**Individual Stock and Market Portfolio.** At this stage, we are going to discuss the relationship between the returns on an individual stock and the market portfolio.

**Individual Stock and Market Portfolio**

**Characteristic Line**

**Characteristic Line**

**The pairs of returns can be plotted in the figure.** The line passing through the observations is the line of best fit. This line helps in describing the relationship between the return from individual stock and that of the market portfolio or market return. If we relate the return from an individual **stock** to the market return in this way, the line of best fit refers to the stock’s characteristic line. The characteristic line shows the return an **investor** expects the stock to produce, given that a particular rate of return appears for the market. Individual Stock and Market Portfolio.

**R _{i}**

**r _{f} ·**

**· · · · · · · · · · R**_{m}

**The straight line in the figure represents the line of best fit between the return on stock I and market return. ****I**n the regression analysis, the term *е _{it}* in the Equation is a random-error term which will have a mean value of zero and is assumed to be uncorrelated with the market returns, the error terms of other securities, and error terms of the same security over time. The most interesting parameters in the line are the intercept and

**beta**coefficient. Individual Stock and Market Portfolio.

*Beta factor*

*Beta factor*

**Beta** picks up the risk that cannot be diversified away. As the effective diversification eliminates almost all of an asset’s unique risk, the relative measure of a single asset’s risk is not its **standard deviation**, but it’s beta. **Beta** indicates an asset’s contribution to the total risk of a portfolio. Since the characteristic line is a straight line, it can be fully described by its slope and the point where it passes through the vertical axis. **The slope of the characteristic line is commonly known as the stock’s beta factor.** The beta factor of an individual stock is an indicator of the degree to which the stock responds to changes in the return produced by the market. Individual Stock and Market Portfolio.

That is beta measures the co-variance of return on stock I with the market divided by the variance of the market return. Since beta indicates the manner in which the returns on security change systematically with changes in the returns on the market, it is frequently referred to as the measure of a security’s systematic o market risk. If the market’s return increased by 10%, then a stock with a beta of .75 is expected to increase its return by 7.5% (.75 x 10%).

#### The formula for an asset’s beta factor and the intercept

The formula for an asset’s beta factor and the intercept are as follows –

*β*_{i} = s * _{i}*,

*/ s*

_{m}^{2 }r

_{m}*A*_{i} = `r<em><sub>i</sub></em> - <em>β</em><sub>i</sub>`

r_{m}

where,

* β*_{i} = beta factor of stock i,

s * _{i}*,

*= the covariance of return on the stock i with the return on the market,*

_{m}s^{2 }r* _{m}* = the variance of return on the market,

*A*_{i} = intercept and

`r<em><sub>i</sub></em> ,`

r* _{m}* = mean return on stock i and market respectively.

**Another method of estimating historical beta**

Another method of estimating historical beta is to use the fact that –

r* _{i,m }*= s

*/ s*

_{i,m}

_{i}*×*s

_{m}Þ s* _{i,m}* = r

*s*

_{i,m }×

_{i }*×*s

_{m}Again, *β*_{i} = r* _{i,m}× *s

_{i}*×*s

*/ s*

_{m}^{2 }

_{m}= r* _{im }*(s

*/s*

_{i}*)*

_{m}**The beta of stock** i is equal to the correlation coefficient for stock i and market portfolio, multiplied by the ratio of the standard deviation of stock i to the standard deviation of the market return. In another way, the beta of stock i is a function of the correlation of the returns on stock i with those of the market (r* _{im}*) and the variability of the returns on stock i relative to the variability of the market returns (s

*/s*

_{i}*). Individual Stock and Market Portfolio.*

_{m}### The co-variance of market return

The co-variance of market return with itself is the variance of market return –

s * _{mm}* = s

^{2}

_{m}**Thus, the beta for the market index would be –**

Β* _{m}* = s

*/ s*

_{mm}^{2}

_{m}= s^{2}* _{m }* / s

^{2}

*= 1*

_{m}We can, now, classify the **systematic or market risk** of securities by using the beta of the market index into two categories. A stock having a beta of greater than 1 has above-average market-related or **systematic risk** and a stock having a beta of less than 1 has below-average market-related or systematic risk.

Hence –

β > 1: stock holds systematic risk more than average

β < 1: stock holds systematic risk less than average

β = 1: Stock holds systematic risk equal to average

β = 0: Stock holds no systematic risk

Individual Stock and Market Portfolio

**Estimating Beta**

**Estimating Beta**

**Beta** is a mathematical value that measures the risk of one asset in terms of its effects on the risk of a group of assets called portfolio. It is concerned solely with market-related risk, as would be a concern for an investor holding stocks and **bonds**. It is derived mathematically so that a high beta indicates a high level of risk, a low beta represents a low level of risk. There are different methods of estimating beta-like historical beta using ex-post return, ex-ante beta using ex-ante return, and ex-ante beta using adjusted historical betas.

