Beta factor coefficient can be calculated using any of the following approaches:
i. Excess Return Approach ii. Linear Regression Approach iii. Probability Approach.
iv. Covariance Approach
v. Standard Deviation Approach 1. Excess Return Approach
Under this method, Beta factor is given as:
B = Excess return on investment
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Excess return on market
B =
Where Rs = Expected return on security Rf = risk-free rate
Rm = market return Illustration 1
Given a security with an expected return of 9% and a market return of 12%. Also given that government security rate is 7%. Calculate the Beta factor.
Solution B =
Given Rs = 9%
Rm = 12%
Rf = 7%
B = = 2/5
B = 0.40
ii. Line Regression Approach
This approach measures Beta as the gradient of the line of best fit when the return on a security and the market return are plotted in a graph. Therefore Beta is given as:
B =
where
B = The Beta Coefficient X = Return from the market
N = number of pairs of data from x and y Example 2
Wisdom Plc wishes to determine its historic beta coefficient in order to decide the cost of capital. Its financial manager has decided to use linear regression using a sample of 6
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months data about the return on Wisdom PLC ordinary shares and the market as a whole.
The sample data for the first 6 months are given below:
Monthly Return On:
Month Market Index (%) Wisdom’s shares (%)
1 +7 +4
2 +5 +3
3 -2 -5
4 0 -3
5 +1 +2
6 +2 +4
A dividend of 15k per share was paid by Wisdom plc in six months. The month-end price is shown ex-dix,
Required
a. use the data above to calculate a beta value for Wisdom Plc
b. If the risk free rate of return is 8% per annum, calculate the required return on the shares of Wisdom Plc.
Solution
Wisdom Plc
x y x2 xy
7 4 49 28
5 3 25 15
-2 -5 4 10
0 -3 0 0
1 2 1 2
2 4 4 8
13 5 83 63
b. =
=
= =
= 0.951Beta = 0.951
B. Required Return is: Rs = Rf + (Rm – Rf) b Rf for 6 months = 4% (Annual rate 8%
Rm for 6 months = 7% (Annual rate 14%)
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Rs = 4 + (7-4) 0.951 Rs = 4 + 2.853 Rs = 6.853%
ii. Probability approach This is given as
b =
where
p = Probability attached to each possibility. R = Expected return on security
Rs = Forecast return on security Rm = Expected return on market Rm = Forecast return on market
The probability approach is an extension of the co-variance approach.
iv. Covariance Approach
Under this approach, Beta is given as:
b = Co-variance x & y Variance of x
Coy.xy Varx
Where: Covxy = covariance of individual security return and market return as a whole.
Var. x = Variance of returns for the market as a whole Illustration 3
Assume that:
a. the risk free rate of return is 6%
b. the market rate of return is 11%
c. the standard deviation of return on the market as a whole is 40%
d. the co-variance of return for the market with returns for the shares of Endurance Ltd over the same period has been 19.2%.
Calculate:
i. The Beta
ii. The security expected return for Endurance Ltd Solution
i) B =
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Since the variance is the square of a standard deviation then
ii) Rs = Rf + (Rm – Rf ) B 6% + (11 – 6%) 1.20
= 6% + (5) 1.20
= 12%
v) Standard Deviation Approach
This approach determines Beta by using the formular B = Standard deviation of security x r
Standard deviation of market dsxr
dm
where r = co-efficient of correlation between the security and the market.
Illustration 4
Given the following information:
The average stock market return on equity = 15%
The risk-free rate of return (pre-tax) = 8%
Company x: dividend yield = 4%
Company x: share price rise = 12%
Standard deviation of total stock market
On equity = 9%
Standard deviation of total return on
Equity of company X 10.8%
Correlation coefficient between company X return on equity and average stock market
return on equity 0.75
Required
i. What is the beta factor for company X share
ii. What does this information imply for the actual returns and actual value of company x shares?
Solution
(i) B =
=
= 0.9(ii) The cost of company X equity should therefore be:
Rs = Rf + (Rm - Rf ) B
= 8% + (15 – 8) 0.9
= 14. 3%
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The Implications:
The actual returns on company X equity are 4% +12% = 16%. This implies either that the actual returns include extra returns due to factors which can be categorized as unsystematic risk factors or if
lower than it should be.
Beta of a portfolio
The Beta factor of an investor's portfolio is the weighted average beta factor of each security in the portfolio. The portfolio Beta is the weight of individual security multiplied by its respective Beta.
ieBp = SnWipi (i = 1 Illustration 5
Justine is considering allocating his portfolio funds to the following securities
Security Weight Beta
A 15% 0.85
B 10% 1.30
C 20% 1.181
D 25% 1.25
E 30% 0.70
If the risk free rate is 12% and the return on the market portfolio is 18%, calculate:
i. Portfolio Beta
ii. Expected Return on Jude's portfolio
Solution
i. Portfolio Beta (Bp) = SnWipi i=1
Bp = 0.15 (0.85) +0.10(1.30) + 0.20 (1.181) + 0.25 (1.25) + 0.30 (0.7) = 1.016
ii. Expected portfolio return:
Rp = Rf (Rm – Rf )Bp
= 0.12 + 1.016 (0.18- 0.12)
= 0.18096
=18.1%
Beta of a Geared company
The gearing of a company will affect the risk of its equity. It then follows that if a
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company is geared; its financial risk will be higher than the risk of an all equity company.
Therefore, the B value of A geared company's equity will be higher than the B value of a similar ungeared company's equity.
There is a direct connection between M & M's views about gearing and weighted average cost of capital and the CAPM M &M argued that as gearing rises, the cost of equity rises to compensate shareholders for the extra financial risk of investing in a geared company.
