Lesson-3

Chapter-5: Bonds, Bond Valuation, and Interest Rates

Chapter-6: Risk, Return, and the Capital Asset Pricing Model

Discussion Questions (DQs)

1. Do Problems 6-1 thru 6-9 on pp. 258-259.

2. Do Thomson ONE Problem on pp. 261-262.

6-1. Solution:

Given,

Investment        Beta

$35,000              0.8

$ 40,000             1.4

Total amount = $75,000

Here,

The required portfolio’s beta, βp=w1β1+w2β2

                                                                               = ($35,000/$75,000) (0.8) + ($40,000/$75,000) (1.4)

                                                     =1.12

6-2 .Solution:

Given,

Risk free rate of return ( r_RF) = 6%

Expected return on market, (r_M) = 13% 

Beta value of stock i, (bi) = 0.7

Required rate of return on stock, (r_i) =?

Here, from the equation of SML,

We get,

r_i = r_RF + b_i (r_M - r_RF)

    = 6% + 0.7(13% - 6%)

  = 10.90%

6-3. Solution:

Given,

Risk free rate (r_RF ) = 5%, 

Market risk premium (RP_M ) = r_M-r_RF=6%

Required return on the market, r_M =?

For the first stock with beta=1.0

      r_M = r_{RF +} RP_{M  X βm}

               =5% + 6% x 1 

          = 11%

 For the second stock with beta=1.2

 R_s= 5% + 6% x 1.2

     = 12.2%

6-4. Solution:

From the given table,

The expected return of the stock, E(R) = (0.1) (-50%) + (0.2) (-5%) + (0.4) (16%) + (0.2) (25%) + (0.1) (60%) = 11.40%

Variance (σ² ) = (-50% - 11.40%) ²(0.1) + (-5% - 11.40%) ²(0.2) + (16% - 11.40%) ²(0.4) + (25% - 11.40%) ²(0.2) + (60% - 11.40%) ²(0.1) = 712.44

Then, Standard deviation can be calculated from the variance as follows:

Standard deviation, σ= square root of variance of 712.44

                                  =26.69%   

Standard deviation measures the overall risk of the firm.

 Now, the required coefficient of variation (CV) = σ/E(R)

                                                                              =26.69%/11.40% 

                                                                              = 2.34

 Coefficient of variation measures the firm’s per unit risk. In other words, a investor must took a risk of 2.34 percent in an investment for each percent of return.

6-5 

Solution:

a) Expected rates of return for the market is,

E(r _m) = (0.3) (15%) + (0.4) (9%) + (0.3) (18%)

      = 13.5%

Return for the stock J is,

 E(r j) = (0.3) (20%) + (0.4) (5%) + (0.3) (12%)

      = 11.6%

b) Variance of market

σ²_M = [(0.3) (15% - 13.5%) 2 + (0.4) (9% - 13.5%) 2 + (0.3) (18% -13.5%) 2]1/2

       =14.85%

Thus,

 Standard deviation for market (σ_M) = Square root of Variance

                                                         = square root of 14.85%

                                                         =3.85%.

And variance of stock J

σ²J = [(0.3) (20% - 11.6%) ² + (0.4) (5% - 11.6%) ² + (0.3) (12% - 11.6%) ²]^1/2

      =38.64%

Thus standard deviation for stock J =σ_J

                                                         =square root of 38.64%

                                                         = 6.22%

c) Coefficient of variation, for market, CV_M = 3.85%/13.5% 

                                                                       = 0.29

                           Coefficient of stock J, CVJ = 6.22%/11.6%

                                                                       = 0.54

6-6. Solution:

We have,

 r_RF=5%

 r_M=10%

 r_A=12%

a) Stock A’s beta, β_A=?

Here,

    r_A = r_RF + (r_M - r_RF)β_A

12% = 5% + (10% - 5%)β_A

12% = 5% + 5% β_A

 7% = 5% β_A

  β_A= 1.4

b) If  β_A=2.0, then

A’s new required rate of return, r_A= 5% + 5 %(β_A)

                                                                                  = 5% + 5 %( 2)

                                                       = 15%

6-7. Solution:

Given,

 r_RF = 9%

 r_M =14%

 b_i=1.3

a) Required rate, r_i = r_RF + (r_M - r_RF)β_i

                               = 9% + (14% - 9%) 1.3

                               = 15.5%

 b) (1) If r_RF increases to 10%, then r_M increases by 1 percentage point, from 14% to 15%.

Then,

 ri = r_RF + (r_M - r_RF)βi

    = 10% + (15% - 10%) 1.3

    = 16.5%

 (2) If r_RF decreases to 8%, then rM decreases by 1%, from 14% to 13%.

 Then,

  ri = r_RF + (r_M - r_RF)βi

     = 8% + (13% - 8%) 1.3

     = 14.5%

 c. (1) If r_M increases to 16%:

 ri = r_RF + (r_M - r_RF)βi

    = 9% + (16% - 9%) 1.3

    = 18.1%

   (2) If r_M decreases to 13%:

 ri = r_RF + (r_M - r_RF)βi

    = 9% + (13% - 9%) 1.3

    = 14.2%

6-8 Solution:

Given: no. of stocks hold=20

            Portfolio beta, b_p= 1.12

            Beta of sold stock=1

            Beta of new stock=1.75

          New portfolio beta=?

We have, portfolio beta, β_p=b₁xw₁+b₂xw₂+…………..+ b₂₀xw₂₀

Old portfolio beta, β_p = $142,500/$150,000 x b + $7,500/$150,000 x 1.00

Where [weight of overall stocks, i.e. 20 stocks=20x$7500=$150,000

And weight of 19 stocks after the sale of one stock=19x $7500=$142,500

                       1.12 = 0.95xβ + 0.05

                       1.07 = 0.95b

                            β =1.13

 Now, portfolio’s new beta, β_p = 0.95(1.13) + 0.05(1.75)

                                                 = 1.161

 6-9. Solution:

Required rate of return on market r_M=14%

                                Risk free rate, r_RF=6%

         Fund’s required rate of return, r_i=?

Here,

Stocks            Investment         Beta

 A                    $400,000         1.50

B                     600,000           -0.50

C                     1,000,000        1.25

D                     2,000,000        0.75

Total                $4,000,000

 We have, Portfolio beta,

 β_p = $400,000/$4,000,000 x (1.50) + $600,000/$4,000,000 x (-0.50) + $1,000,000/$4,000,000 x (1.25) + $2,000,000/$4,000,000 x (0.75)

     = (0.1) (1.5) + (0.15) (-0.50) + (0.25) (1.25) + (0.5) (0.75)

     = 0.15 - 0.075 + 0.3125 + 0.375 = 0.7625

 Now, the required rate of return, r_p= r_RF + (r_M - r_RF)(β_p)

                                                       = 6% + (14% - 6%) (0.7625)

                                                       = 12.1%

References

C. Paramasivan, T. Subramanian. (2015). Financial Management. New Delhi: New Age International Publishers.

Eugene F. Brigham, Michael C. Ehrhardt. (2011). Financial Management: Theory and Practice . Natorp Boulevard Mason, USA: South-Western Cengage Learning.