BETA - CONCEPT

• In finance, the beta (β) of a stock or portfolio is a number describing the relation of its returns with that of the financial market as a whole.

• The beta coefficient is a key parameter in the capital asset pricing model (CAPM). It measures the part of the asset's statistical variance that cannot be mitigated by the diversification provided by the portfolio of many risky assets, because of the correlation of its returns with the returns of the other assets that are in the portfolio. Beta can be estimated for individual companies using regression analysis against a stock market index such as S&P 500.

BETA - CONCEPT

• The formula for the beta of an asset within a portfolio is

• where ra measures the rate of return of the asset, rp measures the rate of return of the portfolio, and cov(ra,rp) is the covariance between the rates of return.

• The portfolio of interest in the CAPM formulation is the market portfolio that contains all risky assets, and so the rp terms in the formula are replaced by rm, the rate of return of the market.

BETA - CONCEPT

• It Is non-diversifiable risk, its systematic risk, or market risk.

• Beta is also referred to as financial elasticity or correlated relative volatility, and can be referred to as a measure of the sensitivity of the asset's returns to market returns.

• measuring beta can give clues to volatility and liquidity in the marketplace

BETA - CONCEPT

• The beta coefficient was born out of linear regression analysis. It is linked to a regression analysis of the returns of a portfolio (such as a stock index) (x-axis) in a specific period versus the returns of an individual asset (y-axis) in a specific year. The regression line is then called the Security characteristic Line (SCL).

• αa is called the asset's alpha and βa is called the asset's beta coefficient. Both coefficients have an important role in Modern portfolio theory (APT).

BETA - CONCEPT

• The SML graphs the results from the capital asset pricing model (CAPM) formula. The x-axis represents the risk (beta), and the y-axis represents the expected return. The market risk premium is determined from the slope of the SML.

• The relationship between β and required return is plotted on the security market line (SML) which shows expected return as a function of β. The intercept is the nominal risk-free rate available for the market, while the slope is E(Rm)− Rf.

BETA - CONCEPT

BETA - CONCEPT

• It is a useful tool in determining if an asset being considered for a portfolio offers a reasonable expected return for risk. Individual securities are plotted on the SML graph. If the security's risk versus expected return is plotted above the SML, it is undervalued because the investor can expect a greater return for the inherent risk. A security plotted below the SML is overvalued because the investor would be accepting a lower return for the amount of risk assumed.

BETA - CONCEPT

• A misconception about beta is that it measures the volatility of a security relative to the volatility of the market. If this were true, then a security with a beta of 1 would have the same volatility of returns as the volatility of market returns. In fact, this is not the case, because beta also incorporates the correlation of returns between the security and the market. The formula relating beta, relative volatility (sigma) and correlation of returns is:

BETA - CONCEPT

• if one stock has low volatility and high correlation, and the other stock has low correlation and high volatility, beta cannot decide which is more "risky".

• This also leads to an inequality (because |r| is not greater than one): explain ->

• In other words, beta sets a floor on volatility. For example, if market volatility is 10%, any stock (or fund) with a beta of 1 must have volatility of at least 10%.

BETA - CONCEPT

• Another way of distinguishing between beta and correlation is to think about direction and magnitude. If the market is always up 10% and a stock is always up 20%, the correlation is one (correlation measures direction, not magnitude). However, beta takes into account both direction and magnitude, so in the same example the beta would be 2 (the stock is up twice as much as the market).

BETA - CONCEPT

• a stock's beta shows its relation to market shifts, it is also an indicator for required returns on investment (ROI) or [E(R)]

• highly rational investors should consider correlated volatility (beta) instead of simple volatility (sigma).

• Beta has no upper or lower bound, and betas as large as 3 or 4 will occur with highly volatile stocks.

BETA VALUES

• Beta can be zero. Some zero-beta assets are risk-free, such as treasury bonds and cash.

• However, simply because a beta is zero does not mean that it is risk-free. A beta can be zero simply because the correlation between that item's returns and the market's returns is zero.

• An example would be betting on horse racing. The correlation with the market will be zero, but it is certainly not a risk-free endeavor.

BETA VALUES

• A negative beta simply means that the stock is inversely correlated with the market.

• Example : Many precious metals and precious-metal-related stocks are beta-negative as their value tends to increase when the general market is down and vice versa.

• A negative beta might occur even when both the benchmark index and the stock under consideration have positive returns. It is possible that lower positive returns of the index coincide with higher positive returns of the stock, or vice versa. The slope of the regression line (i.e., beta) in such a case will be negative.

