What is the Bangladesh standard on Audit(BSA-300) ? Describes in details

Corporate practice bd

                                       Risk and Return

Risk and return are fundamental concepts in the field of financial management. They are closely related and play a crucial role in investment decisions and portfolio management. Here's an overview of these concepts:

 Definition of Risk :

  Risk in financial management refers to the uncertainty or variability of returns associated with an investment or a portfolio of investments. It is the possibility of losing some or all of the invested capital or not achieving the expected returns.

                    Types of Risk: 

 Systematic Risk: 

This is also known as market risk or non- diversifiable risk. It is related to factors affecting the entire market, such as economic conditions, interest rates, and political events. Systematic risk cannot be eliminated through diversification.

 Unsystematic Risk: 

Also known as specific risk or diversifiable risk, this risk is unique to a particular company or industry. It can be reduced or eliminated through diversification. Examples include company-specific events like management changes or product recalls.

 Return:

 Definition: 

Return is the gain or loss generated on an investment over a certain period. It can be expressed as a percentage of the initial investment, and it represents the profit or loss an investor realizes on their investment.

 Types of Return:

 Total Return: 

This considers both the capital gain (or loss) and any income generated from the investment, such as dividends or interest.

 Capital Gain (or Loss): 

This is the change in the value of the investment from its purchase price to its current market value.

 Income Return: 

This includes any income earned from the investment, such as dividends from stocks or interest from bonds.

 Risk-Return Tradeoff:

The risk-return tradeoff is a fundamental principle in finance that suggests a positive relationship between risk and return. In general, investments with higher risk have the potential for higher returns, while investments with lower risk offer lower potential returns. This principle helps investors make decisions that align with their risk tolerance and investment goals.

 Measuring Risk and Return:

 Risk Measurement: 

Common measures of risk include standard deviation, beta, and the Sharpe ratio. Standard deviation quantifies the volatility of an investment's returns. Beta measures an investment's sensitivity to market movements. The Sharpe ratio assesses the risk-adjusted return of an investment.

 Return Measurement:

Return can be measured in various ways, such as the simple return (the difference between the final value and initial investment), annualized return (returns on an annual basis), or cumulative return (total returns over a specific period).

 Portfolio Diversification:

 Investors can reduce risk through diversification, which involves spreading investments across different asset classes or securities. Diversifying a portfolio can help mitigate unsystematic risk and reduce the overall risk without sacrificing potential returns.

In financial management, the key challenge is to find the right balance between risk and return that aligns with an investor's financial goals and risk tolerance. Different investors will have varying risk appetites and return expectations, and financial managers use these concepts to design and manage portfolios that meet their clients' needs.

Here we discuss some definition of risk & return term such as –

1.Risk: The risk of financial loss, danger, or loss of something is called risk.

 2. Uncertain: The possibility of an event not happening in the future is called uncertainty.

3.Income/ Profit/Return: The extra money that can be earned through investment is called return.

4. Portfolio: Investing in multiple profitable assets without investing in a specific asset or single company is called a portfolio or investment group or letter group

Beta: Amount of risk in the market is called single beta. Means to measure market risk by beta co.

Expected Return:

The future income that an investor expects on a particulars asset is called the expected income.

Standard deviation:

Standard deviation Simply put, standard deviation (SD) is a statistical measure that represents instability or risk in a material.

Abbreviation:

               E (R)       =    Expected return

                   SD         =    Standard deviation

                   CV        =    Coefficient of Variation

                     R         =     Return

                     P         =     Probability

                  Rf          =     Risk free Rate

                   Km       =     Market return

                   B           =     Beta

                E (R) _P    =    Portfolio expected return

                     SD _P   =    Portfolio standard deviation

                        W    =   Weight

                         r     =     correlation

Some important formula for risk & return

When given in the question Return & Probability then,

    1. E (R) = (R₁*P₁) +(R₂*P₂) + (R₃*P₃) ….

 

3. $CV=\{\left(\text{Standard Deviation / E( R}\right)\}\times 100$

4. E (R) _X = (R₁*P₁) +(R₂*P₂) + (R₃*P₃) +(R₄*P₄) [When exist double project)

5. When given in the question Return but not exist probability then,

E (R) = 

SD/(σ) =√([Σ(R –E( R)]²*P )*100

6. When given in the question Capital asset pricing Model(CAMP) or Rf, Km, B then,

 E(R ) = Rf+ b (Km-Rf)

