistics for two mutual funds from the Future Generation family: see rows 5-32 Use Excel to graph the combinations of standard deviation of return (x-axis) and expected return (y-axis) by varying the percentage of SP 500 in the portfolio from 0% to 100%.

12.0% 12.5% 13.0% 13.5% 14.0% 14.5%14.0%

15.0%

16.0%

17.0%

18.0%

19.0%

20.0%

A B C D E F G

1

2

3

456789

10111213141516171819202122232425262728293031323334353637383940414243

Page 24: FM-MTE 2 Preparation-S 2012 Solution

0.8 13.27% 18.74%0.9 13.69% 19.05%

1 14.18% 19.35% 12.0% 12.5% 13.0% 13.5% 14.0% 14.5%14.0%

15.0%

16.0%

17.0%

18.0%

19.0%

20.0%

A B C D E F G444546

Page 25: FM-MTE 2 Preparation-S 2012 Solution

The Index 500 fund mimics the SP500 index; the Innovation Fund is an "aggressive" growth fund

12.0% 12.5% 13.0% 13.5% 14.0% 14.5%14.0%

15.0%

16.0%

17.0%

18.0%

19.0%

20.0%

H I J K L

1

2

3

456789

10111213141516171819202122232425262728293031323334353637383940414243

Page 26: FM-MTE 2 Preparation-S 2012 Solution

12.0% 12.5% 13.0% 13.5% 14.0% 14.5%14.0%

15.0%

16.0%

17.0%

18.0%

19.0%

20.0%

H I J K L444546

Page 27: FM-MTE 2 Preparation-S 2012 Solution

FIVE STOCKS

Market cap 17.34 13.15 100.07 122.93 13.89 67.88

Date2/25/2008 25.94 30.29 46.39 45.85 6.52 56.452/19/2008 25.74 29.75 46.07 44.08 6.25 55.692/11/2008 26.87 31.99 47.53 44.19 6.45 55.30

2/4/2008 25.65 31.37 44.53 41.56 6.08 55.641/28/2008 25.90 31.55 45.98 46.34 6.85 54.221/22/2008 25.23 31.09 47.79 46.25 6.58 54.101/14/2008 25.32 31.91 53.32 49.71 5.92 52.40

1/7/2008 26.37 33.54 60.55 52.38 6.06 54.3212/31/2007 25.27 33.03 56.85 49.57 6.13 57.0512/24/2007 26.89 34.95 58.71 50.03 6.70 59.5012/17/2007 26.72 35.32 59.15 50.19 6.79 60.0312/10/2007 26.13 34.99 59.57 52.25 6.89 61.16

12/3/2007 28.31 34.98 60.67 52.51 7.06 60.1611/26/2007 28.67 34.73 58.97 51.93 7.51 58.4711/19/2007 27.92 33.62 57.28 50.07 7.19 57.7211/12/2007 28.57 32.47 58.00 48.25 7.70 58.13

11/5/2007 26.56 30.65 55.54 49.01 8.20 56.8210/29/2007 28.17 32.90 55.68 50.10 8.95 57.5110/22/2007 28.52 33.05 57.21 49.28 8.67 56.9710/15/2007 27.85 31.44 52.76 49.52 8.37 54.97

10/8/2007 29.65 34.88 53.16 50.94 9.20 55.5610/1/2007 28.83 33.09 53.16 52.75 8.37 54.939/24/2007 28.36 33.05 51.35 51.90 8.49 53.079/17/2007 28.38 32.89 51.48 52.90 8.23 53.409/10/2007 27.01 32.48 49.22 52.10 8.03 54.03

9/4/2007 25.30 30.52 49.25 51.77 7.52 47.988/27/2007 26.43 31.60 49.46 50.94 7.81 47.998/20/2007 27.22 32.81 50.11 50.94 7.90 48.848/13/2007 25.94 32.35 48.92 49.45 7.83 46.35

8/6/2007 25.24 32.35 50.17 50.79 8.23 48.147/30/2007 25.61 31.46 49.58 49.76 8.06 47.287/23/2007 26.09 31.00 49.42 47.95 8.23 47.517/16/2007 28.10 34.08 48.33 50.85 8.40 50.76

7/9/2007 28.56 35.48 50.14 50.76 8.97 50.587/2/2007 27.99 34.38 48.76 50.04 9.11 50.11

6/25/2007 27.89 33.89 49.10 50.59 9.42 49.466/18/2007 28.85 34.54 47.87 50.07 9.13 49.13

KrogerKR

SafewaySWY

MerckMRK

GlaxoGSK

FordF

McDonaldsMCD

A B C D E F G

1

2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

33

34

35

36

37

38

39

40

41

42

Page 28: FM-MTE 2 Preparation-S 2012 Solution

6/11/2007 30.39 34.09 50.02 50.96 8.92 50.836/4/2007 29.29 34.54 49.44 49.82 8.24 50.09

5/29/2007 31.14 36.07 51.03 49.54 8.35 49.645/21/2007 29.29 33.90 52.55 50.65 8.45 49.605/14/2007 29.33 34.68 52.11 55.75 8.69 50.99

5/7/2007 28.60 35.64 50.89 54.88 8.37 49.354/30/2007 29.19 35.64 50.81 55.93 8.22 48.644/23/2007 29.40 36.45 50.75 55.51 8.05 47.704/16/2007 29.67 37.54 50.39 56.86 7.75 47.12

4/9/2007 28.34 36.22 49.14 55.66 8.02 46.424/2/2007 28.78 36.10 44.57 53.88 8.01 44.61

3/26/2007 27.93 36.42 43.23 52.94 7.89 43.903/19/2007 27.99 36.57 43.50 52.54 7.89 43.903/12/2007 26.30 34.24 42.17 52.26 7.55 42.37

3/5/2007 25.01 34.01 43.66 53.25 7.93 43.022/26/2007 25.07 33.62 42.88 52.32 7.59 42.552/20/2007 25.38 34.88 41.67 54.53 8.30 44.832/12/2007 26.17 36.23 42.97 55.35 8.53 44.16

2/5/2007 25.53 35.44 42.52 53.50 8.73 43.421/29/2007 25.50 35.86 43.40 52.95 8.23 43.401/22/2007 24.17 34.51 44.63 51.82 8.42 41.831/16/2007 23.79 34.10 44.25 53.09 8.30 43.66

1/8/2007 23.26 33.53 43.46 51.00 7.89 43.091/3/2007 23.28 32.98 42.99 51.20 7.62 42.42

12/26/2006 22.75 34.30 42.31 50.06 7.51 43.1912/18/2006 22.88 34.49 41.56 49.71 7.42 42.4512/11/2006 23.96 33.72 42.69 49.27 7.08 42.30

12/4/2006 22.75 31.26 42.63 49.83 7.23 42.6411/27/2006 21.35 30.50 43.36 50.98 8.04 41.0011/20/2006 21.47 30.07 42.51 49.01 8.52 40.8111/13/2006 21.87 30.36 43.36 49.34 8.89 40.75

11/6/2006 21.72 30.67 41.49 49.30 8.58 39.9210/30/2006 21.24 28.29 43.40 50.57 8.38 39.2310/23/2006 22.42 28.75 44.35 50.13 8.29 39.4410/16/2006 21.91 28.58 43.92 52.87 8.01 39.44

10/9/2006 21.59 28.23 41.57 50.80 8.17 40.0510/2/2006 22.12 28.12 40.28 51.09 8.31 38.159/25/2006 22.76 30.07 40.32 50.08 8.09 37.219/18/2006 22.46 29.86 40.45 50.79 8.00 36.349/11/2006 22.13 29.06 39.55 52.07 8.02 35.89

9/5/2006 23.59 30.01 39.51 51.82 8.77 35.678/28/2006 23.37 30.83 39.41 53.40 8.27 34.968/21/2006 22.89 30.63 38.55 51.32 8.00 33.518/14/2006 22.89 29.71 37.68 51.22 8.00 34.41

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8/7/2006 22.30 28.01 38.71 51.78 7.37 33.007/31/2006 22.46 28.50 39.42 51.41 7.13 33.687/24/2006 22.53 27.83 39.20 51.94 6.84 33.577/17/2006 22.67 27.66 35.62 51.69 6.23 33.017/10/2006 21.54 24.59 34.46 50.49 6.33 31.43

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Page 30: FM-MTE 2 Preparation-S 2012 Solution

<-- This is in billions of dollars

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Page 31: FM-MTE 2 Preparation-S 2012 Solution

FIVE STOCKS

Market cap 17.34 13.15 100.07 122.93 13.89 67.88

Date2/25/2008 25.94 30.29 46.39 45.85 6.52 56.452/19/2008 25.74 29.75 46.07 44.08 6.25 55.692/11/2008 26.87 31.99 47.53 44.19 6.45 55.30

2/4/2008 25.65 31.37 44.53 41.56 6.08 55.641/28/2008 25.90 31.55 45.98 46.34 6.85 54.221/22/2008 25.23 31.09 47.79 46.25 6.58 54.101/14/2008 25.32 31.91 53.32 49.71 5.92 52.40

1/7/2008 26.37 33.54 60.55 52.38 6.06 54.3212/31/2007 25.27 33.03 56.85 49.57 6.13 57.0512/24/2007 26.89 34.95 58.71 50.03 6.70 59.5012/17/2007 26.72 35.32 59.15 50.19 6.79 60.0312/10/2007 26.13 34.99 59.57 52.25 6.89 61.16

12/3/2007 28.31 34.98 60.67 52.51 7.06 60.1611/26/2007 28.67 34.73 58.97 51.93 7.51 58.4711/19/2007 27.92 33.62 57.28 50.07 7.19 57.7211/12/2007 28.57 32.47 58.00 48.25 7.70 58.13

11/5/2007 26.56 30.65 55.54 49.01 8.20 56.8210/29/2007 28.17 32.90 55.68 50.10 8.95 57.5110/22/2007 28.52 33.05 57.21 49.28 8.67 56.9710/15/2007 27.85 31.44 52.76 49.52 8.37 54.97

10/8/2007 29.65 34.88 53.16 50.94 9.20 55.5610/1/2007 28.83 33.09 53.16 52.75 8.37 54.939/24/2007 28.36 33.05 51.35 51.90 8.49 53.079/17/2007 28.38 32.89 51.48 52.90 8.23 53.409/10/2007 27.01 32.48 49.22 52.10 8.03 54.03

9/4/2007 25.30 30.52 49.25 51.77 7.52 47.988/27/2007 26.43 31.60 49.46 50.94 7.81 47.998/20/2007 27.22 32.81 50.11 50.94 7.90 48.848/13/2007 25.94 32.35 48.92 49.45 7.83 46.35

8/6/2007 25.24 32.35 50.17 50.79 8.23 48.14

KrogerKR

SafewaySWY

MerckMRK

GlaxoGSK

FordF

McDonaldsMCD

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7/30/2007 25.61 31.46 49.58 49.76 8.06 47.287/23/2007 26.09 31.00 49.42 47.95 8.23 47.517/16/2007 28.10 34.08 48.33 50.85 8.40 50.76

7/9/2007 28.56 35.48 50.14 50.76 8.97 50.587/2/2007 27.99 34.38 48.76 50.04 9.11 50.11

6/25/2007 27.89 33.89 49.10 50.59 9.42 49.466/18/2007 28.85 34.54 47.87 50.07 9.13 49.136/11/2007 30.39 34.09 50.02 50.96 8.92 50.83

6/4/2007 29.29 34.54 49.44 49.82 8.24 50.095/29/2007 31.14 36.07 51.03 49.54 8.35 49.645/21/2007 29.29 33.90 52.55 50.65 8.45 49.605/14/2007 29.33 34.68 52.11 55.75 8.69 50.99

5/7/2007 28.60 35.64 50.89 54.88 8.37 49.354/30/2007 29.19 35.64 50.81 55.93 8.22 48.644/23/2007 29.40 36.45 50.75 55.51 8.05 47.704/16/2007 29.67 37.54 50.39 56.86 7.75 47.12

4/9/2007 28.34 36.22 49.14 55.66 8.02 46.424/2/2007 28.78 36.10 44.57 53.88 8.01 44.61

3/26/2007 27.93 36.42 43.23 52.94 7.89 43.903/19/2007 27.99 36.57 43.50 52.54 7.89 43.903/12/2007 26.30 34.24 42.17 52.26 7.55 42.37

