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Case Study Analysis Cost Of Capital At Ameritrade Finance Essay


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Capital Asset Pricing Model is a model that describes the relationship between risk and expected return and that is used in the pricing of risky securities.

Description: Capital Asset Pricing Model (CAPM)

The general idea behind CAPM is that investors need to be compensated in two ways: time value of money and risk.

The time value of money is represented by the risk-free(rf) rate in the formula and compensates the investors for placing money in any investment over a period of time. The other half of the formula represents risk and calculates the amount of compensation the investor needs for taking on additional risk. This is calculated by taking a risk measure (beta) that compares the returns of the asset to the market over a period of time and to the market premium (Rm-rf).

The CAPM says that the expected return of a security or a portfolio equals the rate on a risk-free security plus a risk premium. If this expected return does not meet or beat the required return, then the investment should not be undertaken. The security market line plots the results of the CAPM for all different risks (betas).

Using the CAPM model and the following assumptions, we can compute the expected return of a stock in this CAPM example: "if the risk-free rate is 3%, the beta (risk measure) of the stock is 2 and the expected market return over the period is 10%, the stock is expected to return 17% (3%+2(10%-3%))".

CAPM has a lot of important consequences. For one thing it turns finding the efficient frontier into a doable task, because you only have to calculate the co-variances of every pair of classes, instead of every pair of everything.

Another consequence is that CAPM implies that investing in individual stocks is pointless, because you can duplicate the reward and risk characteristics of any security just by using the right mix of cash with the appropriate asset class. This is why followers of MPT avoid stocks, and instead build portfolios out of low cost index funds.

"Cap-M" looks at risk and rates of return and compares them to the overall stock market. If you use CAPM you have to assume that most investors want to avoid risk, (risk averse), and those who do take risks, expect to be rewarded. It also assumes that investors are "price takers" who can't influence the price of assets or markets. With CAPM you assume that there are no transactional costs or taxation and assets and securities are divisible into small little packets. CAPM assumes that investors are not limited in their borrowing and lending under the risk free rate of interest.

How to Calculate the Cost of Equity CAPM

The cost of equity is the amount of compensation an investor requires to invest in an equity investment. The cost of equity is estimable is several ways, including the capital asset pricing model (CAPM). The formula for calculating the cost of equity using CAPM is the risk-free rate plus beta times the market risk premium. Beta compares the risk of the asset to the market, so it is a risk that, even with diversification, will not go away. As an example, a company has a beta of 0.9, the risk-free rate is 1 percent and the expected return on the equity investment is 4 percent.


Determine the market risk premium. The market risk premium equals the expected return minus the risk-free rate. The risk-free rate of return is usually the United States three-month Treasury bill rate. In our example, 4 percent minus 1 percent equals 3 percent.

Multiply the market risk premium by beta. In our example, 3 percent times 0.9 equals 0.027.

Add the risk-free rate to the number calculated in Step 2 to determine the cost of equity. In our example, 0.027 plus 0.01 equals a cost of equity of 0.037 or 3.7 percent.


Combining the risk-free asset and the market portfolio gives the portfolio frontier.

The risk of an individual asset is characterized by its co-variability with the market portfolio.

The part of the risk that is correlated with the market portfolio, the systematic risk, cannot be diversified away.

Bearing systematic risk needs to be rewarded.

The part of an asset's risk that is not correlated with the market portfolio, the non-systematic risk, can be diversified away by holding a frontier portfolio.

Bearing non-systematic risk need not be rewarded.

For any asset i:


We thus have an asset pricing model - the CAPM.

Example. Suppose that CAPM holds. The expected market return is 14% and T-bill rate is 5%.

What should be the expected return on a stock with β = 0?

Answer: Same as the risk-free rate, 5%.

• The stock may have significant uncertainty in its return.

• This uncertainty is uncorrelated with the market return.

What should be the expected return on a stock with β = 1?

