Iapm Solutions

Chapter 3 Mutual Funds

1. 1 Recurring expenser2 = r1 + in percentage terms 1 – Initial expense

in decimals

1 = x 13% + 1.8%

1 – 0.05

= 15.48%

2. 1 Recurring expenser2 = r1 + in percentage terms 1 – Initial expense

in decimals

1 Recurring expense 16.5% = x 14% + in percentage terms

1 – 0.06

Recurring expense in 16.5% = 0.1489 + percentage terms

Recurring expense = 0.1650 – 0.1489 = 0.0161 in percentage terms = 1.61%

Chapter 4SECURITIES MARKET

1.

Share Price in base year (Rs.)

Price in year t (Rs.)

Price Relative

No. of outstanding

shares(in million)

Market capitalisation

in the base year (1 x 4)


in year t(2 x 4)

1 2 3 4 5 6M 12 16 133 10 120 160N 18 15 83 5 90 75O 35 60 171 6 210 360P 20 30 150 40 800 1200Q 15 6 40 30 450 180

577 1670 1975

The equal weighted index 577For year t is : = 115.4

5 (since there are 5 scrips)

The value weighted index 1975For year t is : x 100 = 118.3

1670

2.Share Price in

base year (Rs.)

Price in year t (Rs.)

Price Relative

No .of outstanding

shares


in the base year (1 x 4)


in year t(2 x 4)

1 2 3 4 5 6X 80 100 125 15 1200 1500Y 40 30 75 20 800 600Z 30 50 1500

3500 The value weighted index for year t is: Market capitalisation in year t

x 1003500

Market capitalisation in year t115 = x 100

3500

115 x 3500 = Market capitalisation in year t x 100 115 x 3500

Market capitalisation in year t = 100

= 4025Market capitalisation of z = 4025 – (500 + 600)

= 1925

1925Price of share z in year t =

50

= 38.5

Chapter 5 RISK AND RETURN

1. R1 = 0.20, R2 = - 0.10, R3 = 0.18, R4 = 0.12, R5 = 0.16

(a) Arithmetic mean0.20 – 0.10 + 0.18 + 0.12 + 0.16

= = 0.112 or 11.2%5

(b) Cumulative wealth indexCWI5 = 1(1.20) (0.90) (1.18) (1.12) (1.16) = 1.656

(c) Geometric Mean = [(1.20) (0.90) (1.18) (1.12) (1.16)]1/5 – 1 = 0.106 or 10.6%

2. The standard deviation of returns is calculated belowPeriod Return in %

Ri

Deviation (Ri –R)

Square of deviations(Ri – R)2

1 20 8.8 77.442 -10 21.2 449.443 18 6.8 46.244 12 0.8 0.645 16 4.8 23.04

Sum = 596.8

Σ (Ri – R)2 596.8Variance = = = 149.2

n – 1 5 – 1

Standard deviation = (149.2)1/2 = 12.21

3. The expected rate of return on Alpha stock is:0.4 x 25 + 0.3 x 12 + 0.3 x –6 = 11.8The standard deviation of return is calculated below:

Ri (RI – R) pi pi (Ri – R)2

25 13.2 0.40 69.69612 0.2 0.30 0.012- 6 -17.8 0.30 95.052

Sum = 164.76

Standard deviation of return = [Σ pi (Ri – R)2]1/2 = 12.84%

Chapter 6 THE TIME VALUE OF MONEY

1. Value five years hence of a deposit of Rs.1,000 at various interest rates is as follows:

r = 8% FV5 = 1000 x FVIF (8%, 5 years)= 1000 x 1.469 = Rs.1469




2. Rs.160,000 / Rs. 5,000 = 32 = 25

According to the Rule of 72 at 12 percent interest rate doubling takes place approximately in 72 / 12 = 6 years

So Rs.5000 will grow to Rs.160,000 in approximately 5 x 6 years = 30 years

3. In 12 years Rs.1000 grows to Rs.8000 or 8 times. This is 23 times the initial deposit. Hence doubling takes place in 12 / 3 = 4 years.

According to the Rule of 69, the doubling period is:

0.35 + 69 / Interest rate

Equating this to 4 and solving for interest rate, we get

Interest rate = 18.9%.

4. Saving Rs.2000 a year for 5 years and Rs.3000 a year for 10 years thereafter is equivalent to saving Rs.2000 a year for 15 years and Rs.1000 a year for the years 6 through 15.

Hence the savings will cumulate to:

2000 x FVIFA (10%, 15 years) + 1000 x FVIFA (10%, 10 years)= 2000 x 31.772 + 1000 x 15.937 = Rs.79481.

5. Let A be the annual savings.

A x FVIFA (12%, 10 years) = 1,000,000A x 17.549 = 1,000,000

So A = 1,000,000 / 17.549 = Rs.56,983.

6. 1,000 x FVIFA (r, 6 years) = 10,000

FVIFA (r, 6 years) = 10,000 / 1000 = 10

From the tables we find that

FVIFA (20%, 6 years) = 9.930FVIFA (24%, 6 years) = 10.980

Using linear interpolation in the interval, we get:

20% + (10.000 – 9.930) r = x 4% = 20.3% (10.980 – 9.930)

7. 1,000 x FVIF (r, 10 years) = 5,000FVIF (r,10 years) = 5,000 / 1000 = 5


FVIF (16%, 10 years) = 4.411FVIF (18%, 10 years) = 5.234

Using linear interpolation in the interval, we get:

(5.000 – 4.411) x 2% r = 16% + = 17.4%

(5.234 – 4.411)

8. The present value of Rs.10,000 receivable after 8 years for various discount rates (r ) are:

r = 10% PV = 10,000 x PVIF(r = 10%, 8 years) = 10,000 x 0.467 = Rs.4,670

r = 12% PV = 10,000 x PVIF (r = 12%, 8 years)= 10,000 x 0.404 = Rs.4,040

r = 15% PV = 10,000 x PVIF (r = 15%, 8 years)= 10,000 x 0.327 = Rs.3,270

9. Assuming that it is an ordinary annuity, the present value is:

2,000 x PVIFA (10%, 5years) = 2,000 x 3.791 = Rs.7,582

10. The present value of an annual pension of Rs.10,000 for 15 years when r = 15% is:

10,000 x PVIFA (15%, 15 years)= 10,000 x 5.847 = Rs.58,470

The alternative is to receive a lumpsum of Rs.50,000.

Obviously, Mr. Jingo will be better off with the annual pension amount of Rs.10,000.

11. The amount that can be withdrawn annually is:100,000 100,000

A = ------------------ ------------ = ----------- = Rs.10,608 PVIFA (10%, 30 years) 9.427

12. The present value of the income stream is:

1,000 x PVIF (12%, 1 year) + 2,500 x PVIF (12%, 2 years)+ 5,000 x PVIFA (12%, 8 years) x PVIF(12%, 2 years)= 1,000 x 0.893 + 2,500 x 0.797 + 5,000 x 4.968 x 0.797 = Rs.22,683.

13. The present value of the income stream is:

2,000 x PVIFA (10%, 5 years) + 3000/0.10 x PVIF (10%, 5 years)= 2,000 x 3.791 + 3000/0.10 x 0.621= Rs.26,212

14. To earn an annual income of Rs.5,000 beginning from the end of 15 years from now, if the deposit earns 10% per year a sum of Rs.5,000 / 0.10 = Rs.50,000

is required at the end of 14 years. The amount that must be deposited to get this sum is:Rs.50,000 / PVIF (10%, 14 years) = Rs.50,000 / 3.797 = Rs.13,165

15. Rs.20,000 =- Rs.4,000 x PVIFA (r, 10 years)

PVIFA (r,10 years) = Rs.20,000 / Rs.4,000 = 5.00

From the tables we find that:

PVIFA (15%, 10 years) = 5.019PVIFA (18%, 10 years) = 4.494

Using linear interpolation we get:

5.019 – 5.00r = 15% + ---------------- x 3%

5.019 – 4.494

= 15.1%

16. PV (Stream A) = Rs.100 x PVIF (12%, 1 year) + Rs.200 xPVIF (12%, 2 years) + Rs.300 x PVIF(12%, 3 years) + Rs.400 xPVIF (12%, 4 years) + Rs.500 x PVIF (12%, 5 years) +Rs.600 x PVIF (12%, 6 years) + Rs.700 x PVIF (12%, 7 years) +Rs.800 x PVIF (12%, 8 years) + Rs.900 x PVIF (12%, 9 years) +Rs.1,000 x PVIF (12%, 10 years)

= Rs.100 x 0.893 + Rs.200 x 0.797 + Rs.300 x 0.712 + Rs.400 x 0.636 + Rs.500 x 0.567 + Rs.600 x 0.507 + Rs.700 x 0.452 + Rs.800 x 0.404 + Rs.900 x 0.361 + Rs.1,000 x 0.322

= Rs.2590.9

Similarly,PV (Stream B) = Rs.3,625.2PV (Stream C) = Rs.2,851.1

17. FV5 = Rs.10,000 [1 + (0.16 / 4)]5x4

= Rs.10,000 (1.04)20

= Rs.10,000 x 2.191= Rs.21,910

18. FV5 = Rs.5,000 [1+( 0.12/4)] 5x4

= Rs.5,000 (1.03)20

= Rs.5,000 x 1.806= Rs.9,030

19. A B C

Stated rate (%) 12 24 24

Frequency of compounding 6 times 4 times 12 times

Effective rate (%) (1 + 0.12/6)6- 1 (1+0.24/4)4 –1 (1 + 0.24/12)12-1

= 12.6 = 26.2 = 26.8

Difference between theeffective rate and statedrate (%) 0.6 2.2 2.8

20. Investment required at the end of 8th year to yield an income of Rs.12,000 per year from the end of 9th year (beginning of 10th year) for ever:

Rs.12,000 x PVIFA(12%, ∞ )

= Rs.12,000 / 0.12 = Rs.100,000

To have a sum of Rs.100,000 at the end of 8th year , the amount to be deposited now is:

Rs.100,000 Rs.100,000 = = Rs.40,388

PVIF(12%, 8 years) 2.476

21. The interest rate implicit in the offer of Rs.20,000 after 10 years in lieu of Rs.5,000 now is:

Rs.5,000 x FVIF (r,10 years) = Rs.20,000

Rs.20,000 FVIF (r,10 years) = = 4.000

Rs.5,000


FVIF (15%, 10 years) = 4.046

This means that the implied interest rate is nearly 15%.

I would choose Rs.20,000 for 10 years from now because I find a return of 15% quite acceptable.

22. FV10 = Rs.10,000 [1 + (0.10 / 2)]10x2

= Rs.10,000 (1.05)20

= Rs.10,000 x 2.653= Rs.26,530

If the inflation rate is 8% per year, the value of Rs.26,530 10 years from now, in terms of the current rupees is:

Rs.26,530 x PVIF (8%,10 years)= Rs.26,530 x 0.463 = Rs.12,283

23. A constant deposit at the beginning of each year represents an annuity due. PVIFA of an annuity due is equal to : PVIFA of an ordinary annuity x (1 + r) To provide a sum of Rs.50,000 at the end of 10 years the annual deposit should be

Rs.50,000A =

FVIFA(12%, 10 years) x (1.12)

Rs.50,000 = = Rs.2544

17.549 x 1.12

24. The discounted value of Rs.20,000 receivable at the beginning of each year from 2005 to 2009, evaluated as at the beginning of 2004 (or end of 2003) is:

Rs.20,000 x PVIFA (12%, 5 years)= Rs.20,000 x 3.605 = Rs.72,100.