Practical Example-01

Probability	Return
0.30	20%
0.40	5%
0.30	12%

Requirement:

1. Expected Return

2. Standard deviation

3. Co-efficient

Solution:

   1.Expected Return = (R₁*P₁) +(R₂*P₂) + (R₃*P₃)

                                      = (0.20*0.30) +(0.05*0.40) + (0.12*0.30)

                                       = 0.06+0.02+0.036

                                        = 0.116 or,11.60% (Ans)

                               Standard deviation

Given, R₁=0.20 R₂ =0.05, R₃=0.12, P₁=0.30, P₂=0.40, P₃=0.30

R	E(R)	{R-E( R )}2	P	{R-E( R )}2*P
0.20	0.116	0.007056	0.30	0.0021168
0.05	0.116	0.004356	0.40	0.0017424
0.12	0.116	0.000016	0.30	0.0000048
			E{R-E(R)}2*p=	0.003864

  2.SD/σ = √[Σ(xi - x̄)² / (n - 1)]

              = √0.003864

              = 0.0621 

               or,6.21(Ans)

Coefficient of Variation (CV) = (Standard Deviation / Mean) * 100  = *100

                                               = 0.0621/0.116  *100

                                                = 53.58%

Practical Example 2

Security X and Y have the following distribution

Probability	10%	30%	40%	20%
X	15%	20%	10%	5%
Y	25%	30%	35%	40%

An investor seeks your opinion as to which security he should invest? Give your opinion on the basis of co- efficient of variation.

Solution:

Expected return for security X:

Given, R₁₌0.15, R₂₌0.20, R₃₌0.10, R₄₌0.05, (for X)

                 P₁₌0.10, P₂₌0.30, P₃₌0.40, P₄₌0.20

E (R) _X = (R₁*P₁) +(R₂*P₂) + (R₃*P₃) +(R₄*P₄)

            = (0.15*0.10) +(0.20*0.30) + (0.10*0.40) +(0.05*0.20)

             = 0.015+0.06+0.04+0.01

             = 0.125 or,12.50%(Ans)

Expected return for security Y:

Given, For X: R_{1 =} 0.15, R₂₌0.20, R₃₌0.10, R₄₌0.05,

                   For Y:  R_{1 =} 0.25, R₂₌0.30, R₃₌0.35, R₄₌0.40,

                          P₁₌ 0.10, P₂₌0.30, P₃₌0.40, P₄₌0.20

E (R) _y = (R₁*P₁) +(R₂*P₂) + (R₃*P₃) +(R₄*P₄)

            = (0.25*0.10) +(0.30*0.30) + (0.35*0.40) +(0.40*0.20)

            = 0.025+0.09+0.14+0.08

             =0.3350 or, 33.50(Ans)

 Standard deviation:

Given, For X: R_{1 =} 0.15, R₂₌0.20, R₃₌0.10, R₄₌0.05,

                   For Y:  R_{1 =} 0.25, R₂₌0.30, R₃₌0.35, R₄₌0.40,

                          P₁₌ 0.10, P₂₌0.30, P₃₌0.40, P₄₌0.20

Table for calculation of SD (for X)

R	E(R )	{R-E9R )}2	P	{R-E9R )}2*P
0.15	0.125	0.000625	0.10	0.0000625
0.20	0.125	0.000625	0.30	0.0016875
0.10	0.125	0.000625	0.40	0.00025
0.05	0.125	0.000625	0.30	0.001125
			Σ{R-E(R)}2*p  =	0.003125

2.SD/σ =  2.SD/σ = √[Σ(xi - x̄)² / (n - 1)]

              = √0.003125

              = 0.056

Practical Example 3

When given in the question Return but not exist probability then,

E (R) =  R1+R2+r3 /N

SD/(σ) =√[Σ(xi - x̄)² / (n - 1)]

     CV = SD((σ)/E(R )*100

Security “A “and” B” have the following distribution

Year	Return of security -A	Return of security -B
2014	25%	30%
2015	22%	25%
2016	19%	20%

Requirement:

(a). Calculate average return, Standard deviation, and Co efficient of variation of two securities.