3/5/2007 25.01 34.01 43.66 53.25 7.93 43.022/26/2007 25.07 33.62 42.88 52.32 7.59 42.552/20/2007 25.38 34.88 41.67 54.53 8.30 44.832/12/2007 26.17 36.23 42.97 55.35 8.53 44.16

2/5/2007 25.53 35.44 42.52 53.50 8.73 43.421/29/2007 25.50 35.86 43.40 52.95 8.23 43.401/22/2007 24.17 34.51 44.63 51.82 8.42 41.831/16/2007 23.79 34.10 44.25 53.09 8.30 43.66

1/8/2007 23.26 33.53 43.46 51.00 7.89 43.091/3/2007 23.28 32.98 42.99 51.20 7.62 42.42

12/26/2006 22.75 34.30 42.31 50.06 7.51 43.1912/18/2006 22.88 34.49 41.56 49.71 7.42 42.4512/11/2006 23.96 33.72 42.69 49.27 7.08 42.30

12/4/2006 22.75 31.26 42.63 49.83 7.23 42.6411/27/2006 21.35 30.50 43.36 50.98 8.04 41.0011/20/2006 21.47 30.07 42.51 49.01 8.52 40.8111/13/2006 21.87 30.36 43.36 49.34 8.89 40.75

11/6/2006 21.72 30.67 41.49 49.30 8.58 39.9210/30/2006 21.24 28.29 43.40 50.57 8.38 39.2310/23/2006 22.42 28.75 44.35 50.13 8.29 39.4410/16/2006 21.91 28.58 43.92 52.87 8.01 39.44

10/9/2006 21.59 28.23 41.57 50.80 8.17 40.0510/2/2006 22.12 28.12 40.28 51.09 8.31 38.15

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Page 33: FM-MTE 2 Preparation-S 2012 Solution

9/25/2006 22.76 30.07 40.32 50.08 8.09 37.219/18/2006 22.46 29.86 40.45 50.79 8.00 36.349/11/2006 22.13 29.06 39.55 52.07 8.02 35.89

9/5/2006 23.59 30.01 39.51 51.82 8.77 35.678/28/2006 23.37 30.83 39.41 53.40 8.27 34.968/21/2006 22.89 30.63 38.55 51.32 8.00 33.518/14/2006 22.89 29.71 37.68 51.22 8.00 34.41

8/7/2006 22.30 28.01 38.71 51.78 7.37 33.007/31/2006 22.46 28.50 39.42 51.41 7.13 33.687/24/2006 22.53 27.83 39.20 51.94 6.84 33.577/17/2006 22.67 27.66 35.62 51.69 6.23 33.017/10/2006 21.54 24.59 34.46 50.49 6.33 31.43

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Page 34: FM-MTE 2 Preparation-S 2012 Solution

KR SWY MRK GSK F<-- This is in billions of dollars

Mean -0.22% -0.25% -0.35% 0.11% -0.03%Variance 0.0013 0.0014 0.0012 0.0009 0.0023Sigma 3.54% 3.78% 3.52% 3.01% 4.79%

85 #VALUE!

Date KR SWY MRK GSK F2/25/20082/19/2008 -0.0077 -0.0180 -0.0069 -0.0394 -0.04232/11/2008 0.0430 0.0726 0.0312 0.0025 0.0315

2/4/2008 -0.0465 -0.0196 -0.0652 -0.0614 -0.05911/28/2008 0.0097 0.0057 0.0320 0.1089 0.11921/22/2008 -0.0262 -0.0147 0.0386 -0.0019 -0.04021/14/2008 0.0036 0.0260 0.1095 0.0721 -0.1057

1/7/2008 0.0406 0.0498 0.1272 0.0523 0.023412/31/2007 -0.0426 -0.0153 -0.0631 -0.0551 0.011512/24/2007 0.0621 0.0565 0.0322 0.0092 0.088912/17/2007 -0.0063 0.0105 0.0075 0.0032 0.013312/10/2007 -0.0223 -0.0094 0.0071 0.0402 0.0146

12/3/2007 0.0801 -0.0003 0.0183 0.0050 0.024411/26/2007 0.0126 -0.0072 -0.0284 -0.0111 0.061811/19/2007 -0.0265 -0.0325 -0.0291 -0.0365 -0.043511/12/2007 0.0230 -0.0348 0.0125 -0.0370 0.0685

11/5/2007 -0.0730 -0.0577 -0.0433 0.0156 0.062910/29/2007 0.0589 0.0708 0.0025 0.0220 0.087510/22/2007 0.0123 0.0045 0.0271 -0.0165 -0.031810/15/2007 -0.0238 -0.0499 -0.0810 0.0049 -0.0352

10/8/2007 0.0626 0.1038 0.0076 0.0283 0.094510/1/2007 -0.0280 -0.0527 0.0000 0.0349 -0.09459/24/2007 -0.0164 -0.0012 -0.0346 -0.0162 0.01429/17/2007 0.0007 -0.0049 0.0025 0.0191 -0.03119/10/2007 -0.0495 -0.0125 -0.0449 -0.0152 -0.0246

9/4/2007 -0.0654 -0.0622 0.0006 -0.0064 -0.06568/27/2007 0.0437 0.0348 0.0043 -0.0162 0.03788/20/2007 0.0295 0.0376 0.0131 0.0000 0.01158/13/2007 -0.0482 -0.0141 -0.0240 -0.0297 -0.0089

8/6/2007 -0.0274 0.0000 0.0252 0.0267 0.0498

Number of data points

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Page 35: FM-MTE 2 Preparation-S 2012 Solution

7/30/2007 0.0146 -0.0279 -0.0118 -0.0205 -0.02097/23/2007 0.0186 -0.0147 -0.0032 -0.0371 0.02097/16/2007 0.0742 0.0947 -0.0223 0.0587 0.0204

7/9/2007 0.0162 0.0403 0.0368 -0.0018 0.06577/2/2007 -0.0202 -0.0315 -0.0279 -0.0143 0.0155

6/25/2007 -0.0036 -0.0144 0.0069 0.0109 0.03356/18/2007 0.0338 0.0190 -0.0254 -0.0103 -0.03136/11/2007 0.0520 -0.0131 0.0439 0.0176 -0.0233

6/4/2007 -0.0369 0.0131 -0.0117 -0.0226 -0.07935/29/2007 0.0612 0.0433 0.0317 -0.0056 0.01335/21/2007 -0.0612 -0.0620 0.0294 0.0222 0.01195/14/2007 0.0014 0.0227 -0.0084 0.0959 0.0280

5/7/2007 -0.0252 0.0273 -0.0237 -0.0157 -0.03754/30/2007 0.0204 0.0000 -0.0016 0.0190 -0.01814/23/2007 0.0072 0.0225 -0.0012 -0.0075 -0.02094/16/2007 0.0091 0.0295 -0.0071 0.0240 -0.0380

4/9/2007 -0.0459 -0.0358 -0.0251 -0.0213 0.03424/2/2007 0.0154 -0.0033 -0.0976 -0.0325 -0.0012

3/26/2007 -0.0300 0.0088 -0.0305 -0.0176 -0.01513/19/2007 0.0021 0.0041 0.0062 -0.0076 0.00003/12/2007 -0.0623 -0.0658 -0.0311 -0.0053 -0.0440

3/5/2007 -0.0503 -0.0067 0.0347 0.0188 0.04912/26/2007 0.0024 -0.0115 -0.0180 -0.0176 -0.04382/20/2007 0.0123 0.0368 -0.0286 0.0414 0.08942/12/2007 0.0307 0.0380 0.0307 0.0149 0.0273

2/5/2007 -0.0248 -0.0220 -0.0105 -0.0340 0.02321/29/2007 -0.0012 0.0118 0.0205 -0.0103 -0.05901/22/2007 -0.0536 -0.0384 0.0279 -0.0216 0.02281/16/2007 -0.0158 -0.0120 -0.0086 0.0242 -0.0144

1/8/2007 -0.0225 -0.0169 -0.0180 -0.0402 -0.05071/3/2007 0.0009 -0.0165 -0.0109 0.0039 -0.0348

12/26/2006 -0.0230 0.0392 -0.0159 -0.0225 -0.014512/18/2006 0.0057 0.0055 -0.0179 -0.0070 -0.012112/11/2006 0.0461 -0.0226 0.0268 -0.0089 -0.0469

12/4/2006 -0.0518 -0.0758 -0.0014 0.0113 0.021011/27/2006 -0.0635 -0.0246 0.0170 0.0228 0.106211/20/2006 0.0056 -0.0142 -0.0198 -0.0394 0.058011/13/2006 0.0185 0.0096 0.0198 0.0067 0.0425

11/6/2006 -0.0069 0.0102 -0.0441 -0.0008 -0.035510/30/2006 -0.0223 -0.0808 0.0450 0.0254 -0.023610/23/2006 0.0541 0.0161 0.0217 -0.0087 -0.010810/16/2006 -0.0230 -0.0059 -0.0097 0.0532 -0.0344

10/9/2006 -0.0147 -0.0123 -0.0550 -0.0399 0.019810/2/2006 0.0243 -0.0039 -0.0315 0.0057 0.0170

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Page 36: FM-MTE 2 Preparation-S 2012 Solution

9/25/2006 0.0285 0.0670 0.0010 -0.0200 -0.02689/18/2006 -0.0133 -0.0070 0.0032 0.0141 -0.01129/11/2006 -0.0148 -0.0272 -0.0225 0.0249 0.0025

9/5/2006 0.0639 0.0322 -0.0010 -0.0048 0.08948/28/2006 -0.0094 0.0270 -0.0025 0.0300 -0.05878/21/2006 -0.0208 -0.0065 -0.0221 -0.0397 -0.03328/14/2006 0.0000 -0.0305 -0.0228 -0.0020 0.0000

8/7/2006 -0.0261 -0.0589 0.0270 0.0109 -0.08207/31/2006 0.0071 0.0173 0.0182 -0.0072 -0.03317/24/2006 0.0031 -0.0238 -0.0056 0.0103 -0.04157/17/2006 0.0062 -0.0061 -0.0958 -0.0048 -0.09347/10/2006 -0.0511 -0.1176 -0.0331 -0.0235 0.0159

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Page 37: FM-MTE 2 Preparation-S 2012 Solution

MCD

-0.69%0.00072.69%

MCD

-0.0136-0.00700.0061

-0.0259-0.0022-0.03190.03600.04900.04200.00890.0186

-0.0165-0.0285-0.01290.0071

-0.02280.0121

-0.0094-0.03570.0107

-0.0114-0.03440.00620.0117

-0.11880.00020.0176

-0.05230.0379

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Page 38: FM-MTE 2 Preparation-S 2012 Solution

-0.01800.00490.0662

-0.0036-0.0093-0.0131-0.00670.0340

-0.0147-0.0090-0.00080.0276

-0.0327-0.0145-0.0195-0.0122-0.0150-0.0398-0.01600.0000

-0.03550.0152

-0.01100.0522

-0.0151-0.0169-0.0005-0.03680.0428

-0.0131-0.01570.0180

-0.0173-0.00350.0080

-0.0392-0.0046-0.0015-0.0206-0.01740.00530.00000.0153

-0.0486

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-0.0249-0.0237-0.0125-0.0061-0.0201-0.04240.0265

-0.04180.0204

-0.0033-0.0168-0.0490

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Page 40: FM-MTE 2 Preparation-S 2012 Solution

KR SWY MRK GSK F MCDKR 0.0013 0.0009 0.0004 0.0002 0.0004 0.0003

SWY 0.0009 0.0014 0.0003 0.0003 0.0005 0.0004MRK 0.0004 0.0003 0.0012 0.0005 0.0002 0.0001GSK 0.0002 0.0003 0.0005 0.0009 0.0003 0.0002

F 0.0004 0.0005 0.0002 0.0003 0.0023 0.0004MCD 0.0003 0.0004 0.0001 0.0002 0.0004 0.0007

Constant 0.10% #VALUE!

KR -0.0419

#VALUE!SWY -0.0620MRK 0.4209GSK -0.4061

F -0.1013MCD 1.1904

Check 1.000 #VALUE!

Weekly statistics for optimal portfolioPortfolio mean -0.985% #VALUE!Portfolio variance 0.0011 #VALUE!Portfolio sigma 3.272%

Annual statistics for optimal portfolioPortfolio mean -51.25% #VALUE!Portfolio sigma 23.59% #VALUE!