This financial risk is an aspect of systematic risk and ought to be reflected in a company's Beta factor.
The connection between M&M theory and CAPM means that it is possible to establish a mathematical relationship between the B value of an ungeared company and the B value of a similar
We should expect the B value of a geared company to be higher; because of the extra financial risk, and the formulae to learn are:
Bu = Bg
1 + D (1 –t)………..(1) veg
Where:
Bu = the beta factor of an ungeared company i.e. the ungeared beta
Bg = the beta factor of a similar, but geared company i.e. the geared beta D = the market value of the debt capital in the geared company.
Veg = the market value of the equity capital in the geared company t = the rate of company's income tax
Re-arranging this, we have
Bg = Bu (1+ D (1-+) ... (2) Veg
Which is also = Bu + Bu (D (1-t) Veg
Notice especially in formular 2 that the geared beta is equal to the ungeared beta plus a premium for financial risk which equals.
Bu {D(1 - 1 )}
Veg
This is the geared Company's gearing ratio, multiplied by a tax adjustment factor (1-t) and also multiplied by the beta of an ungeared company
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Illustration 6
Suppose that two companies are identical in every respect except for their capital structure. Their market values are in equilibrium, as follows:
Geared Ungeared
Ltd. Ltd.
N’000 N’000
Annual profit b/fInt& Tax 1,000 1,000
Interest (4,000 x 8%) 320 -
680 1,000
Tax at 35% 238 350
Profit after Tax = dividends 442 650 N’000 N’000
Market value of equity 3,900 6,500
Market value of debt 4,000 -
The total value of geared Ltd is higher than the total value of ungeared , which is consistent with MM's proposition that:
Vg =Vu + Dt.
The beta value of ungeared Ltd has been calculated as 1.0.
The debt capital of Geared Ltd can be regarded as risk-free Required:
Calculate
a. the cost of equity in Geared Ltd b. the market Return Rm
c. the beta value of Geared Ltd
Solution
a. The cost of equity in Geared Ltd is
=
= 11.33%This can be checked using the MM formula
Kg = Ku = +
Since kg
=
= 10%, and kd=
= 8%kg = 10% + { (1-0.35) (10-8) 4000) % 3,900
= 11.33%
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b. The beta value of ungeared Ltd is 1.0 which means that the expected returns from ungeared Ltd are exactly the same as the market returns and so Rm = 10%
This allows us to reverify cost of equity in Geared Ltd as:
Kg = Rf + Bu (Bu (Rm – Rf) {1 + (1-+) D
= 8%+ 10 (10-8)%(1+0.65x400 veg 3,900
= 11.33%
c. Bg = Bu {f 1+ (1-t) D}
veg
=1.67 Illustration 7
Musagift Ltd has an opportunity to invest in a project lasting one year.
The net cash flows and the beta factor for each of the projects are as follows:
Musagift N’000 B
Ltd 500 1.20
Sure Success Ltd 200 1.25
100 0.80
200 1.35
The market returns is 12% and the risk free rate of interest is 7%.
Required:
a. Calculate the total present value of the project that can be undertaken by 1. Musagift Ltd
2. Sure success Ltd
b. Calculate the overall beta factor for sure success Ltd; Project, assuming that all three are undertaken.
c. Using the information, discuss which company is likely to be valued more highly by investors and suggest how portfolio diversification by a company can reduce the risk experienced by an investor.
Solution
a. Project discount rates
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Musagift Ltd 7% + 1.2 (12-7)% = 13%
Sure success Ltd
i. 7% + 1.25 (12 – 7)% = 13.25%
ii. 7% + 0.8 (12 – 7)% = 11%
iii. 7% + 1.35 (12 – 7)% = 13.75%
Project net present values, assuming the cash flows all occur at the end of year 1, are N’000
Musagift 500 = 442.48
1.13 Sure success Ltd. i. 200
1.1325 176.60
ii. 100
1.11 = 90.09
iii. 200
1.1375 = 442.51
Allowing for rounding errors, the PV of the three projects of sure success Ltd added up to the same amount as the PV of the project of Musagift
b. Sure Success’ overall beta factor is a weighted average of the beta factor of the three projects
Project Value B Weighting
(i) 200 1.25 250
(ii) 100 0.80 80
(iii) 200 1.35 270
500 600
Overall beta factor = = 1.2
This is the same as MusagiftLtd's project Beta factor.
c. This information shows that for the projects under review both companies have the same PV and the same systematic risk (ie the same beta factors). It therefore follows that on the basis of these projects alone, investors should value both companies equally.
It might be tempting to assume that since sure success Ltd is divesting into three separate projects, whereas Musagift is putting all its eggs in one basket and investing in one project, that investors should show a preference for the low risk sure success Ltd because Musa gift's unsystematic risk will be higher. But with
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CAPM theory, it is assumed that investor can eliminate unsystematic risk by diversifying their own investment portfolio, and do not have to rely on companies to do the diversifying on their behalf.
Portfolio diversification reduces risk beta use the returns from projects will not be perfectly positively correlated, and diversification reduces risk more when project returns show little or no positive correlation (or preferably a negatively correlation when only this is achievable). However, diversification by a company reduces the risk of bankruptcy for the company itself As stated earlier, investors can diversify themselves without having to rely on a company to do it for them, and provided that bankruptcy brings, no added costs to the investor, CAPM theory states that diversification by a company should have no effect on the risk experienced by a well diversified investor.
The Alpha Factor
The alpha factor in CAPM theory is another term for abnormal return due to the specific (unsystematic) risk of an individual security that can be "eliminated" by diversifying. It is the return on a share that is not due to movements in the general market. Alpha factors is the recorded difference between the actual return and Rf+ B(Rm – Rf).