BETA VALUES

• If beta is a result of regression of one stock against the market where it is quoted, betas from different countries are not comparable.

• Staple stocks are thought to be less affected by cycles and usually have lower beta. Procter & Gamble, which makes soap, is a classic example. Other similar ones are Philip Morris (tobacco) and Johnson & Johnson(Health & Consumer Goods). Utility stocks are thought to be less cyclical and have lower beta as well.

BETA VALUES

• 'Tech' stocks typically have higher beta.

• An example is the dot-com bubble. Although tech did very well in the late 1990s, it also fell sharply in the early 2000s, much worse than the decline of the overall market.

• Beta has no upper or lower bound, and betas as large as 3 or 4 will occur with highly volatile stocks.

Analysis of Common Stock Betas

• Negative Beta Shows an inverse relation to the stock market and is highly unlikely. Gold Stocks though fall into this category.

• Beta of 0 Value of current cash (with no inflation) has a Beta of 0. No matter how the market performs, idle cash sitting always remains the same (with no inflation).

• Beta 0 - 1 These stocks are less volatile than the stock market in general. Commonly includes utility company stocks.

• Beta of 1 A Beta of 1 means the stock market is moving in the same direction as the Market Index such as S&P 500.

• Beta >1 Stocks with a Beta of >1 are more volatile than the stock market. This commonly includes high-tech stocks. Why? This is because as technology becomes rapidly advanced, outdated technology is useless. Many companies are thus wiped out due to out-dated technology. Beta >100 This is impossible. A stock can never be 100 times more riskier than the stock market in general. This is because a small change in the returns of the stock will make the stock price go to $0.

CRITICISM OF BETA

• Beta views risk solely from the perspective of market prices, failing to take into consideration specific business fundamentals or economic developments.

• Beta fails to allow for the influence that investors themselves can exert on the riskiness of their holdings through such efforts as proxy contests, shareholder resolutions, communications with management, or the ultimate purchase of sufficient stock to gain corporate control and with it direct access to underlying value.(qualitative things)

• Beta also assumes that the upside potential and downside risk of any investment are essentially equal, being simply a function of that investment's volatility compared with that of the market as a whole.

BETA ESTIMATION

• DIRECT METHOD: RATIO OF COV. B/W MARKET RETURN AND SECURITY’S RETURN TO THE MARKET RETURN VARIANCE.

• β = COV(S,M)/ VAR(M) ; S=SECURITY, M=MKT.

STEPS

1. CALCULATE AVERAGE RETURN ON MARKET AND THE SECURITY.

2. CALCULATE DEVIATIONS ON RETURN ON MARKET FROM AVG RETURN.

3. CALCULATE DEVIATIONS ON RETURN ON SECURITY FROM AVG RETURN.

4. MULTIPLY DEVIATIONS OF MARKRT RETURN AND DEVIATION OF SECURITY’S RETURN.TAKE SUM AND DIVIDE BY NO. OF OBSERVATIONS.

STEPS CONTD..

5. CALCULATE SQUARED DEVIATIONS OF MARKET RETURNS. NOW DIVIDE THE SUM OF SQUARED DEVIATIONS BY NO. OF OBSERVATIONS (AVG) TO GET VARIANCE OF MARKET RETURN.

6. DIVIDE THE COV OF MARKET AND SECURITY WITH THE VARIANCE OF THE MARKET. THIS IS BETA.

BETA ESTIMATION

• MARKET(INDEX) MODEL: REGRESS RETURNS ON A SECURITY AGAINST RETURNS OF MARKET INDEX

• R j= α + β J R m + e J

R j= E(R)ON SECURITY J

R m IS EXPECTED MKT RETURN

β J IS SLOPE OF REGRESSION

e J IS ERROR TERM

BETA- DETERMINANTS

• Nature of business (cycles). Eg commodity goods vs utility.

• Operating leverage – use of fixed costs Dol = (∆ebit/ebit)/(∆sales/sales) VC ά sales FC const. so variability in ebit is due to FC Higher the dol higher the risk, ie high beta.

BETA- DETERMINANTS

• Financial leverage: refers to debt in a firjms cap. Structure. Any firm with debt- levered firm. There are fixed interests (financial FC) and cause ∆ in ebit

• Dfl = (∆ eps)/ (∆ ebit/ebit)

• As fl of a firm increases the βe of firm also increases.

example

• Standard & Poor's 500 Index (S&P 500) has a beta coefficient (or base) of 1. That means if the S&P 500 moves 2% in either direction, a stock with a beta of 1 would also move 2%.