Constant -0.4061 #VALUE!-0.2 -0.0019

-0.15 -0.0014-0.1 -0.0006

-0.05 0.00220 -0.6039

6. On the spreadsheet with this exam you will find the weekly stock prices and market capitalizations of equity for five American stocks. 6.1. For each stock, compute the mean returns, standard deviation of returns.6.2. Compute the variance-covariance matrix for the five stocks.6.3. Suppose that the annual interest rate is 5%. Find an optimal portfolio. Give the portfolio proportions and the portfolio’s annualized mean and annualized standard deviation.6.4. Is there an all positive portfolio on the efficient frontier?

Optimalportfolio

All positive portfolio? Below I do a data table for the constant.For all constants, the minimum portfolio position is negative

Page 41: FM-MTE 2 Preparation-S 2012 Solution

0.05 -0.00780.1 -0.0056

0.15 -0.00480.2 -0.0044

0.25 -0.00410.3 -0.0040

0.35 -0.00380.4 -0.0038

0.45 -0.00370.5 -0.0036

0.55 -0.00360.6 -0.0036

0.65 -0.00350.7 -0.0035

0.75 -0.00350.8 -0.0035

0.85 -0.00340.9 -0.0034

0.95 -0.00341 -0.0034

1.05 -0.00341.1 -0.0034

1.15 -0.00341.2 -0.0033

1.25 -0.00331.3 -0.0033

1.35 -0.00331.4 -0.0033

1.45 -0.00331.5 -0.0033

1.55 -0.00331.6 -0.0033

Page 42: FM-MTE 2 Preparation-S 2012 Solution

Means-0.22% #VALUE!

#VALUE!

-0.25%-0.35%0.11%

-0.03%-0.69%

Page 43: FM-MTE 2 Preparation-S 2012 Solution

KR SWY MRK GSK F MCD

KR 0.0013 0.0009 0.0004 0.0002 0.0004 0.0003SWY 0.0009 0.0014 0.0003 0.0003 0.0005 0.0004MRK 0.0004 0.0003 0.0012 0.0005 0.0002 0.0001GSK 0.0002 0.0003 0.0005 0.0009 0.0003 0.0002

F 0.0004 0.0005 0.0002 0.0003 0.0023 0.0004MCD 0.0003 0.0004 0.0001 0.0002 0.0004 0.0007

Risk-free 0.0008 #VALUE!

0.23% #VALUE!Parameter 3.13

<-- Weekly, remember!KR 0.0019

#VALUE!

9.86%SWY 0.0020 10.36%MRK 0.0027 13.91%GSK 0.0025 12.76%

F 0.0019 10.03%MCD 0.0017 8.88%

0.0023 #VALUE!

Weekly statistics for optimal portfolioPortfolio mean 77.110% #VALUE!Portfolio variance 0.0000 #VALUE!Portfolio sigma 0.028%

Annual statistics for optimal portfolioPortfolio mean 4009.71% #VALUE!Portfolio sigma 0.20% #VALUE!

7. Refer to the same data and variance-covariance matrix as in the previous problem.7.1. Compute the annualized expected returns for each stock under the following assumptions:• These five stocks constitute a benchmark portfolio. • The risk-free rate is 4% annually.• The expected annual benchmark return is 12%.7.2. Ms NNN is an eminent stock analyst. She thinks that the annualized return for Kroger will be 12% annually, but has no opinions about any of the other stock’s returns. What should be her optimal portfolio holdings of the five stocks?

Benchmark return

Benchmarkreturns

Annualreturns

Benchmark return

Page 44: FM-MTE 2 Preparation-S 2012 Solution

Constant 0.0017 #VALUE!-0.2 0.0017

-0.15 0.0017-0.1 0.0017

-0.05 0.00170 0.0017

0.05 0.00170.1 0.0017

0.15 0.00170.2 0.0017

0.25 0.00170.3 0.0017

0.35 0.00170.4 0.0017

0.45 0.00170.5 0.0017

0.55 0.00170.6 0.0017

0.65 0.00170.7 0.0017

0.75 0.00170.8 0.0017

0.85 0.00170.9 0.0017

0.95 0.00171 0.0017

1.05 0.00171.1 0.0017

1.15 0.00171.2 0.0017

1.25 0.00171.3 0.0017

1.35 0.00171.4 0.0017

1.45 0.00171.5 0.0017

1.55 0.00171.6 0.0017

All positive portfolio? Below I do a data table for the constant.For all constants, the minimum portfolio position is positive

Page 45: FM-MTE 2 Preparation-S 2012 Solution

17.34

#VALUE!#VALUE!

13.15100.07122.93

13.8967.88

#VALUE!

Asset values

Page 46: FM-MTE 2 Preparation-S 2012 Solution

5.17%#VALUE!3.92%

29.85%36.67%

4.14%20.25%

Benchmarkproportions

Page 47: FM-MTE 2 Preparation-S 2012 Solution

A FOUR-ASSET PORTFOLIO PROBLEM

Variance-covariance Mean returns St. dev.0.06 0.01 0.03 0.05 6% 24.49%0.04 0.45 0.06 -0.04 10% 67.08%0.03 0.06 0.77 0.02 12% 87.75%0.04 -0.04 0.02 0.89 15% 94.34%

Constant 0.05

Portfolio 1 Portfolio 2 Both these portfolios are efficientz x z y

0.8069 67.20% 0.0132 4.17%0.1480 12.33% 0.1096 34.63%0.1093 9.10% 0.0789 24.91%0.1365 11.37% 0.1149 36.30%

Mean 0.08062 0.12146Variance 0.06714 0.22571Covariance 0.09672

Proportion of 1 -0.9Portfolio mean 15.82%Portfolio var. 53.84% 0.054385 0.814807 -0.33077775540131Portfolio st. dev. 73.38% 0.538414

Data table of portfolios

Proportion of 1 0.7338 15.82%-1.4 88.42% 17.86%

-1.15 80.86% 16.84%-0.9 73.38% 15.82%

-0.65 65.98% 14.80%-0.4 58.72% 13.78%

-0.15 51.64% 12.76%0.1 44.82% 11.74%

0.35 38.42% 10.72%0.6 32.67% 9.70%

0.85 27.97% 8.67%1.1 24.94% 7.65%

8. This problem returns to the four-asset problem Calculate the envelope set for these four assets and show that the individual assets all lie within this envelope set. You should get a graph.

20% 30% 40% 50% 60% 70% 80% 90% 100%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 48: FM-MTE 2 Preparation-S 2012 Solution

1.35 24.21% 6.63%1.6 25.97% 5.61%

1.85 29.78% 4.59%2.1 34.98% 3.57%

2.35 41.04% 2.55%2.6 47.64% 1.53%

2.85 54.58% 0.51%Stock A 24.49% 6.00%Stock B 67.08% 10.00%Stock C 87.75% 12.00%Stock D 94.34% 15.00%

20% 30% 40% 50% 60% 70% 80% 90% 100%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 49: FM-MTE 2 Preparation-S 2012 Solution

Mean minusconstant

1%5%7%

10%

Calculate the envelope set for these four assets and show that the individual assets all lie within this

20% 30% 40% 50% 60% 70% 80% 90% 100%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 50: FM-MTE 2 Preparation-S 2012 Solution

20% 30% 40% 50% 60% 70% 80% 90% 100%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 51: FM-MTE 2 Preparation-S 2012 Solution

DOW-JONES 30 INDUSTRIALS, RETURN DATA for Dec. 93 - April 99

1 2 3 4 5 6 7 8Index AA DIS JNJ PG ALD EK JPM S

Dec-93 3754.1 15.7954 13.7441 20.5433 25.8483 18.1703 38.7584 56.8164 23.1166Jan-94 3978.4 18.1096 15.2573 19.3988 27.1296 18.0841 38.4493 58.7615 23.991Feb-94 3832 17.2228 15.4995 18.4815 26.2177 17.631 37.8293 55.7927 20.1182Mar-94 3636 16.3931 13.562 17.3876 24.3938 16.8519 39.0389 51.9298 19.0158Apr-94 3681.7 15.5634 13.7482 19.0572 26.0828 15.9285 36.5096 50.8953 20.7245

May-94 3758.4 16.2599 14.0312 20.5245 25.8535 16.2939 41.5923 54.7228 22.5034Jun-94 3625 16.8354 13.4651 19.8867 24.4777 16.0619 42.7014 51.8464 21.3366Jul-94 3764.5 18.0153 13.732 21.8 25.7313 17.7435 42.9232 52.6826 21.0032

Aug-94 3913.4 19.4385 13.3674 23.3911 28.0968 17.4192 44.5038 55.0868 21.2385Sep-94 3843.2 19.6121 12.5573 24.1495 27.5198 15.9045 46.2929 51.3587 21.5187Oct-94 3908.1 19.8209 12.8253 25.4911 29.1826 16.1375 43.0502 52.4155 22.1912Nov-94 3739.2 18.9781 14.1241 25.0445 29.0086 15.2823 41.0561 49.6679 21.3663Dec-94 3834.4 20.1406 14.9358 25.6897 28.7765 15.9264 43.0864 48.0786 20.8011Jan-95 3843.9 18.3827 16.5456 27.2733 30.3982 16.7461 44.2143 53.9679 20.0097Feb-95 4011.1 18.2366 17.3587 26.7622 31.04 17.8333 46.3827 55.2529 22.4588Mar-95 4157.7 19.4056 17.3993 28.0591 30.9233 18.4807 48.429 52.9051 24.3399Apr-95 4321.3 20.9838 18.039 30.6527 32.7841 18.6573 52.2942 56.9164 24.7389

May-95 4465.1 21.8532 18.0797 31.3388 33.7225 19.1005 55.2699 61.4697 25.9474Jun-95 4556.1 23.5568 18.0797 31.9904 33.7225 21.0519 55.4988 61.4473 27.3252Jul-95 4708.5 26.7878 19.129 34.0047 32.502 22.1163 52.7524 64.1856 30.3641

Aug-95 4610.6 26.9548 18.3541 32.8557 32.7379 21.0826 53.2332 63.857 30.3351Sep-95 4789.1 24.9494 18.7211 35.2961 36.3362 20.9639 54.6159 68.452 34.5516Oct-95 4755.5 24.0647 18.8331 38.8078 38.4158 20.1918 57.7269 68.2309 31.8577Nov-95 5074.5 27.726 19.6501 41.4161 41.0243 22.6059 63.2775 69.4473 37.1143Dec-95 5117.1 25.06 19.2708 40.8783 39.3644 22.6655 62.1185 71.7194 36.7609Jan-96 5395.3 26.4098 21.0301 45.8984 40.019 23.7988 68.0291 72.6131 39.1173Feb-96 5485.6 27.0641 21.4393 44.8547 39.0662 26.6518 66.6572 73.1717 42.983Mar-96 5587.1 29.8002 20.9074 44.2551 40.3763 28.3288 66.191 74.8961 46.1801Apr-96 5569.1 29.6812 20.3288 44.375 40.4467 27.7898 71.3185 75.9112 47.3642

May-96 5643.2 29.4305 19.9189 46.9028 42.0622 26.3332 69.7105 78.3927 48.413Jun-96 5654.6 27.4008 20.6157 47.6855 43.3785 27.4755 72.8738 77.0895 46.2719Jul-96 5528.9 27.8072 18.2735 45.9997 42.9491 28.2571 69.9448 77.4312 39.0159

Aug-96 5616.2 29.7848 18.7252 47.615 42.7686 29.8078 68.323 79.7085 42.0817Sep-96 5882.2 28.2866 20.7784 49.5486 46.9191 31.799 73.9773 81.6862 42.799Oct-96 6029.4 28.2627 21.6782 47.615 47.8624 31.6179 75.0375 79.3884 46.266Nov-96 6521.7 30.6732 24.3109 51.6752 52.5762 35.4694 76.8253 86.6264 47.8049Dec-96 6448.3 30.7334 22.9534 48.2787 52.0323 32.443 75.9967 90.551 44.2015Jan-97 6813.1 33.2644 24.0202 56.0421 56.1255 34.0773 82.1521 95.5365 46.1233Feb-97 6877.7 34.4627 24.4734 55.9742 58.3098 35.1086 85.4084 97.3916 52.3483Mar-97 6583.5 32.8908 24.0202 51.4719 55.7007 34.6226 72.3235 91.8891 48.3679Apr-97 7009 33.9192 26.9937 59.503 61.2664 35.1086 79.1039 95.2794 46.3174