(b). Which security is more  risky?

Solution:

      For the Security-A = R_{1 =} 0.25, R₂₌0.22, R₃₌0.19, N₌3, E (R)=?

      For the Security- B = R_{1 =} 0.30, R₂₌0.25, R₃₌0.20, N₌3, E (R)=?

   _{We know that,}

          E (R)_A = R1+R2+r3 /N

     = 0.25+0.22+0.19/3

      =0.22

        or,22%

Given,

      For the Security-A = R_{1 =} 0.25, R₂₌0.22, R₃₌0.19, N₌3, E (R)=0.22

      For the Security- B = R_{1 =} 0.30, R₂₌0.25, R₃₌0.20, N₌3, E (R)=?

   _{We know that,}

          E (R)    = R1+R2+r3 /N

                       = 0.25+0.22+0.19/3

                       = 0.22

                        = 0.25 or,25%

SD/(σ)_A   = √[Σ(xi - x̄)² / (n - 1)]

                   =  √(0.25 - 0.22)²+(0.22-0.22)2+(0.19-0.22)2 / (3 - 1)]

                = √{(0.0009+0+0.0009) }/2

                = √(0.0018) /2}

                =  √0.0009

                = 0.03 or,3%(Ans)

Given,

      For the Security-A = R_{1 =} 0.25, R₂₌0.22, R₃₌0.19, N₌3, E (R)=0.22

      For the Security- B = R_{1 =} 0.30, R₂₌0.25, R₃₌0.20, N₌3, E (R)=?

 

SD/(σ)_B   = √[Σ(xi - x̄)² / (n - 1)]

                   =  √(0.30 - 0.25)²+(0.25-0.25)2+(0.20-0.25)2 / (3 - 1)]

                = √0.0025+0+0.0025 /2

                = √0.0050 /2

                =  √0.0025

                = 0.05 or, 5%(Ans)

Given,

 For the Security-A = R_{1 =} 0.25, R₂₌0.22, R₃₌0.19, N₌3, E (R)=0.22,SD=0.03

     For the Security- B = R_{1  =}  0.30, R₂₌0.25, R₃₌0.20, N₌3, E (R)=0.25 ,SD=0.05

 

 CV_A =  SD/ER*100

         =  0.03/0.22*100

         = 13.64%

CV_B = SD/ER*100 *

         = 0.05/0.25SD *100

         = 20%

Comment:

Security –B is more, risky because it has higher CV

Practical Example 4

From the return of stock –A and B over the past five years, determination of co-efficient of variance. Which stock would you like to invest considering co-efficient of variance.

Year	Return-A	Return-B
2011	7%	12%
2012	9%	15%
2013	10%	-16%
2014	-12%	20%
2015	16%	18%

Requrement :

Try yourself

Ans

           E (R)_A = 6%, E (R)_B = 9.8%,

           SD_{A = 10.61%,} SD_{B =14.74%}

            _{CVA = 176.83%, CVB = 150.41%,}

Decision:

I would like to invest in stock B because co-efficiencies of variation to Stock-B is lower than other stock

When given in the question Capital asset pricing Model(CAMP) or Rf, Km, B then,

E(R )= Rf+ b (Km-Rf)

Practical Example 5

The wimax company Ltd. Wishes to calculate its cost of equity capital using the CAMP Approach. The following information are available.

Rf =10% b =1.5 Km=12.50%

Solution:

E(R )= Rf+ b (Km-Rf)

=10%+1.50(12.50%-10%)

=10%+1.50*2.50%

=10%+3.75%

=13.75%(Ans)

Practical Example 6

Assume that risk free rate is 5% and market risk premium is 6%. What is the expected return for the overall stock market? What is the required rate of return on a stock that has beta of 1.2?

Solution

 Here,

[ risk free rate(Rf)=5%, Market risk premium(Km-Rf)=6%,Beta=1,E(R )?    We know that,                                        

E(R )= Rf+ b (Km-Rf)

= 5%+ 1 *6%

=11% (Ans)

Again,

Here,

risk free rate(Rf)=5%, Market risk premium(Km-Rf)=6%,Beta=1.2,E(R )?    We know that,                                        

E(R )= Rf+ b (Km-Rf)

= 5%+ 1 .2*6%

=12.20% (Ans)

Practical Example 7

At present, suppose the risk free rate is 12% and the expected return on the market portfolio is 16%. The expected return for the four stocks enlisted together with their expected beta.