The discounted value of Rs.72,100 evaluated at the end of 2000 is

Rs.72,100 x PVIF (12%, 3 years)= Rs.72,100 x 0.712 = Rs.51,335

If A is the amount deposited at the end of each year from 1995 to 2000 then

A x FVIFA (12%, 6 years) = Rs.51,335A x 8.115 = Rs.51,335A = Rs.51,335 / 8.115 = Rs.6326

25. The discounted value of the annuity of Rs.2000 receivable for 30 years, evaluated as at the end of 9th year is:

Rs.2,000 x PVIFA (10%, 30 years) = Rs.2,000 x 9.427 = Rs.18,854

The present value of Rs.18,854 is:Rs.18,854 x PVIF (10%, 9 years)

= Rs.18,854 x 0.424= Rs.7,994

26. 30 per cent of the pension amount is 0.30 x Rs.600 = Rs.180

Assuming that the monthly interest rate corresponding to an annual interest rate of 12% is 1%, the discounted value of an annuity of Rs.180 receivable at the end of each month for 180 months (15 years) is:

Rs.180 x PVIFA (1%, 180)

(1.01)180 - 1Rs.180 x ---------------- = Rs.14,998

.01 (1.01)180

If Mr. Ramesh borrows Rs.P today on which the monthly interest rate is 1%

P x (1.01)60 = Rs.14,998P x 1.817 = Rs.14,998

Rs.14,998P = ------------ = Rs.8254

1.817

27. Rs.300 x PVIFA(r, 24 months) = Rs.6,000 PVIFA (4%,24) = Rs.6000 / Rs.300 = 20

From the tables we find that:PVIFA(1%,24) = 21.244PVIFA (2%, 24) = 18.914

Using a linear interpolation

21.244 – 20.000r = 1% + ---------------------- x 1%

21.244 – 18,914

= 1.53%Thus, the bank charges an interest rate of 1.53% per month.The corresponding effective rate of interest per annum is

[ (1.0153)12 – 1 ] x 100 = 20%

28. The discounted value of the debentures to be redeemed between 8 to 10 years evaluated at the end of the 5th year is:

Rs.10 million x PVIF (8%, 3 years) + Rs.10 million x PVIF (8%, 4 years) + Rs.10 million x PVIF (8%, 5 years) = Rs.10 million (0.794 + 0.735 + 0.681) = Rs.22.1 million If A is the annual deposit to be made in thesinking fund for the years 1 to 5, then A x FVIFA (8%, 5 years) = Rs.22.1 million A x 5.867 = Rs.22.1 million A = 5.867 = Rs.22.1 million A = Rs.22.1 million / 5.867 = Rs.3.77 million

29. Let `n’ be the number of years for which a sum of Rs.20,000 can be withdrawn annually.Rs.20,000 x PVIFA (10%, n) = Rs.100,000PVIFA (15%, n) = Rs.100,000 / Rs.20,000 = 5.000

From the tables we find thatPVIFA (10%, 7 years) = 4.868PVIFA (10%, 8 years) = 5.335

Thus n is between 7 and 8. Using a linear interpolation we get 5.000 – 4.868

n = 7 + ----------------- x 1 = 7.3 years 5.335 – 4.868

30. Equated annual installment = 500000 / PVIFA(14%,4)= 500000 / 2.914= Rs.171,585

Loan Amortisation Schedule

Beginning Annual Principal RemainingYear amount installment Interest repaid balance------ ------------- --------------- ----------- ------------- ------------- 1 500000 171585 70000 101585 398415 2 398415 171585 55778 115807 282608 3 282608 171585 39565 132020 150588 4 150588 171585 21082 150503 85*

(*) rounding off error

31. Define n as the maturity period of the loan. The value of n can be obtained from the equation.

200,000 x PVIFA(13%, n) = 1,500,000PVIFA (13%, n) = 7.500

From the tables or otherwise it can be verified that PVIFA(13,30) = 7.500Hence the maturity period of the loan is 30 years.

32. Expected value of iron ore mined during year 1 = Rs.300 million

Expected present value of the iron ore that can be mined over the next 15 years assuming a price escalation of 6% per annum in the price per tonne of iron

1 – (1 + g)n / (1 + i)n

= Rs.300 million x i - g

1 – (1.06)15 / (1.16)15

= Rs.300 million x 0.16 – 0.06

= Rs.300 million x (0.74135 / 0.10)= Rs.2224 million

MINICASE

1. How much money would Ramesh need 15 years from now?

500,000 x PVIFA (10%, 15years)+ 1,000,000 x PVIF (10%, 15years)= 500,000 x 7.606 + 1,000,000 x 0.239= 3,803,000 x 239,000 = Rs.4,042,000

2. How much money should Ramesh save each year for the next 15 years to be able to meet his investment objective?

Ramesh’s current capital of Rs.600,000 will grow to :

600,000 (1.10)15 = 600,000 x 4.177 = Rs 2,506,200

This means that his savings in the next 15 years must grow to :

4,042,000 – 2,506,200 = Rs 1,535,800

So, the annual savings must be : 1,535,800 1,535,800

= = Rs.48,338FVIFA (10%, 15 years) 31.772

3. How much money would Ramesh need when he reaches the age of 60 to meet his donation objective?

200,000 x PVIFA (10% , 3yrs) x PVIF (10%, 11yrs)

= 200,000 x 2.487 x 0.350 = 174090

4. What is the present value of Ramesh’s life time earnings?

400,000 400,000(1.12) 400,000(1.12)14

46 1 2 15

1.12 15

1 – 1.08

= 400,000 0.08 – 0.12

= Rs.7,254,962

Chapter 7 Financial Statement Analysis

1.

(a) Assets Rs.- Fixed assets ( Net ) 150 million- Cash and bank 20- Marketable securities 10- Receivables 70- Inventories 110- Prepaid expenses 10

Liabilities- Equity capital 90- Preference capital 20- Reserves and surplus 50- Debentures (secured) 60- Term loans (secured) 70- Short term bank borrowing (unsecured) 40- Trade creditors 30- Provisions 10

Balance Sheet of Mahaveer Limited as on March 31, 2001

Liabilities Assets

Share capital Fixed assets- Equity 90 - Net fixed assets 150- Preference 20Reserve & surplus 50 Investments

Secured loans Current assets, loans & advances- Debentures 60- Term loans 70 - Marketable securities 10

- Pre-paid expenses 10Unsecured loans - Inventories 110- Short term bank borrowing 40 - Receivables 70Current liabilities & provisions - Cash & Bank 20- Trade creditors 30- Provisions 10

------ -----370 370

------ ------

2.(a)

Sources & Uses of Cash Statement for the Period 01.04.2000 to 31.03.2001

(Rs. in million)----------------------------------------------------------------------------------------------------------

Sources Uses

Net profit 30 Dividend payment 20Depreciation 20Decrease in inventories 10 Purchase of fixed assets 30Increase in short term

bank borrowings 10 Increase in debtors 10Increase in other assets 5

Increase in trade creditors 10 Decrease in long term debt 15Decrease in provisions 5

Total sources 80 Total uses 85

Net decline in Cash balance 5

(b)Classified cash flow statement for the Period 01.04.2000 to 31.03.2001

(Rs. in million)---------------------------------------------------------------------------------------------------------- A. Cash flow from operating activities

- Net profit before tax and extraordinary items 100- Adjustments for

Interest paid 30Depreciation 20

- Operating profit before working capital changes 150- Adjustments for

Inventories 10Debtors (10)Short term bank borrowings 10Trade creditors 10Provisions (5)Increase in other assets (5)

- Cash generated from operations 160Income tax paid (20)

- Cash flow before extraordinary items 140Extraordinary item (50)

- Net cash flow from operating activities 90B. Cash flow from investing activities

- Purchase of fixed assets (30)- Net cash flow from investing activities (30)

C. Cash flow from financing activities- Interest paid (30)- Repayment of term loans (15)- Dividends paid (20)Net cash flow from financing activities (65)

D. Net increase in cash and cash equivalents (5)- Cash and cash equivalents as on 31.03.2000 20- Cash and cash equivalents as on 31.03.2001 15

Note : It has been assumed that “other assets” represent “other current assets”.

Net profit3. Return on equity =

Equity

= Net profit Net sales Total assets

x x Net sales Total assets Equity

1 = 0.05 x 1.5 x = 0.25 or 25 per cent 0.3

Debt EquityNote : = 0.7 So = 1-0.7 = 0.3

Total assets Total assets

Hence Total assets/Equity = 1/0.3

4. PBT = Rs.40 million PBIT

Times interest covered = = 6 Interest

So PBIT = 6 x InterestPBIT – Interest = PBT = Rs.40 million 6 x Interest = Rs.40 millionHence Interest = Rs. 8 million

5. Sales = Rs. 7,000,000Net profit margin = 6 percent

Net profit = Rs. 7000000 x 0.06 = 420,000Tax rate = 60 per cent

420,000 So Profit before tax = = Rs. 1,050,000

(1-.6)Interest charge = Rs.150,000

So Profit before interest and taxes = Rs. 1,200,000Hence

1,200,000 Times interest covered ratio = = 8

150,000

6. CA = 1500 CL = 600

Let BB stand for bank borrowing

CA+BB = 1.5

CL+BB

1500+BB = 1.5

600+BB

BB = 1200

7. Accounts receivableACP =

Sales / 365

120,000= = 43.8 days 1,000,000 / 365

So the receivables must be collected in 43.8 days

Current assets8. Current ratio = = 1.5

Current liabilities

Current assets - InventoriesAcid-test ratio = = 1.2

Current liabilities

Current liabilities = 800,000 Sales

Inventory turnover ratio = = 5 Inventories

Current assets - InventoriesAcid-test ratio = = 1.2

Current liabilities

Current assets InventoriesThis means - = 1.2

Current liabilities Current liabilities

Inventories1.5 - = 1.2

800,000 Inventories

= 0.3 800,000

Inventories = 240,000Sales

= 5 So Sales = 1,200,0002,40,000

9. Debt/equity = 0.60Equity = 50,000 + 60,000 = 110,000So Debt = 0.6 x 110,000 = 66,000Hence Total assets = 110,000+66,000 = 176,000Total assets turnover ratio = 1.5So Sales = 1.5 x 176,000 = 264,000Gross profit margin = 20 per centSo Cost of goods sold = 0.8 x 264,000 = 211,200Day’s sales outstanding in accounts receivable = 40 days

SalesSo Accounts receivable = x 40

360

264,000 = x 40 = 29,333

360

Cost of goods sold 211,200Inventory turnover ratio = = = 5

Inventory InventorySo Inventory = 42,240

Assuming that the debt of 66,000 represent current liabilitiesCash + Accounts receivable

Acid-test ratio = Current liabilities

Cash + 29,333 = = 1.2

66,000So Cash = 49867

Plant and equipment = Total assets - Inventories - Accounts receivable - Cash = 176,000 - 42240 – 29333 - 49867 = 54560

Pricing together everything we get

Balance SheetEquity capital 50,000 Plant & equipment 54,560Retained earnings 60,000 Inventories 42,240Debt(Current liabilities) 66,000 Accounts receivable 29,333

Cash 49,867

176,000 176,000

Sales 264,000Cost of goods sold 211,200

Cash & bank balances + Receivables + Inventories + Pre-paid expenses10.(i) Current ratio =

Short-term bank borrowings + Trade creditors + Provisions

5,000,000+15,000,000+20,000,000+2,500,000

= 15,000,000+10,000,000+5,000,000

42,500,000 = = 1.42

30,000,000

Current assets – Inventories 22,500,000(ii) Acid-test ratio = = = 0.75

Current liabilities 30,000,000

Long-term debt + Current liabilities (iii) Debt-equity ratio =

Equity capital + Reserves & surplus

12,500,000 + 30,000,000 = = 1.31 10,000,000 + 22,500,000

Profit before interest and tax (iv) Times interest coverage ratio =

Interest

15,100,000 = = 3.02

5,000,000

Cost of goods sold 72,000,000 (v) Inventory turnover period = = = 3.6

Inventory 20,000,000

365 (vi) Average collection period =

Net sales/Accounts receivable

365 = = 57.6 days

95,000,000/15,000,000

Net sales 95,000,000 (vii) Total assets turnover ratio = = = 1.27

Total assets 75,000,000

Profit after tax 5,100,000

(ix) Net profit margin = = = 5.4% Net sales 95,000,000

PBIT 15,100,000 (x) Earning power = = = 20.1%

Total assets 75,000,000

Equity earning 5,100,000 (xi) Return on equity = = = 15.7%

Net worth 32,500,000

The comparison of the Omex’s ratios with the standard is given below

Omex Standard

Current ratio 1.42 1.5Acid-test ratio 0.75 0.80Debt-equity ratio 1.31 1.5Times interest covered ratio 3.02 3.5Inventory turnover ratio 3.6 4.0Average collection period 57.6 days 60 daysTotal assets turnover ratio 1.27 1.0Net profit margin ratio 5.4% 6%Earning power 20.1% 18%Return on equity 15.7% 15%

MINICASE

cash and bank + receivables + inventories 12.4a. Current ratio = ----------------------------------------------------- = -------- = 0.67

current liabilities+ short- term bank borrowing 18. 4

current assets – inventories 12.4 – 9.3 Acid-test ratio = --------------------------------- = ----------- = 0.17

current liabilities 18.4

cash and bank balance + current investments 1.1+0 Cash ratio = ------------------------------------------------------ = ------- = 0.06

current liabilities 18.4

Debt-equity ratio = debt / equity =( 3.8+ 11.7 ) / 15.8 = 15.5 / 15.8 = 0. 98 Interest coverage ratio = PBIT / Interest = 5.0 / 2.0= 2.5