May-97 7331 35.7396 27.0349 58.6167 67.1738 37.422 79.2844 100.774 47.6131Jun-97 7672.8 36.5891 26.4984 62.8908 68.8182 40.957 73.4247 98.4039 52.0957Jul-97 8222.6 42.9603 26.7293 60.6927 74.3627 44.9796 64.0972 109.246 61.1822

Aug-97 7622.4 40.0402 25.4063 55.5878 65.075 40.3795 62.9529 101.35 55.2243Sep-97 7945.3 39.9185 26.6673 56.5684 67.5191 41.5717 62.5316 107.98 55.4067Oct-97 7442.1 35.5372 27.2878 56.262 66.7107 35.2137 57.6567 104.892 40.7492Nov-97 7823.1 32.8507 31.4493 61.9439 74.7429 36.4466 58.8035 108.515 44.8024Dec-97 7908.3 34.3772 32.795 64.8351 78.2992 38.1033 58.7429 108.102 44.2523Jan-98 7906.5 37.3081 35.4513 65.8808 77.1297 38.226 63.2895 96.9092 44.9858

Page 52: FM-MTE 2 Preparation-S 2012 Solution

Feb-98 8545.7 36.018 37.1305 74.4217 83.5264 41.9341 64.0829 114.447 52.0514Mar-98 8799.8 33.7783 35.4098 72.5087 83.0343 41.3799 63.3506 129.559 56.4095Apr-98 9063.4 38.0428 41.3787 70.5958 81.1268 43.1656 70.4912 126.605 58.2509

May-98 8900 34.165 37.6207 68.4261 82.7925 42.2658 70.1363 119.792 60.9295Jun-98 8952 32.4722 34.9009 73.3181 89.8872 43.8724 71.7946 113.822 60.1902Jul-98 8883.3 34.1342 34.3669 76.5381 78.6044 43.0073 82.4194 122.634 50.025

Aug-98 7539.1 29.5993 27.3813 68.5905 75.7573 34.057 77.1713 90.9121 44.9177Sep-98 7842.6 35.099 25.323 77.7856 70.4345 35.1116 75.9366 83.1117 43.7421Oct-98 8592.1 39.1774 26.9375 81.0163 88.1321 38.6476 76.554 92.5646 44.4846Nov-98 9116.55 36.8237 32.1875 81.0076 87.0763 43.8271 72.1615 104.964 47.1769Dec-98 9181.43 36.9786 30 83.6248 90.7407 44.1383 71.5405 104.206 42.2666Jan-99 9358.83 41.4421 33 84.8711 90.6047 38.8467 64.9577 104.64 39.9046Feb-99 9306.58 40.3666 35.1875 85.375 89.2338 41.375 66.1875 110.529 40.625Mar-99 9786.16 41.0518 31.125 93.5 97.6462 49.1875 63.875 123.375 45.1875Apr-99 10878.4 62.0449 31.75 97.5 93.8125 58.75 74.75 138 46

Page 53: FM-MTE 2 Preparation-S 2012 Solution

9 10 11 12 13 14 15 16 17 18AXP GE KO T BA GM MCD UK C GT24.3631 23.3923 20.9735 20.9341 21.625 45.0597 13.7669 20.0961 12.8788 40.95225.9542 24.0336 19.211 22.6287 21.625 50.2944 14.6726 22.9028 13.9554 43.301723.2693 23.5039 20.0335 20.9341 23.375 47.9845 14.6684 21.6049 12.299 40.631

22.076 22.4581 19.1848 20.5637 22.25 44.3805 13.7611 20.3606 11.6779 36.365923.7819 21.3913 19.716 20.5637 22.5 46.7489 14.5172 23.8672 11.5122 35.01925.2682 22.3458 19.0077 21.9179 23.1875 44.4373 15.0011 24.4934 10.891 34.978623.5532 21.1025 19.2745 21.9486 23.125 41.5437 13.9999 24.1516 10.684 32.496324.4529 22.7998 21.0537 22.0496 22.3125 42.4738 13.1514 25.7465 10.9739 32.157825.9523 22.5169 21.8247 22.0496 22.75 41.7106 13.7276 31.5076 12.2576 31.778528.0285 21.9434 23.1681 21.9313 21.625 38.9091 12.8165 31.1639 10.891 30.303128.7025 22.2854 24.0019 22.3374 21.9375 32.7874 14.0313 30.2473 11.5122 31.89227.5405 20.9745 24.4514 19.9514 22.4375 31.8097 13.817 26.3993 10.8496 30.931927.6359 23.4447 24.6307 20.5399 23.5 35.1472 14.243 27.091 10.7254 30.703729.5096 23.6745 25.109 20.3866 22.25 32.4355 15.8865 23.5173 12.2162 32.872331.5003 25.1686 26.3046 21.1019 23.0625 35.7471 16.2202 26.5746 12.8788 33.858132.6713 25.0111 27.0696 21.2865 26.875 36.9002 16.647 28.4313 12.7959 33.743332.7656 25.9374 27.9099 20.8752 27.5 37.8437 17.0739 29.8239 13.707 34.891133.4728 26.8638 29.5905 20.8752 29.4375 40.5359 18.5111 27.2011 13.9554 38.9218

33.237 26.3037 30.723 21.9376 31.3125 39.5859 19.122 31.2871 14.4938 37.997936.5355 27.5285 31.6266 21.8341 33.5 41.1693 18.8776 32.4545 15.6947 40.076838.3148 27.4702 30.964 23.4381 32 40.0453 17.8725 33.3309 15.9018 37.177842.1107 29.9358 33.3667 27.3558 34.125 39.8329 18.7294 37.3212 17.5996 36.596938.7554 29.701 34.757 26.6277 32.75 37.1774 20.0759 35.5608 16.73 35.318940.5441 31.5206 36.7381 27.4078 36.4375 41.4904 21.8847 37.3939 19.7115 39.629839.4709 34.0269 36.0106 27.078 39.1875 45.2331 22.1299 35.3885 20.7468 42.435544.1223 36.2717 36.5562 27.9667 38.75 45.0192 24.6432 39.7531 21.7821 44.773544.0024 35.6809 39.163 26.6076 40.5625 44.1771 24.5535 42.6474 22.1548 44.534947.3595 37.0249 40.2569 25.6972 43.3125 45.9011 23.5714 47.0306 21.8649 47.942546.7295 36.7277 39.6488 25.7498 41.0625 46.7631 23.51 43.1213 20.3741 49.000144.0799 39.3427 44.7569 26.2227 42.625 47.8778 23.6691 41.0433 20.6226 47.703542.9959 41.2444 47.803 26.2043 43.5625 45.4894 22.9928 37.8312 22.6724 45.34242.3635 39.3134 45.7299 22.0834 44.25 42.3409 22.8084 39.9726 20.9953 41.799642.3635 39.7316 48.7785 22.1891 45.25 43.4287 22.8445 41.3415 21.5543 43.338644.7843 43.7155 49.7509 22.2208 47.25 42.0066 23.3371 43.6117 24.4117 43.813545.7277 46.4777 49.3842 20.8044 47.6875 46.9292 21.9208 40.7441 26.9584 43.576150.8356 49.9605 50.1173 23.3306 49.6875 50.7992 23.066 44.272 29.8158 46.3434

55.187 47.739 51.5878 25.9819 53.25 49.1463 22.3876 39.2329 30.0643 49.090660.6812 49.972 56.7343 24.8377 53.5625 52.0114 22.4493 43.5522 34.7023 52.076664.0999 49.6702 59.7977 25.1531 50.875 51.4604 21.373 45.5385 35.5305 50.664758.4835 48.1623 54.7748 22.4413 49.3125 49.2375 23.3497 42.6471 31.8035 50.184564.4607 53.8641 62.5121 21.3272 49.3125 51.4604 26.4382 48.0684 36.7728 50.544768.1372 58.5954 67.3018 23.3963 52.6875 51.4537 24.8726 45.2292 36.4415 56.352373.0392 63.0841 66.9477 22.5228 53.0625 49.9964 23.9136 45.5316 41.7835 61.118782.3565 68.3252 68.0552 23.647 58.6875 55.4893 26.605 53.5737 47.6639 62.446176.4564 60.9568 56.4255 25.0521 54.5 56.7362 23.4582 49.8163 42.0734 59.748680.5127 66.5744 60.2022 28.6348 54.4375 60.5224 23.6132 47.2679 45.262 66.656676.9136 63.2121 55.8845 31.6277 48 58.0359 22.2187 44.3554 46.3801 60.718177.7764 72.2599 61.8176 36.1575 53.125 55.5262 24.0869 43.0558 50.5626 59.131988.2357 72.0761 65.9594 39.8849 48.9375 58.6689 23.7144 41.8971 53.5442 61.994182.7364 76.1281 64.043 40.7388 47.625 55.9528 23.404 42.7509 49.2582 61.0198

Page 54: FM-MTE 2 Preparation-S 2012 Solution

89.039 76.3737 67.8757 39.6817 54.25 67.1049 27.1407 45.526 55.2834 67.776290.7691 84.996 76.7505 42.9822 52.125 65.9489 29.8454 49.1411 59.6316 74.1378101.244 84.0098 75.2019 39.305 50.0625 65.5839 30.7781 47.548 60.8118 68.5102101.677 82.2224 77.6797 39.7953 47.75 70.4579 32.6882 49.164 60.8739 70.65112.927 89.6187 84.8982 37.5561 44.5625 65.4952 34.3693 52.5483 60.2527 63.3393109.576 88.493 79.9334 39.8571 38.8125 70.8867 33.2797 47.2565 66.8371 59.898977.4356 79.1552 64.6666 32.9541 30.9375 57.3827 28.1201 39.5737 44.1025 48.430977.0633 79.0076 57.3608 38.6328 34.3125 54.1742 29.7724 42.6654 37.2697 50.901887.7343 86.8898 67.2527 41.3185 37.5625 62.3806 33.4511 38.0897 46.7114 53.249299.6542 89.7447 69.8842 41.1119 40.625 69.5008 34.9912 44.5074 49.9414 56.4017102.082 101.638 66.8295 50.2994 32.625 71.1793 38.3624 42.2696 49.3824 50.1279102.687 104.503 65.1463 60.2596 34.6875 89.2694 39.3612 39.348 55.9046 48.6992108.302 99.957 63.7125 54.5325 35.625 82.5625 42.4514 44 58.5845 46.0625

117.75 110.625 61.375 53.2083 34 87 45.3125 45.1875 63.6951 49.8125135.375 107.625 68.0625 53.125 40.625 89.0625 42.375 51.875 74.875 57.1875