Portfolio	Average return	Beta
A	18%	1.35
B	15%	0.85
C	16%	1.20
D	20%	1.75

On the basis of these expectations which stocks are overvalued and undervalued?

Here,

 Risk free rate(Rf)=12%, Km=16%, B=1.35,0.85,1.20,1.75

Average return=18%, 5%, 16%, 20%

Portfolio	E(R )=Rf+b(Km-Rf)	Av.Return	Ac.Return	Difference
A	E(R )= Rf+ b (Km-Rf) =12%+1.35*4% =17.40%

Portfolio	E(R )=Rf+b(Km-Rf)	Av.Return	Ac.Return	Difference
B	E(R )= Rf+ b (Km-Rf) =12%+0.85(16%-12%) =12%+0.85*4% =15.40%
C	E(R )= Rf+ b (Km-Rf) =12%+1.20(16%-12%) =12%+1.20*4% =16.80%
D	=12%+1.35(16%-12%) =12%+1.75*4% =19%

 

Comments:

Security A & B Provide more than the expected return and hence may be assumed to be undervalued. Security B and C may be assumed overvalued as each of them provides lower return compared to the expected return.

 

Practical Example 8

ABC Company LTD has a beta of 1.60. The risk free rate of return is 8% and risk premium is 6%. Determine the cost of common stock of ABC Company LTD. Using CAMP.

Solution

Here given,

Rf=8%,b=1.6,(Km-Rf)=6%, E(R )=?

We know that

E(R ) = Rf+ b (Km-Rf)

          = 8%+1.60*6%

          = 17.60%

Practical Example 9

A Kazi Firm risk free rate is 7%. The rate of return on the market is 14% and beta is 0.65. What is expected return based on the CAPM. If another stock has an expected return on 24%, what will be its Beta?

Solution

Here given,

Rf=7%,b=0.65, Km=14%, E(R )=?

We know that

E(R ) = Rf+ b (Km-Rf)

          = 7%+0.65*(14%-7%)

          = 7%+0.65*7%

          = 11.55 %

Again

Here given,

Rf=7%, Km=14%, E(R )=24% b=?

We know that

         E (R) = Rf+ b (Km-Rf)

   Or, 24% = 7%+b (14%-7%)

    Or, 24% = 7%+b *7%

    Or, 24% - 7%=b*7%

    Or, b=17%/7%

              =2.429(Ans)

Practical Example 10

You have the following two investment opportunities of X and Y: -

Economic condition	probability	Return-X	Return-X
Good	0.50	50%	0%
Bad	0.50	0%	50%

Solution

Here,

[ P₁₌0.50, P₂₌0.50, R₁₌0.50, R₂₌0, E (R) =?

We know that,

E (R) _{X =} (R₁*P₁) +(R₂*P₂)                   

            = (0.50*0.50) +(0*0.50)

             =25% (Ans)

Here,

[ P₁₌0.50, P₂₌0.50, R₁₌0, R₂₌0.50, E (R) =?

We know that,

E (R) _{y =} (R₁*P₁) +(R₂*P₂)                  

            = (0*0.50) +(0.50*0.50)

             =25% (Ans)

Practical Example 11

Standard deviation and expected rate of return of stock –X is 0.05 and 0.15 respectively, whereas standard deviation and expected rate of return Stock-Y is 0.08 and 0.18 respectively. Compute the co- efficient of variation and which stock do you recommended to buy?

Solution

Here,

SD =0.05, E (R)=0.15 CV=?

We know that,

 Co –efficient of variation for X stock:

   CV  =SD/ER *100

             = 0.05/0.15 *100

             = 33.33 % (Ans)

We know that,

 Co –efficient of variation for Y stock:

Here,

SD =0.05, E (R)=0.15 CV=?

  CV  =SD/E(R )*100

             =  0.08/0.18*100

             = 44.44 % (Ans)

Comments: I recommend to buy Stock-X for its Lower CV.

