anthony ford i1676485 business analysis - quants assignment dec 2011

8/2/2019 Anthony Ford i1676485 Business Analysis - Quants Assignment Dec 2011

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A Critical Evaluation of the use of CAPM and Beta

If there is one lesson that the investment market has learned over the last 4 years, probably

more than any other time in their recent history is that financial models used to reduce risk in

forecasting share performances is risky business.

When Markowitz (Markowitz, 1952) first suggested through his portfolio theory that investors

could select a portfolio of companies that would return a high likelihood of expected return for

a known amount of risk, it spawned an enormous interest from academia and the investor

market as a whole.

Capital Asset Pricing Model (CAPM) emerged from the work of William Sharpe (Sharpe,

1964) who finally published his paper introducing the risk free rate as an additional element to

portfolio theory in 1964 from a paper written in 1962. When Lintner (Lintner , 1965) and

Mossin (Mossin ,1966) independently arrived at similar conclusions to Sharpe it helped

establish CAPMs acceptance in economic theory circles as a major breakthrough to managing

market risk through a diversified portfolio, incredibly earning Markowitz and Sharpe a Nobel

Memorial Prize in Economic Sciences in 1990.

CAPm is a model to calculate the expected return on a capital asset in relation to a related

market, a risk free rate and the Beta () (Beta is an individual relationship of the security

market line (SML) risk and expected return of the asset).

To demonstrate the use of this model and to comment on the statistical information extracted

from the raw data, I have taken five years of monthly share values of HSBC PLC and the FTSE

All share index. To ensure that I follow the CAPm requirement for a geometric mean I have


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Converted the arithmetical share values to geometric by the use of natural logs (figure 1).

Figure 1.

Table of Share prices and Calculated Natural Logs


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Descriptives

Descriptive Statistics

N Range Minimum Maximum Sum

Statistic Statistic Statistic Statistic Statistic

RF 59 .3936374000 -.1976959400 .1959414600 -.4950981500

RM 59 .2350537950 -.1441181700 .0909356250 -.1274840621


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N Range Minimum Maximum Sum

Statistic Statistic Statistic Statistic Statistic

RF 59 .3936374000 -.1976959400 .1959414600 -.4950981500

RM 59 .2350537950 -.1441181700 .0909356250 -.1274840621

Valid N (listwise) 59


Mean Std. Deviation Variance Skewness Kurtosis

Statistic Std. Error Statistic Statistic Statistic Std. Error Statistic Std. Error

-.008391494068 .0100262307260 .0770129395083 .006 .224 .311 .680 .613

-.002160746815 .0065947846935 .0506555024063 .003 -.506 .311 .232 .613

Correlations

Correlations

RF RM

RF Pearson Correlation 1 .633**

Sig. (2-tailed) .000

Sum of Squares and Cross-products .344 .143

Covariance .006 .002

N 59 59

RM Pearson Correlation .633**

1

Sig. (2-tailed) .000

Sum of Squares and Cross-products .143 .149

Covariance .002 .003

N 59 59

**. Correlation is significant at the 0.01 level (2-tailed).

Regression


Mean Std. Deviation N

RF -.008391494068 .0770129395083 59

RM -.002160746815 .0506555024063 59

Correlations

RF RM


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Pearson Correlation RF 1.000 .633

RM .633 1.000

Sig. (1-tailed) RF . .000

RM .000 .

N RF 59 59

RM 59 59

Variables Entered/Removedb

Model Variables Entered Variables Removed Method

1 RMa

. Enter

a. All requested variables entered.

b. Dependent Variable: RF

Model Summaryb

Model R R Square Adjusted R Square

Std. Error of the

Estimate

1 .633a

.400 .390 .0601590807227

a. Predictors: (Constant), RM


Model Summaryb

Change Statistics

Durbin-WatsonR Square Change F Change df1 df2 Sig. F Change

.400 38.050 1 57 .000 2.066



ANOVAb

Model Sum of Squares df Mean Square F Sig.