• Under the same market conditions, however, a stock with a beta of 1.5 would move 3% (2% increase x 1.5 beta = 0.03, or 3%). But a stock with a beta lower than 1 would be expected to be more stable in price and move less

PORTFOLIO BETA

• The beta of a portfolio is the weighted sum of the individual asset betas, According to the proportions of the investments in the portfolio.

• A measure of a portfolio's volatility. A beta of 1 means that the portfolio is neither more nor less volatile or risky than the wider market. A beta of more than 1 indicates greater volatility while a beta of less than 1 indicates less. Beta is an important component of the Capital Asset Pricing Model, which attempts to use volatility and risk to estimate expected returns.

• Example: if 50% of the money is in stock A with a beta of 2.00, and 50% of the money is in stock B with a beta of 1.00,the portfolio beta is 1.50.

PORTFOLIO BETA

• Portfolio betas tend to be more stable than individual security betas.

• βp = w1 β1 + ……+ wn βn

PROJECT BETA

• For evaluation of the projects institutional discount rates are established and applied to the analysis of all projects.

• Institutional discount rates are determined by calculating an organization's weighted average cost of capital (WACC)Often the weighted average of the cost of equity and the cost of debt The weights are determined by the relative proportions of equity and debt.

• It is determined by adding the weighted cost of debt to the weighted cost of equity to determine the required rate of return.

PROJECT BETA

• For example, assume that the cost of debt is 5 percent and that debt makes up 20 percent of your capital structure, while the cost of equity is 15 percent and equity constitutes 80 percent of your capital structure.

The weighted average cost of capital (WACC) is then:

WACC= (5%)(20%) + (15%)(80%) = 1% + 12% = 13%.

• Therefore, your required rate of return (discount rate, hurdle rate is 13%

PROJECT BETA

• Cost of debt is determined by how much an organization pays in interest on its debt multiplied by one minus the tax rate.

For example, if your debt rate is 5 percent, and your corporate tax rate is 22 percent, then your cost of debt is: Cost of Debt = I (1-t)Cost of Debt = 5% x (1 - 22%) = 4%

PROJECT BETA

• The cost of equity is more difficult to determine. The most commonly accepted method in finance is the capital asset pricing model (CAPM)

• An economic theory that describes the relationship between risk and expected return, and serves as a model for the pricing of risky securities.

• three major elements to the CAPM- risk free rate, expected rate of return and beta, which represents the measure of a company's risk relative to the overall market.

PROJECT BETA

• Taking a company's returns and plotting them against the returns of the S & P 500 determines beta. 

• analysis is then used to determine the slope of the "best fit" line through the plotted points. Beta is the slope of the "best fit" line.

PROJECT BETA

• The beta used to determine risk, and from that, cost of equity, WACC, and the discount rate for an institution is a reflection of the risk of the entire organization's cash flow.

• It does not reflect the risk associated with the Incremental cash flows associated with a new project, new building, or new piece of equipment.

PROJECT BETA

• Estimating BetaIn risk analysis, estimating the beta of a project is quite important. But like many estimations, it can be difficult to determine.

The two most widely used methods of estimating beta are:

1.Pure-play method 2.Accounting-beta method

PROJECT BETA

• Pure-Play MethodWhen using the pure-play method, a company seeks out companies with a product line that is similar to the line for which the company is trying to estimate the beta. Once these companies are found, the company would then take an average of those betas to determine its project beta.

Example : Newco would like to add beer to its existing product line of soda. Newco is quite familiar with the beta of making soda given its history. However, determining the beta for beer is not as intuitive for Newco as it has never produced it.

Thus, to determine the beta of the new beer project, Newco can take the average beta of other beer makers, such as Anheuser Busch and Coors.

PROJECT BETA

• When using the accounting-beta method, a company would run a regression using the company's return on assets (ROA) against the ROA for market benchmark, such as the S&P 500.

The determination of cost of capital under the CAPM approach involves the estimation of Beta, riskfree rate and market return. Beta is generally determined by comparing the return of the firm or the project as the

case may be with the market return and ascertaining the relationship.

The historical Beta is the first step in the determination of the ex-ante Beta. Either the historical Beta can be accepted as the proxy for the future Beta or

modifications can be made to it to conform to the future. If we are thinking of a new Company for a single project, we will have no historical records to go by. We would then compute the Beta of companies of the same size and

about the same lines of business and after making necessary adjustments to it; take it as the Beta for the firm. In case the Company has been existence for some time, first taking the historical records of this as well as other similar Companies and then by modifying the findings, we can determine Beta.

