Project

Submitted To:

Mr. Khalid Sohail

Submitted By:

Waseem Ahmed

SP14-RBA-008/ISB

Topic:

Variability of Beta Over time

Program:

RBA-2

Date:02-01-2015

Table of Contents

Page 1

Introduction………………………………………………………………………03

Beta (Systematic Risk) …………………………………………………….......04

Introduction of Stocks…………………………………………………….........05

Variability of beta over time……………………………………………...........05

Methodologies for beta calculations………………………………………….06

BOP beta estimation summary………………………………………………07

BOP monthly data regression model…………………………………………08

BOP monthly data Scatter diagram…………………………………………09

HBL beta estimation summary………………………………………………09

HBL monthly data regression model…………………………………………10

HBL monthly data Scatter diagram………………………………………….10

Analysis………………………………………………………………………….11

Conclusions ……………………………………………………………………12

References………………………………………………………………………14

Page 2

Introduction:A security’s beta may very substantially depending upon whether it is estimated

on the basis of daily, weekly, fortnightly or monthly returns of that security.

Basically the betas of securities with smaller market value than the average of the

market will decrease as the return interval is shortened, whereas the betas of securities

with a large market value relative to the market will increase. This suggest that betas

measured over return intervals of arbitrary length will tend to be less risky than truly are,

whereas securities with relatively large market values may appear to be more risky than

they truly are.

Security’s historical rate of returns can be used to estimate its systematic risk. Rate of

returns can be calculated as daily, weekly, fortnightly and monthly basis then with the

help of those returns on different time interval betas can be estimated and its variability

over time.

Page 3

Beta Definition:“A measure of the volatility, or systematic risk, of a security and a portfolio in

comparison to the market as a whole. Beta is used in the capital asset pricing model

(CAPM), a model that calculates the expected return of an asset based on its beta and

expected market returns”

Beta is calculated using regression analysis, and you can think of beta as the

tendency of a security's returns to respond to swings in the market. A beta of 1 indicates

that the security's price will move with the market. A beta of less than 1 means that the

security will be less volatile than the market. A beta of greater than 1 indicates that the

security's price will be more volatile than the market. For example, if a stock's beta is

1.2, it's theoretically 20% more volatile than the market.

Many utilities stocks have a beta of less than 1. Conversely, most high-tech, NASDAQ-

based stocks have a beta of greater than 1, offering the possibility of a higher rate of

return, but also posing more risk.

Systematic Risk

Risk is a consideration in every investment decision and, for a stock, risk is quantified

by beta. Fortunately, the widespread availability of published betas on many

financial websites makes the acquisition of this risk metric conveniently available.

On the other hand, the calculation of beta does not include consistent factors, thus

making the interpretation of published betas and the ensuing risk difficult and

incomplete.

If you want to know how much systematic risk a particular security, fund or portfolio has,

you can look at its beta, which measures how volatile that investment is compared to

the overall market. A beta of greater than 1 means the investment has more systematic

risk than the market, less than 1 means less systematic risk than the market, and equal

to one means the same systematic risk as the market.

Page 4

Whereas this type of risk affects a broad range of securities, unsystematic risk affects a

very specific group of securities or an individual security. Unsystematic risk can be

mitigated through diversification.

Introduction of stocks:Habib Bank Limited (HBL):

HBl (formerly Habib Bank Limited) now referred to as "HBL Pakistan" and

headquartered in Habib Bank Plaza,Karachi, Pakistan, is the largest bank in Pakistan.

The bank has a network of over 1500 branches and over 1000 ATM(S) in Pakistan and

55 branches worldwide. It has a domestic market share of over 40%. It continues to

dominate the commercial banking sector with a major market share in inward foreign

remittances (55%) and loans to small industries, traders and farmers.

Bank of Punjab (BOP):

The Bank of Punjab is a Pakistani bank headquartered at BOP Tower, Main

Gulberg, Lahore Pakistan. It serves Pakistan and functions as an international bank and

is one of the prominent financial institutions of the country holding AA ratings

from PACRA. The bank was established in 1989, pursuant to The Bank of Punjab Act

1989, and was given the status of a retail bank in 1994.