1 Regression .138 1 .138 38.050 .000a

Residual .206 57 .004

Total .344 58



Coefficientsa

Model

Unstandardized Coefficients

Standardized

Coefficients

t Sig.B Std. Error Beta

1 (Constant) -.006 .008 -.805 .424

RM .962 .156 .633 6.168 .000


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Variables Entered/Removedb

Model Variables Entered Variables Removed Method

1 RMa

. Enter

a. Dependent Variable: RF

Coefficientsa

95.0% Confidence Interval for B Correlations Collinearity Statistics

Lower Bound Upper Bound Zero-order Partial Part Tolerance VIF

-.022 .009

.650 1.274 .633 .633 .633 1.000 1.000


Coefficient Correlationsa

Model RM

1 Correlations RM 1.000

Covariances RM .024


Collinearity Diagnosticsa

Model Dimension Eigenvalue Condition Index

Variance Proportions

(Constant) RM

1 1 1.043 1.000 .48 .48

2 .957 1.044 .52 .52


Residuals Statisticsa

Minimum Maximum Mean Std. Deviation N

Predicted Value -.144943192601 .081159777939 -.008391494068 .0487265471040 59

Residual -.1328763663769 .1851410716772 .0000000000000 .0596382130771 59

Std. Predicted Value -2.802 1.838 .000 1.000 59

Std. Residual -2.209 3.078 .000 .991 59

Scatter Graph and Trend line


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CAPM formula takes the form of Ri = Rf+ (Rm - Rf) i

Where Ri = the return on the asset, Rf = The risk free rate, which I have decided will be the

1yr 5% Tr 12 RED which is currently .40 % which equates to .033% RFR (Monthly) using

Monthly RFR = ( 1 + .0040)(1/12)

1 = 0.00033. Rm = the average return on the market

(Geometric Mean) which has been calculated to be -0.002. i = The Beta which has been

calculated using SPSS for the HSBC share value to be 0.962, however Thompson analytics was

1.12.

Ri = Rf+ (Rm - Rf) i

0.00033 + (-0.0020.00033)0.962 = -0.00191, however due to the low Rm we are using (Ri - Rf

)i to calculate the expected rate of return (ERR) of 0.00152

Results

RMarket = 0.4162(Ret HSBC) + 0.0013

R = 0.4003


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The CAPm calculation and statistical analysis resulted in a mixed bag of results, due to the very

low return on the market of -0.002 the Market premium of -0.0023 is a negative value to the

risk free rate, indicating that investment in the FTSE market overall is barely worth the risk,

and that there is a good argument not to do anything other than purchase gilts.For the expectedreturn on HSBC I have used beta x Equity Risk Premium, the Risk Premium calculates at -0.25

providing a expected rate of return for January at -0.16 (Annual non compounded rate = -1.92

(compounded) = -1.9)

The analysis returned a beta for HSBC of 0.962 indicating that this is a lower risk portfolio

company but very close to the market combined with a negative ERR would not excite short

term investors at all, and barely turn the heads of long term investors who would normally be

attracted by this company, perhaps the dividend is amazing.

Higher kurtosis of the tails in the HSBC shares than that of the market, suggesting that the

market has a more evenly probability of distribution than that of the company, and that there is

more chance that good and bad days occur within the company share price than that in the

general FTSE market. However there is also skewness in both distributions, negative in the

market and positive in HSBC suggesting that there are more bad days occurring in the market

than of the company. There is also a high correlation to the market and HSBC share prices,

suggesting that changes within the market affect the company shares, and vice vesa.

Roll's critique

mean-variance efficient

Assumption The risk free rate is Arithmetical, the other rates were converted to geometric


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The combination of risky assets and a risk free asset gave investors a basis to calculate a

capital market line that provides lending and borrowing and when Markowitz

As the efficient frontier only includes the portfolio of risky assets, a risk free asset with a zero

risk return can be combined with risky assets added to the efficient portfolio and investors can

then go beyond the frontier by borrowing and lending at risk free rate. A capital market line that

used to linearly predict the market return can be drawn by lining up a point of a portfolio with

only risk free rate to the other point that touches the efficient frontier, known as a tangency

portfolio. The tangency portfolio explains that investors separate their decisions in investing

and financing the investment as suggested by

Investors were CAPM

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Michael Gordon was previously Chief Investment Officer for Fidelity International

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Hedge Funds CAPM Issues Financial Planning

Benchmark Issue in CAPM

Jeffry Merril Liando (2007)

Capital Asset Pricing Model (CAPM) has been widely used for finding a suitable required rate

of return of a share or portfolio. However, the assumption of using a major share market index

as the benchmark becomes an issue. This essay aims to discuss this issue related to the good

theoretical assumptions, the problems arising in the CAPM, the distortion in its application, the


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practice of seeking alpha in responding the distortion and the impact of distortion for the New

Zealand context.