**The following Table represents historical returns on stock i and market –**

Return on | (r_{it} -`r_{i})^{2} | ( r_{M t} -`r_{M})^{2} | (r_{it} -`r_{i})( r_{M t} -`r_{M}) | |

r_{i} | r_{M} | |||

.03 .09 .12 -.04 .08 .14 | .02 -.04 .08 .03 .02 .13 | (.03-.07) =(-.04)^{2}=.0016(.09 -.07) = (.02) (.12 -.07) =(.05) (-.04-.07)=(-.11) (.08 -.07) =(.01) (.14 -.07) = (.07) | (.02-.04)^{2}=(-.02)^{2}= .0004(-.04 -.04) (.08-.04) (.03-.04) (.02-.04) (.13-.04) | (-.04)(-.02)= .0008 (.02)(00) = 0000 (.05)(.04) = .0020 (-.1)(-.01) = .0011 (.01)(-.02)=-.0002 (.07)(.09)=.0063 |

`¯r_{i} = .07 | ˉ`r_{M} = .04 | ∑(r_{it} -`r_{i}) = .0216 | ∑( r_{M t} -`r_{M})^{2} = .017 | ∑(r_{it}–r_{i})(r_{M t}-`r_{M})=.01 |

**The co-variance between return on stock i and market portfolio and variance of the return on market are, therefore –**

_{ N}

s _{i},_{m }= 1/(N-1) ∑ (*r*_{it}–`<em>r</em><sub>i</sub>)(<em>r</em><sub>mt</sub>-`

*r*_{m}) = 1/4(.01) = .0025

^{ t}^{ = 1}

_{ N}

s^{2}* _{m }*= 1/ (N-1) ∑ (

*r*

_{m t}-`

*r*

_{m})

^{2}=1/4(.017) = .00425

^{t= }^{1}

s* _{m }* = .065 = 6.5 %

^{ }_{N}

s^{2}* _{i }*= 1/ (N-1) ∑(

*r*

_{i,t}-`

*r*

_{i}) = 1/4 (.0216) = .0054

^{ t }^{= 1}

s* _{i }*= .073 = 7.3 %

**The correlation coefficient between return on stock i and market portfolio is-**

r* _{im}* = s

*/s*

_{ im }

_{i}*×*s

_{m }= .0025/(.065)×(.073) = .53

**Thus, the beta factor and intercept using ex post return can be estimated as-**

* β*_{i} = s * _{i m }* / s

^{2 }

_{m}* * = .0025 /.00425 = .59

Again,

* β*_{i} = r* _{im }*(s

*/s*

_{i}*)*

_{m}= .53 *×* (.073/.065)

= .53 *×* 1.1231 = .59

**The intercept is-**

* A*i =`<em>r</em><sub>it </sub>- <em>β</em><sub>i</sub>`

*r*_{m} = .07 -.59 (.04) =.07 -.0236 =.0464

In the above case, the beta factor for stock i is .59 which indicates that if the market return goes to be higher by 1 percent, the return for stock i tends to increase by .59 percent.

**The ex-ante or expected beta can be estimated from the probability distribution. **

**The following information is given to find beta.**

Probability | .20 | .25 | .30 | .25 |

Return on stock-i | -.18 | .16 | .12 | .40 |

Return on market portfolio | -.09 | .08 | .16 | .20 |

**From the above information, the following estimations are made –**

Prob. | Return | h r_{t}_{i t} | h_{t }r_{mt} | h_{t }[ r_{it }–E(r)]_{i}^{2} | h_{t }[ r_{mt }–E(r)]_{m}^{2} | h_{t }[ r_{it }–E(r)] [ r_{i}_{m}_{t }–E(r)]_{m} | |

r_{i} | R_{m} | ||||||

.20 .25 .30 .25 | -.18 .16 .12 .40 | -.09 .08 .16 .20 | -.036 .040 .036 .100 | -.018 .020 .048 .050 | .20(-.32)^{2 }=.0205.25(.02) .30(-.02) .25(.26) | .20(-.188)^{2}=.0071.25(-.018) .30(.062) .25(.102) | .20(-.32) (-.188)=.0120 .25(.02) (-.018)= -.00009 .30(-.02) (.062)= -.00037 .25(.26) (.102)=.0066 |