Answer: The same as the market return, 14%.

What should be the expected return on a portfolio made up of 50% T-bills and 50% market portfolio?

Answer: the expected return should be

¯r = (0.5)(0.05)+(0.5)(0.14) = 9.5%.

Multifactor CAPM

In CAPM, investors care about returns on their investments over the next short

horizon - they follow myopic investment strategies.

In practice, however:

• Investors do invest over long horizons

• Investment opportunities do change over time.

In equilibrium, an asset's premium is given by a multi-factor CAPM :

Limitations of CAPM

Based on highly restrictive assumptions i.e. no tax, transaction costs etc

Serious doubts about its testability.

Market factor is not the sole factor influencing stock returns.

Summary of CAPM

CAPM is attractive:

1. It is simple and sensible:

is built on modern portfolio theory

distinguishes systematic risk and non-systematic risk

provides a simple pricing model.

2. It is relatively easy to implement.

CAPM is controversial:

1. It is difficult to test:

difficult to identify the market portfolio

difficult to estimate returns and betas.

2. Empirical evidence is mixed.

3. Alternative pricing models might do better.

Multi-factor CAPM.

Consumption CAPM (C-CAPM).


Other Methods for calculating cost of equity

There are 3 methods which are mainly used for calculating Cost of equity other than CAPM

Arbitrage Pricing theory

3 factor method

Dividend Growth Method

Arbitrage Pricing Theory

APT assumes that returns on securities are generated by number of industry-wide and market-wide factors. Correlation between a pair of securities occurs when these securities are affected by the SAME factor or factors.

Return on any stock traded in a financial market consists of two parts.

R = Re + U

Where, R = return on any stock

Re = Expected or Normal return (depends on all of information shareholders have on the stock for next month.)

U = Uncertain or Risky return (this comes from information revealed in the month)

U = m + ¥€ 


m = Systematic risk or market risk (it influences all assets of market)

¥€ € €½ Unsystematic risk (it affects single asset or small group interrelated of assets, it is specific to company)

The capital asset pricing theory begins with an analysis of how investors construct efficient portfolios. But in real life scenarios, it isn't necessary that every time portfolios will be efficient.

It is developed by Stephen Ross.

Moreover, the return is assumed to obey the following simple relationship:

Where b1, b2 and b3 are sensitivities associated with factor 1, factor 2 and factor 3 which can be interest rate or other price factors.

Noise = ¥ is event unique to the company.

APT states that the expected risk premium on a stock should depend on the expected risk premium associated with each factor and the stock's sensitivity to each of the factors. Thus, formula modifies to:

Where, rf = risk free rate is subtracted from each return to give risk premium associated from each factor.

Analysis of the formula:

If we put value for b = 0, the expected risk premium will be zero. It will create a diversified portfolio which has zero sensitivity to macroeconomic factor which offers risk free rate of interest. Portfolio offered a higher return, investors could make a risk-free (or "arbitrage") profit by borrowing to buy the portfolio. If it offered a lower return, you could make an arbitrage profit by running the strategy in reverse; in other words, you would sell the portfolio and invest the proceeds in U.S. Treasury bills.

Consider portfolio A and B are sensitive to factor 1, A is twice sensitive to factor1 as then portfolio Therefore, if you divided your money equally between U.S. Treasury bills and portfolio A, combined portfolio would have exactly the same sensitivity to factor 1 as portfolio B and would offer the same risk premium.

Steps of Arbitrage Pricing Theory

The various steps during Arbitrage Pricing Theory can be stated as:

Identify the macroeconomic factors: APT doesn't indicate which factors are to be considered. But there are 6 principle factors which are:

Yield spread

interest rate,

exchange rate,



portion of the market return

Estimate the risk premium of each factor

Estimate the factor sensitivity

Net Return = risk free interest rate + expected risk premium

3 factor model

It is a special case of APT

It considers 3 major factors called as

market factor

size factor

book to market factor.