PBIT + depreciation Fixed charges coverage ratio = --------------------------------------------------- =

interest + repayment of loan / ( 1- tax rate)

cost of goods sold 45.8Inventory turnover ratio = ----------------------- = --------------- = 5.23

average inventory ( 8.2 + 9.3)/ 2

Debtors turnover ratio = net credit sales / average debtors = 57.4 / ( 2.9+2.0) / 2 = 23.43

Average collection period = 365 / debtors turnover = 365 / 23.43 = 15.6 days

Fixed assets turnover = net sales/average total assets = 57.4/ ( 34 + 38) / 2 = 1.59

Gross profit margin = gross profit / net sales = 11.6 / 57.4 = 20.21 %Net profit margin = net profit / net sales = 3.0 / 57.4 = 5.22 %Return on assets = net profit / average total assets = 3.0/ ( 34+38) /2 = 8.3 %Earning power = PBIT / average total assets = 5.0/ ( 34+38) /2 = 13.89 %Return on equity = Net profit / average equity = 3.0 / ( 13.9 +15.8)/2 = 20.20 %

b. net profit net profit net sales Dupont equation : -------------------------- = ------------- x --------------

average total assets net sales average total assets

Dupont chart

Return on total assets

8.3%

Net profitmargin5.22%

Total asset turnover 1.59

Net profit3.0

Net sales57.4

Net sales57.4

Average total assets

36

Net sales +/-Non-op. surplus

deficit 57.8

Total costs54.8

Average fixed assets

21.4

Average other assets

2.55

Average current assets 12.05

X

—

+

+

c. Common size statements

Balance sheet

Regular ( Rs. in million) Common Size ( %)20x4 20x5 20x4 20x5------------------------------ --------------------------

Shareholder’ funds 13.9 15.8 63 62Long term debt 5.2 3.8 23 15Net current liabilities 3.2 6.0 14 23

---------------------------- ---------------------------Total 22.3 25.6 100 100

------------------------------ ---------------------------Fixed assets 19.6 23.2 88 91Other assets 2.7 2.4 12 9

------------------------------ --------------------------- Total 22.3 25.6 100 100

------------------------------ ---------------------------

Profit and loss account

Regular ( Rs. in million) Common Size ( %)20x4 20x5 20x4 20x5------------------------------ -------------------------

Net sales 39.0 57.4 100 100Cost of goods sold 30.5 45.8 78 80Gross profit 8.5 11.6 22 20PBIT 4.1 5.0 11 9Interest 1.5 2.0 4 4PBT 2.6 3.0 7 5Tax ----- ------ ------ ------PAT 2.6 3.0 7 5

Common base financial statements

Balance sheet

Regular ( Rs. in million) Common base year( %)20x4 20x5 20x4 20x5------------------------------ --------------------------

Shareholder’ funds 13.9 15.8 100 114Long term debt 5.2 3.8 100 73Net current liabilities 3.2 6.0 100 187

---------------------------- Total 22.3 25.6 100 115

------------------------------Fixed assets 19.6 23.2 100 118Other assets 2.7 2.4 100 89

------------------------------ Total 22.3 25.6 100 115

------------------------------

Profit and loss account

Regular ( Rs. in million) Common base year ( %)20x4 20x5 20x4 20x5------------------------------ -------------------------

Net sales 39.0 57.4 100 147Cost of goods sold 30.5 45.8 100 150Gross profit 8.5 11.6 100 136PBIT 4.1 5.0 100 122Interest 1.5 2.0 100 133PBT 2.6 3.0 100 115Tax ----- ------PAT 2.6 3.0 100 115

d. Financial strengths : leverage position is satisfactory.

Interest repayment capacity is good. Inventory is efficiently managed. Credit management is efficient. Margin on sales is satisfactory.

Financial weaknesses : liquidity position is very bad.

return on assets is low. fixed assets do not seem to be efficiently employedl.

e. The problems in analyzing financial statements are generally as follows: lack of underlying theory. conglomerate firms. window dressing. price level changes. variations in accounting policies. interpretation of results. correlation among ratios.

f. The qualitative factors relevant for evaluating the performance and prospects of a company are mainly the following:

Are the company’s revenues tied to one key customer? To what extent are the company’s revenues tied to one key product ? To what extent does the company rely on a single supplier ? What percentage of the company’s business is generated overseas ? Competition. Future prospects. Legal and regulatory environment.

Chapter 8PORTFOLIO THEORY

1. (a)E (R1) = 0.2(-5%) + 0.3(15%) + 0.4(18%) + .10(22%)

= 12%E (R2) = 0.2(10%) + 0.3(12%) + 0.4(14%) + .10(18%)

= 13%σ(R1) = [.2(-5 –12)2 + 0.3 (15 –12)2 + 0.4 (18 –12)2 + 0.1 (22 – 12)2]½

= [57.8 + 2.7 + 14.4 + 10]½ = 9.21%

σ(R2) = [.2(10 –13)2 + 0.3(12 – 13)2 + 0.4 (14 – 13)2 + 0.1 (18 – 13)2] ½

= [1.8 + 0.09 + 0.16 + 2.5] ½ = 2.13%

(b) The covariance between the returns on assets 1 and 2 is calculated belowState of nature

Probability Return on asset 1

Deviation of return on asset 1 from its mean

Return on asset 2

Deviation of the

return on asset 2 from its mean

Product of deviation

times probability

(1) (2) (3) (4) (5) (6) (2)x(4)x(6)1 0.2 -5% -17% 10% -3% 10.22 0.3 15% 3% 12% -1% -0.9%3 0.4 18% 6% 14% 1% 2.44 0.1 22% 10% 18% 5% 5

Sum = 16.7

Thus the covariance between the returns of the two assets is 16.7.

(c) The coefficient of correlation between the returns on assets 1 and 2 is:Covariance12 16.7

= = 0.85 σ1 x σ2 9.21 x 2.13

2. (a) For Rs.1,000, 20 shares of Alpha’s stock can be acquired. The probability distribution of the return on 20 shares is

Economic Condition Return (Rs) Probability

High Growth 20 x 55 = 1,100 0.3Low Growth 20 x 50 = 1,000 0.3Stagnation 20 x 60 = 1,200 0.2Recession 20 x 70 = 1,400 0.2

Expected return = (1,100 x 0.3) + (1,000 x 0.3) + (1,200 x 0.2) + (1,400 x 0.2)

= 330 + 300 + 240 + 280= Rs.1,150

Standard deviation of the return = [(1,100 – 1,150)2 x 0.3 + (1,000 – 1,150)2 x 0.3 + (1,200 – 1,150)2 x 0.2 + (1,400 – 1,150)2 x 0.2]1/2 = Rs.143.18

(b) For Rs.1,000, 20 shares of Beta’s stock can be acquired. The probability distribution of the return on 20 shares is:

Economic condition Return (Rs) Probability

High growth 20 x 75 = 1,500 0.3Low growth 20 x 65 = 1,300 0.3Stagnation 20 x 50 = 1,000 0.2Recession 20 x 40 = 800 0.2

Expected return = (1,500 x 0.3) + (1,300 x 0.3) + (1,000 x 0.2) + (800 x 0.2) = Rs.1,200

Standard deviation of the return = [(1,500 – 1,200)2 x .3 + (1,300 – 1,200)2 x .3 + (1,000 – 1,200)2 x .2 + (800 – 1,200)2 x .2]1/2 = Rs.264.58

(c ) For Rs.500, 10 shares of Alpha’s stock can be acquired; likewise for Rs.500, 10 shares of Beta’s stock can be acquired. The probability distribution of this option is:

Return (Rs) Probability(10 x 55) + (10 x 75) = 1,300 0.3(10 x 50) + (10 x 65) = 1,150 0.3(10 x 60) + (10 x 50) = 1,100 0.2(10 x 70) + (10 x 40) = 1,100 0.2

Expected return = (1,300 x 0.3) + (1,150 x 0.3) + (1,100 x 0.2) + (1,100 x 0.2)= Rs.1,175

Standard deviation = [(1,300 –1,175)2 x 0.3 + (1,150 – 1,175)2 x 0.3 + (1,100 – 1,175)2 x 0.2 + (1,100 – 1,175)2 x 0.2 ]1/2

= Rs.84.41

d. For Rs.700, 14 shares of Alpha’s stock can be acquired; likewise for Rs.300, 6 shares of Beta’s stock can be acquired. The probability distribution of this

option is:

Return (Rs) Probability

(14 x 55) + (6 x 75) = 1,220 0.3(14 x 50) + (6 x 65) = 1,090 0.3(14 x 60) + (6 x 50) = 1,140 0.2(14 x 70) + (6 x 40) = 1,220 0.2

Expected return = (1,220 x 0.3) + (1,090 x 0.3) + (1,140 x 0.2) + (1,220 x 0.2) = Rs.1,165

Standard deviation = [(1,220 – 1,165)2 x 0.3 + (1,090 – 1,165)2 x 0.3 + (1,140 – 1,165)2 x 0.2 + (1,220 – 1,165)2 x 0.2]1/2

= Rs.57.66

The expected return to standard deviation of various options are as follows :

OptionExpected return

(Rs)Standard deviation

(Rs)Expected / Standardreturn deviation

a 1,150 143 8.04b 1,200 265 4.53c 1,175 84 13.99d 1,165 58 20.09

Option `d’ is the most preferred option because it has the highest return to risk ratio.

3. Expected rates of returns on equity stock A, B, C and D can be computed as follows:

A: 0.10 + 0.12 + (-0.08) + 0.15 + (-0.02) + 0.20 = 0.0783 = 7.83%6

B: 0.08 + 0.04 + 0.15 +.12 + 0.10 + 0.06 = 0.0917 = 9.17%6

C: 0.07 + 0.08 + 0.12 + 0.09 + 0.06 + 0.12 = 0.0900 = 9.00%6

D: 0.09 + 0.09 + 0.11 + 0.04 + 0.08 + 0.16 = 0.095 = 9.50%6

(a) Return on portfolio consisting of stock A = 7.83%

(b) Return on portfolio consisting of stock A and B in equalproportions = 0.5 (0.0783) + 0.5 (0.0917)

= 0.085 = 8.5%

(c ) Return on portfolio consisting of stocks A, B and C in equalproportions = 1/3(0.0783 ) + 1/3(0.0917) + 1/3 (0.090)

= 0.0867 = 8.67%

(d) Return on portfolio consisting of stocks A, B, C and D in equalproportions = 0.25(0.0783) + 0.25(0.0917) + 0.25(0.0900) +

0.25(0.095)

= 0.08875 = 8.88%

4. The standard deviation of portfolio return is:

p= [w121

2 + w222

2 + w323

2 + 424

2 + 2 w1 w2 12 1 2 + 2 w1 w3 13 1 3 + 2 w1 w4

14 14 + 2 w2 w3 23 2 3 + 2 w2 w4 24 2 4 + 2 w3 w4 34 3 4 ]1/2

= [0.22 x 42 + 0.32 x 82 + 0.42 x 202 + 0.12 x 102 + 2 x 0.2 x 0.3 x 0.3 x 4 x 8 + 2 x 0.2 x 0.4 x 0.5 x 4 x 20 + 2 x 0.2 x 0.1 x 0.2 x 4 x 10 + 2 x 0.3 x 0.4 x 0.6 x 8 x 20 + 2 x 0.3 x 0.1 x 0.8 x 8 x 10 + 2 x 0.4 x 0.1 x 0.4 x 20 x 10]1/2

= 10.6%

5. (i) Since there are 3 securities, there are 3 variance terms and 3 covariance terms. Note that if there are n securities the number of covariance terms are: 1 + 2 +…+ (n – 1) = n (n –1)/2. In this problem all the variance terms are the same (2

A) all the covariance terms are the same (AB) and all the securities are equally weighted (wA⅓)So,

2p = [3 w2

A 2A + 2 x 3 AB]

2p = [3 w2

A 2A + 6 wA wBAB]

1 2 1 1 = 3 x x 2

A + 6 x x x AB

3 3 3 1 2 = 2

A + AB

3 3

(ii) Since there are 9 securities, there are 9 variance terms and 36 covariance terms. Note that if the number of securities is n, the number of covariance terms is n(n – 1)/2.In this case all the variance terms are the same (2

A), all the covariance terms are 1the same (AB) and all the securities are equally weighted wA

9 So,

n(n-1)2

p = 9 w2A 2

A t 2 x wA wBAB

2

1 2 1 1 = 9 x x 2

A + 9(8) x x AB

9 9 9

1 72

= 2A + AB

9 81

6. Let us arrange the portfolio in the order of ascending expected returns.

Portfolio Expected return(%) Standard deviation(%)

4 8 14

3 9 15

5 10 20

1 11 21

7 12 21

2 14 24

8 14 28

6 16 32

Examining the above we find that (i) portfolio 7 dominates portfolio 1 because it offers a higher expected return for the same standard deviation and (ii) portfolio 2 dominates portfolio 8 as it offers the same expected return for a lower standard deviation. So, the efficient set consists of all the portfolios except portfolio 1 and portfolio 8.