Variability of beta over timeA security’s beta may very substantially depending upon whether it is estimated on the

basis of daily, weekly, fortnightly or monthly returns of that security.

Betas are not constant:

A stable beta implies that the systematic risk of a firm does not change; that is to say

that the relationship between the stock and the market is continuous. A company (fund)

is like a living entity that changes through time, bringing in managers with a

higher(lower) risk appetite, developing new products, expanding into new markets,

Page 5

and being exposed to new regulations and new competition. Consequently, the firm is

not static and should not be expected to behave and perform at a constant

measureable level.

Basically the betas of securities with smaller market value than the average of the

market will decrease as the return interval is shortened, whereas the betas of securities

with a large market value relative to the market will increase. This suggest that betas

measured over return intervals of arbitrary length will tend to be less risky than truly are,

whereas securities with relatively large market values may appear to be more risky than

they truly are.

Security’s historical rate of returns can be used to estimate its systematic risk. Rate of

returns can be calculated as daily, weekly, fortnightly and monthly basis then with the

help of those returns on different time interval betas can be estimated and its variability

over time.

Here in this study we used two different stocks (HBL and BOP) and there returns in

different time intervals like daily, weekly, fortnightly and monthly. We used KSE-100

index as market proxy. We have three-year period july 2011 to june 2014, and

estimated betas of Habib Bank limited and Bank of Panjab on daily, weekly, fortnightly

and monthly returns.

Methodologies for Beta calculations:

Different methodologies and formulas used for Beta calculation. Those formulas are as

follows;

Average formula:

Beta manual calculations (By excel):

Page 6

Computational formula:

Conceptual formula:

N (sum XY)-sum XY/N (sum X^2)-(sum X)^2

Slop function

Covariance and variance function

Beta Estimation:Here, betas for the three-year period July 2011 to June 2014 have been estimated

using 36 monthly returns, 250 daily returns, 72 fortnightly and 156 weekly returns.

BOP Monthly Beta Calculations:

DateShare price Index RF Ri Rm

Mkt model beta CAPM

6/1/2010 6.65 12496.03 y x y1 x17/29/2011 6.38 12,190.37 0.011258 -0.0406 -0.02446 -0.05186 -0.035728/29/2011 5.29 10,903.88 0.011258 -0.17085 -0.10553 -0.1821 -0.116799/29/2011 6.04 11,642.46 0.011258 0.141777 0.067736 0.130519 0.056477

10/28/2011 5.77 11,561.67 0.011258 -0.0447 -0.00694 -0.05596 -0.018211/30/2011 5.55 11,532.83 0.011258 -0.03813 -0.00249 -0.04939 -0.0137512/30/2011 5.41 11,347.66 0.011258 -0.02523 -0.01606 -0.03648 -0.027311/27/2012 5.85 11,960.21 0.011258 0.081331 0.05398 0.070073 0.0427222/24/2012 8.78 12706.52 0.011258 0.500855 0.062399 0.489596 0.0511413/26/2012 10.47 13286.73 0.011258 0.192483 0.045662 0.181225 0.0344044/23/2012 9.95 14083.44 0.011258 -0.04967 0.059963 -0.06092 0.048705

Page 7

5/22/2012 8.94 14142.08 0.011258 -0.10151 0.004164 -0.11277 -0.007096/19/2012 7.96 13682.99 0.011258 -0.10962 -0.03246 2.42857 -0.12088 -0.04372 2.428577/17/2012 8.3 14445.28 0.011258 0.042714 0.055711 0.031455 0.0444528/15/2012 8.19 14970.92 0.011258 -0.01325 0.036388 -0.02451 0.025139/18/2012 8.16 15517.19 0.011258 -0.00366 0.036489 -0.01492 0.02523