There is also evidence that these factors are related to company profitability and therefore may be picking up risk factors that are left out of the simple CAPM.

The practical application of this model is to estimate the betas for the three factors and then use them to predict where returns should fall, much like the CAPM.

It was researched by Fama and French.

Dividend Growth Method

Dividend Discount Model.

It is useful when the growth rate of dividend is forecasted constantly.

The present value of stocks is given as


r = discount rate,

g = rate of growth,

DIV = annual cash payment,

This formula can be used when growth rate g < rate of return r.

When growth rate = rate of return, the present value becomes infinite.

For perpetual growth, r > g.

Growing perpetuity formula,

Where,P0 in terms of next year's expected dividend DIV

g = the projected growth trend

r = expected rate of return on other securities of comparable risk.

We can estimate cost of equity from this formula by re-arranging.

Let's understand by an example:

Suppose that your company is expected to pay a dividend of $1.50 per share next year. There has been a steady growth in dividends of 5.1% per year and the market expects that to continue. The current price is $25. Then cost of equity r is given as:

When the growth rate isn't constant but varies from year to year, then average can be calculated. Growth rate for current year is calculated using the formula:

For example,


Dividend (in Rs. Million)

Percent change (g)






(1.30 - 1.23) / 1.23 = 5.7%



(1.36 - 1.30) / 1.30 = 4.6%



(1.43 - 1.36) / 1.36 = 5.1%



(1.50 - 1.43) / 1.43 = 4.9%

Growth rate is average of all percent changes and equals

This model serves the major advantage of being easy to understand and use but has a major drawback total dependence on dividend and it cannot be used where company isn't paying any dividend. Also, it doesnot consider any risk and is highly sensitive to the change in growth rate.

Estimating Beta

Beta is an important term in Capital Asset Pricing Method. Beta is the non-diversified risk of holding a single stock. But it turns out that companies in similar markets have similar risks.

Interpretation of beta

Beta = 1,it matches market portfolio

Beta > 1, higher risk.

Beta < 1, less risky and returns are highly predictable.

Methods for calculation of beta

It is calculated as:

\beta_{i} = \frac {\mathrm{Cov}(R_i,R_m)}{\mathrm{Var}(R_m)}

Where, Ri = rate of return of asset and Rm is rate of return of market. Thus, beta is dependent on regression analysis. Beta is found by statistical analysis of individual, daily share price returns, in comparison with the market's daily returns over precisely the same period. We need to gather a lengthy time-series of observations for the market return and the individual asset return. Then required co-variances and variances can be calculated. If coefficient of correlation P is known then

The alternative method of calculating beta is (by rearranging terms from CAPM equation):

In practice, an additional constant alpha is also added in the above equation which tells how much better (or worse) the funds did than what the CAPM predicted. Alpha is a risk-adjusted measure of the so-called active return on an investment.

Here, E(Ri) - Rf is estimated return on asset portfolios and E(Rm) - Rf is estimated return on market index.

In order to check that there are no serious violations of the linear regression model assumptions. The slope of the fitted line from the linear least-squares calculation is the estimated Beta. The vertical intercept of this curve is called as the alpha.

For a portfolio of assets, we have the relation:

Given that beta is a linear risk measure, the beta of a portfolio of assets as simply the weighted average of all the individual betas that comprise the portfolio.


Estimate of Risk Premium

We don't have reliable estimate where stock market will move in future. So we are using long term historical spreadsheets for estimate & large stock than small stocks because they are more closer to proper estimate of market

We are considering all values after Second World War because after that laws became stable in U.S.

Risk premium = Rm - Rf

U.S. government securities rate = 6.69% (20 years bond, Exhibit 3)

Average annual return for Large company stocks = 14 % (Exhibit 3)

So Risk premium for Ameritrade

= 14 % - 6.69 %

=7.31 %

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