Individual Stock and Market Portfolio
Individual Stock and Market Portfolio. At this stage we are going to discuss about the relationship between the returns on a individual stock and the market portfolio.
Characteristic Line
The pairs of returns can be plotted in 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 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 as 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. In the regression analysis, the term е_{it} in the Equation is a randomerror 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 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 its 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 covariance of return on stock i with market divided by the variance of market return. Since beta indicates the manner in which the returns on security change systematically with changes in the returns on 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 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 are as follows –
β_{i} = s _{i},_{m} / s^{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},_{m} = the covariance of return on the stock i with the return on the market,
s^{2 }r_{m} = the variance of return on 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 is to use the fact that –
r_{i,m }= s_{i,m} / s_{i}× s_{m}
Þ s_{i,m} = r_{i,m }×s_{i }×s_{m}
Again, β_{i} = r_{i,m}× s_{i}× s_{m} / s^{2 }_{m}
= r_{im }(s_{i}/s _{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_{i}/s _{m}). Individual Stock and Market Portfolio.
The covariance 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 _{mm} / s^{2}_{m}
= s^{2}_{m } / s^{2}_{m} = 1
We can, now, classify the systematic or market risk of securities by using the beta of market index into two categories. A stock having a beta of greater than 1 has aboveaverage marketrelated or systematic risk and a stock having beta of less than 1 has belowaverage marketrelated 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
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 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 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)^{2}=.0004 (.12 .07) =(.05)^{2}=.0025 (.04.07)=(.11)^{2}=.0121 (.08 .07) =(.01)^{2}=.0001 (.14 .07) = (.07)^{2}=.0049 
(.02.04)^{2}=(.02)^{2}= .0004
(.04 .04)^{2} =(.08)^{2}=. 0064 (.08.04)^{2} =(.04)^{2}= .0016 (.03.04)^{2} =(.01)^{2}=.0001 (.02.04)^{2}=(.02)^{2}=.0004 (.13.04)^{2} =(.09)^{2}=.0081 
(.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 covariance between return on stock i and market portfolio and variance of the return on market are, therefore –
_{ N}
s _{i},_{m }= 1/(N1) ∑ (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/ (N1) ∑ ( r_{m t} `r_{m})^{2} =1/4(.017) = .00425
^{t= }^{1}
s_{m } = .065 = 6.5 %
^{ }_{N}
s^{2}_{i }= 1/ (N1) ∑(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_{ im }/s_{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_{i}/s _{m})
= .53 × (.073/.065)
= .53 × 1.1231 = .59
The intercept is
Ai =<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 per cent, the return for stock i tends to increase by .59 per cent.
The ex ante or expected beta can be estimated from probability distribution.
The following information are given to find beta.
Probability  .20  .25  .30  .25 
Return on stocki  .18  .16  .12  .40 
Return on market portfolio  .09  .08  .16  .20 