Practical Example 12

 When ask to the question to find out Portfolio Expected Return then,

E(R )_p={ E(R )_A*W_A) += E(R )_B*W_B}….

When ask to the question to find out Portfolio standard deviation then,

SD_P=√  σ² _{A *W}([Σ(R –E( R)]²*P )

When ask to the question to find out co variation where exist, correlation then,

Co variation(COV) =r* σ_A* σ_B

When ask to the question to find out co variation where no exist, correlation then,

Co variation(COV) =

Correlation(r )=

The following information of AR limited are presented to you to calculate –

(i)                Return on portfolio

(ii)              Portfolio standard deviation

Details	Stock-X	Stock-Y
E (r )	0.12	0.18
Weight(W)	0.40	0.60
SD	0.06	0.12

Solution

Here given,

E (R )_X,=0.12, E( R )_X,=0.12, E( R )_X,=0.18, SD_X,=0.06, SD_y,=0.12, W_X,=0.40, , W_y=0.60,

Req-(i)

E (R )_P  = E (R )_A*W_A+ E (R )_B *W_B

              = (0.12*0.40) +(0.18*0.60)

                =0.048+0.108

                = 0.056 or,15.60%

Co variation(COV)  = r* σ_A* σ_B

                                   = 1*0.06*0.12

                                   = 0.0072

Here given,

E (R )_X,=0.12, E( R )_X,=0.12, E( R )_X,=0.18, SD_X,=0.06, SD_y,=0.12, W_X,=0.40, , W_y=0.60,

SD_P       =  √{(σx2*Wx2) +( σy2*Wy2) +2 * WA* WB*)Covariance}

              =  √{(0.062*0.0402) +( 0.122*0.602) +2 * 0.40* 0.60*)(0.0072)}

              =  √{(0.000576+0.0144+0.003456)}

             =  √{(0.009216)}

             =  0.096

             or .9.6% (Ans)

Practical Example 13

The standard deviation of A and B companies are 7 and 6 respectively. Correlation between returns of two companies are -080.

Requirement:

Find covariance between stocks of two companies.

Solution

 Here,

 SD =SD_A=7, SD_B=6, Correlation=-0.80

We know that,

Co variation (COV) = r* σ_A* σ_B  

                             = -0.80*7*6

                             = -0.00504 (Ans)

Practical Example 14

The covariance of returns between of stock X and Y is 0.10. Stock X, s variance of return is 12 and Y, s Variance of return is 8.

Requirement:

(i)Find out the correlation between stock X and Y.

Solution

 Here,

 Covariance=0.10

Variance of X (SD²) = 12

Standard deviation  (SD_X) =   = 0.3464

Variance of Y (SD²) = 8

Standard deviation  (SD_X)   = 0.2828

We know that,

Correlation (R)= (co -variance)/( σ2x*σ2y)

                          = (0.10) / ( 0.3464*0.2828)

                          = 1.02 (Ans)

Practical Example 15

Your considering two securities with the relevant.

Securities	expected return	Standard deviation
Padma	15%	12%
Meghna	20%	20%

Requirement: Yourself

Calculate the portfolio expected return and portfolio standard deviation of the securities if r=0.60

Ans

E(R )_P =17.50, SD_P =8$

Practical Example 16

You have the following information on two Stock: -

State of economics	Probability	Return-X	Return-Y
Boom	0.35	0.08	0.05
Good	0.30	0.10	0.10
Bad	0.35	0.12	0.30

Your required to calculate

(i)                Expected return of t6he above two stocks

(ii)              Standard deviation of two stock

(iii)            Portfolio standard deviation of X and Y assuming those two are equally weighted.

