Lecture Slides I

Stock Market Analysis & Equity Valuation

Rates of Return

The total holding-period return (HPR) measures by how much the value of initial investment has grown over the investment period:

Yield Gains CapitalYield Dividend

Price Beginning

DividendPrice Beginning-Price Ending

Price Beginning

Return DollarHPR

The Beauty of Compounding

There are different conventions for quoting returns on different assets. Mortgage payments and bond coupon payments, for instance, are quoted as annual nominal percentage rate (APR):

APR = Per-period rate Periods per year

Suppose that your money earns interest at an APR of 6% per year compounded semiannually. What is your effective annual rate of return, accounting for compound interest?

The general formula for the effective annual rate is

where n is the number of compounding periods per year. 

%09.61)03.1( 2 EFFr

11

n

EFF n

APRr

Annual Effective Rates for APR of 6%

Note 1: As the compounding frequency increases (1+APR/n)n gets closer to eAPR, where e is a magic number approximately equal to 2.718. This is exactly the same as exp(APR). Your pocket calculator should have the exp() or ex function.

Note2 : The difference between the Effective Annual Rate and APR has oftentimes been exploited by bankers trying to evade part of the capital gain tax. For instance, with weekly compounding we get almost 18 basis points that are potentially tax-free.

Estimating Expected Value and Variance From a Sample of Historical Returns

Sample average can be used as estimate of expected value, i.e.

Estimate of sample variance

For a technical reason the sum of squared deviations is divided by (N-1), rather than N.

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rErN t

Returns on stocks, bonds and T-bills

Cumulative Real Returns in the UK

Frequency Distribution of Annual Holding Period Returns in the U.S., 1926-2006

Risk Premium Puzzle

The size of equity premium cannot be easily accounted for by simple economic models. This fact has been first highlighted by Mehra and Prescott.

Quantitative Review: Covariance

Consider the two following random variables; bungee jumps and the number of bungee-related accidents. The covariance between the two is:

A positive covariance would imply that the more you jump on bungee the more likely you are to injure yourself. A negative covariance would be difficult to interpret.

Note: Due to the adjustment for the lost degree of freedom we have (N-1) in the denominator rather than N.

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N

BAEBABJEBJBABJCov t tt

Quantitative Review: Covariance and Correlation Coefficient

Correlation Coefficient

Sign aside, the covariance itself is rather uninformative. The correlation coefficient, on the other hand, always varies between –1 and 1, which gives us a deeper insight into the strength of the relationship.

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4330),cov(,

BABJBABJ

BABJ

Quantitative Review: Regression Analysis

Regression Analysis Suppose that the relationship between bungee-related accidents and the number of bungee jumps is linear:

The parameters a and b can be estimated as follows:

As predicted, the estimate of b is positive and a is close to zero.

0.153-BJEbBAEa

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4330BJBACovb

BJ

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2

Quantitative Review: Regression Analysis

Quantitative Review: Regressions in Excel

Markowitz Portfolio Theory

The exact relationship between portfolio’s risk and return was firstly described in a rigorous, mathematical manner by Harry Markowitz, a Nobel prize laureate.Let us start with a

Simple Two Asset Case Imagine you invested a proportion of your wealth w1 in Asset 1 and w2=(1- w1) in Asset 2, then

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portfolio

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Markowitz Portfolio Theory

Markowitz Portfolio Theory

Markowitz Portfolio Theory: Multiple Risky Assets

If the number of securities is raised the area representing the whole population of potential risk-return combinations comes to resemble an umbrella battling against the wind.

All risk-return combinations within the shaded region are available to an investor. However, most of them will be unattractive because they are dominated by other alternatives. In fact, the rational investor will be interested in portfolios lying on the upper part of the frontier (part of hyperbola above apex).

The efficient frontier is a solution to a maximisation problem: For any given portfolio maximise E(Rportfolio).

Don‘t Put Your Eggs Into One Basket

The firm-specific factors diffuse in a portfolio, which makes the variability of portfolio return small. The remaining risk can be attributed to macroeconomic fluctuations.

Diversification: Practical Issues

Solnik (1974) investigated the issue of how many securities should be included in a portfolio to achieve a reasonable degree of risk reduction. He looked at the extend to which risk is reduced as additional securities are added to a portfolio.

Solnik randomly generated portfolios containing between one and 50 shares. Risk is reduced in a dramatic manner by the addition of the first four securities to the portfolio. Most of the benefits of diversification are generated by a portfolio of 10-15 securities. Thus up to 90 per cent of the benefits of diversification can be gained by holding a relatively small portfolio. Beyond this level the marginal risk reduction becomes relatively small.

Solnik (1974). Why not Diversify Internationally, rather than Domestically?, Financial Analysts Journal, July-August, 48-54.

Solnik (1974). Why not Diversify Internationally, rather than Domestically?, Financial Analysts Journal, July-August, 48-54.