PROJECT BETA

Typical procedure for developing a risk-adjusted discount rate is as follows:1. A company first begins with its cost of capital for the firm. 2. The cost of capital then must be adjusted for the riskiness of the project, by adjusting the company's cost of capital either up or down depending on the risk of the project relative to the firm.

• For projects that are riskier, the company's WACC would be adjusted higher and if the project is less risky, the company's WACC is adjusted lower ( because average cost of capital may not always reflect the extreme cases)

The main issue in this procedure is that it is subjective.

numerical• An illustrationLet us assume that a Company has a Beta of 1.2. Therisk-free rate of interest is 5% and the expected return onthe market is 12%. The cost of capital of equity of theCompany under the CAPM model would work out to13.4% as follows:5%+(12-5)1.2=13.4%Let us also assume that the Debt-Equity ratio is60:40 and that the cost of debt post-tax is 10%. Theweighted average cost of capital would then be(10*60/100)+(13.4*40/100)=11.36%Now a new project is being considered for expansionof activities. That project has a Beta of 3.6 and onbeing taken up would require 25% of the total resourcesof the Company.

We can thus gather that the Beta of the Company asa whole after taking up the project would be(1.2*75%)+(3.6*25%)=1.8So, the new cost of equity would be5%+(12-5)1.8=17.6%Consequently the new WACC would be10*60%+17.6*40%=13.04%In order to meet the WACC of 13.04%, the new projectmust generate 18.08% as follows:Let the return from the new project be x0.75*11.36+0.25*x=13.04%Solving, we get x=18.08%

• An alternative approachThe required return of 18.08% can also be ascertainedby another method. Taking each project as a separateentity, we can arrive at their respective costs of capital.The equity component of the new project then willhave a cost of 5% +(12-5)3.6=30.2%. The weightedaverage cost of capital of the new project alone will thusbe 30.2*40%+10*60%=18.08

The Hamada formula for adjusting theLeverage factor

• The Beta that we impute to a project is likely to undergo changes with the change in the capital structure of the Company.

• If the Company is entirely equity based, its Beta is likely to be lower than if it undertakes a borrowing. A number of factors like default risk, bankruptcy risk and agency costs contribute to this phenomenon.

• For the sake of convenience, let us call the Beta of a firm which is levered as Levered Beta and that of a firm on an all-equity structure as Unlevered Beta. Robert Hamada brought out formulae for ascertaining Levered Beta given the unlevered beta and also to find out unlevered Beta given the levered Beta.

GEARED AND UNGEARED BETA

• Also called levered and unlevered beta.• Levered Beta (or equity beta)= Risk of Equity • Unlevered Beta (Asset beta) = Risk of Entire

Firm (Assets)• Firm Value (V) = Debt Value (D) + Equity Value

(E)

GEARED AND UNGEARED BETA

Beta of a Unlevered Firm

Beta of an Unlevered Firm

BU=BL(1+(1-T)D/S), where

BU= Beta of an unlevered firm

BL=Beta of a levered firm

T=tax rate

D=component of Debt in capital structure

S=component of Equity in capital structure

GEARED AND UNGEARED BETA

Beta of a Levered FirmBL=BU(1+(1-T)D/S), where

BL=Beta of a levered firm

BU= Beta of an unlevered firm

T=tax rate

D=component of Debt in capital structure

S=component of Equity in capital structure

GEARED AND UNGEARED BETA

Given the Beta of a firm which is already levered, we can ascertain what its Beta would be if it chooses on all-equity structure. This also means that if the target firm has leverage different from the structure assumed in estimating the levered Beta, this can first be converted into an unlevered Beta and then re-converted into a levered Beta using the leverage parameters relevant to the firm.

example

Suppose there are three firms P, Q and R, which closely resemble project X that is to be embarked upon. The stock Betas of the three firms are taken and found to be 2.73, 2.23 and 1.73 respectively for P, Q and R. The Ratio of Debt to Equity for the three firms averages to 0.67. The marginal tax rate is 36%. The average stock Beta works out to 2.23.

Translating these into the Hamada formula for unlevered firms we get:

BU=BL/(1+(1-T)(D/S)) =2.23/(I+0.64)(0.67)=1.56

If the project is proposed to be financed by 50% equity and 50% debt, we can modify the above Beta by applying the Hamada formula for Levered firms:

BL=BU((I-T)D/S) =1.56(1+0.60)(0.5/0.5)=2.5 So, on a 1:1 debt equity ratio, the Beta will be 2.5. This Beta can be used now for determining the cost of

equity for the project and its weighted average cost of capital, so that a more meaningful appraisal can be had.