Solution

We know that,

E (R) _X = (R₁*P₁) +(R₂*P₂) + (R₃*P₃) [When exist double project)

           = (0.08*0.35) +(0.10*0.30) +(0.12*0.35)

           = 0.028+0.03+0.042

           = 0.10 or,10%

E (R) _Y = (R₁*P₁) +(R₂*P₂) + (R₃*P₃) [When exist double project]

           = (0.05*0.35) +(0.10*0.30) +(0.30*0.35)

           = 0.0175+0.03+0.105

           = 0.1525 or,15.25%

 Standard deviation

Here, P₁₌0.35, P₂₌0.30, P₃₌0.35, E (R) _X =0.10, E (R) _Y=0.1525

Return for X = R₁₌0.08, R₂₌0.10, R₃₌0.12,

Return for Y= R₁₌0.05, R₂₌0.10, R₃₌0.30,

Table for calculation of standard deviation(SD) for -X

R	E (R )	{R-E(R )}2	P	{R-E(R )}2*P
0.08	0.10	0.0004	0.5	0.00014
0.10	10.10	0.0	0.30	0
0.12	0.10	0.0004	0.35	0.00014
			=Σ{R-E(R)}2*p =	0.00028

      SDA/(σ)  = √ {([Σ (R –E(R)] ²*P)}

                     = √ {(0.00028)}

                     =  0.016733 

                     or 1.673% (Ans)

  Variance ==Σ{R-E(R)}²*p  = 0.00028

Standard deviation (For Y)

Here, P₁₌0.35, P₂₌0.30, P₃₌0.35, E (R) _X =0.10, E (R) _Y=0.1525

Return for X = R₁₌0.08, R₂₌0.10, R₃₌0.12,

Return for Y= R₁₌0.05, R₂₌0.10, R₃₌0.30,

Table for calculation of standard deviation(SD) for -Y

R	E (R )	{R-E(R )}2	P	{R-E(R )}2*P
0.05	0.1525	0.0105063	0.35	0.00.6771875
0.10	0.1525	0.0027563	0.30	0.000826875
0.30	0.1525	0.0217563	0.35	0.0076146875
			==Σ{R-E(R)}2*p =	0.01211875

      SDy/(σ)  = √ {([Σ (R –E(R)] ²*P)}

                     = √ {(0.01211875)}

                     =  0.110085

                      or 11%% (Ans)

  Variance ==Σ{R-E(R)}²*p  =0.01211875

Here,

E (R )_X = 0.10, E(R )_Y = 0.1525,  SD_X = 0.01673, SD_Y = 0.11

W_X=.50, W_Y=.50, Cov =0

E (R) p= {E (R )_A*W_A) += E(R )_B*W_B}….

= (01.10*0.50) +(0.1525*0.50)

= 0.05+0.07625

 = 0.1263, or ,12.63%

SD_P = √{(σx2*Wx2) +( σy2*Wy2) +2 * WA* WB*)Covariance}

          =√{(0.01672*0.502) +( 0.11*0.502) +2 * 0.50* 0.50*)(0)}

           =√{(0.00028+0.25+0.01211875*0.25+0)}

              = =√{(0.0030996875)}

               = 0.05567

                or 5.57 % (Ans)

   Practical Example 17

Four securities have the following expected return:

A=12%, B=20%, C=-15%, D=10%

Calculate the expected return for a portfolio consisting of all four securities under the following condition-

 (i)  The fort folio weights are equal

(ii)  The fort folio weights are 10% in A, with the reminder equally divided among others three stocks

(iii)  The fort folio weights are equal 10% in each A and b with the reminder equally divided among C & D

Solution

Here,

E (R)_A = 0.12, E (R)_B = 0.20, E (R)_C = 0.15, E (R)_D = 0.10, W_ABC =1/4 =0.25

(i)The fort folio weights are equal

E (R) p= {E (R )_A*W_A) += E(R )_B*W_B}+ E (R )_C*W_C) + E(R )_D*W_D

= (0.12*0.25) +(0.20*0.25) + (0.15*0.25) +(0.10*0.25)

= (0.03+0.05+0.375+0.025)

 = 0.1425, or ,14.25%(Ans)

Req-(ii)

Here,

E (R)_A = 0.12, E (R)_B = 0.20, E (R)_C = 0.15, E (R)_D = 0.10

W_A=0.10, W_BCD = (1-0.10) = (0.90/3) =0.30

Solution:

E (R) p= {E (R )_A*W_A) += E(R )_B*W_B}+ E (R )_C*W_C) + E(R )_D*W_D

= (0.12*0.10) +(0.20*0.30) + (0.15*0.30) +(0.10*0.30)

= (0.012+0.06+0.045+0.03)

 = 0.1470, or ,14.70% (Ans)

Req-(iii)