CAPM and efficient portfolio

CAPM assumes that a major stock index can be used as a benchmark to determine riskpremium and beta for calculating the required rate of return of a stock. The formulae can be

shown as follows, Rs = Rf + Beta ( RmRf ), or by noticing the future expectation as this:

E(Rs) = Rf + Beta ( E(Rm)Rf ). To see the relationship between share and market returns, it

can be formulated as follows: E(Rs)Rf = Beta ( E(Rm)Rf ).

The benchmark index assumed in the CAPM is promoted as the market proxy of the efficient

portfolio of risky assets. The benchmark index is then set artificially to be a manifestation of a

whole market reflecting the acceptable portfolio chosen by investors within the efficient

frontier. Markowitz (1952)[1] suggests that rational investors would choose minimum risk and

maximum return in diversification and with any combination of weight the optimal portfolio

lies on the efficient frontier.

This is such an excellent portfolio theory so that Sharpe (1964)[2], Lintner (1965)[3] and

Mossin (1966)[4] further modelled it to include the risk free rate. As the efficient frontier only

includes the portfolio of risky assets, a risk free asset with a zero risk return can be combined

with risky assets added to the efficient portfolio and investors can then go beyond the frontier

by borrowing and lending at risk free rate. A capital market line that used to linearly predict the

market return can be drawn by lining up a point of a portfolio with only risk free rate to the

other point that touches the efficient frontier, known as a tangency portfolio. The tangency

portfolio explains that investors separate their decisions in investing and financing the

investment as suggested by Tobin (1958)[5].

In a sense of linear prediction of an individual share return, CAPM is then modelled with a

security market line, in which a shares rate of return can be predicted given the market return,

risk free rate and beta. The line is plotted by the expected rate of return of a share with its beta

to the market return. The issue of benchmark index can be seen at this point. If there is a

benchmark error, CAPM cannot estimate the correct beta and risk premium properly thus

cannot calculated the expected rate of share return correctly.

Benchmark error

Benchmark error using CAPM for evaluating portfolio performance according to Ross

(1980,1981)[6] can be seen in two ways when the market index produces an incorrect beta for

the share and when it produces incorrect estimation for the market premium optimised to the

risk free rate (see figure 1 and 2). The problem is not due to statistical variation but rather to the


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cause that the market index is not a good predictor of mean/variance efficient portfolio.

Figure-1. Incorrect beta

Figure-2. Incorrect market premium

A further study by Green (1986)[7] shows that benchmark errors are continuous behaviour and

different for different indexes, thus, share or portfolio performance is sensitive to the choice of

benchmark for the market index. We may now say that as beta assumed equals to one, the

expected rate of return should be higher as we choose any benchmark that produces a higher

market risk premium, and lower for any benchmark that produces a lower premium.

Benchmark errors are also considered in the context of global investment. Reilly and Akhtar

(1995)[8] found that there is a variation of beta when using a domestic index, global index or a

diversified global stock and bond portfolio. The beta of domestic equity index is lower than the

world equity index and much larger for the diversified global stock and bond portfolio.

Market proxy, beta and risk premium problems

The root of benchmark error was based fundamentally on Rolls critique (1977)[9] that found

that market index is efficient per se, not for the individual shares or portfolios. Ross (1978)[10]

also added that market proxy is not ex ante mean-variance efficient and individual preference

in portfolio selection may be judged with a different market index and then will be penalised by

shares beta according to the different market index. Roll and Ross (1994)[11] found that the

market proxy may be located within 22 bps below the efficient frontier (Figure-3).

Figure-3. Market proxy and efficient frontier

Using the true market proxy of the value weighted portfolios of all US shares, Fama and French

(2004)[12] tested CAPM by plotting the annualised monthly return and beta of every stock in

NYSE, AMEX and NASDAQ from 1928 to 2003 and to be compared to the returns predicted

with CAPM. The result is telling us further problem other than benchmark error and market

proxy problem, that is, beta inconsistency.