Page 55: FM-MTE 2 Preparation-S 2012 Solution

19 20 21 22 23 24 25 26 27 28MMM UTX CAT HWP MO WMT CHV IBM MRK XON

46.828 27.7036 20.0575 18.6978 14.7072 12.0448 36.1975 28.218 15.2539 26.233446.1821 30.1053 23.505 20.1771 15.93 12.7675 38.7942 28.218 16.1968 27.63645.7475 30.5833 24.4644 21.4493 14.8063 13.6709 36.346 26.4076 14.3664 27.253642.9799 27.941 25.3674 19.4914 13.5521 12.4851 35.2446 27.2816 13.3181 26.413442.5457 28.6719 24.8367 19.0464 14.5895 12.1835 37.3425 28.7175 13.2621 26.413444.6825 30.0639 24.1586 18.631 13.1841 11.3391 36.8951 31.4644 13.6538 25.976343.4778 29.1014 22.6045 17.9285 13.9778 11.7217 35.5168 29.3418 13.442 24.11746.5443 27.2896 24.5652 18.4944 14.9278 12.0843 37.6372 30.9025 13.3855 25.285748.6825 29.0531 26.1519 21.4428 16.5563 11.9234 36.3278 34.2112 15.4187 25.593548.7929 28.5964 24.5369 20.8867 16.8167 11.3182 35.6848 34.7731 16.2382 24.733248.9032 28.7677 27.1602 23.3967 16.8511 11.3787 38.5782 37.2079 16.2382 27.045245.6392 26.9392 24.6033 23.3668 16.4384 11.2776 37.7918 35.335 16.9789 26.290547.5316 28.9539 25.0578 23.95 16.0502 10.3075 38.6581 36.7084 17.5193 26.453746.6411 29.5871 23.4565 24.0999 16.6782 11.0957 38.6581 36.0217 18.4958 27.215849.1862 30.7963 23.5707 27.577 16.9574 11.5201 41.6687 37.5824 19.4723 28.15152.2182 32.0723 25.3399 28.9451 18.49 12.4547 41.9968 41.016 19.7272 29.36353.5657 33.9282 26.8286 31.8005 19.1617 11.5433 41.45 47.259 19.8429 30.630154.3169 35.4396 27.6312 31.7404 20.6112 12.0901 43.3881 46.4474 21.752 31.791651.9405 36.4905 29.4656 35.9233 21.2828 13.0271 40.9593 47.9457 22.8808 31.457651.2616 39.2346 32.4282 37.5507 20.4959 12.9662 43.6089 54.3759 24.0452 32.292749.8743 39.1772 30.9306 38.5754 21.3543 11.9553 43.1673 51.629 23.2301 30.949951.4721 41.5267 26.2075 40.3029 24.2147 12.0773 43.5019 47.1965 26.2606 32.525651.9286 41.7029 26.3289 44.7743 24.4322 10.5524 41.7173 48.57 26.964 34.382660.1258 44.3294 28.4746 40.1217 25.4472 11.7353 44.5198 48.2578 29.0156 35.177161.0455 44.8614 27.2568 40.5905 26.4642 10.8796 47.2248 45.6358 30.9401 36.881959.3211 48.526 30.0604 41.0752 27.1973 9.9627 46.774 54.1886 33.0617 36.484160.3223 51.0455 31.1694 48.8298 29.0299 10.3906 50.6061 61.2431 31.2348 36.478859.8591 53.3631 31.7531 45.7104 25.9849 11.2711 51.061 55.5621 29.4945 37.396560.9012 52.5312 30.0991 51.4166 26.6882 11.6999 52.7668 53.814 28.6653 39.002563.6441 52.2582 30.8032 51.7201 29.4274 12.68 54.8215 53.3146 30.6198 39.258764.3435 54.9458 31.8006 48.4987 31.097 12.4599 54.1334 49.444 30.7781 40.243160.6134 53.7513 31.1098 42.8395 31.2839 11.7847 53.1012 53.6892 30.5995 38.100664.5723 54.1274 32.5266 42.5961 26.8361 12.976 54.5231 57.1228 31.2543 38.117765.5115 57.7879 35.5962 47.5807 27.1908 12.976 57.9959 62.1796 33.7237 38.936171.7339 61.8085 32.5829 43.0667 27.9861 13.0375 60.8899 64.427 35.4009 41.4579.1315 67.6083 37.5682 52.5828 31.2808 12.5455 62.5583 79.5973 39.7736 44.536378.4228 63.8723 35.7284 49.1579 34.5841 11.2163 60.6909 75.6643 38.3462 46.24780.5488 67.2467 37.0472 51.4813 36.3822 11.7093 61.9748 78.3488 43.6436 48.901487.4679 72.8436 37.3455 54.9052 41.3938 13.0035 60.7035 71.7937 44.366 47.674980.4562 73.3276 38.3001 52.2029 35.2348 13.7756 65.5268 68.5474 40.7611 51.241682.7142 73.2066 42.6778 51.4676 36.4698 13.8991 64.468 80.1592 43.7245 53.857187.8566 78.1192 46.8137 50.4873 40.7535 14.764 66.4233 86.4021 43.4826 56.75597.7778 80.6705 51.4891 55.0435 41.3688 16.744 70.1596 90.1479 49.734 58.670890.6059 82.1891 53.9448 68.8044 42.1868 18.57 74.9635 105.63 50.4936 61.5445

86.437 76.1604 55.9316 60.4495 40.8429 17.5796 74.0371 101.26 44.63 59.001188.9015 79.0263 51.958 68.5135 39.2154 18.169 79.4151 105.88 48.8091 61.773487.9998 68.2943 49.5757 60.6957 37.3873 17.3629 79.2956 98.3886 43.5894 59.242294.2409 73.4204 46.3714 60.2032 41.0434 19.8743 77.2014 109.376 46.3137 59.216779.3703 71.3384 46.9155 61.575 43.0775 19.597 74.1326 104.507 52.0161 59.398780.7606 79.6051 46.6835 59.3538 39.5076 19.7833 71.9664 98.6383 57.598 57.5785

Page 56: FM-MTE 2 Preparation-S 2012 Solution

83.0319 87.8189 53.066 66.1406 41.4115 23.0132 78.4815 104.319 62.5972 62.298788.5673 90.7687 53.5523 62.7019 40.0455 25.2877 77.9357 103.757 63.1279 66.085591.8521 96.7913 55.6215 74.5744 35.6627 25.1633 80.2404 115.744 59.342 71.399290.6633 92.7796 53.6677 61.6507 35.9029 27.4339 78.0714 117.367 57.6184 69.283280.4468 91.299 51.6834 59.4079 38.2225 30.2744 81.8589 114.683 66.1257 70.143173.5339 94.6302 47.6539 55.067 42.5301 31.458 80.7593 132.35 61.089 69.037567.5285 71.9086 41.2673 48.1836 40.346 29.4023 72.939 112.498 57.3192 64.700372.6424 75.7487 43.7237 52.6896 45.3604 27.256 82.7873 128.355 64.339 69.829478.4957 94.3917 44.432 59.9678 50.1417 34.4599 80.2637 148.332 67.0702 70.818179.9102 106.609 48.8814 62.2695 54.8616 37.5784 82.9833 164.938 77.033 74.571170.6039 108.163 45.4825 68.1583 52.9161 40.6763 82.3011 184.166 73.4993 72.706877.0563 118.793 43.1019 78.1981 46.3634 42.9552 73.9284 183.043 73.1256 69.848274.0625 123.573 45.3409 66.2875 38.698 43.0176 76.875 169.558 81.223 66.5625

70.75 135.107 45.7141 67.8125 35.1875 46.0938 88.75 177.049 80.125 70.562589 143.836 64.375 78.875 35.0625 46.875 99.75 208.951 70.375 83.0625

Page 57: FM-MTE 2 Preparation-S 2012 Solution

29 30DD IP20.9803 30.041524.4045 33.311723.3416 32.3809

23.232 30.263125.0402 29.092727.3835 31.080925.7272 29.734526.2241 32.707926.9319 34.815

25.819 35.435726.5424 33.630124.1885 32.463325.1987 34.222623.9079 32.29325.4169 34.866527.3982 34.238829.8323 35.151930.9758 36.079131.3751 39.348530.5765 38.774930.0755 37.803831.6281 38.784928.6954 34.167730.8523 35.567332.4181 35.218635.6078 37.891935.7218 33.465

38.757 36.975337.4729 37.326337.5092 37.545937.2152 34.721137.9795 35.662738.8929 37.893241.7344 40.261643.9247 40.3844.9021 40.494344.8425 38.588752.2269 38.94651.3607 39.892550.7621 37.376850.8219 40.491552.3171 46.1264

60.565 46.788864.5987 54.075160.3075 51.056859.5816 53.2346

55.045 43.555658.9206 46.219258.4342 41.962255.0898 44.4556

Page 58: FM-MTE 2 Preparation-S 2012 Solution

59.9518 45.602966.4909 45.847471.1966 51.043475.8064 45.207673.3512 42.259360.8907 43.856357.1733 36.5762

55.568 46.09157.0498 45.905658.4921 43.178652.7174 44.545450.8546 39.326751.3125 4258.0625 42.187570.6875 56.25

Page 59: FM-MTE 2 Preparation-S 2012 Solution

DOW-JONES 30 INDUSTRIALS, RETURN DATA for Dec. 93 - April 99

AA PG GEMeans 0.031168 0.022528 0.031426

Variance-covariance matrixAA PG GE

AA 0.019579 0.001062 0.001512PG 0.001062 0.005990 0.001351GE 0.001512 0.001351 0.005003

Return data Price dataAA PG GE AA PG GE

Dec-03 16.4445 28.5569 18.4578Jan-04 0.101256 -0.04998 0.302084 18.1096 27.1296 24.0336Feb-04 -0.04897 -0.03361 -0.02204 17.2228 26.2177 23.5039Mar-04 0.122094 0.042647 -0.04449 19.3256 27.3358 22.4581Apr-04 -0.19467 -0.04584 -0.0475 15.5634 26.0828 21.3913

May-04 0.044752 -0.00879 0.044621 16.2599 25.8535 22.3458Jun-04 0.035394 -0.05322 -0.05564 16.8354 24.4777 21.1025Jul-04 0.070084 0.051214 0.080431 18.0153 25.7313 22.7998

Aug-04 0.079 0.091931 -0.01241 19.4385 28.0968 22.5169Sep-04 0.060509 -0.02054 -0.02547 20.6147 27.5198 21.9434Oct-04 -0.03851 0.060422 0.015586 19.8209 29.1826 22.2854Nov-04 -0.04252 -0.00596 -0.05882 18.9781 29.0086 20.9745Dec-04 0.061255 -0.008 0.203156 20.1406 28.7765 25.2356Jan-05 -0.08728 0.056355 -0.06186 18.3827 30.3982 23.6745Feb-05 -0.00795 -0.22835 0.06311 18.2366 23.4569 25.1686Mar-05 0.064102 0.318303 -0.00626 19.4056 30.9233 25.0111Apr-05 0.569593 0.060175 0.037036 30.4589 32.7841 25.9374

May-05 -0.28253 0.028624 0.035717 21.8532 33.7225 26.8638Jun-05 0.077957 0 -0.02085 23.5568 33.7225 26.3037Jul-05 0.137158 -0.03619 0.046564 26.7878 32.502 27.5285

Aug-05 0.006234 0.007258 -0.00212 26.9548 32.7379 27.4702Sep-05 -0.0744 0.109912 0.089755 24.9494 36.3362 29.9358Oct-05 -0.03546 0.057232 -0.00784 24.0647 38.4158 29.701Nov-05 0.152144 0.067902 0.061264 27.726 41.0243 31.5206Dec-05 -0.09616 -0.04046 0.079513 25.06 39.3644 34.0269Jan-06 0.053863 0.016629 0.065971 26.4098 40.019 36.2717Feb-06 0.024775 -0.02381 -0.01629 27.0641 39.0662 35.6809Mar-06 0.101097 0.033535 0.037667 29.8002 40.3763 37.0249Apr-06 -0.00399 0.001744 -0.00803 29.6812 40.4467 36.7277

May-06 -0.00845 0.039941 0.0712 29.4305 42.0622 39.3427Jun-06 -0.06897 0.031294 0.048337 27.4008 43.3785 41.2444

9. Using the data base of the DJ Industrials—For American Airlines (AA), Procter & Gamble (PG), and GeneralElectric (GE)—compute:a. The average monthly returnsb. The covariances of the monthly returns: σAA,AA = Covariance(RAA, RAA), σPG,PG = Covariance(RPG, RPG), σGE,GE = Covariance (RGE,RGE)—these are equal to the variance of AA, PG, and GE respectively. σAA, GE = Covariance (RAA, RGE), σAA,PG = Covariance (RAA, RPG),σPG, GE = Covariance(RPG, RGE).c. What are the monthly expected return and monthly standard deviation of a portfolio which is invested in the three stocks respectively:0,2 and 0,4 and 0,4?

Part a. Average monthly returns.

Part b. Variance-covariance matrix.