7. The weights that drive the standard deviation of portfolio to zero, when the returns are perfectly correlated, are:

σB 35wA = = = 0.614

σA + σB 22 + 35wB = 1 - wA = 0.386

The expected return of the portfolio is :0.614 x 14% + 0.386 x 20% = 16.316

8. (a) Covariance (P,Q) = PPQ x σP x σQ

= 0.4 x 14 x 20 = 112 (b) Expected return = 0.5 x 14 + 0.5 x 20 = 17%

Risk (standard deviation) = [w2P 2

P + w2Q 2

Q + 2 Cov (P,Q)]½

= [0.52 x 625 + 0.52 x 1600 + 2 x 112] ½

= 27.93%

Chapter 9CAPITAL ASSET PRICING MODEL AND ARBITRAGE PRICING THEORY

1. Define RA and RM as the returns on the equity stock of Auto Electricals Limited a and Market portfolio respectively. The calculations relevant for calculating the beta of the stock are shown below:

Year RA RM RA-RA RM-RM (RA-RA) (RM-RM) RA-RA/RM-RM

1 15 12 -0.09 -3.18 0.01 10.11 0.292 -6 1 -21.09 -14.18 444.79 201.07 299.063 18 14 2.91 -1.18 8.47 1.39 -3.434 30 24 14.91 8.82 222.31 77.79 131.515 12 16 0-3.09 0.82 9.55 0.67 -2.536 25 30 9.91 14.82 98.21 219.63 146.877 2 -3 -13.09 -18.18 171.35 330.51 237.988 20 24 4.91 8.82 24.11 77.79 43.319 18 15 2.91 -0.18 8.47 0.03 -0.5210 24 22 8.91 6.82 79.39 46.51 60.7711 8. 12 -7.09 -3.18 50.27 10.11 22.55

RA = 15.09 RM = 15.18

(RA – RA)2 = 1116.93 (RM – RM) 2 = 975.61 (RA – RA) (RM – RM) = 935.86

Beta of the equity stock of Auto Electricals

(RA – RA) (RM – RM)

(RM – RM) 2

= 935.86 = 0.96975.61

Alpha = RA – βA RM

= 15.09 – (0.96 x 15.18)= 0.52

Equation of the characteristic line is

RA = 0.52 + 0.96 RM

2. The beta for stock B is calculated below:

Period Return of stock B, RB (%)

Return on market

portfolio, RM (%)

Deviation of return on

stock B from its mean(RB - RB)

Deviation of return on market portfolio from its mean

(RM – RM)

Product of the

deviation (RB – RB)(RM – RM)

Square of the

deviation of return on market portfolio, from its mean

(RM – RM)2

1 15 9 6 -1 -6 12 16 12 7 2 14 43 10 6 1 -4 -4 164 -15 4 -24 -6 144 365 -5 16 -14 6 -84 366 14 11 5 1 5 17 10 10 1 0 0 08 15 12 6 2 12 49 12 9 3 -1 -3 110 -4 8 -13 -2 26 411 -2 12 -11 2 -22 412 12 14 3 4 12 1613 15 -6 6 -16 -96 25614 12 2 3 -8 -24 6415 10 8 1 -2 -2 416 9 7 0 -3 0 917 12 9 3 -1 -3 118 9 10 0 0 0 019 22 37 13 27 351 72920 13 10 4 0 0 0

180 200 Σ(RB – RB) Σ(RB – RB)2

Σ RB = 180 ΣRM = 200 (RM – RM) = 1186RB = 9% RM = 10% = 320

Beta of stock B is equal to:Cov (RB, RM)

2M

Σ (RB - RB) (RM – RM) 320Cov (RB, RM) = = = 16.84

n –1 19

Σ (RM – RM)2 11862

M = = = 62.42 n –1 19

So the beta for stock B is: 16.84

= 0.27062.42

3.a. The slope of the capital market line is:

E(RM) – Rf 15 – 8 λ = = = 0.28

M 25

b. The expected return for various mutual funds is:Omega: 8 + 0.28 x 16 = 12.48%Pioneer: 8 + 0.28 x 20 = 13.60%Monarch: 8 + 0.28 x 24 = 14.72%Zenith: 8 + 0.28 x 30 = 16.40%

4. E(RM) = 14% M = 20% Rf = 6%A,M = 0.7 B,M = 0.8 A = 24% B = 32%

(a) Beta for stock A:A,M A,M A M 0.7 x 24 x 20

βA = = = = 0.842

M 2M 20 x 20

Beta for stock B:B,M B,M B M 0.8 x 32 x 20

βB = = = = 1.282

M 2M 20 x 20

(b) Required return for A = Rf + βA [E(RM) - Rf ] = 6 + 0.84 [14 – 6] = 12.72%

Required return for B = Rf + βB [E(RM) - Rf ] = 6 + 1.28[14 – 6] = 16.24%

5. (a) Market portfolio has an expected return of 13% and standard deviation of 20% Riskless asset has an expected return of 7% and standard deviation of 0%

The expected return of a portfolio which has 60% of market portfolio and 40% of riskless asset is :

0.6 x 13 + 0.4 x 7 = 10.6%

The standard deviation of a portfolio which has 60% of market portfolio and 40% of riskless asset is :

0.6 x 20 + 0.4 x 0 = 12%

(b) The expected return of a portfolio which has 125% of market portfolio and –25% of riskless asset is :

1.25 x 13 – 0.25 x 7 = 14.5%

The standard deviation of a portfolio which has 125% of market portfolio and –125% of riskless portfolio is:

1.25 x 20 – 0.25 x 0 = 25%

6.(a) The beta of the aggressive stock is:

40% - (-5%) 45% = = 2.25

25% - 5% 20%

The beta of the defensive stock is:18% - 8% 10%

= = 0.5025% - 5% 20%

(b) The expected return on the two stocks is:Aggressive stock: 0.5(-5) + 0.5(40) = 17.5%Defensive stock : 0.5(8) + 0.5(18) = 13.0%

(c) The expected return on the market portfolio is:0.5 x 5 + 0.5 x 25 = 15%If the risk-free rate is 8%, the market risk premium is: 15% - 8% = 7%So, the SML is:Required returni = 8% + βi x 7%

(d) The alphas of the two stocks are calculated below:Aggressive Stock

Expected return = 17.5% Beta = 2.25Required return = 8 + 2.25% x 7 = 23.75% Alpha = 17 – 23.75 = - 6.75%

Defensive StockExpected return = 13.0%

Beta = 0.50Required return = 8 + 0.5 x 7 = 11.5% Alpha = 13.0 – 11.5 = 1.5%

MINICASE

a. For stock A :Expected return = ( 0.2x 15) +( 0.5x20) +( 0.3x 40) =3+10 +12 =25Standard deviation= [ 0.2( 15-25)2 + 0.5( 20-25)2 + 0.3(40-25)2]1/2

= [ 20 + 12.5+67.5] ½ = 10 For stock B:Expected return = ( 0.2x 30) + ( 0.5x 5) + [ 0.3x (-) 15] = 6+2.5- 4.5= 4

Standard deviation= [ 0.2( 30-4)2 + 0.5(5-4)2 + 0.3(-15-4)2]1/2

= ( 135.2 + 0.5 + 108.3) ½ = 15.62 For stock C: Expected return = [ 0.2x(-)5] +( 0.5x15) +( 0.3x 25) =-1+7.5 +7.5 =14

Standard deviation= [ 0.2(-5-14)2 + 0.5(15-14)2 + 0.3(25-14)2]1/2

= [ 72.2 + 0.5+36.3] ½ = 10.44

For market portfolio:

Expected return = [ 0.2x(-)10] +( 0.5x16) +( 0.3x 30) =-2+8 +9 =15

Standard deviation= [ 0.2(-10-15)2 + 0.5(16-15)2 + 0.3(30-15)2]1/2

= ( 125+ 0.5 + 67.5)1/2 = 13.89

b. State of the Proba Return on Return on RA-E(RA) RB-E(RB) pEconomy -bility(p) A(%) ( RA) B (%) ( RB) x[RA-E(RA)]

x[RB-E(RB)] -------------- --------- ----------- ------------- ---------- --------- ------------- Recession 0.2 15 30 -10 26 -52.0

Normal 0.5 20 5 -5 1 - 2.5Boom 0.3 40 -15 15 -19 - 85.5

------------total = - 140.0

-------------Covariance between the returns of A and B is (-) 140

State of the Proba Return on Return on RA-E(RA) RC-E(RC) pEconomy -bility(p) A(%) ( RA) C (%) ( RC) x[RA-E(RA)]

x[RC-E(RC)] -------------- --------- ----------- ------------- ---------- --------- ------------- Recession 0.2 15 -5.0 -10 - 19.0 38.0

Normal 0.5 20 15.0 -5 1.0 - 2.5Boom 0.3 40 25.0 15 11.0 49.5

------------total = 85.0

-------------Covariance between the returns of A and C is 85

(-) 140c. Coefficient of correlation between the returns of A and B =------------- = (-)0.90

10x15.62 85 . Coefficient of correlation between the returns of A and C =------------- = 0. 81

10x 10.44

d Portfolio in which stocks A and B are equally weighted:

Economic condition Probability Overall expected return------------------------ --------------- ----------------Recession 0.2 0.5x15 +0.5x30 = 22.5Normal 0.5 0.5x20 +0.5x5 =12.5Boom 0.3 0.5x40 + 0.5x(-)15=12.5

Expected return of the portfolio=( 0.2x22.5)+( 0.5x12.5)+( 0.3x12.5) = 4.5 +6.25 + 3.75 = 14.5 Standard deviation of the portfolio

= [ 0.2(22.5-14.5)2 + 0.5(12.5-14.5)2 + 0.3(12.5-14.5)2]1/2

= [ 12.8 + 2 + 1.2] ½ = 4Portfolio in which weights assigned to stocks A , B and C are 0.4, 0.4 and 0.2 respectively: Expected return of the portfolio = ( 0.4x25) + ( 0.4x4) +( 0.2x14)

= 10 +1.6 +2.8= 14.4For calculating the standard deviation of the portfolio we also need covariance betweenB and C, which is calculated as under:

State of the Proba Return on Return on RB-E(RB) RC-E(RC) pEconomy -bility(p) B(%) ( RB) C (%) ( RC) x[RB-E(RB)]

x[RC-E(RC)] -------------- --------- ----------- ------------- ---------- --------- ------------- Recession 0.2 30 -5.0 26 - 19.0 (-)98.8

Normal 0.5 5 15.0 1 1.0 0.5Boom 0.3 (-)15 25.0 (-)19 11.0 (-)62.7

------------total = (-)161.0

-------------Covariance between the returns of B and C is (-) 161

We have the following values:wA =0.4 wB = 0.4 wC = 0.2 A=10 B =15.62 C =10.44

AB=(-)140 AC= 85 BC = (-) 161Standard deviation=[(0.4x10)2+(0.4x15.62)2+(0.2x10.44)2+{2x0.4x0.4x(-)140}+{2x0.4x0.2x85}

+{ 2x0.4x0.2x(-)161}]1/2

= ( 16+ 39.04 + 4.36 – 44.8 + 13.6 – 25.76)1/2 = 1.56

e. (i) Risk-free rate is 6% and market risk premium is 15-6= 9%The SML relationship isRequired return = 6% + bx 9%

(ii) For stock A: Required return = 6% + 1.2x9% = 16.8 %; Expected return = 25 %Alpha = 25-16.8 = 8.2 %

For stock B:Required return =6 %-0.70x9%=(-)0.3% ; Expected return = 4 %Alpha = 4- (-) 0.3 = 4.3 %

For stock C:Required return = 6% + 0.9x 9% = 14.1 %; Expected return= 14%Alpha = 14 – 14.1 = (-) 0.1 % _ _ _ _ _

f. Period RD(%) RM(%) RD-RD RM-RM ( RM-RM)2 (RD-RD)(RM-RM)

--------- --------- --------- ---------- ------------ ------------- ---------------------------