Here,

E (R)_A = 0.12, E (R)_B = 0.20, E (R)_C = 0.15, E (R)_D = 0.10

W_A=0.10, W_B = 0.10, W_CD = (0.80/2) =0.40

Solution:

E (R) p= {E (R )_A*W_A) += E(R )_B*W_B}+ E (R )_C*W_C) + E(R )_D*W_D

= (0.12*0.10) +(0.20*0.10) + (0.15*0.40) +(0.10*0.40)

= (0.03+0.05+0.375+0.25)

 = 0.132, or ,13.20% (Ans)

Practical Example 18

 The expected rate of return on market fort folio is 15%, The standard deviation for the market portfolio is 20% and risk free rate is 95. Determined the slope of CML.

Solution:

Here,

Expected rate of return of market(Km)  15%, or, 0.15

standard deviation for the market (Rf)= 20%, or 0.20

Risk free rate =9% or 0.09

We know that,

Slope of CML = KM-Rf /σ_m

                           =0.15-0.09/0.20

                       = 0.06/0.20

                       = 0.30 (Ans)

Practical Example 19

You have been asked for your advice in selecting a portfolio of an assets and have been given the following data.

Year	E(R ) Asset -A	E(R ) Asset -B	E(R ) Asset -C
2010	0.12	0.16	0.12
2011	0.14	0.14	0.14
2012	0.16	0.12	0.16

No probabilities have been supplied. You can create two portfolios: one consisting of asset –A and B and the others consisting of A and c by investing equal portion in each of two components assets.

Requirement:

     (a) What is the expected return for each asset over 3 periods?

     (b) What is the standard deviation for each asset return?

(C) What is the expected return for each of the two portfolio?

Solution:

Here,

For asset –A: R₁=0.12, R₂=0.14, R₃=0.16,

For asset –B: R₁=0.16, R₂=0.14, R₃=0.12,

For asset –C: R₁=0.12, R₂=0.14, R₃=0.16,

Solution:

(a)  Calculation of expected return

                 E(R )_A   =  (R₁ + R₂₊ R₃₎ / N = 0.14 or ,14% ( Ans)

E(R )_B    =  (R₁ + R₂₊ R₃₎ / N 0.14 or ,14% ( Ans)

E(R )_C   =  (R₁ + R₂₊ R₃₎ / N 0.14 or ,14% ( Ans)

 (b) Calculate the standard deviation:

SDA   = √ ([Σ (R –E(R)] ²/N-1

          = √ {(0.12-0.14)2 + (0.14-0.14)2 +(0.16-0.14)2 /3-1

          = √ {0.0004+0+0.0004) /2)}

          = √ {0.0008)/2}

          = √ {0.0004)}

          = 0.02

          or,2% 

Here,

  E(R )_A =(R₁ + R₂₊ R₃₎ / N = 0.14 or ,14% ( Ans)

E(R )_B = =(R₁ + R₂₊ R₃₎ / N 0.14 or ,14% ( Ans)

E(R )_C =  =(R₁ + R₂₊ R₃₎ / N 0.14 or ,14% ( Ans)

 (b) Calculate the standard deviation:

SDB   = √ ([Σ (R –E(R)] ²/N-1

          = √ {(0.16-0.14)2 + (0.14-0.14)2 +(0.12-0.14)2 /3-1

          = √ {0.0004+0+0.0004) /2)}

          = √ {0.0008)/2}

          = √ {0.0004)}

          = 0.02

          or,2% 

 Here,

  E(R )_A = (R₁ + R₂₊ R₃₎ / N = 0.14 or ,14% ( Ans)

E(R )_B  =  (R₁ + R₂₊ R₃₎ / N 0.14 or ,14% ( Ans)

E(R )_C   = (R₁ + R₂₊ R₃₎ / N 0.14 or ,14% ( Ans)

 (b) Calculate the standard deviation:

SDC   = √ ([Σ (R –E(R)] ²/N-1

          = √ {(0.12-0.14)2 + (0.14-0.14)2 +(0.16-0.14)2 /3-1

          = √ {0.0004+0+0.0004) /2)}

          = √ {0.0008)/2}

          = √ {0.0004)}

          = 0.02

          or,2%

     
"The End"