10/17/2012 8.06 15654.62 0.011258 -0.01225 0.008857 -0.02351 -0.002411/19/2012 8.82 16251.38 0.011258 0.094293 0.03812 0.083034 0.02686212/17/2012 9.2 16801.02 0.011258 0.043084 0.033821 0.031826 0.0225631/16/2013 7.83 16181.47 0.011258 -0.14891 -0.03688 -0.16017 -0.048132/15/2013 8.82 17797.22 0.011258 0.126437 0.099852 0.115178 0.0885943/15/2013 8.26 17664.83 0.011258 -0.06349 -0.00744 -0.07475 -0.01874/12/2013 8.27 18714.28 0.011258 0.001211 0.059409 -0.01005 0.0481515/13/2013 9.31 20244.82 0.011258 0.125756 0.081785 0.114497 0.0705266/10/2013 14.73 22150.74 0.011258 0.58217 0.094144 3.224428 0.570911 0.082885 3.2244287/8/2013 14.62 22365.72 0.011258 -0.00747 0.009705 -0.01873 -0.001558/6/2013 12.43 22621.93 0.011258 -0.14979 0.011455 -0.16105 0.0001979/6/2013 10.77 22765.87 0.011258 -0.13355 0.006363 -0.14481 -0.0049

10/4/2013 10.88 22085.96 0.011258 0.010214 -0.02987 -0.00104 -0.0411211/7/2013 10.31 23220.21 0.011258 -0.05239 0.051356 -0.06365 0.04009812/9/2013 11.95 24998.89 0.011258 0.159069 0.076601 0.147811 0.0653421/7/2014 11.54 26259.57 0.011258 -0.03431 0.050429 -0.04557 0.0391712/6/2014 11.04 26862.51 0.011258 -0.04333 0.022961 -0.05459 0.0117023/6/2014 11.32 26842.53 0.011258 0.025362 -0.00074 0.014104 -0.0124/3/2014 10.28 28336.36 0.011258 -0.09187 0.055652 -0.10313 0.0443935/2/2014 8.88 28921.13 0.011258 -0.13619 0.020637 -0.14745 0.009378

5/30/2014 8.75 29167.54 0.011258 -0.01464 0.00852 -0.0259 -0.002746/27/2014 8.97 28987.75 0.011258 0.025143 -0.00616 0.522213 0.013885 -0.01742 0.522213

BOP Beta estimation Summary:

Market

model

Daily weekly Fortnightly Monthly

1st year 1.7221 1.7600 1.4107 2.4285

2nd year 0.4738 1.8917 1.8645 3.2244

3rd year 0.3452 0.1137 -0.2496 0.5222

Page 8

Regression model of BOP monthly returns:

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.608439R Square 0.370198Adjusted R Square 0.352204Standard Error 0.125277Observations 37

ANOVA

Df SS MS FSignifican

ce F

Regression 1 0.32288 0.3228820.5730

1 6.46E-05

Residual 35 0.5493020.01569

4Total 36 0.872181

Coefficients

Standard Error t Stat P-value

Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept -0.03544 0.023728-

1.493430.14428

6 -0.083610.01273

4-

0.083610.01273

4

X Variable 1 2.239374 0.4937174.53574

86.46E-

05 1.2370763.24167

31.23707

63.24167

3

Page 9

CAPM Daily weekly Fortnightly Monthly

1st year 1.7222 1.7593 1.4115 2.4285

2nd year 0.4686 1.8917 1.8645 3.2244

3rd year 0.3452 0.1134 -0.2496 0.5222

Scatter Diagram of BOP Monthly data:

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Series2

HBL Monthly Beta Calculations:

DateShare price Index RF Ri Rm

Mkt model beta CAPM

6/1/2010 116.04 12496.03 y x y1 x17/28/2011 119.68 12,098.05 0.011258 0.031368 -0.03185 0.02011 -0.043118/25/2011 116.56 10,901.76 0.011258 -0.02607 -0.09888 -0.03733 -0.110149/28/2011 118.32 11,625.69 0.011258 0.0151 0.066405 0.003841 0.055147