1 -12 -5 -18.4 -11.2 125.44 206.08 2 6 4 - 0.4 - 2.2 4.84 0.88 3 12 8 5.6 1.8 3.24 10.08 4 20 15 13.6 8.8 77.44 119.68 5 6 9 -0.4 2.8 7.84 - 1.12 ----------------------------------------------------------------------------------------- _ _ _ S RD=32 S RM=31 S( RM-RM)2

=218.80 S(RD-RD)(RM-RM)=335.6 _ _

RD =6.4 RM =6.2 2m =218.8/ 4 = 54.7 Cov( D,M)=335.6/4=83.9

b = 83.9/54.7 = 1.53Interpretation: The change in return of D is expected to be 1.53 times the

expected change in return on the market portfolio.

g. The linear relationship between expected return and standard deviation for efficient portfolios is called the Capital Market Line ( CML) and the same is given by the equation

E (Rm-Rf) E ( Rj) = Rf + [ ---------- ] j

m

where E(Rj) = expected return on portfolio j Rf = risk-free rate E(Rm) = expected return on the market portfolio m =standard deviation of the market portfolio j = standard deviation of the portfolio j

Linear relationship between expected return and standard deviation of individual securities and inefficient portfolios is called Security Market Line (SML) and the equation for it is

E( Rm)-Rf

E(Ri) = Rf + [ ------------- ] Ci,m 2

m where E(Ri) and E(Rm) are the expected returns on the security/ portfolio i and market respectively.Rf = risk-free rate

m =standard deviation of the market portfolio Ci,m = Covariance of the return on security/ portfolio i with the market portfolio.CML is a special case of SML as seen from the following.As per SML

E( Rm)-Rf

E(Ri) = Rf + [ ------------- ] Ci,m 2

m

Since Ci,m = i,m i m ,the above equation can be rewritten as

E( Rm)-Rf

E(Ri) = Rf + [ ------------- ] i,m i m

For efficient portfolios , as returns on i and m are perfectly positively correlated, i,m =1Therefore,

E( Rm)-Rf

E(Ri) = Rf + [ ------------- ] i , which is nothing but the CML.

m

h. Systematic risk refers to the risk associated with the responsiveness of the return of the investment arising from economy-wide factors, which have a bearing on the fortunes of all firms.Unsystematic risk refers to the risk associated with the responsiveness of the return of the investment arising from firm-specific factors.Systematic risk is usually represented by beta ( b), which is given by the formula i,m

bi = ------------ 2

m

where bi = beta of the security/ portfolio ii,m = covariance between the returns on investment i and the market portfolio2

m = variance of the return on the market portfolio.

Unsystematic risk: Being firm specific there is no generalised formula for this risk.

i. CAPM assumes that return on a stock/ portfolio is solely influenced by the market factor whereas the APT assumes that the return is influenced by a set of factors called risk factors.

Chapter 12 BOND PRICES AND YIELDS

1. 5 11 100P = +

t=1 (1.15) (1.15)5

= Rs.11 x PVIFA(15%, 5 years) + Rs.100 x PVIF (15%, 5 years)= Rs.11 x 3.352 + Rs.100 x 0.497= Rs.86.7

2.(i) When the discount rate is 14%7 12 100

P = +t=1 (1.14) t (1.15)7

= Rs.12 x PVIFA (14%, 7 years) + Rs.100 x PVIF (14%, 7 years)= Rs.12 x 4.288 + Rs.100 x 0.4= Rs.91.46

(ii) When the discount rate is 12%7 12 100

P = + = Rs.100t=1 (1.12) t (1.12)7

Note that when the discount rate and the coupon rate are the same the value is equal to par value.

3. The yield to maturity is the value of r that satisfies the following equality. 7 120 1,000

Rs.750 = + = Rs.100 t=1 (1+r) t (1+r)7

Try r = 18%. The right hand side (RHS) of the above equation is:Rs.120 x PVIFA (18%, 7 years) + Rs.1,000 x PVIF (18%, 7 years)= Rs.120 x 3.812 + Rs.1,000 x 0.314= Rs.771.44

Try r = 20%. The right hand side (RHS) of the above equation is:Rs.120 x PVIFA (20%, 7 years) + Rs.1,000 x PVIF (20%, 7 years)= Rs.120 x 3.605 + Rs.1,000 x 0.279= Rs.711.60

Thus the value of r at which the RHS becomes equal to Rs.750 lies between 18% and 20%.

Using linear interpolation in this range, we get

771.44 – 750.00

Yield to maturity = 18% + 771.44 – 711.60 x 2%

= 18.7%

4. 10 14 100

80 = + t=1 (1+r) t (1+r)10

Try r = 18%. The RHS of the above equation isRs.14 x PVIFA (18%, 10 years) + Rs.100 x PVIF (18%, 10 years)= Rs.14 x 4.494 + Rs.100 x 0.191 = Rs.82

Try r = 20%. The RHS of the above equation isRs.14 x PVIFA(20%, 10 years) + Rs.100 x PVIF (20%, 10 years)= Rs.14 x 4.193 + Rs.100 x 0.162= Rs.74.9

Using interpolation in the range 18% and 20% we get:

82 - 80Yield to maturity = 18% + ----------- x 2%

82 – 74.9= 18.56%

5.12 6 100

P = +t=1 (1.08) t (1.08)12

= Rs.6 x PVIFA (8%, 12 years) + Rs.100 x PVIF (8%, 12 years)= Rs.6 x 7.536 + Rs.100 x 0.397= Rs.84.92

6. The post-tax interest and maturity value are calculated below:Bond A Bond B

* Post-tax interest (C ) 12(1 – 0.3) 10 (1 – 0.3)=Rs.8.4 =Rs.7

* Post-tax maturity value (M) 100 - 100 -[ (100-70)x 0.1] [ (100 – 60)x 0.1]=Rs.97 =Rs.96

The post-tax YTM, using the approximate YTM formula is calculated below

8.4 + (97-70)/10Bond A : Post-tax YTM = --------------------

0.6 x 70 + 0.4 x 97

= 13.73%

7 + (96 – 60)/6Bond B : Post-tax YTM = ----------------------

0.6x 60 + 0.4 x 96

= 17. 47%

7.14 6 100

P = +t=1 (1.08) t (1.08)14

= Rs.6 x PVIFA(8%, 14) + Rs.100 x PVIF (8%, 14)= Rs.6 x 8.244 + Rs.100 x 0.341= Rs.83.56

8. . The YTM for bonds of various maturities isMaturity YTM(%) 1 12.36 2 13.10

3 13.21

4 13.48

5 13.72Graphing these YTMs against the maturities will give the yield curve

The one year treasury bill rate , r1, is

1,00,000 - 1 = 12.36 %

89,000

To get the forward rate for year 2, r2, the following equation may be set up :

12500 112500 99000 = +

(1.1236) (1.1236)(1+r2)

Solving this for r2 we get r2 = 13.94%

To get the forward rate for year 3, r3, the following equation may be set up :

13,000 13,000 113,000 99,500 = + +

(1.1236) (1.1236)(1.1394) (1.1236)(1.1394)(1+r3)


To get the forward rate for year 4, r4 , the following equation may be set up :

13,500 13,500 13,500100,050 = + +

(1.1236) (1.1236)(1.1394) (1.1236)(1.1394)(1.1349)

113,500 +

(1.1236)(1.1394)(1.1349)(1+r4)


To get the forward rate for year 5, r5 , the following equation may be set up :

13,750 13,750 13,750 100,100 = + +

(1.1236) (1.1236)(1.1394) (1.1236)(1.1394)(1.1349)

13,750 +

(1.1236)(1.1394)(1.1349)(1.1454)

113,750 +

(1.1236)(1.1394)(1.1349)(1.1454)(1+r5)


9. The pre-tax rate to the debenture holder is the value of the r in the following equation:

n It ai Pi n Fj

Subscription = + + price t=1 (1+r)t (1+r)i j=m (1+r)

j

where: It = interest receivable at the end of period tn = life of the debenturea = number of equity shares receivable when part-conversion occurs at the end

of period i Pi = expected price per equity share at the end of period iFj = instalment of principal repayment at the end of period j

For the given problem, r is obtained by solving the following equation:

60 40 40 40 40 20600 = + + + + +

(1+r) (1+r)2 (1+r)3 (1+r)4 (1+r)5 (1+r)6

2 x 150 200 200+ + + (1+r)1 (1+r)5 (1+r)6

r works out to 15.5%

10. Annual interest receipt will be Rs.100 for 4 years the future value at the end of 4 years. The future value at the end of 4 years, given the re-investment rate of 9 percent

will be:

100 (1.09) + 100 (1.09) + 100 (1.09) + 100 + 1,000 = 100 x FVIFA (r = 9%, n = 4) + 1,000= 100 x 4.641 + 1,000 = Rs.1464.1

Since the present market price of the bond is Rs.1020, the realised yield to maturity is the value of r* in the following equation.

1020 (1+r*)4 = 1464.1

1464.1

(1+r*)4 = = 1.435 1020

r* = 0.946 or 9.46 percent

MINICASE

a. Value of a bond is calculated as the present value of all future cash flows associated with it.

Value of a bond (V) carrying an annual coupon payment of C ( in rupees) maturing after n years with maturity value of M is given by n C MV = S -------- + -------- t=1 ( 1+r)t (1+r)n

where r is the required periodic rate of return and t is the time period for receipt of periodic payments.

b. V = 100 PVIFA8%,9yrs + 1000 PVIF8%, 9yrs

= 100 x 6.247 + 1000 x 0.5 = 624.7 + 500 = Rs. 1124.7

c. V= 50 PVIFA 4%, 18 yrs + 1000 PVIF 4 %, 18 yrs

= 50 x 12. 659 +1000 x 0. 494 = 632. 95 + 494 = Rs. 1126. 95d. Let the YTM be r % . We have

100 PVIFA r, 6yrs + 1000 PVIFr, 6 yrs = 1050Trying r = 8%, LHS = 100 x 4. 623 + 1000 x 0. 630 = 1092.3Trying r= 9%, LHS = 100x 4.486 + 1000 x 0. 596 = 1044.6By linear interpolationr= 8% + ( 9-8) ( 1092. 3- 1050) / ( 1092.3 – 1044.6) = 0.8868 i.e. 8.87 %

100+ (1000- 1050)/6 100- 8.33 e. V = --------------------------- = ------------ = 0.089 i.e. 8.9 %

0.4 x 1000 + 0.6 x 1050 1030

f. Let r be the yield to call. We then have100 PVIFA r%, 3yrs +1050 PVIF r%, 3yrs =1050

Trying r= 9%, LHS = 100x 2.531 + 1050 x 0. 772 =1063.7Trying r=10%, LHS = 100x 2.487 + 1050x 0.751 = 1037. 25By linear interpolation,

( 1063.7- 1050) 13.7r= 9% + ( 10- 9)----------------------- = 9 + ------- =9.52 %

( 1063.7- 1037.25) 26.45

g. If future cash flows are reinvested at 8% p.a. the terminal value will be 100 PVIFA 8%, 6 yrs + 1000 = 100x 7.336 + 1000 = 1733.6 Let r* be the realized yield to maturity. We have 1050 ( 1+ r *)6 = 1733.6

( 1+r*) 6 = 1733.6/ 1050 = 1.6510 1+r* = 1.0872 r* = 0.0872 or 8.72 %

100 + ( 1000 – 1050) / 6h. Stated YTM = -------------------------------- = 0.089 or 8.9 %

0.4 x 1000 + 0.6 x 1050

100+ ( 900 – 1050)/ 6 75 Expected YTM = ---------------------------- = ----- = 0.0758 or 7.58%

0.4 x 900 + 0.6 x 1050 990Difference between the expected and stated YTM = 8.9 – 7.58 = 1.32%

i. Annual percentage rate of a bond refers to the stated coupon rate per annum. If m is the frequency of coupon payment per year,

annual parentage rate Effective annual coupon interest rate = (1+ --------------------------- )m - 1

m Effective annual coupon interest rate x maturity value

Effective annual yield= ----------------------------------------------------------------- Current market price

j. Interest rate risk: Interest rates tend to vary over time, causing fluctuations in bond prices. A rise in interest rates will depress the market price of outstanding bonds. This is called interest rate risk.