.140 | .098 | =.03762 | =.01098 | = .01814 |

**Expected return-**_{ }

_{ }

_{ N }

E(r_{i}) = ∑ h_{t }r* _{m,t}* = .140

^{ t}^{ = 1}

_{ N}

E(r* _{M}*) = ∑ h

_{t }r

*= .098*

_{m,t}^{ t}^{ = 1}

### Variance:_{ N}

_{ N}

s^{2}(r* _{i}*) = ∑ h

*[r*

_{t }

_{it }*–*

*E*(r

*)]*

_{i}^{2}= .03762

^{ t}^{= 1}

_{ N}

s^{2}(r* _{m}*) = ∑ h

*[r*

_{t }

_{mt }*–*

*E*(r

*)]*

_{m}^{2}= .01098

^{ t }^{= 1}

**Standard deviation**:_{ N}

_{ N}

s(r* _{i}*) = Ö ∑ h

*[r*

_{t }

_{it }*–*

*E*(r

*)]*

_{i}^{2}= Ö.03762= .1940

^{ t}^{= 1}

s(r* _{m}*) = Ö ∑ h

*[r*

_{t }

_{mt }*–*

*E*(r

*)]*

_{m}^{2}= Ö.01098 = .1048

^{ t }^{= 1}

### Covariance:_{ N}

s * _{i}*,

*= ∑ h*

_{m}*[r*

_{t}*–*

_{i t}*E*(r

*)][ r*

_{i}*–*

_{m t}*E*(r

*)] = .01814*

_{m}^{ t = 1}

### Correlation coefficient:

* *r* _{A,B}* = s

*,*

_{i}*/ s(r*

_{m}*)s(r*

_{i}*)*

_{m}= .01814/ [(.1940)(.1048)] = .89

### Beta coefficient:

* β*_{i} = s * _{i}*,

*/ s*

_{m}^{2 }

_{m}* * = .01814 /.01098 = 1.65

Again,

* β*_{i} = r* _{im }*(s

*/s*

_{i}*)*

_{m}= .89 × (.1940/.1048) = 1.65

**Individual Stock and Market Portfolio**

**Problem Set**

**Suppose you are given the following observation:**

Return on | Jan | Feb | Mar | Apl | May | Jun |

Stock-A | .02 | .04 | -.02 | .08 | -.04 | .04 |

Stock-B | .02 | .03 | .06 | .03 | -.04 | .08 |

**You are required to-**

- find out the sample mean return for each of the stock
- find out the variance and standard deviation for the stocks

- compute the covariance and correlation coefficient between the return on the stocks

- find out the coefficient of determination and comment on the result

**Solution:**

Month | Return on | (r_{At} -`r_{A}) | r_{At}-`r_{A})^{2} | ( r_{Bt} -`r_{B}) | ( r_{Bt} -`r_{B})^{2} | (r_{At} -`r_{A}) ×( | |

Stock-A | Stock-B | ||||||

Jan Feb Mar Apl May Jun | .02 .04 -.02 .08 -.04 .04 | .02 .03 .06 .03 -.04 .08 | 00 .02 -.04 .06 -.06 .02 | 0000 .0004 .0016 .0036 .0036 .0004 | -.01 00 .03 00 -.07 .05 | .0001 0000 .0009 0000 .0049 .0025 | 0000 0000 -.0012 0000 .0042 .0010 |

Toal | .12 | .18 | .0096 | .0084 | .0040 |

**Sample mean:**

**Stock-A**_{ N}

`r* _{A}* = 1/ N ∑ r

_{At}= 1/6 × .12 = .02

^{ t}^{ = 1}

**Stock-B ** _{ N}

`r* _{B }*= 1/ N ∑ r

*= 1/6 × .18 = .03*

_{Bt }^{ t}^{ = 1}

**Sample variance:**

**Stock-A**_{ N}

s^{2}_{(rA)} = 1/N-1∑(r* _{At}* -`r

*)*

_{A}^{2 }= 1/5 x .0096 = .00192

_{ t}_{ =1}

**Stock-B ** _{ N}

s^{2}_{(rB)} = 1/N-1∑(r* _{Bt}* -`r

*)*

_{B}^{2 }= 1/5 x .0084 = .00168

_{ t =1}

**Standard deviation:**

**Stock-A**_{ N}

s_{(rA)} = Ö1/N-1∑(r* _{At}* -`r

*)*

_{A}^{2}

_{ t}_{ =1}

= Ö1/5 × .0096 = Ö .00196 =.044 = 4.40%

**Stock-B**_{ N}

s_{(rB)} = Ö1/N-1∑(r* _{Bt}* -`r

_{B})