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Figure-3. Beta inconsistency.

Figure-3 shows an inconsistency of beta in CAPM. Low beta shares that are predicted to have

low returns are in fact too high whereas high beta shares that are predicted to have high returns

are in fact too low, as seemingly the line rotates.The inconsistency of beta was identified a decade before by Fama and French (1992)[13] as

suggested a three-factor model by adding size and book to market ratio (B/M) to CAPM. They

concluded that value shares with high dividend yield, high B/M, low P/E tend to have bigger

expected return than growth shares with low dividend yield, low B/M, high P/E.

Again using the true market proxy and the three-factor CAPM, Fama and French (2006)[14]

tested whether the value premium exists in CAPM pricing. The result is rejecting CAPM

pricing for portfolio based on size, B/M and beta as concluding that size and B/M, not beta,

reward the expected return. The evidence shows that expected return does not compensate beta

variation unrelated with size and B/M for both small stocks and big stocks. We may say that if

CAPM is fine a good benchmark index should contain both small stocks and big stocks.

Alpha

Market proxy distortion opens an opportunity to have a better portfolio performance than the

market itself which was earlier suggested by Jensen (1969)[15] with alpha as a performance

indicator: Alpha = Rs - Rf + Beta ( RmRf). However, one can say that even a passive

portfolio can beat the benchmark. Bowden (2000)[16] further argued that alpha relates to

market timing and cannot be observed by conventional performance measures and suggested

ordered mean difference (OMO) as an alternative measure, which is a function of a running

mean of the difference between asset/portfolio and benchmark returns that is ordered by values

of the benchmark.

Recently, Fama and French (2006)[17] discussed about a portable alpha as a way to add a

portfolio consisting risk free rate and index funds with an additional hedge position that

generates alpha. Moreover, an alternative indexing has been suggested by Arnott (2005)[18] in

fundamental indexation to solve the distortion in value weighted index with some alternative

weighting constraints, such as book value, cash flow, revenues, gross sales, gross dividend and

total employment.

New Zealand context

In order to see the impact of benchmark distortion in the New Zealand context, there are some

market characteristics need to be considered (Bowden, 2005, p.133-168)[19], as follows,

narrow true market proxy, large foreign capitalisation, high dividend yield, high risk free rate,

low holding period return and low prospect dominance.


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A consensus for share market proxy in NZ is the NZX-50 index that can be seen in every

business page and news and the NZX-All for the true market proxy. However, there is a

significant proportion of investment taking in the form of farming shares as the grass root of the

NZ economy as a whole. For example, a rich Waikato Fonterra farmer may see a comparableinvestment choice between investing in Fonterra shares and NZX-50 shares, thus she needs a

broader efficient frontier for the true market proxy other than just NZX-All. In fact, Fonterra

only issues capital notes and none of its capitalisation in the share market.

The top ten capitalisations in the NZX-50 hold around 37.5% of foreign capitalisation. Since

beta domestic equity market after influenced by beta foreign equity market may change, then

the return of individual shares may be predicted below or above the original security market

line.

High dividend yield and low P/E make NZ shares considered as value shares. Value shares may

have a higher expected rate of return, thus a higher cost of capital, than growth shares

according to the CAPM test by Fama and French (1992). If the CAPM holds, it is likely that the

correct benchmark used would be the gross index, not the capital index which is widely used in

the other market. A promotion to use the gross index to international investors may attract them

to choose NZ equity.

High risk free rate may increase the cost of capital with a condition that there is no false market

risk premium estimation in the benchmark. However, if benchmark error deviates to a lower

market return then the market risk premium would be narrow and the security market line may

rotate.

Low holding period return for NZ shares may be compensated by high risk free rate so that at

some period the return of NZ shares is in fact is lower than government bond return. Under

CAPM prediction, if NZ benchmark consists mainly of such shares, of course the market risk

premium would be negative and the expected share return would be below the risk free rate.

The NZX-50 benchmark is dominated by larger companies with low growth prospect. That is

why investors may have the opportunity of seeking alpha and beat the index with some shares

like Michael Hill, Fisher and Paykel, Cavalier, Dorchester, etc. If tested with CAPM, there

would be a big deviation from the predicted return of such shares.