Reinvestment risk: When a bond pays periodic interest, there is a risk that these interest payments may have to be reinvested at a lower interest rate. This is called reinvestment risk.

k. Key financial ratios that have a bearing on debt rating are:Interest coverage ratio = EBIT/ Interest

EBIT + DepreciationFixed charges coverage ratio = ---------------------------------------

Interest + Repayment of loan ------------------------ 1 – tax rate

PAT+ Depreciation+ Other non- cash charges +Interest on term loans + Lease rentals

Debt Service Coverage Ratio = ----------------------------------------------------------Interest on term loans +Lease rentals+ Repayment of term loans

l. Yield curve shows how yield to maturity is related to term to maturity for bonds that are similar in all respects, excepting maturity.

m. Factors that determine interest rates are:a) Short-term risk-free interest rate, which is given by

Expected real rate of return+ Expected inflationb) Maturity premium: It is the difference between the YTM on a short-term ( one year) risk-free security and the YTM on a risk-free security of a

longer duration and depends on (i) expectation of the market participants, (ii) liquidity preference of the market participants and (iii) supply and demand for funds in different maturity ranges( called habitats)c) Default premium: An additional default premium will have to be paid

when there exists a possibility of default on interest / principal payment.d) Special features: Interest rates are affected when a bond has some special

features like call or put option, conversion option, floating rate, zero coupon etc.

Chapter 13BOND PORTFOLIO MANAGEMENT

9+( 100-105)/5 81. Yield to maturity =--------------------------------- = --------- = 0.0767 or 7.77 %

( 0.4x100) + ( 0.6x 105) 40+63

Duration is calculated below

Year Cash flow Present Value Proportion of Proportion of bond’s at 18% bond’s value Value x Time

1 9 8.35 0.080 0.080

2 9 7.75 0.074 0.148

3 9 7.19 0.068 0.204

4 9 6.67 0.064 0.256

5 109 74.98 0.714 3.570------------------------

4.258 ------------------------

Duration of the bond is 4.258 years.

2. . a. Issue price (1.10)8 = Rs.10,000

Rs.10,000Issue price = = Rs.4670

(1.10)8

b. The duration of the bond is 8 years. Note that the term to maturity and the duration of a zero coupon bond are the same.

c. The modified duration of the bond is:Duration 8

= = 7.273(1+ yield) (1.10)

d. The percentage change in the price of the bond, if the yield declines by 0.5 percent is:

∆P/ P = - Modified duration x 0.5 = - 3.637 percent

3. . a. The duration of a coupon bond is:1 + y (1 + y) + T(c –y)

- y c [(1 +y)T – 1] + y

y = 10%, c = 10%, T = 10 years

So, the duration of the bond is:1.10 (1.10) + 10 (0.10 – 0.10) -0.10 0.10 [(1.10) – 1] + 0.10

1.10 11 - = 6.759

0.2594

b. Because the bond carries a coupon c. (i) A decrease in coupon rate from 10% to 8% will increase the duration.

(ii) An increase in yield from 10% to 12% reduces the duration because the duration of a coupon bearing bond varies inversely with its yield.

(iii) A decrease in maturity period from 10 years to 8 years decreases the duration.

4. The duration of a level annuity is:1 + yield Number of payments

- yield (1 + yield) No. of payments – 1

Yield = 8.5%; No. of payments = 15So, the duration is:

1.085 15 -

.085 (1.085) 15 – 1

1.085 15 - = 12.76 – 6.25 = 6.51 years

.085 3.400 – 1

5. The duration is:1 + y (1 + y) + T(c –y)

- y c [(1 +y)T – 1] + y

1.05 (1.05) + 10 (.06 – .05) -.05 .06 [(1.05)10 – 1] + 0.05

= 7.89 half year periods.

6. The liability has a duration of ten years. The duration of the zero coupon bond is 6 years and the duration of the perpetuities is : 1.07/ 0.07 = 15.29 years.As the portfolio duration is 10 years, if w is the proportion of investment in zero coupon bonds, we have( wx6) + ( 1-w)x 15.29 = 106w + 15.29 – 15.29= 10 w=0.569 and 1-w= 0. 431Therefore the amount to be invested in zero coupon bonds is 100,000 x 0.569 =Rs. 56,900 and the amount to be invested in perpetuitiesis Rs. 43,100

7. Current price = 9000 PVIFA( 8 %, 8 years) + 100,000 PVIF ( 8%, 8 years) = 9000 x 5.747 + 100,000 x 0. 540 = 51,723 + 54,000 = 105,723

Forecast price = 9000 PVIFA( 7 %, 5 years) + 100,000 PVIF ( 7%, 5 years)= 9000 x 4.100 + 100,000x 0.713 = 36,900 + 71,300= 108,200

Future value of reinvested coupon = 9000 ( 1.065)2 + 9000 ( 1.065) + 9000 = 10,208 + 9585 + 9000 = 28,793

28,793 + ( 108,200- 105, 723)Three year return = ---------------------------------------- = 0. 2958

105, 723The expected annualized return over the three year period will be ( 1. 2958 )1/3 – 1 = 0.0902 or 9.02 %

CHAPTER 14 EQUITY VALUATION

1. Do = Rs.2.00, g = 0.06, r = 0.12

Po = D1 / (r – g) = Do (1 + g) / (r – g)

= Rs.2.00 (1.06) / (0.12 - 0.06)= Rs.35.33

Since the growth rate of 6% applies to dividends as well as market price, the market price at the end of the 2nd year will be:

P2 = Po x (1 + g)2 = Rs.35.33 (1.06)2

= Rs.39.70

2. Po = D1 / (r – g) = Do (1 + g) / (r – g)= Rs.12.00 (1.10) / (0.15 – 0.10) = Rs.264

3. Po = D1 / (r – g)

Rs.32 = Rs.2 / 0.12 – gg = 0.0575 or 5.75%

4. Po = D1/ (r – g) = Do(1+g) / (r – g)Do = Rs.1.50, g = -0.04, Po = Rs.8So8 = 1.50 (1- .04) / (r-(-.04)) = 1.44 / (r + .04)

Hence r = 0.14 or 14 per cent

5. The market price per share of Commonwealth Corporation will be the sum of three components:

A: Present value of the dividend stream for the first 4 yearsB: Present value of the dividend stream for the next 4 yearsC: Present value of the market price expected at the end of 8 years.

A = 1.50 (1.12) / (1.14) + 1.50 (1.12)2 / (1.14)2 + 1.50(1.12)3 / (1.14)3 ++ 1.50 (1.12)4 / (1.14)4

= 1.68/(1.14) + 1.88 / (1.14)2 + 2.11 / (1.14)3 + 2.36 / (1.14)4

= Rs.5.74B = 2.36(1.08) / (1.14)5 + 2.36 (1.08)2 / (1.14)6 + 2.36 (1.08)3 / (1.14)7 +

+ 2.36 (1.08)4 / (1.14)8

= 2.55 / (1.14)5 + 2.75 / (1.14)6 + 2.97 / (1.14)7 + 3.21 / (1.14)8

= Rs.4.89

C = P8 / (1.14)8

P8 = D9 / (r – g) = 3.21 (1.05)/ (0.14 – 0.05) = Rs.37.45So

C = Rs.37.45 / (1.14)8 = Rs.13.14Thus,

Po = A + B + C = 5.74 + 4.89 + 13.14 = Rs.23.77

6. The intrinsic value of the equity share will be the sum of three components:

A: Present value of the dividend stream for the first 5 years when the growth rate expected is 15%.

B: Present value of the dividend stream for the next 5 years when the growth rate is expected to be 10%.

C: Present value of the market price expected at the end of 10 years.

2.00 (1.15) 2.00 (1.15)2 2.00 (1.15)3 2.00(1.15)4 2.00 (1.15)5

A = + + + + (1.12) (1.12)2 (1.1.2)3 (1.1.2)4 (1.12)5

= 2.30 / (1.12) + 2.65 / (1.12)2 + 3.04 / (1.12)3 + 3.50 / (1.12)4 + 4.02/(1.12)5

= Rs.10.84

4.02(1.10) 4.02 (1.10)2 4.02(1.10)3 4.02(1.10)4 4.02 (1.10)5

B = + + + + (1.12)6 (1.12)7 (1.12)8 (1..12)9 (1.12)10

4.42 4.86 5.35 5.89 6.48 = + + + +

(1.12)6 (1.12)7 (1.12)8 (1.1.2)9 (1.12)10

= Rs.10.81 D11 1 6.48 (1.05)

C = x = x 1/(1.12)10

r – g (1 +r)10 0.12 – 0.05 = Rs.97.20The intrinsic value of the share = A + B + C = 10.84 + 10.81 + 97.20 = Rs.118.85

7. Intrinsic value of the equity share (using the 2-stage growth model)

(1.18)6

2.36 x 1 - ----------- 2.36 x (1.18)5 x (1.12) (1.16)6

= + 0.16 – 0.18 (0.16 – 0.12) x (1.16)6

- 0.10801= 2.36 x + 62.05

- 0.02

= Rs.74.80

8. Intrinsic value of the equity share (using the H model)

4.00 (1.20) 4.00 x 4 x (0.10)= +

0.18 – 0.10 0.18 – 0.10

= 60 + 20= Rs.80

9.Price Po = D1 / (r – g) Dividend yield

D1/ Po

Capital yield(Po - Po)/ Po

Price earnings

ratio Po/ E1

Low growth firm

Po = 2 / (0.16 - .04) = 16.67 12.0% 4.0% 4.17

Normal growth firm

Po = 2 / (0.16 - .08) = 25.00 8.0% 8.0% 6.25

Supernormal growth firm

Po = 2 / (0.16 - .12) = 50.00 4.0% 12.0% 12.5

10.E1/Po = 2.50/30.00 r = 0.16

PVGOE1/Po = r 1 -

Po

2.50 PVGO

= 0.16 1 -30.00 Po

PVGO= 0.48

30.00

So, 48 percent of the price is accounted for by PVGO.

MINICASE

Dr

a. The general formula is P0 = S ------------------ t=1 ( 1+ r)t

where Dt = dividend expected t years hence r= expected return

D1

b. Value of a constant growth stock P0= ------------ r- g

where D1 is the dividend expected a year hence, r the expected return and g the growth rate in dividends.

c. Required rate of return = 7 % + 1.2 x 6 % = 14.2 % 5 x 1.10 x 1.10 d. (i) Expected value of the stock a year hence = 0.142 – 0.10 = Rs. 144.05 ( ii) Expected dividend in the first year = 5x 1.10 = Rs.5.50

5x 1.10 Intrinsic price of the stock at present = P0 = ------------ = Rs. 130. 95

0.142- 0.10

5.50Expected dividend yield = ---------- = 0.042 or 4.2 %

130.95

5x 1.10x 1.10 Expected price of the stock one year hence=P1 = ----------------- = Rs. 144.05

0.142- 1.10144.05-130.95

Capital gains yield in the first year = ------------------- = 10 %

130.95

e Let r be the expected rate of return on the stock. We then have 5x1.10 5x 1.10

110 = ---------- i.e. r = ----------- + 0.10 = 0.05 + 0.10 = 0.15 or 15% r – 0.10 110

f. Let us assume that the required rate of return is 15 percent. Year Expected dividend PV factor @15% PV of dividend

1 5x 1.25 = 6.25 0.870 5.44 2 5x ( 1.25)2= 7.81 0. 756 5.90 3 5x ( 1.25)3= 9.77 0. 658 6.43

4 5x ( 1.25)4= 12.21 0. 572 6.98--------------

total = Rs. 24.75 (A)--------------

Price of the stock at the beginning of the 5th year 12.21x 1.10= ---------------- = Rs. 268.62 0.15- 0.10

Present value of the above is 268.62x 0.572 = Rs. 153. 65 (B) Present value of the stock = A+B = 24.75 + 153.65 = Rs. 178. 40

The expected dividend in the second year= Rs. 7.81

Expected price of the stock at the beginning of the second year:

7.81 9.77 12.21 268.62= ------ + ------- + --------- + ---------- 1.15 ( 1.15)2 ( 1.15 )3 ( 1.15 )3

=6.7913 + 7.3875 + 8.0283 + 176. 6220 = 198. 8291Dividend yield in the second year = 7.81/ 198. 8291 = 0.0393

Expected price of the stock at the end of the second year,

9.77 12.21 268.62= ------- + ------- + -------- = 8.4956 + 9. 2325 + 203.1153 = 220.8434

(1.15) (1.15)2 (1.15)2

220.8434 – 198. 8291 Capital gain in the second year = --------------------------- = 0. 1107

198. 8291

The total return for the second year = 3.93 + 11.07 = 15 %

Expected dividend in the fifth year = 12.21x1.10= Rs. 13.43Expected price of the stock in the beginning of the 5th year = Rs.268.62Expected dividend yield in the 5th year = 13.43/268.62 =0.05 or 5 %

Expected price of the stock at the end of 5th year

13.43x1.10------------- =295.460.15-0.10

Expected capital gains yield in the 5th year =(295.46-268.62)/268.62 = 0.10 or 10 %

.

g. . Year Expected dividend PV factor @15% PV of dividend 1 5.00 0. 870 4. 35 2 5.00 0. 756 3.78

------------ total = Rs. 8.13 (A)

-------------

Expected price of the stock at the beginning of the 3rd year 5x 1.10= ---------- = Rs. 110 0.15-0.10

Present value of which is 110x 0.756 = Rs. 83. 16 ( B)Present value of the stock = A+B = 8.13 + 83.16 = Rs. 91.29

5 [ ( 1+ 0.10) + 2 ( 0.30- 0.10 ) ]h. Present value of the stock = ---------------------------------------- = Rs. 150

0. 15-0.10 5x (1- 0.05) 5x0.95 i. Present value of the stock= --------------- = --------- = Rs. 23.75

0.15- (-) 0.05 0.20 Dividend expected after one year = 5x0.95 = Rs. 4.75

Dividend yield per year = 4.75/23.75 = 0.2 or 20 %.Expected price of the stock at the end of the first year 4.75x0.95= ---------------- = Rs.22.56 0.15-(-)0.05Capital gains yield per year =( 22.56-23.75) / 23.75 = (-) 0.05 or (-)5%

j. The question is incomplete. Let us assume that the decline in growth rate to 10

percent will occur linearly over 4 years.