Page 60: FM-MTE 2 Preparation-S 2012 Solution

Jul-06 0.014832 -0.0099 -0.04682 27.8072 42.9491 39.3134Aug-06 0.071118 -0.0042 0.010638 29.7848 42.7686 39.7316Sep-06 -0.0503 0.097045 0.10027 28.2866 46.9191 43.7155Oct-06 -0.00084 0.020105 0.063186 28.2627 47.8624 46.4777Nov-06 0.085289 0.098486 0.074935 30.6732 52.5762 49.9605Dec-06 0.001963 -0.01034 -0.04447 30.7334 52.0323 47.739Jan-07 0.082353 0.078667 0.046775 33.2644 56.1255 49.972Feb-07 0.036023 0.038918 -0.00604 34.4627 58.3098 49.6702Mar-07 -0.04561 -0.04475 -0.03036 32.8908 55.7007 48.1623Apr-07 0.031267 0.099922 0.118387 33.9192 61.2664 53.8641

May-07 0.053669 0.096422 0.087838 35.7396 67.1738 58.5954Jun-07 0.023769 0.02448 0.076605 36.5891 68.8182 63.0841Jul-07 0.174128 0.080567 0.083081 42.9603 74.3627 68.3252

Aug-07 -0.06797 -0.1249 -0.10784 40.0402 65.075 60.9568Sep-07 -0.00304 0.037558 0.092157 39.9185 67.5191 66.5744Oct-07 -0.10976 -0.01197 -0.0505 35.5372 66.7107 63.2121Nov-07 -0.0756 0.120403 0.143134 32.8507 74.7429 72.2599Dec-07 0.046468 0.04758 -0.00254 34.3772 78.2992 72.0761Jan-08 0.085257 -0.01494 0.056218 37.3081 77.1297 76.1281Feb-08 -0.03458 0.082934 0.003226 36.018 83.5264 76.3737Mar-08 -0.06218 -0.00589 0.112896 33.7783 83.0343 84.996Apr-08 0.12625 -0.02297 -0.0116 38.0428 81.1268 84.0098

May-08 -0.10193 0.020532 -0.02128 34.165 82.7925 82.2224Jun-08 -0.04955 0.085693 0.089955 32.4722 89.8872 89.6187Jul-08 0.051182 -0.12552 -0.01256 34.1342 78.6044 88.493

Aug-08 -0.13286 -0.03622 -0.10552 29.5993 75.7573 79.1552Sep-08 0.185805 -0.07026 -0.00186 35.099 70.4345 79.0076Oct-08 0.116197 0.251263 0.099765 39.1774 88.1321 86.8898Nov-08 -0.06008 -0.01198 0.032857 36.8237 87.0763 89.7447Dec-08 0.004207 0.042083 0.132524 36.9786 90.7407 101.638Jan-08 0.120705 -0.0015 0.028188 41.4421 90.6047 104.503Feb-08 -0.02595 -0.01513 -0.0435 40.3666 89.2338 99.957Mar-09 0.016974 0.094274 0.106726 41.0518 97.6462 110.625Apr-09 0.716125 0.014991 0.044429 70.45 99.11 115.54

Page 61: FM-MTE 2 Preparation-S 2012 Solution

of a nonequal-proportion portfolio:

Proportions 0.16 0.22 0.52

Return 2.63% <-- =MMULT(L8:N8,TRANSPOSE(B5:D5))Sigma 5.27% <-- =SQRT(MMULT(MMULT(L8:N8,B9:D11),TRANSPOSE(L8:N8)))

must be entered with [ctrl]+[shift]+[enter].

Using the data base of the DJ Industrials—For American Airlines (AA), Procter & Gamble (PG), and General

= Covariance(RPG, RPG), σGE,GE = Covariance AA, GE = Covariance (RAA, RGE), σAA,PG =

c. What are the monthly expected return and monthly standard deviation of a portfolio which is invested in the three stocks

Part c. Expected return and standard deviation

Note: Since the two formulas above use the array functions, they

Page 62: FM-MTE 2 Preparation-S 2012 Solution

Ex 10

Page 62

10

A B

Expected return 3% 5%Variance of return 0.0025 0.0045Standard deviation of return 0.05 0.067082Correlation 0.44

Proportion of A 0.37

Portfolio expected return 4.260% <-- =B8*B4+(1-B8)*C4Portfolio standard deviation 5.307% <-- =SQRT(B8^2*B5+(1-B8)^2*C5+2*B8*(1-B8)*B6*SQRT(B5)*SQRT(C5))

Sigma for Correlation(A,B) Expected

5.307% 1 0 -1 return

0 6.71% 6.71% 6.71% 5.00%Proportion 0.1 6.54% 6.06% 5.54% 4.80%

of A 0.2 6.37% 5.46% 4.37% 4.60%0.3 6.20% 4.93% 3.20% 4.40%0.4 6.02% 4.49% 2.02% 4.20%0.5 5.85% 4.18% 0.85% 4.00%

0.572949 5.73% 4.05% 0.00% 3.85%0.7 5.51% 4.04% 1.49% 3.60%0.8 5.34% 4.22% 2.66% 3.40%0.9 5.17% 4.55% 3.83% 3.20%

1 5.00% 5.00% 5.00% 3.00%

corr = +1 corr = 0 corr = -1

Sigma for correlation = 1 6.71% 5.00%6.54% 4.80%

To graph: Line up all the sigmas, put expected returns in different columns:

Page 63: FM-MTE 2 Preparation-S 2012 Solution

Ex 10

Page 63

6.37% 4.60%6.20% 4.40%6.02% 4.20%5.85% 4.00%5.73% 3.85%5.51% 3.60%5.34% 3.40%5.17% 3.20%5.00% 3.00%

Sigma for correlation = 0 6.71% 5.00%6.06% 4.80%5.46% 4.60%4.93% 4.40%4.49% 4.20%4.18% 4.00%4.05% 3.85%4.04% 3.60%4.22% 3.40%4.55% 3.20%5.00% 3.00%

Sigma for correlation = -1 6.71% 5.00%5.54% 4.80%4.37% 4.60%3.20% 4.40%2.02% 4.20%0.85% 4.00%0.00% 3.85%1.49% 3.60%2.66% 3.40%3.83% 3.20%5.00% 3.00%

Page 64: FM-MTE 2 Preparation-S 2012 Solution

Ex 10

Page 64

10

<-- =SQRT(B8^2*B5+(1-B8)^2*C5+2*B8*(1-B8)*B6*SQRT(B5)*SQRT(C5))

<-- Note that proportion = 0.572949 gives sigma = 0 for corr = -1.This proportion is determined by:

prop(A) = sigma(B)/(sigma(A)+sigma(B))

0% 1% 2% 3% 4% 5% 6% 7% 8%0%

1%

2%

3%

4%

5%

6%

corr = +1

corr = 0

corr = -1

Page 65: FM-MTE 2 Preparation-S 2012 Solution

Ex 10

Page 65

0% 1% 2% 3% 4% 5% 6% 7% 8%0%

1%

2%

3%

4%

5%

6%

corr = +1

corr = 0

corr = -1

Page 66: FM-MTE 2 Preparation-S 2012 Solution

Ex 10

Page 66

0% 1% 2% 3% 4% 5% 6% 7% 8%0%

1%

2%

3%

4%

5%

6%

corr = +1

corr = 0

corr = -1

Page 67: FM-MTE 2 Preparation-S 2012 Solution

Ex 10

Page 67

0% 1% 2% 3% 4% 5% 6% 7% 8%0%

1%

2%

3%

4%

5%

6%

corr = +1

corr = 0

corr = -1

Page 68: FM-MTE 2 Preparation-S 2012 Solution

UN-5H

Page 68

KAZZINK AND KAZAKHMYS

KazZink KazakhMysStock price Dividend Stock pric Dividend

1995 12.85 5.051996 16.35 0.44 3.25 0.161997 19.6 0.58 5.15 0.161998 17.95 0.61 4.85 0.171999 23.45 0.74 6.35 0.182000 29.75 0.81 10.95 0.212001 29.25 0.9 18.55 0.272002 26.9 1.05 17.95 0.432003 32.45 1.2 22.85 0.552004 34.1 1.32 26.35 0.642005 61.45 1.3 44.36 0.752006 80.05 1.5 55.15 0.84

RETURN CALCULATIONKazZink KazakhMys

1996 0.26744 -0.39268 ### 19961997 0.21046 0.49094 19971998 -0.05452 -0.02557 19981999 0.29835 0.29743 19992000 0.26482 0.56388 20002001 0.01336 0.54158 20012002 -0.04546 -0.00921 20022003 0.22389 0.26515 20032004 0.08758 0.16652 20042005 0.60986 0.53764 20052006 0.28299 0.23283 2006

Mean return 19.63% 24.26% ### Mean returnReturn variance 0.036632975 0.08896225 ### Return varianceStandard deviati 19.14% 29.83% ### Standard deviationCovariance 0.0172 ### CovarianceCorrelation 0.3318 ### Correlation

11. KazZink and KazakhMys are two kazakh exploration firms. The following table (see rows 3-16) gives the end-of-year stock prices for each of the firms for the years 1995-2006 as well as the dividends paid in the years 1995-2006.a. For the decade 1995-2006 calculate the following:1.Annual returns from each of the shares. (Don't forget the dividends!)2.The mean, variance, and standard deviation of each stock's return.3.The covariance and the correlation coefficient of the returns.b. Graph the mean portfolio return (y-axis) against the standard deviation of the portfolio return (x-axis) for portfolios of the two shares in which the weight of KazZink goes from 0 to 1.4.

NOTE: you can also use

A B C D E F G

1

2

345

6

7

8

9

10

11

12

13

14

15

161718192021222324252627282930313233343536

Page 69: FM-MTE 2 Preparation-S 2012 Solution

UN-5H

Page 69

Proportion of KazZ 0.24Portfolio mean 23.15% ###Portfolio variance 5.98% ###Portfolio s. dev. 24.45% ###

S. dev. MeanProportion <-- Contains the data table header, which has been hidden

0 29.83% 24.26%0.1 27.48% 23.80%0.2 25.28% 23.33%0.3 23.26% 22.87%0.4 21.48% 22.41%0.5 20.00% 21.94%0.6 18.89% 21.48%0.7 18.22% 21.02%0.8 18.03% 20.55%0.9 18.35% 20.09%

1 19.14% 19.63%1.1 20.35% 19.16%1.2 21.92% 18.70%1.3 23.77% 18.23%1.4 25.84% 17.77% 16% 18% 20% 22% 24% 26% 28% 30% 32%

17%

19%

21%

23%

25%

27%

Portfolio Returns--Pfizer and Merck

Standard deviationM

ea

n r

etu

rn

16% 18% 20% 22% 24% 26% 28% 30% 32%

17%

19%

21%

23%

25%

27%

Portfolio Returns-KazZink and KazakhMys

Standard deviationM

ea

n r

etu

rn

A B C D E F G37383940414243444546474849505152535455565758596061

Page 70: FM-MTE 2 Preparation-S 2012 Solution

UN-5H

Page 70

KazZink KazakhMys0.30661 -0.32475 ###0.23425 0.63385

-0.05306 -0.025240.34763 0.346390.30320 0.757480.01345 0.71872

-0.04444 -0.009160.25093 0.303620.09153 0.181180.84018 0.711950.32710 0.2621723.79% 32.33% ###0.06281 0.12692031 ###25.06% 35.63% ###0.0310 ###0.3818 ###

KazZink and KazakhMys are two kazakh exploration firms. The following table (see rows 3-16) gives the end-of-year stock prices for each of the firms for the years 1995-2006 as well as

1.Annual returns from each of the shares. (Don't forget the dividends!)

b. Graph the mean portfolio return (y-axis) against the standard deviation of the portfolio return (x-axis) for portfolios of the two shares in which the weight of KazZink goes from 0 to