^{2}

_{ t =1}

= Ö1/5 × .0084 = Ö.00168 =.042 = 4.20%

**Covariance:**_{ N}

s **r*** _{A}*,

**r**

*= 1*

_{B}**/**(N-1) ∑ [(r

*–*

_{A,t}`r<em><sub>A</sub></em>)( r<em><sub>B,t</sub></em> -`

r*)]*

_{B}^{ t}^{ = 1}

= 1/5 ×.0040 = .0008

**Correlation coefficient:**

* ***r*** _{A,B}* = s (r

*, r*

_{A}*) / s(r*

_{B}*)s(r*

_{A}*)*

_{B}= .0008/ (.044)(.041) = .44

Again,

s **r*** _{A}*,

**r**

*= r*

_{B}*s(*

_{A,B}*r*)s(r

_{A}*)*

_{B}= (.44) × (.044) × (.041) = .0008

Coefficient od determination:

CD = (.44)^{2}

= .20

**Suppose following information are the infor**mation about ex ante data:

Probability | .20 | .25 | .30 | .15 | .10 |

Stock-A | .16 | .12 | .08 | .04 | .02 |

Stock-B | .02 | .07 | .10 | .13 | .21 |

**You are required to-**

- find out the expected rate of return for each of the stocks
- compute the variance and standard deviation for the stocks

- compute the covariance and correlation coefficient between the return on the stocks

**find out the coefficient of determination of the stocks and comment on the result**

**Solution –**

Prob. | Return | h_{t }r_{At} | h_{t }r_{Bt} | h_{t }[r_{At}–E(r)]_{A}^{2} | h_{t}[r_{B}_{t}–E(r)]_{B}^{2} | h_{t }[ r_{At }–E(r)] _{A}×[ r | |

r_{A} | r_{B} | ||||||

.20 .25 .30 .15 .10 | .16 .12 .08 .04 .02 | .02 .07 .10 .13 .21 | .032 .030 .024 .006 .002 | .004 .018 .030 .020 .021 | .00087 .00017 .000059 .00044 .00055 | .0011 .00013 .000015 .00021 .0014 | -.00096 -.00015 -.00003 -.0003 -.00087 |

1.00 | .094 | .093 | .0021 | .0029 | -.0023 |

**Expected rate of return-**

**Stock-A **_{ N}

* E*(r* _{A}*) = ∑ h

*r*

_{t}*= .094 = 9.4%*

_{A t }^{ t}^{ = 1}

**Stock-B**_{ N}

* E*(r* _{B}*) = ∑ h

*r*

_{t}*= .093 = 9.3%*

_{B t }^{ t}^{ = 1}

**Variance:**

**Stock-A**_{ N}

s^{2}(r* _{A}*) = ∑ h

*[r*

_{t }

_{At }*–*

*E*(r

*)]*

_{A}^{2 }= .0021

^{ t}^{= 1}

**Stock-B**

_{ N}

s^{2}(r* _{B}*) = ∑ h

*[r*

_{t }

_{Bt }*–*

*E*(r

*)]*

_{B}^{2 }= .0029

^{ t }^{= 1}

**Standard deviation:**

**Stock-A **_{ N}

s(r* _{A}*) = Ö ∑ h

*[r*

_{t }

_{At }*–*

*E*(r

*)]*

_{A}^{2 }= Ö.0021= .046 = 4.6%

^{ t}^{= 1}

**Stock-B**_{ N}

s(r* _{B}*) = Ö ∑ h

*[r*

_{t }

_{Bt }*–*

*E*(r

*)]*

_{B}^{2 }= Ö.0029 = .054 = 5.4%

^{ t }^{= 1}

**Covariance:**_{ N}

s (r* _{A}*,r

*) = ∑ h*

_{B}*[r*

_{t}*–*

_{A t}*E*(r

*)][ r*

_{A}*–*

_{B t}*E*(r

*)] = -. 0023*

_{B}^{ t = 1}

**Correlation coefficient:**

* *r* _{A,B}* = s (r

*, r*

_{A}*) / s(r*

_{B}*)s(r*

_{A}*)*

_{B}= -. 0023/ [(.046)(.054)] = -. 93

Again,

s (r* _{A}*, r

*) = r*

_{B}*s(r*

_{A,B}*)s(r*

_{A}*)*

_{B}= -. 93 [(.046)(.054)] = -. 0023

**Coefficient of determination:**

CD = (-.93)^{2} = .86

**Individual Stock and Market Portfolio**

Individual Stock and Market Portfolio

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