From the above information the following estimations are made –
Prob.  Return  h_{t} r_{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_{i})] [ r_{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)^{2}=.0001 .30(.02)^{2}=.00012 .25(.26)^{2}=.0169 
.20(.188)^{2}=.0071
.25(.018)^{2=.}.00008 .30(.062)^{2=.}0012 .25(.102)^{2=.}0026 
.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_{m,t} = .098
^{ t}^{ = 1}
Variance:_{ N}
s^{2}(r_{i}) = ∑ h_{t }[r_{it }– E(r_{i})]^{2} = .03762
^{ t}^{= 1}
_{ N}
s^{2}(r_{m}) = ∑ h_{t }[r_{mt }– E(r_{m})]^{2} = .01098
^{ t }^{= 1}
Standard deviation:_{ N}
s(r_{i}) = Ö ∑ h_{t }[r_{it }– E(r_{i})]^{2} = Ö.03762= .1940
^{ t}^{= 1}
s(r_{m}) = Ö ∑ h_{t }[r_{mt }– E(r_{m})]^{2} = Ö.01098 = .1048
^{ t }^{= 1}
Covariance:_{ N}
s _{i},_{m} = ∑ h_{t} [r_{i t} – E (r_{i})][ r_{m t} – E (r_{m})] = .01814
^{ t = 1}
Correlation coefficient:
r_{A,B} = s _{i},_{m} / s(r_{i})s(r_{m})
= .01814/ [(.1940)(.1048)] = .89
Beta coefficient:
β_{i} = s _{i},_{m} / s^{2 }_{m}
= .01814 /.01098 = 1.65
Again,
β_{i} = r_{im }(s_{i}/s _{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 
StockA  .02  .04  .02  .08  .04  .04 
StockB  .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}) ×
( r_{Bt} `r_{B}) 

StockA  StockB  
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:
StockA_{ N}
`r_{A} = 1/ N ∑ r_{At} = 1/6 × .12 = .02
^{ t}^{ = 1}
StockB _{ N}
`r_{B }= 1/ N ∑ r_{Bt }= 1/6 × .18 = .03
^{ t}^{ = 1}
Sample variance:
StockA_{ N}
s^{2}_{(rA)} = 1/N1∑(r_{At} `r_{A})^{2 }= 1/5 x .0096 = .00192
_{ t}_{ =1}
StockB _{ N}
s^{2}_{(rB)} = 1/N1∑(r_{Bt} `r_{B})^{2 }= 1/5 x .0084 = .00168
_{ t =1}
Standard deviation:
StockA_{ N}
s_{(rA)} = Ö1/N1∑(r_{At} `r_{A})^{2}
_{ t}_{ =1}
= Ö1/5 × .0096 = Ö .00196 =.044 = 4.40%
StockB_{ N}
s_{(rB)} = Ö1/N1∑(r_{Bt} `r_{B})^{2}
_{ t =1}
= Ö1/5 × .0084 = Ö.00168 =.042 = 4.20%
Covariance:_{ N}
s r_{A}, r_{B} = 1/ (N1) ∑ [(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_{A}, r_{B}) / s(r_{A})s(r_{B})
= .0008/ (.044)(.041) = .44
Again,
s r_{A}, r_{B} = r_{A,B} s(r_{A})s(r_{B})
= (.44) × (.044) × (.041) = .0008
Coefficient od determination:
CD = (.44)^{2}
= .20

Suppose following information are the information about ex ante data:
Probability  .20  .25  .30  .15  .10 
StockA  .16  .12  .08  .04  .02 
StockB  .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_{B}_{t}–E(r_{B})]  
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
StockA _{ N}
E(r_{A}) = ∑ h_{t} r_{A t }= .094 = 9.4%
^{ t}^{ = 1}
StockB_{ N}
E(r_{B}) = ∑ h_{t} r_{B t }= .093 = 9.3%
^{ t}^{ = 1}
Variance:
StockA_{ N}
s^{2}(r_{A}) = ∑ h_{t }[r_{At }– E(r_{A})]^{2 }= .0021
^{ t}^{= 1}
StockB
_{ N}
s^{2}(r_{B}) = ∑ h_{t }[r_{Bt }– E(r_{B})]^{2 }= .0029
^{ t }^{= 1}
Standard deviation:
StockA _{ N}
s(r_{A}) = Ö ∑ h_{t }[r_{At }– E(r_{A})]^{2 }= Ö.0021= .046 = 4.6%
^{ t}^{= 1}
StockB_{ N}
s(r_{B}) = Ö ∑ h_{t }[r_{Bt }– E(r_{B})]^{2 }= Ö.0029 = .054 = 5.4%
^{ t }^{= 1}
Covariance:_{ N}
s (r_{A},r_{B}) = ∑ h_{t} [r_{A t} – E (r_{A})][ r_{B t} – E (r_{B})] = . 0023
^{ t = 1}
Correlation coefficient:
r_{A,B} = s (r_{A}, r_{B}) / s(r_{A})s(r_{B})
= . 0023/ [(.046)(.054)] = . 93
Again,
s (r_{A}, r_{B}) = r_{A,B} s(r_{A})s(r_{B})
= . 93 [(.046)(.054)] = . 0023
Coefficient of determination:
CD = (.93)^{2} = .86
Individual Stock and Market Portfolio
Individual Stock and Market Portfolio