NOTE: you can also use discrete returns

H I J

1

2

345

6

7

8

9

10

11

12

13

14

15

161718192021222324252627282930313233343536

Page 71: FM-MTE 2 Preparation-S 2012 Solution

UN-5H

Page 71

<-- Contains the data table header, which has been hidden

16% 18% 20% 22% 24% 26% 28% 30% 32%

17%

19%

21%

23%

25%

27%

Portfolio Returns--Pfizer and Merck

Standard deviation

Me

an

re

turn

16% 18% 20% 22% 24% 26% 28% 30% 32%

17%

19%

21%

23%

25%

27%

Portfolio Returns-KazZink and KazakhMys

Standard deviation

Me

an

re

turn

H I J37383940414243444546474849505152535455565758596061

Page 72: FM-MTE 2 Preparation-S 2012 Solution

Answer to Problem 1

Leggett Herman Shaw

La-Z-Boy Kimball Flexsteel & Platt Miller Industries

1982 36.67% 0.20% 41.54% 21.92% 26.13% 22.50%1983 122.82% 61.43% 195.09% 62.27% 73.38% 117.89%1984 14.44% 63.51% -38.38% -1.27% 45.15% 7.80%1985 21.39% 28.42% 1.30% 81.17% 24.27% 38.14%1986 45.36% -7.44% 21.89% 19.83% 10.73% 54.48%1987 20.19% 48.27% 9.11% -10.21% -11.92% 26.82%1988 -8.94% -11.28% 12.65% 13.77% 7.06% -6.24%1989 27.02% 12.85% 12.08% 32.55% -7.55% 123.03%1990 -11.64% 2.42% -17.13% -6.48% 1.31% 15.48%1991 20.29% 6.90% 3.62% 50.12% -5.54% 19.92%1992 34.08% 22.21% 33.46% 84.40% 5.71% 62.76%

mean 29.24% 20.68% 25.02% 31.64% 15.34% 43.87%sigma 33.94% 25.47% 57.74% 32.13% 24.38% 40.66%

excess return matrix

La-Z-Boy Kimball Flexsteel Leggett Miller Shaw

1982 7.42% -20.48% 16.52% -9.73% 10.79% -21.37%1983 93.58% 40.75% 170.07% 30.63% 58.04% 74.02%1984 -14.81% 42.83% -63.40% -32.92% 29.81% -36.07%1985 -7.85% 7.74% -23.72% 49.53% 8.94% -5.73%1986 16.12% -28.12% -3.13% -11.81% -4.61% 10.61%1987 -9.05% 27.59% -15.91% -41.86% -27.26% -17.05%1988 -38.18% -31.96% -12.37% -17.87% -8.27% -50.11%1989 -2.23% -7.83% -12.94% 0.91% -22.89% 79.16%1990 -40.89% -18.26% -42.15% -38.12% -14.03% -28.39%1991 -8.95% -13.78% -21.40% 18.48% -20.88% -23.95%1992 4.84% 1.53% 8.44% 52.75% -9.63% 18.89%

average 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% this line included as check only!

transpose of excess return matrix

1982 1983 1984 1985 1986 1987 1988 1989

La-Z-Boy 7.42% 93.58% -14.81% -7.85% 16.12% -9.05% -38.18% -2.23%Kimball -20.48% 40.75% 42.83% 7.74% -28.12% 27.59% -31.96% -7.83%

Flexsteel 16.52% 170.07% -63.40% -23.72% -3.13% -15.91% -12.37% -12.94%Leggett -9.73% 30.63% -32.92% 49.53% -11.81% -41.86% -17.87% 0.91%Miller 10.79% 58.04% 29.81% 8.94% -4.61% -27.26% -8.27% -22.89%

12.In the following table you will find annual return data for six furniture companies between the years 1982 and 1992. Use these data to calculate the variance-covariance matrix of the returns in the next sheet.

Page 73: FM-MTE 2 Preparation-S 2012 Solution

Shaw -21.37% 74.02% -36.07% -5.73% 10.61% -17.05% -50.11% 79.16%

Variance-covariance matrix

La-Z-Boy Kimball Flexsteel Leggett Miller Shaw Means

La-Z-Boy 0.1152 0.0398 0.1792 0.0492 0.0568 0.0989 29.24%Kimball 0.0398 0.0649 0.0447 0.0062 0.0349 0.0269 20.68%

Flexsteel 0.1792 0.0447 0.3334 0.0775 0.0886 0.1487 25.02%Leggett 0.0492 0.0062 0.0775 0.1033 0.0191 0.0597 31.64%Miller 0.0568 0.0349 0.0886 0.0191 0.0594 0.0243 15.34%Shaw 0.0989 0.0269 0.1487 0.0597 0.0243 0.1653 43.87%

Page 74: FM-MTE 2 Preparation-S 2012 Solution

this line included as check only!

1990 1991 1992

-40.89% -8.95% 4.84%-18.26% -13.78% 1.53%-42.15% -21.40% 8.44%-38.12% 18.48% 52.75%-14.03% -20.88% -9.63%

Page 75: FM-MTE 2 Preparation-S 2012 Solution

-28.39% -23.95% 18.89%

Page 76: FM-MTE 2 Preparation-S 2012 Solution

Single-Index Model

variance-covariance matrix calculated from return data :

La-Z-Boy Kimball Flexsteel Leggett Miller Shaw

La-Z-Boy 0.1152 0.0398 0.1792 0.0492 0.0568 0.0989Kimball 0.0398 0.0649 0.0447 0.0062 0.0349 0.0269Flexsteel 0.1792 0.0447 0.3334 0.0775 0.0886 0.1487Leggett 0.0492 0.0062 0.0775 0.1033 0.0191 0.0597Miller 0.0568 0.0349 0.0886 0.0191 0.0594 0.0243Shaw 0.0989 0.0269 0.1487 0.0597 0.0243 0.1653

market variance 0.0324

equity betas on both borders0.8 0.95 0.65 0.85 0.85 1.4

0.8 0.0207 0.0246 0.0168 0.0220 0.0220 0.03630.95 0.0246 0.0292 0.0200 0.0262 0.0262 0.04310.65 0.0168 0.0200 0.0137 0.0179 0.0179 0.02950.85 0.0220 0.0262 0.0179 0.0234 0.0234 0.03860.85 0.0220 0.0262 0.0179 0.0234 0.0234 0.0386

1.4 0.0363 0.0431 0.0295 0.0386 0.0386 0.0635

difference between two matrices

La-Z-Boy Kimball Flexsteel Leggett Miller Shaw

La-Z-Boy 0.0945 0.0152 0.1624 0.0271 0.0348 0.0626Kimball 0.0152 0.0356 0.0247 -0.0199 0.0087 -0.0162Flexsteel 0.1624 0.0247 0.3197 0.0596 0.0707 0.1193Leggett 0.0271 -0.0199 0.0596 0.0799 -0.0043 0.0212Miller 0.0348 0.0087 0.0707 -0.0043 0.0360 -0.0142Shaw 0.0626 -0.0162 0.1193 0.0212 -0.0142 0.1018

13. For the given variance- covariance matrix below : Supposing that the standard deviation of the market index is 18 percent,calculate the variance-covariance matrix using the single-index model.

Page 77: FM-MTE 2 Preparation-S 2012 Solution

Variance-covariance matrix

La-Z-Boy Kimball Flexsteel Leggett Miller Shaw

La-Z-Boy 0.1152 0.0398 0.1792 0.0492 0.0568 0.0989Kimball 0.0398 0.0649 0.0447 0.0062 0.0349 0.0269Flexsteel 0.1792 0.0447 0.3334 0.0775 0.0886 0.1487Leggett 0.0492 0.0062 0.0775 0.1033 0.0191 0.0597Miller 0.0568 0.0349 0.0886 0.0191 0.0594 0.0243Shaw 0.0989 0.0269 0.1487 0.0597 0.0243 0.1653

constant 10%

raw normalized

portfolio weights

3.1708 0.72030.4861 0.1104

-2.4405 -0.55441.3533 0.30740.0614 0.01401.7712 0.4023

1.0000

c = 0% c = 10% transposes

42.02% 72.03% c = 0%

15.79% 11.04% 42.02% 15.79% -41.86% 28.09%-41.86% -55.44% c = 10%

28.09% 30.74% 72.03% 11.04% -55.44% 30.74%25.12% 1.40%30.85% 40.23%

mean 31.35% 37.07%variance 4.10% 6.15%covariance 0.0485016

sample portfolio calculation

proportion 0.3mean 35.35%variance 5.42%

14. DATA FOR 6 FURNITURE COMPANIES. In Chapter 8 you were asked to calculate the variance-covariance matrix of returns for six furniture companies. The calculated variance-covariance matrix and mean returns for these firms are as follows:(see rows 4-10) a. Given this matrix, and assuming that the risk-free rate is 0 percent, calculate the efficient portfolio of these six firms. b. Repeat, assuming that the risk-free rate is 10 percent. c. Use these two portfolios to generate an efficient frontier for the six furniture companies. Plot this frontier. d. Is there an efficient portfolio with only positive proportions of all the assets?[

Page 78: FM-MTE 2 Preparation-S 2012 Solution

sigma mean23.28% 35.35%

1 1 20.25% 31.35%0.1 1.1 19.90% 30.78%

1.2 19.56% 30.21%1.3 19.24% 29.64%1.4 18.95% 29.07%1.5 18.69% 28.49%1.6 18.45% 27.92%1.7 18.23% 27.35%1.8 18.05% 26.78%1.9 17.89% 26.21%

2 17.77% 25.64%2.1 17.67% 25.07%2.2 17.61% 24.49%2.3 17.57% 23.92%2.4 17.57% 23.35%2.5 17.60% 22.78%2.6 17.66% 22.21%

A crude test to see if we can generate positive portfolio weights we vary r tosee what the portfolio proportions are (in a data table). No portfolioproportions seem to give a positive weight to the third asset.

Portfolio weightsLa-Z-Boy Kimball Flexsteel & Platt Miller

0.7203 0.1104 -0.5544 0.3074 0.0140-10% 0.298805 0.177103 -0.36367144 0.27012646 0.347149

-9% 0.307591 0.175712 -0.36764699 0.27090391 0.340203-8% 0.316934 0.174234 -0.37187426 0.27173058 0.332817-7% 0.326887 0.172659 -0.37637795 0.27261132 0.324948-6% 0.337513 0.170978 -0.38118608 0.27355158 0.316547-5% 0.348882 0.169179 -0.38633059 0.27455763 0.307558-4% 0.361076 0.16725 -0.39184807 0.27563662 0.297918-3% 0.374187 0.165175 -0.39778061 0.27679677 0.287553-2% 0.388322 0.162939 -0.40417687 0.27804761 0.276377-1% 0.403608 0.16052 -0.41109342 0.27940019 0.2642920% 0.42019 0.157896 -0.41859646 0.28086747 0.2511831% 0.43824 0.15504 -0.42676387 0.28246467 0.2369132% 0.457962 0.15192 -0.43568798 0.28420985 0.2213213% 0.4796 0.148496 -0.44547907 0.28612457 0.2042144% 0.503448 0.144723 -0.45626992 0.2882348 0.185365% 0.529862 0.140543 -0.46822191 0.29057211 0.1644776% 0.55928 0.135888 -0.48153309 0.29317521 0.141227% 0.592245 0.130672 -0.49644931 0.29609219 0.1151588% 0.629439 0.124787 -0.51327949 0.29938346 0.085752

17.0% 17.5% 18.0% 18.5% 19.0% 19.5% 20.0% 20.5%

20.0%

22.0%

24.0%

26.0%

28.0%

30.0%

32.0%

Efficient Frontier

sigma

me

an

Page 79: FM-MTE 2 Preparation-S 2012 Solution

means

minus

Means constant

29.24% 19.24%20.68% 10.68%25.02% 15.02%31.64% 21.64%15.34% 5.34%43.87% 33.87%

25.12% 30.85%

1.40% 40.23%

In Chapter 8 you were asked to calculate the variance-covariance matrix of returns for six furniture companies. The calculated variance-covariance matrix and mean returns for these firms are as follows:(see rows 4-10) a. Given this matrix, and assuming that the risk-free rate is 0 percent, calculate the efficient portfolio of these six firms. b. Repeat, assuming that the risk-free rate is 10 percent. c. Use these two portfolios to generate an efficient frontier for the six furniture companies. Plot this frontier. d. Is there an efficient portfolio with only positive proportions of all the assets?[

Page 80: FM-MTE 2 Preparation-S 2012 Solution

Number of non-negativeShaw proportions

0.40230.270488 5 <-- =COUNTIF(D67:I67,">=0")0.273237 50.276159 50.279273 50.282597 50.286153 50.289968 50.294069 50.298491 50.303273 5

0.30846 50.314106 50.320276 50.327045 50.334505 50.342768 5

0.35197 50.362282 50.373917 5

17.0% 17.5% 18.0% 18.5% 19.0% 19.5% 20.0% 20.5%

20.0%

22.0%

24.0%

26.0%

28.0%

30.0%

32.0%

Efficient Frontier

sigma

me

an

Page 81: FM-MTE 2 Preparation-S 2012 Solution

epsilon 1 <-- Use solver to find epsilon such that minimum portfolio weight = 0.