Conclusion

CAPM assumes a major share market index as the best market proxy for efficient portfolio to

predict the required rate of return. Despite the good theory background of mean (return)

variance (risk) efficient portfolio, the linear CAPM prediction using such proxy could be far

from the reality of historical returns. As Fama and French (2004, p.44) suggested, But we also


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warn students that despite its seductive simplicity, the CAPMs empirical problems probably

invalidate its use in applications.

References

[1] Markowitz, Harry. (1952). Portfolio selection. Journal of Finance, Mar 1952, Vol. 7 Issue

1, p77-91.

[2] Sharpe, William F. (1964). Capital asset prices: a theory of market equilibrium under

conditions of risk. Journal of Finance, Sep 1964, Vol. 19 Issue 3, p425-442.

[3] Lintner, John. (1965). The valuation of risk assets and the selection of risky investments in

stock portfolios and capital budgets. Review of Economics & Statistics, Feb 1965, Vol. 47

Issue 1, p13-37.

[4] Mossin, Jan. (1966). Equlibrium in a capital asset market. Econometrica, Oct 1966, Vol. 34

Issue 4, p768-783.

[5] Tobin, James (1958). Liquidity preference as behaviour towards risk. The Review of

Economic Studies, Vol. 25, No. 2, p65-86.

[6] Roll, Richard. (1980). Performance evaluation and benchmark errors (I). Journal of

Portfolio Management, Summer 1980, Vol. 6 Issue 4, p7-14.

Roll, Richard. (1981). Performance evaluation and benchmark errors (II). Journal of Portfolio

Management, Winter 1981, Vol. 7 Issue 2, p17-22.

[7] Green, Richard C. (1986). Benchmark Portfolio Inefficiency and Deviations from the

Security Market Line. Journal of Finance, Jun 1986, Vol. 41 Issue 2, p295-312.


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[8] Reilly, Frank K and Rashid A Akhtar. (1995). The benchmark error problem with global

capital markets. Journal of Portfolio Management, Fall 1995, Vol. 22 Issue 1, p33-50.

[9] Roll, Richard. (1977). A critique of the asset pricing theory's tests: part I: on past andpotential testability of the theory. Journal of Financial Economics, Mar 1977, Vol. 4 Issue 2,

p129-176.

[10] Ross, Stephen A. (1978). The Current Status of the Capital Asset Pricing Model (CAPM).

Journal of Finance, Jun 1978, Vol. 33, Issue 3, p885-901.

[11] Roll, Richard and Stephen A Ross. (1994). On the cross-sectional relation between

expected returns and betas. Journal of Finance, Mar 1994, Vol. 49 Issue 1, p101-121.

[12] Fama, Eugene F and Kenneth R French. (2004). The capital asset pricing model: theory

and evidence. Journal of Economic Perspectives, Summer 2004, Vol. 18 Issue 3, p25-46.

[13] Fama, Eugene F and Kenneth R French. (1992). The Cross-Section of Expected Stock

Returns. By: Journal of Finance, Jun 1992, Vol. 47 Issue 2, p427-465.

[14] Fama, Eugene F and Kenneth R French. (2006). The Value Premium and the CAPM.

Journal of Finance, Oct 2006, Vol. 61 Issue 5, p2163-2185.

[15] Jensen, Michael C. (1969). Risk, the pricing of capital assets, and the evaluation of

investment portfolios. Journal of Business, Apr 1969, Vol. 42 Issue 2, p167-247.

[16] Bowden, Roger and Jennifer Zhu. (2005). Kiwicap: an introduction to New Zealand

capital markets. (2nd ed.). Wellington: Kiwicap Education.

[17] Fama, Eugene F and Kenneth R French. (2006). Tilted Portfolios, Hedge Funds, and

Portable Alpha. Chicago GSB Magazine, Winter 2007.

[18] Arnott, Robert D, Jason Hsu and Philip Moore. (2005). Fundamental Indexation. Financial

Analysts Journal, Mar/Apr 2005, Vol. 61 Issue 2, p83-99.


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[19] Bowden, Roger and Jennifer Zhu. (2005). Kiwicap: an introduction to New Zealand

capital markets. (2nd ed.). Wellington: Kiwicap Education.

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