The benefits of international diversification shown in the exhibit are significant. An internationally diversified portfolio can reduce risk to less than half the level of the USA domestically held portfolio.

The benefits to international diversification will be transparent as long as the correlation between national stock markets is less than +1.

With international portfolio your definition of the systematic (non-diversifiable) risk changes. You are concerned with the global, rather than domestic trends.

Capital Asset Pricing Model and Diversification

The Capital Asset Pricing Model states (CAPM) that:

Expressed as a regression equation it reads:

)RFR)R(E(RFR)R(E tt,Mtt,i

ttt,Mtt,i )RFRR()RFRR(

Capital Asset Pricing Model and Diversification II

The systematic risk cannot be eliminated, as it pertains to risk factors common to all firms (e.g. GDP growth, monetary policy). These common risk factors determine the market movement. The unsystematic (firm-specific) risk is diversifiable, i.e. it diffuses out in a large portfolio.

riskicUnsystematriskSystematicRiskTotal

Technical Problems with the CAPM

Measuring Beta / The Impact of the Time Interval

The theory does not provide any hints as to the frequency of data that should be used to estimate Beta, i.e. should we use daily, weekly or monthly observations on returns?

The approach to estimate Beta varies between different institutions:

Value Line Investment Services - weekly observations

Merrill Lynch – 60 monthly observations

Apparently these two procedures produce “somewhat” different results, see: Statman, M. (1981). Betas Compared: Merrill Lynch vs. Value Line, Journal of Portfolio Management 7, 41-44.

Technical Problems with the CAPM

The Benchmark Error / Market Portfolio is UnobtainableRoll (1977) criticised the tests of the CAPM. In fact he claimed that CAPM is, in general, not testable. This is because we cannot construct the real market portfolio which should include all risky assets. Although there are world stock indices (e.g. Morgan Stanley World Stock Index), there is no comprehensive index which will include assets such as real estate, human capital, precious metals, stamp collections, or antiques. Some of the Assumptions may be UnrealisticE.g no taxes or transaction costs, borrowing at risk-free rate of interest.

Empirical Evidence on the CAPM

Ironically, up until the time of development of the CAPM in the mid-1960s, the model seems to work reasonably well; but following its development the relationship breaks down.

Consider dividing all stocks according to their beta into decile portfolios (i.e. portfolio 1 consists of the 10 per cent of shares with the highest betas). The average risk premiums on portfolios are plotted against beta on the graph for the period 1931-1991. Generally, this picture supports CAPM.

Empirical Evidence on the CAPM

The problem starts when we split the data into two periods. The post-1965 data shows complete absence of a relationship. Both the high beta portfolio and the low beta portfolio show average annual return over the risk-free rate of 6 per cent.

Possible Alternative to CAPM - Multifactor Models

Multi-factor models are based on the notion that portfolio’s return may be sensitive to a variety of factors:

Factors could either represent macro-variables (GDP growth, inflation, changes in exchange rates) or firm-specific characteristics. CAPM can be viewed as a single factor model.

 

The terminology changes slightly:Systematic risk Factor Risk

Unsystematic Risk non-factor risk

Multifactor Model of Fama and French

The model of Fama and French (1993) has beencreated to improve on the CAPM. Although it seems to provide much better description of therisk and return relationship, it does not have such a strong theoretical background like the CAPM.

SMB (i.e., small minus big) is the return to a portfolio of small capitalisation stocks less the return to a portfolio of large capitalisation stocksHML (i.e., high minus low) is the return to a portfolio of stocks with high book-to-market ratios less the return to a portfolio of low book-to-market stocks.

The data on three factors for the U.S. market can be found on:http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

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The Fama and French Three Factor Model: An Illustration

Consider the Fama-French model for 3 stocks. First, all stocks are positively correlated with the general stock market movement. The coefficients on the SMB factor confirm that Intel and JP Morgan Chase produce returns consistent with the large-cap stocks. Finally, JP Morgan Chase and Whole Foods Market are more likely to be considered value stocks.

The data given in the two tables can be used to estimate the expected return for each of the stocks.

Regression Estimates of Risk Factor Sensitivities

Stock Market SMB HML

Intel 0.62 -0.64 -1.48

JP Morgan Chase 1.37 -0.39 0.58

Whole Foods Market 1.93 0.82 1.68

Risk Factor Premia Estimates in the U.S.: 1927-2012

Risk Factor Risk Premium

Market 8.18%

SMB 3.76%

HML 4.26%

Weighted Average Cost of Capital

Weighted Average Cost of Capital (WACC) (no taxes):

E – equityD – debt

Weighted Average Cost of Capital (WACC) (with taxes):

t – tax rate

DebtofCost*DE

DEquityofCost*

DE

EWACC

)t1(*DebtofCost*DE

DEquityofCost*

DE

EWACC