Variance-covariance matrix

La-Z-Boy Kimball Flexsteel & Platt Miller Shaw

La-Z-Boy 0.1152 0.0398 0.1792 0.0492 0.0568 0.0989 La-Z-Boy

Kimball 0.0398 0.0649 0.0447 0.0062 0.0349 0.0269 Kimball

Flexsteel 0.1792 0.0447 0.3334 0.0775 0.0886 0.1487 Flexsteel

& Platt 0.0492 0.0062 0.0775 0.1033 0.0191 0.0597 & Platt

Miller 0.0568 0.0349 0.0886 0.0191 0.0594 0.0243 Miller

Shaw 0.0989 0.0269 0.1487 0.0597 0.0243 0.1653 Shaw

constant 10%Explanation

raw normalized When epsilon = 1, the variance-covariance matrix is the originalportfolio weights matrix (as in Exercise 1). For this case the optimized portfolio

3.1708 0.7203 contains at least one short-selling position (illustrated here).0.4861 0.1104 minimum

-2.4405 -0.5544 port. weight1.3533 0.3074 -0.5544 is zero. You can determine manually that when epsilon is larger0.0614 0.0140 than this, there are short-selling positions.1.7712 0.4023

diagonal. (Try setting epsilon = 0 to see what this means.)

15. A sufficient condition to produce positively weighted efficient portfolios is that the variance-covariance matrix be diagonal, that is, that σij = 0, for i≠j. By continuity, positively weighted portfolios will result if the offdiagonal elements of the variance-covariance matrix are sufficiently small compared to the diagonal. Consider a transformation of this matrix in which

We use Solver to determine epsilon so that the entry in cell E22

Epsilon "shrinks" the variance-covariance matrix towards the

Here are the Solver settings:

Page 82: FM-MTE 2 Preparation-S 2012 Solution
Page 83: FM-MTE 2 Preparation-S 2012 Solution

minus

Means constant

29.24% 19.24%20.68% 10.68%25.02% 15.02%31.64% 21.64%15.34% 5.34%43.87% 33.87%

When epsilon = 1, the variance-covariance matrix is the originalmatrix (as in Exercise 1). For this case the optimized portfoliocontains at least one short-selling position (illustrated here).

is zero. You can determine manually that when epsilon is largerthan this, there are short-selling positions.

diagonal. (Try setting epsilon = 0 to see what this means.)

to determine epsilon so that the entry in cell E22

"shrinks" the variance-covariance matrix towards the

Page 84: FM-MTE 2 Preparation-S 2012 Solution
Page 85: FM-MTE 2 Preparation-S 2012 Solution

A FOUR-ASSET PORTFOLIO PROBLEM

Variance-covariance Mean returns St. dev.0.10 0.01 0.03 0.05 6% 31.62%0.01 0.30 0.06 -0.04 8% 54.77%0.03 0.06 0.40 0.02 10% 63.25%0.05 -0.04 0.02 0.50 15% 70.71%

Constant 0.05

Portfolio 1 Portfolio 2 Both these portfolios are efficientz x z y

0.3861 35.53% -0.0460 -12.31%0.2567 23.62% 0.1091 29.18%0.1688 15.53% 0.1016 27.17%0.2752 25.32% 0.2093 55.96%

Mean 0.09372 0.12707Variance 0.08624 0.20608Covariance 0.11692

Proportion of 1 -0.9Portfolio mean 15.71%Portfolio var. 41.39% 0.069854 0.743945 -0.39986641Portfolio st. dev. 64.34% 0.413933

Data table of portfolios

Proportion of 1 0.6434 15.71%-1.4 75.52% 17.37%

-1.15 69.89% 16.54%-0.9 64.34% 15.71%

-0.65 58.88% 14.87%-0.4 53.55% 14.04%

-0.15 48.39% 13.21%0.1 43.45% 12.37%

0.35 38.84% 11.54%0.6 34.66% 10.71%

0.85 31.11% 9.87%1.1 28.41% 9.04%

16. This problem returns to the four-asset problem considered in section 7.5:Calculate the envelope set for these four assets and show that the individual assets all lie within this envelope set. You should get a graph that looks something like the following:(see part in right of the calculations)

20% 30% 40% 50% 60% 70% 80%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 86: FM-MTE 2 Preparation-S 2012 Solution

1.35 26.82% 8.21%1.6 26.55% 7.37%

1.85 27.63% 6.54%2.1 29.92% 5.70%

2.35 33.16% 4.87%2.6 37.12% 4.04%

2.85 41.58% 3.20%Stock A 31.62% 6.00%Stock B 54.77% 8.00%Stock C 63.25% 10.00%Stock D 70.71% 15.00%

20% 30% 40% 50% 60% 70% 80%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 87: FM-MTE 2 Preparation-S 2012 Solution

Mean minusconstant

1%3%5%

10%

20% 30% 40% 50% 60% 70% 80%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 88: FM-MTE 2 Preparation-S 2012 Solution

20% 30% 40% 50% 60% 70% 80%

0%

2%

4%

6%

8%

10%

12%

14%

16%

18%

20%

Efficient Frontier Showing the Individual Stocks

Standard deviation of portfolio

Ex

pe

cte

d p

ort

foli

o r

etu

rn

Stock B

Stock A

Stock C

Stock D

Page 89: FM-MTE 2 Preparation-S 2012 Solution

KR 0.0052 0.0033 0.0015 0.0039 0.0068 0.0010F 0.0033 0.0120 0.0034 0.0072 0.0063 0.0015

TGT 0.0015 0.0034 0.0046 0.0058 0.0039 0.0015JNPR 0.0039 0.0072 0.0058 0.0379 0.0073 0.0023AHO 0.0068 0.0063 0.0039 0.0073 0.0389 0.0023KEY 0.0010 0.0015 0.0015 0.0023 0.0023 0.0018

Global mean variance portfolio (GMVP)

KR 0.0999 #VALUE!F 0.1554

TGT 0.0952JNPR 0.2983AHO 0.3031KEY 0.0481Sum 1 #VALUE!

Mean -0.33% #VALUE!Variance 1.15% #VALUE!Sigma 10.73% #VALUE!

Efficient portfolio

Risk-free 0.45%

KR 0.1067 #VALUE!F -0.4642

TGT -0.0641JNPR 0.0252AHO -0.1830KEY 1.5794Sum 1 #VALUE!

Mean 2.33% #VALUE!Variance 0.56% #VALUE!Sigma 7.50% #VALUE!

17. The following table shows the var-covar matrix and the mean return for six stocks: a) compute the global minmum variance portfolio(GMVP), b) compute the efficient portfolio aasuming a monthly trisk-free rate of 0.45%, c) show the frontier as the expected return and standard deviation.

KrogerKR

FordF

TargetTGT

Juniper Networks

JNPRAholdAHO

KeyCorpKEY

Note that the book formula for the GMVP is for a row vector; here we want a column vector, hence Transpose.

Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give us the whole frontier. We do this below.

Page 90: FM-MTE 2 Preparation-S 2012 Solution

Covar -0.00233966 #VALUE!

Proportion of GMVP 0.3Proportion of efficient 0.7 #VALUE!

1.53% #VALUE!Portfolio sigma 5.30% #VALUE!

Data table: varying proportion of GMVP

Sigma Mean5.30% 1.53% #VALUE!

-1 20.83% 4.99%-0.8 17.99% 4.46%-0.6 15.18% 3.92%-0.4 12.45% 3.39%-0.2 9.85% 2.86%

0 7.50% 2.33%0.2 5.76% 1.80%0.4 5.24% 1.27%0.6 6.26% 0.73%0.8 8.28% 0.20%

1 10.73% -0.33%1.2 13.39% -0.86%1.4 16.15% -1.39%1.6 18.97% -1.93%1.8 21.83% -2.46%

2 24.71% -2.99%

Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give us the whole frontier. We do this below.

Expected portfolioreturn

0% 5% 10% 15% 20% 25%

-3%

-2%

-1%

0%

1%

2%

3%

4%

5% Portfolio Returns & Sigma

Standard deviation

Exp

ecte

d r

etu

rn

Page 91: FM-MTE 2 Preparation-S 2012 Solution

0.24% 1-0.89% 10.48% 10.44% 1

-1.46% 11.04% 1

combination of GMVP and the eficient portfolio.

The following table shows the var-covar matrix and the mean return for six stocks: a) compute the global minmum variance portfolio(GMVP), b) compute the efficient portfolio aasuming a monthly trisk-free rate of 0.45%, c) show the

Mean returns

Note that the book formula for the GMVP is for a row vector; here we want a column vector, hence Transpose.

Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give us the whole frontier. We do

Page 92: FM-MTE 2 Preparation-S 2012 Solution

Drawing the efficient frontier: By Proposition 2 of Chapter 9, the efficient frontier is the convex combination of any two frontier portfolios. Thus combining the GMVP and the efficient portfolio will give us the whole frontier. We do

0% 5% 10% 15% 20% 25%

-3%

-2%

-1%

0%

1%

2%

3%

4%

5% Portfolio Returns & Sigma

Standard deviation

Exp

ecte

d r

etu

rn

Page 93: FM-MTE 2 Preparation-S 2012 Solution

Solving an Unconstrained Portfolio Problem

Variance-covariance matrix Means

0.10 0.03 -0.08 0.05 8%0.03 0.20 0.02 0.03 9%

-0.08 0.02 0.30 0.20 10%0.05 0.03 0.20 0.90 11%

c 8.0% 3.00%

Optimal portfolio without short sale restrictions (Chapter 9, Proposition 1)

0.2004 0.6219 #VALUE!0.2587 0.08040.4219 0.35420.1190 -0.0565

Total 1 1

Portfolio mean 9.46% 8.62% #VALUE!Portfolio sigma 31.91% 19.39% #VALUE!

4.32% #VALUE!

Proportion of first Sigma Mean

-1 28.17% 7.78%-0.9 26.53% 7.86%-0.8 25.00% 7.95%-0.7 23.60% 8.03%-0.6 22.34% 8.12%-0.5 21.26% 8.20%-0.4 20.39% 8.28%-0.3 19.74% 8.37%-0.2 19.35% 8.45%-0.1 19.23% 8.54%

0 19.39% 8.62%0.1 19.81% 8.70%0.2 20.48% 8.79%0.3 21.38% 8.87%

18.To set the scene, consider the following optimization problem, which is solved without any short-sale constraints. The spreadsheet shows a four asset variance-covariance matrix and associated expected returns. Given a constants c = 8 percent, find the optimal portfolio . By changing the value of c in the spreadsheet,to c = 3%, compute other portfolio; show the frontier as the expected return and standard deviation.

PORTFOLIO OPTIMIZATION ALLOWING SHORT SALESFollows Proposition 1, Chapter 9

x1

x2

x3

x4

Covariance between portfolios

Page 94: FM-MTE 2 Preparation-S 2012 Solution

0.4 22.48% 8.96%0.5 23.76% 9.04%0.6 25.18% 9.12%0.7 26.72% 9.21%0.8 28.37% 9.29%0.9 30.10% 9.38%

1 31.91% 9.46%1.1 33.77% 9.54%1.2 35.69% 9.63%1.3 37.65% 9.71%1.4 39.65% 9.80%1.5 41.67% 9.88%1.6 43.73% 9.96%1.7 45.81% 10.05%1.8 47.91% 10.13%1.9 50.02% 10.22%

2 52.15% 10.30%2.1 54.30% 10.38%2.2 56.46% 10.47%2.3 58.63% 10.55%2.4 60.81% 10.64%2.5 63.00% 10.72%2.6 65.19% 10.80%2.7 67.39% 10.89%2.8 69.60% 10.97%2.9 71.82% 11.06%

3 74.04% 11.14%3.1 76.27% 11.22%3.2 78.49% 11.31%3.3 80.73% 11.39%3.4 82.97% 11.48%3.5 85.21% 11.56%3.6 87.45% 11.64%3.7 89.70% 11.73%3.8 91.95% 11.81%

Portfolio with C=8% 31.91% 9.46%Portfolio with C=3% 19.39% 8.62%

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%7%

8%

9%

10%

11%

12%

13% Portfolio Returns & Sigma

Standard deviation

Exp

ecte

d r

etu

rn

C=3% portfolio

C=3% portfolio

C=8% portfolio

Page 95: FM-MTE 2 Preparation-S 2012 Solution

10% 20% 30% 40% 50% 60% 70% 80% 90% 100%7%

8%

9%

10%

11%

12%

13% Portfolio Returns & Sigma

Standard deviation

Exp

ecte

d r

etu

rn

C=3% portfolio

C=3% portfolio

C=8% portfolio