10/27/2011 115.63 11,283.49 0.011258 -0.02273 -0.02943 -0.03399 -0.0406911/29/2011 111.99 11,506.94 0.011258 -0.03148 0.019803 -0.04274 0.00854512/29/2011 108 11,435.67 0.011258 -0.03563 -0.00619 -0.04689 -0.017451/26/2012 114.78 11,883.92 0.011258 0.062778 0.039198 0.051519 0.0279392/23/2012 125.17 12515.92 0.011258 0.090521 0.053181 0.079263 0.0419233/22/2012 100.7 13273.29 0.011258 -0.19549 0.060513 -0.20675 0.0492544/20/2012 108.81 13936.48 0.011258 0.080536 0.049964 0.069278 0.0387065/21/2012 109.51 13875.74 0.011258 0.006433 -0.00436 -0.00483 -0.015626/18/2012 107.75 13754.13 0.011258 -0.01607 -0.00876 0.107273 -0.02733 -0.02002 0.1072737/16/2012 113.72 14384.58 0.011258 0.055406 0.045837 0.044148 0.0345798/13/2012 114.97 14911.97 0.011258 0.010992 0.036664 -0.00027 0.0254059/17/2012 107.09 15398.68 0.011258 -0.06854 0.032639 -0.0798 0.021381

10/16/2012 105.77 15674.3 0.011258 -0.01233 0.017899 -0.02358 0.00664111/16/2012 109.32 16197.74 0.011258 0.033563 0.033395 0.022305 0.02213612/14/2012 116.97 16845.09 0.011258 0.069978 0.039965 0.05872 0.0287071/15/2013 113.88 16107.89 0.011258 -0.02642 -0.04376 -0.03768 -0.055022/14/2013 116.93 17765.82 0.011258 0.026783 0.102927 0.015524 0.0916683/14/2013 113.25 17740.69 0.011258 -0.03147 -0.00141 -0.04273 -0.012674/11/2013 96.3 18764.55 0.011258 -0.14967 0.057713 -0.16093 0.0464545/10/2013 92.22 19916.27 0.011258 -0.04237 0.061377 -0.05363 0.050119

Page 10

6/7/2013 111.94 22358.96 0.011258 0.213836 0.122648 0.957694 0.202578 0.11139 0.9576947/5/2013 126.56 22178.34 0.011258 0.130606 -0.00808 0.119347 -0.019348/5/2013 167.41 22701.3 0.011258 0.322772 0.02358 0.311513 0.0123219/5/2013 157.98 22451.46 0.011258 -0.05633 -0.01101 -0.06759 -0.02226

10/3/2013 150.08 22152.35 0.011258 -0.05001 -0.01332 -0.06126 -0.0245811/6/2013 146.54 23165.21 0.011258 -0.02359 0.045722 -0.03485 0.03446412/6/2013 162.99 24870.55 0.011258 0.112256 0.073616 0.100998 0.0623581/6/2014 163.79 26169.83 0.011258 0.004908 0.052242 -0.00635 0.0409832/4/2014 166.76 26751.45 0.011258 0.018133 0.022225 0.006875 0.0109663/5/2014 170.88 26521.99 0.011258 0.024706 -0.00858 0.013448 -0.019844/2/2014 175.55 27932.02 0.011258 0.027329 0.053165 0.016071 0.041906

4/30/2014 184.85 28912.98 0.011258 0.052976 0.03512 0.041718 0.0238615/29/2014 184.2 29006.78 0.011258 -0.00352 0.003244 -0.01477 -0.008016/26/2014 182.69 29002.76 0.011258 -0.0082 -0.00014 1.080389 -0.01946 -0.0114 1.080389

HBL Beta estimation summary:

Market

model

Daily Weekly Fortnightly Monthly

1st year 0.7265 0.6536 0.5767 0.1072

2nd year 0.9576 0.8346 0.7742 0.2136

3rd year 1.3554 1.0803 0.9756 0.5297

CAPM Daily Weekly Fortnightly Monthly

1st year 0.7263 0.6532 0.5765 0.1072

2nd year 0.9579 0.8374 0.0.7751 0.2173

3rd year 1.3534 1.0803 0.9767 0.5138

Page 11

Regression model of HBL monthly returns:

SUMMARY OUTPUT

Regression StatisticsMultiple R 0.226015R Square 0.051083Adjusted R Square 0.023971Standard Error 0.087659Observations 37

ANOVA

Df SS MS FSignificanc

e F

Regression 1 0.0144780.01447

81.88414

3 0.178596

Residual 35 0.2689450.00768

4Total 36 0.283423

Coefficients

Standard Error t Stat P-value Lower 95%

Upper 95%

Lower 95.0%

Upper 95.0%

Intercept 0.004623 0.0166150.27824

10.78246

4 -0.029110.03835

4 -0.02911 0.038354

X Variable 1 0.475535 0.3464381.37264

10.17859

6 -0.227771.17884

2 -0.22777 1.178842

Scatter Diagram of HBL Monthly Data:

Page 12

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

Series2

Analysis:We recognize that high risk stocks should experience either very good or very

bad returns more frequently compared to low risk stocks, i.e. high risk stocks

should cluster in the tails of the cross-sectional return distribution. Building on

this intuition, we test the risk interpretation of the CAPM’s beta by examining if

high beta stocks are more likely than low beta stocks to experience either very

high or very low returns. The departure point for our tests is the intuition that

risky stocks should experience very good or very bad returns more frequently

compared to low risk stocks, i.e. risky stocks should concentrate in the tails of

the cross-sectional return distribution. Building on this insight, a test of whether

high beta stocks are more risky is equivalent to testing if high beta stocks tend to

experience very high and very low returns more often than low beta stocks.

BOP betas based on weekly, fortnightly and monthly returns are 1.7600, 1.4107 and

2.4285 whereas the daily beta is 1.722 in 1st year, as its smaller market value than the

average of all securities outstanding. In 2nd and 3rd year BOP betas on weekly,

fortnightly and monthly are more than its daily beta. And in the case of estimation of

HBL on the other hand, weekly, fortnightly and monthly betas are 0.6536, 0.5767 and

o.1071 which are less than its daily beta 0.7265 in 1st year. When the return interval is

Page 13

shortened, the following occurs, securities with a smaller market value than the average

of all securities outstanding (the market) will generally have decreased beta. Whereas

securities with a larger market value than the average of all securities outstanding will

generally have an increasing beta.

Market value of Shares outstanding:

Market value of shares outstanding is another fact about why betas shift upward or

downward over time. Market value of shares outstanding can also use as a proxy for

security’s relative market thinness to determine the direction of the shift in beta.

The price an asset would fetch in the marketplace. Market value is also

commonly used to refer to the market capitalization of a publicly-traded company,

and is obtained by multiplying the number of its outstanding shares by the current

share price. Market value is easiest to determine for exchange-traded

instruments such as stocks and futures, since their market prices are widely

disseminated and easily available, but is a little more challenging to ascertain for

over-the-counter instruments like fixed income securities. 

Number of shares outstanding:

HBL 1,466,852,508

BOP 1,555,113,165

MVSO of HBL= 8.97*1,466,852,508

= 267,979,284,686.52

MVSO of BOP= 182.34*1,555,113,165

=13949365090

Page 14

Here, HBL estimation with a large market value of shares outstanding will have a

decreasing beta when the return interval is shortened. On the other hand BOP has

small market value of shares outstanding that’s why having increasing beta when return

interval is shortened.

Conclusion:Stock's price variability is important to consider when assessing risk where risk as the

possibility of a stock losing its value as beta has appeal as a proxy for risk. Intuitively, it

makes sense that stock with a price that bounces up and down more than the market

because securities with large market value of shares outstanding will have estimated

betas that are biased upward, whereas securities with small market value of shares

outstanding will have estimated betas that are biased downward considered as market

of arbitrary length are biased so high market values will appear to be more risky as

compare to the lower market values so other way round we can say that it's hard not to

think that stock will be riskier than say a safe-haven utility industry stock with a low beta.

Finally because of temporal cross- correlation between security’s price movements,

beta estimates will generally depend upon the return intervals, implying that betas

measured over return intervals of arbitrary length are biased. In Particular securities

with large market value of shares outstanding will have estimated betas that are biased

upward, whereas securities with small market value of shares outstanding will have

estimated betas that are biased downward. Hence securities with relatively small market

values may appear to be less risky than they truly are, whereas securities with relatively

large market values may appear to be more risky than truly are.

Page 15

References http://www.brecorder.com/market-data/karachi-stocks/

www.studyfinance.com/jfsd/pdffiles/v7n2/weinraub.pdf

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Betas Are Much Too Low, ”Journal of Portfolio Management 23, no. 4 (Summer 1997):

104–111.

Meir Statman, “Betas Compared: Merrill Lynch vs. Value Line,” Journal of Portfolio

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Page 17