Year Expected dividend PV factor @15% PV of dividend 1 5x1.30 = 6.50 0. 870 5.66

2 5x( 1.30)2 =8.45 0. 756 6.393 5x( 1.30)3= 10.98 0.658 7.22

----------- total Rs.19.27 ( A)

-----------Expected price of the stock at the beginning of the 4th year

10.98[ ( 1+ 0.10) + 2 ( 0.30- 0.10) ] = ------------------------------------------- = Rs. 329.40

0. 15 – 0. 10Present value of this is 329.40 x 0.658 = Rs. 216.75 ( B)

Present value of the stock = A+ B = 19.27 + 216.75 = Rs 236. 02

Chapter 16 COMPANY ANALYSIS

1.Return on equity = Profit after tax / Shareholders’ fundsBook value per share = Shareholders’ funds / Number of sharesEPS = Profit after tax / Number of shares

Capital after bonus issue

Bonus adjustment factor = Capital before bonus issue

Price per share at the beginning of the yearPE ratio (prospective) =

Earnings per share for the year

Price per share at the end of the yearPB ratio (retrospective) =

Book value per share at the end of the year

Sales for 20X5 1/4

CAGR in sales = - 1 Sales for 20X1

EPS for 20X5 1/4

CAGR in EPS = - 1 EPS for 20X1

Range of ROE over the period of 20X1 – 20X5Volatility of ROE =

Average ROE over the period 20X1 – 20X5

Sustainable growth rate = Retention ratio x ROEReturn on equity = Profit after tax / Shareholders’ funds

Book value per share = Shareholders’ funds / Number of sharesEPS = Profit after tax / Number of shares

Capital after bonus issueBonus adjustment factor =

Capital before bonus issue





Sales for 20X5 1/4

CAGR in Sales = - 1 Sales for 20X1

EPS for 20X5 1/4




Sustainable growth rate = Retention ratio x ROE(a)

20X1 20X2 20X3 20X4 20X5Return on equity

26 / 120 29 / 137 32 / 157 42 / 183 49 / 216

= 21.7% = 21.2% = 20.4% = 23% = 22.7%

Book value 120/ 16 137/ 16 157/ 16 183/ 24 216/ 24per share = Rs.7.5 = Rs.8.6 = Rs.9.8 = Rs.7.6 = Rs.9

EPS 26/ 16 29/ 16 32/ 16 42/ 24 49/ 24= Rs.1.63 = Rs.1.81 = Rs.2 = Rs.1.75 = Rs.2.04

Bonus adjustment factor

1 1 1 1.5 1.5

Adjusted EPS Rs.1.63 Rs.1.81 Rs.2 Rs.2.63 Rs.3.06PE ratio 17.50/ 1.81 21/ 2 24.5/ 1.75 21.6/ 2.04(prospective) = 9.7 = 10.5 = 14 = 10.6

PB ratio (retrospective)

17.50/ 7.5 21/ 8.6 24.5/ 9.8 21.6/ 7.6 24.2/ 9

= 2.3 = 2.4 = 2.5 = 2.8 = 2.7

Retention ratio

16/ 26 17/ 29 20/ 32 26/ 42 33/ 49

= 0.62 = 0.59 = 0.63 = 0.62 = 0.67

(b) 520 1/4

CAGR of Sales = -1 = 0.201 = 20.1% 250

3.06 1/4

CAGR of EPS = -1 = 0.171 = 17.1%1.63

23 – 20.4 Volatility of ROE = = 0.12

21.8

(c) 0.63 + 0.62 + 0.67 20.4 + 23 + 22.7Sustainable growth rate =

3 3

= 0.64 X 22.03 = 14.09%

(d) PBIT Sales Profit before tax Profit after tax Assets ROE = x x x x

Sales Assets PBIT Profit before tax Net worth

The decomposition of ROE for the last two years, viz., 20X4 and 20X5 is shown below:

PBIT Sales Profit before tax Profit after tax Assets x x x x


20X4 0.167 x 1.758 x 0.70 x 0.75 x 1.49220X5 0.181 x 1.646 x 0.702 x 0.742 x 1.463

MINICASE

Return on equity = Profit after tax / Shareholders’ fundsBook value per share = Shareholders’ funds / Number of sharesEPS = Profit after tax / Number of shares







Sales for 20X5 1/4

CAGR in sales = - 1 Sales for 20X1

EPS for 20X5 1/4




Sustainable growth rate = Retention ratio x ROEReturn on equity = Profit after tax / Shareholders’ funds

Book value per share = Shareholders’ funds / Number of sharesEPS = Profit after tax / Number of shares







Sales for 20X5 1/4

CAGR in Sales = - 1 Sales for 20X1

EPS for 20X5 1/4




Sustainable growth rate = Retention ratio x ROE

(a)20X1 20X2 20X3 20X4 20X5

Return on equity

60/ 500 60/ 540 110/ 620 150/ 730 240/ 920

= 12% = 11.1% = 17.7% = 20.5% = 26.1%

Book value 500/ 30 540/ 30 620/ 30 730/ 30 920/ 30per share = Rs.16.7 = Rs.18 = Rs.20.7 = Rs.24.3 = Rs.30.7

EPS 60/ 30 60/ 30 110/ 30 150/ 30 240/ 30= Rs.2 = Rs.2 = Rs.3.7 = Rs.5 = Rs.8

PE ratio 20/ 2 22/ 3.7 45/ 5 56/ 8(prospective) = 10 = 5.9 = 9 = 7

PB ratio (retrospective)

20/ 16.7 22/ 18 45/ 20.7 56/ 24.3 78/ 30.7

= 1.2 = 1.2 = 2.2 = 2.3 = 2.5

Retention ratio

40/ 60 40/ 60 80/ 110 110/ 150 190/ 240

= 0.67 = 0.67 = 0.73 = 0.73 = 0.79

(b)1780 1/4

CAGR of sales = -1 = 0.229 = 22.9%

780

8 1/4

CAGR of EPS = -1 = 0.414 = 41.4% 2

26.1 – 11.1Volatility of ROE = = 0.86

17.5

(c) 0.73 + 0.73 + 0.79 17.7 + 20.5 + 26.1Sustainable growth rate =

3 3

= 0.75 x 21.4 = 16.05%

(d) The decomposition of ROE for the last two years, viz., 20X4 and 20X5 is shown below:

PBIT Sales Profit before tax Profit after tax Assets x x x x


20X4 0.193 x 0.979 x 0.704 x 0.789 x 1.95920X5 0.230 x 0.937 x 0.707 x 0.828 x 2.065

(e) EPS estimate for 20X6 is

20X5 20X6 RemarksNet sales 1780 2047 Increase by 15%Cost of goods sold 1210 1403.60 Increase by 16%Operating expenses 170 204 Increase by 20%Non-operating surplus/deficit 10 10 Remains samePBIT 410 449.4Interest 120 132 Increase by 10%Profit before tax 290 317.4Tax 50 70.59 Effective tax rate

increases by 5%Profit after tax 240 246.81EPS 8.23

(f) Average retention ratio for the period 20X3 – 20X5 was 0.75. So the average payout ratio was 1 – 0.75 = 0.25

Required rate of return= 10% + 1.1 x 8% = 18.8%

Expected growth rate in dividends Average retention ratio Average return on equity

= in the last three years x in the last three years

Average return on equity in 17.7 + 20.5 + 26.1the last three years = = 21.4%

3

So, the expected growth rate in dividends is:0.75 x 21.4 = 16.05%

The PE ratio as per the constant growth model is: 0.25

= 9.09 0.188 – 0.1605

(g) The value anchor is:Expected EPS x PE ratio= Rs.8.23 x 9.09 = Rs.74.8

Chapter 18 OPTIONS

1. S = 100 , uS = 150, dS = 90u = 1.5 , d = 0.9, r = 1.15 R = 1.15 E = 100

Cu = Max (uS – E, 0) = Max (150 – 100,0) = 50

Cd = Max (dS – E, 0) = Max (90 – 100,0) = 0

Cu – Cd 50 = = = 0.833

(u-d)S 0.6 x 100

u Cd – d Cu 0 – 0.9 x 50 B = = = - 65.22

(u-d)R 0.6 x 1.15

C = S + B = 0.833 x 100 – 65.22 = 18.08

2. S = 60 , dS = 45, d = 0.75, C = 5r = 0.16, R = 1.16, E = 60

Cu = Max (uS – E, 0) = Max (60u – E, 0) Cd = Max (dS – E, 0) = Max (45 – 60, 0) = 0

Cu – Cd 60u – 60 u – 1 = = =

(u-d)S (u – 0.75)60 u – 0.75

u Cd – d Cu – 0.75 (60u – 60) 45 (1 – u) B = = =

(u-d)R (u – 0.75) 1.16 1.16 (u – 0.75)

C = S + B

(u – 1) 60 45 (1 – u) 5 = + u – 0.75 1.16 (u – 0.75)

Multiplying both the sides by u – 0.75 we get

455(u – 0.75) = (u – 1) 60 + (1 – u)

1.16

Solving this equation for u we getu = 1.077

So Beta’s equity can rise to

60 x 1.077 = Rs.64.62

3. EC0 = S0 N(d1) - N (d2)

ert

S0 = 70, E = 72, r = 0.12, 0.3, t = 0.50

S0 1 ln + r + 2 t

E 2d1 =

t 70

ln + (0.12 + 0.5 x .09) x 0.50 72

= 0.30 0.50

- 0.0282 + 0.0825 = = 0.2560

0.2121

d2 = d1 - t = 0.2560 – 0.30 0.50 = 0.0439

N (d1) = 0.6010N (d2) = 0.5175

E 72= = 67.81

ert e0.12x 0.50

C0 = S0 x 0.6010 – 67.81 x 0.5175 = 70 x 0.6010 – 67.81 x 0.5175 = Rs.6.98

4. EC0 = S0 N(d1) - N (d2)

ert

E = 50, t = 0.25, S = 40, 0.40, r = 0.14

S0 1 ln + r + 2 t

E 2d1 =

t

40 ln + (0.14 + 0.5 x 0.16) 0.25

50 d1 =

0.40 0.25

- 0.2231 + 0.055 = = - 0.8405

0.20

d2 = d1 - t = - 0.8405 – 0.40 0.25 = -1.0405

N (d1) = 0.2003N (d2) = 0.1491

E 50= = 48.28

ert e0.14 x 0.25

C0 = S0 x 0.2003 – 48.28 x 0.1491 = 40 x 0.2003 – 48.28 x 0.1491 = 0.8135

5. S = 100 u = 1.5 d = 0.8

E = 105 r = 0.12 R = 1.12

The values of ∆ (hedge ratio) and B (amount borrowed) can be obtained as follows:

Cu – Cd

∆ =(u – d) S

Cu = Max (150 – 105, 0) = 45

Cd = Max (80 – 105, 0) = 0

45 – 0 45 9∆ = = = = 0.6429

0.7 x 100 70 14

u.Cd – d.Cu

B =(u-d) R

= (1.5 x 0) – (0.8 x 45)0.7 x 1.12

= -36 = -45.920.784

C = ∆ S + B= 0.6429 x 100 – 45.92= Rs.18.37

Value of the call option = Rs.18.37

6. S = 40 u = ? d = 0.8R = 1.10 E = 45 C = 8

We will assume that the current market price of the call is equal to the fair value of the call as per the Binomial model.

Given the above data

Cd = Max (32 – 45, 0) = 0

∆ Cu – Cd R= x

B u Cd – d Cu S

∆ Cu – 0 1.10= x

B -0.8Cu 40

= (-) 0.034375

∆ = - 0.34375 B (1)C = ∆ S + B8 = ∆ x 40 + B (2)

Substituting (1) in (2) we get

8 = (-0.034365 x 40) B + B8 = -0.375 Bor B = - 21.33

∆ = - 0.034375 (-21.33) = 0.7332

The portfolio consists of 0.7332 of a share plus a borrowing of Rs.21.33 (entailing a repayment of Rs.21.33 (1.10) = Rs.23.46 after one year). It follows

that when u occurs either u x 40 x 0.7332 – 23.46 = u x 40 – 45

-10.672 u = -21.54 u = 2.02

or

u x 40 x 0.7332 – 23.46 = 0u = 0.8

Since u > d, it follows that u = 2.02.Put differently the stock price is expected to rise by 1.02 x 100 = 102%.

7. E

C0 = S0 N(d1) - N (d2) ert

S0 = 120, E = 110, r = 0.14, t1.0, = 0.4

S0 1 ln + r + 2 t

E 2d1 =

t

120 1 ln + 0.14 + x 0.42 1 110 2d1 =

0.4 1

.0870 + 0.22 = = 0.7675

0.4

d2 = d1 - t = 0.7675 – 0.40 = 0.3675

N (d1) = 0.2003 N (d2) = 0.6434E 110

= = 99.63 ert 1.1503

C0 = 120 x 0.7786 – 99.63 x 0.6434 = Rs.29.33

8. EC0 = S0 N(d1) - N (d2)

ert

S0 = Rs.80, E = Rs.82, ert = 1.1503, 0.20, t = 1, r = ln (1.1503) = 0.14

S0 1 ln + r + 2 t

E 2d1 =

t

80 1 ln + 0.14 + x 0.4 1 82 2d1 =

0.20 1

- 0.0247 + 0.1600 = = 0.6765

0.20

d2 = d1 - t = 0.6765 – 0.20 = 0.4765

N (d1) = 0.751 N (d2) = 0.683E 82

= = 71.29 ert 1.1503C0 = Rs.80 x 0.751 – Rs.71.29 x 0.683 = Rs.11.39

9. According to the put-call parity

C0 = S0 + P0 – E/ ert

S0 = Rs.75, P0 = Rs.0.70, E = Rs.80, r = 0.08, t = 0.25

So C0 should be 80

Rs.75 + Rs.0.70 - = - 2.716 e0.08 x 0.25

C0 is given to be Rs.7.

Clearly the put-call parity is not working in this case.

10.S0 = Rs.60, u = 1.30, d = 0.95, r = 8%, E = Rs.50

If investors are risk-neutral, the expected return on the stock is 8%.

Since Bharat’s stock can either rise by 30 percent to Rs.78 or fall by 5 percent to Rs.57, we can calculate the probability of a price rise in the hypothetical risk-neutral world.

Expected return = [Probability of rise x 30%] + [1 – Probability of rise] x – 5% = 8%

Therefore the probability of rise is 0.3714

If the stock price rises the call option has a value of Rs.28 (Rs.78 – 50) and if the stock price falls the call option has a value of Rs.7 (Rs.57 – 50).

Hence, if investors are risk-neutral, the call option has an expected future value of:Probability of rise x Rs.28 + (1- Probability of rise) x Rs.7= 0.3714 x 28 + (1 – 0.3714) x 7= 10.40 + 4.40 = Rs.14.80The current value of the call option is:

Expected future value 14.80 = = Rs.13.70

1 + Risk-free rate (1.08)

MINICASE a. Call option : A call option gives the option holder the right to buy an asset at a fixed

price during a certain period.Put option : : A put option gives the option holder the right to sell an asset at a fixed price during a certain period.Strike price ( exercise price ) : The fixed price at which the option holder can buy and /or sell the underlying asset is called the strike price or the exercise price .Expiration date : The date when the option expires is called the expiration date.

b. Call options with strike prices 280, 300 and 320 and put options with strike prices 340

and 360 are in - the - money . Call options with stike prices 340 and 360 and put options with strike prices 280, 300 and 320 are out of – the – money.

c. (i) If Pradeep Sharma sells Jan/340 call on 1000 shares, he will earn a call premium of Rs.5000 now. However, he will forfeit the gains that he would have enjoyed if the price of Newage Hospitals rises above Rs.340.

(ii) If Pradeep Sharma sells Mar/300 call on 1000 shares, he will earn a call premium of Rs.41,000 now. However, he will forfeit the gains he would have enjoyed if the price of Newage Hospital remains above Rs.300.

d. Let s be the stock price, p1 and p2 the call premia for March/ 340 and March/ 360 calls respectively. When s is greater than 360, both the calls will be exercised and the profit will be { s-340-p1} – { s-360-p2 } = Rs. 11

The maximum loss will be the initial investment , i.e. p1-p2 =Rs. 9 The break even will occur when the gain on purchased call equals the net premium paid i.e. s-340 = p1 – p2

=9 Therefore s= 349

e. If the stock price goes below Rs.300, Mr. Sharma can execute the put option and ensure that his portfolio value does not go below Rs. 300 per share. However , if stock price goes above Rs. 340, the call will be exercised and the stocks in the portfolio will have to be delivered/ sold to meet the obligation, thus limiting the upper value of the portfolio to Rs. 340 per share. So long as the share price hovers between R. 300 and Rs. 340, Mr. Sharma will be gainer by Rs. 8 ( net premium received ).

f.

g. Other things remaining constant, value of a call option - increases when the current price of the stock increases. - decreases when the exercise price increases. - increases when option term to maturity increases. - increases when the risk-free interest rate increases. - increases when the variability of the stock price increases.

h. The assumptions underlying the Black-Sholes option pricing model are as follows:

1. The call option is the European options 2. The stock price is continuous and is distributed lognormally.3. There are no translation costs and taxes.4. There are no restrictions on or penalties for short selling.5. The stock pays no dividend.6. The risk-free interest rate is known and constant.

0

Profit

Stock price·

305 375340

Pay off

i. The three equations are

E C0 = S0 N(d1) - ------ N (d2)

ert

S0 σ2

ln ----- + r + --- E 2 d1 = σ t

d2 = d1 - σ √¯t¯¯

j. S0 = 325 E =320 t =0.25 r = 0.06 σ =0.30

325 (0.30)2

ln + 0.06 + x 0.25 320 2d1 =

0.30 x 0.25

= ( 0.0155 + 0.02625 ) / 0.15 = 0. 2783

d2 = 0.2783 -0.30 √¯0.25¯¯ = 0.2783 – 0.15 = 0.1283 Using normal distribution table

N (d1) = 1 – [ 0.3821 + ( 0.4013- 0. 3821) ( 0.30 – 0.2783 ) /( 0.30 – 0.25) ]=1- [ 0.3821 + 0. 0192 x 0.0217 / 0.05 ] = 0.6096

N ( d2 ) = 1- [ 0. 4404 + ( 0. 4602- 0.4404) ( 0. 15 – 0. 1283 ) / ( 0. 15- 0.10 ) ]= 1- [ 0.4404 + 0.0198 x 0.0217 / 0.05 ] = 0. 5510

E / ert = 320 / e0.06 x 0. 25 = 320 / 1. 0151 = 315. 24

C0 = 325 x 0.6096 – 315.24 x 0. 5510 = 198.12 – 173. 70 = Rs. 24.42

k. A collar is an option strategy that limits the value of a portfolio within two bounds. For example the strategy adopted in ( e ) above is a collar.

Chapter 19FUTURES

1. Cash flow to the buyer

March 2 1128 – 1125 = 3March 3 1127 – 1128 = -1March 4 1126 – 1127 = -1 March 5 1128 – 1126 = 2

2. F0 = S0 (1+rf)t

= Rs.40 (1.08)0.25 = Rs.40.78

3. F0 = S0 (1+rf - d)t

= 1200 (1 + 0.10 - .03)1 = 1284

4. If the 6-months futures contract for gold is $432.8 and the interest rate is 8 percent; the appropriate value for the one-year gold futures contract is :

$432.8 (1.08) 0.5 = $449.8If the one-year gold futures has a price of $453 it means that it is over-priced relative to

the 6-months futures contract.A profitable strategy would be to :

Sell a one-year futures contract for $453 Buy a 6-months futures contract for $432.8 Take delivery of the 6-months futures contract after 6-months with the help of borrowed

money, hold the gold for 6 months, and give delivery of the one-year futures contract.

5. The appropriate value of the 3-months futures contract is 1,000 (1.01)3 = Rs.1030.3

Since the 3-months futures price of Rs.1035 exceeds Rs.1030.3, it pays to buy the share in the spot market with borrowed money and sell the futures contract. Such an action produces a riskless profit of Rs.4.7 as shown below :

Action Initial cash flow Cash flow at time T (3 months) Borrow Rs.1,000 now and + Rs.1,000 - Rs.1,000 (1.01)3

repay with interest at time T = - Rs.1030.3

Buy a share - Rs.1,000 ST

Sell a futures contract 0 Rs.1035 - ST

(F0 = Rs.1035) 0 Rs.4.7

Chapter 20PORTFOLIO MANAGEMENT FRAMEWORK

1.Rp – Rf

Treynor Measure: βp

15 – 10 Fund P: = 5.55%

0.9

17 – 10 Fund Q: = 6.36%

1.1

19 – 10 Fund R: = 7.50%

1.2

16 – 10Market index: = 6%

1.0

Rp – Rf Sharpe Measure:

σp

15 – 10 Fund P: = 0.25

20

17 – 10 Fund Q: = 0.29

24

19 – 10 Fund R: = 0.33

27

16 – 10

Market index: = 0.25 20

Jensen Measure: Rp – [Rf + βp (RM – Rf )]

Fund P: 15 – [10 + 0.9 (6)] = -0.4%

Fund Q: 17 – [10 + 1.1 (6)] = 0.4%

Fund R: 19 – [10 + 1.2 (6)] = 1.8%

Market Index: 0 ( By definition)

2.(a) The arithmetic average return is: (5 + 12 + 16 + 3)/ 4 = 9% (b) The time-weighted (geometric average) return is:

[(1.05) (1.12) (1.16) (1.03)]1/4 - 1 = .089 = 8.9%

(c) The rupee-weighted average (IRR) return is computed below:

Period 1 2 3 4

Rate of return earned 5% 12% 16% 3%Beginning value of assets 200 220 296.4 373.82Investment profit during theperiod (Rate of return x Assets) 10 26.4 47.42 11.21Net inflow at the end 10 50 30 -Ending value of assets 220 296.4 373.82 385.03

Time 0 1 2 3 4

Net cash flow -200 -10 -50 -30 385.03

The IRR of this sequence is 10 50 30 385.03

200 + + + =(1 + r) (1 + r)2 (1 + r)3 (1 + r)4

r = 8.81%

Appendix 20A SOLUTION

Market Level is 100 Portfolio

Stocks Bonds Total Buy and Hold Policy 60,000 40,000 100,000 Constant Mix Policy 60,000 40,000 100,000 Constant Proportion

Portfolio Insurance Policy 60,000 40,000 100,000

Market Level Falls to 80 Portfolio Portfolio

(before rebalancing) (after rebalancing) Stocks Bonds Total Stocks Bonds Total

Buy and Hold Policy 48,000 40,000 88,000 48,000 40,000 88,000 Constant Mix Policy 48,000 40,000 88,000 52,800 35,000 88,000 Constant Proportion

Portfolio Insurance Policy 48,000 40,000 88,000 24,000 64,000 88,000

Market Level Falls to 100 Portfolio Portfolio (before rebalancing) (after rebalancing) Stocks Bonds Total Stocks Bonds Total

Buy and Hold Policy 60,000 40,000 100,000 60,000 40,000 100,000

Constant Mix Policy 66,000 35,200 101,000 60,720 40,480 101,000 Constant Proportion

Portfolio Insurance Policy 28,800 64,000 92,800 38,400 54,400 92,800

APPENDIX 20 B

1. The portfolio return is decomposed into four components as follows

1. Risk- free return, . Rf = 10 %2. The impact of systematic return, β ( Rm – Rf ): 1.2 ( 18 – 10 ) = 9.63. The impact of imperfect diversification,

( σp/σm – βp ) (Rm – Rf ) : ( 14/16- 1.2) ( 18- 10) = - 2.64. The net superior return due to selectivity, Rp – { Rf + σp/σm – βp ) (Rm – Rf ) }: 16 – { 10 + 14/ 16 ( 8) } = - 1.00

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