CAPM Handout

FIN 402Capital Budgeting and Corporate Objectives

Yunjeen Kim

6. The CAPM.

Agenda

• The Nature of Risk

• The Capital Asset Pricing Model

• How to Get a Discount Rate Using the CAPM

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The Nature of Risk

• Is variance (or standard deviation) the relevant measure of risk to you? Why or why not?

• Total fluctuation is not what matters but how the fluctuation is related to the fluctuation in our wealth.

• What should be the relevant measure?

Review of Portfolio Theory

• When you start to combine stocks into a portfolio, you start to reduce the overall risk to your investment. This phenomenon is called diversification.

• What makes it possible & what are the limits of diversification?

• Diversification cannot eliminate total risk:

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Even More Efficient Portfolios

• Introducing a risk free asset. Lend/borrow money in the market at the risk free rate.

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Even More Efficient Portfolios

• You can create an even better set of efficient portfolios: improving return without increasing risk or decreasing risk without diminishing return.

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Capital Market Line

• CML is a linear combination of the market portfolio and the risk-free rate.

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Std. Dev.

Exp. Ret.

M

Capital Market Line (CML)

Capital Market Line

• What if an asset or a portfolio is not on the CML?

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Adding a Risk-Free Asset (Many Risky)

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Many Risky Assets & 1 Risk-Free Asset

• The line from the risk-free rate that is tangent to the efficient frontier is sometimes called the capital allocation line.

• The unique portfolio of risky assets that is also on this line is called the tangency portfolio.

• It is the best efficient portfolio as it offers the highest ratio of risk premium to standard deviation (the Sharpe Ratio)

• Sharpe ratio = Risk premium / SD = (rp-rf)/σp

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Sharpe-Ratio

• A risk-free asset has zero variance and zero covariance with any other asset.

• Thus, a portfolio with one risky asset and one risk-free asset has an expected return of:

and a portfolio standard deviation of:

• The risk-return tradeoff is:

• The slope is called the Sharpe-Ratio.

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The CAPM

• Assumptions– Financial Markets are efficient with no frictions.– Investors are risk averse.– Investors’ beliefs about correlations, returns, and risk are

identical.

• The results:– All investors build their portfolios using the same risky asset:

The Value Weighted Market Portfolio– The Value Weighted Market Portfolio is efficient– The expected return on any security is determined by its beta.

The CAPM

• Excess return on an asset:

• Risk measure of the asset:

• Excess return per unit of risk:

• This should be the same for all:

• The Security Market Line:– relationship between risk and expected return for individual assets.

E

Security Market Line

Ret

urn

1.0β

Risk Free Return = Rf

Security Market Line (SML)

Market Return = E(RM)

The CAPM

• In equilibrium, any two assets with the same marginal contribution to the risk of the market portfolio must have the same expected return.

• The relationship is of the specific functional form:

• In the CAPM world, all efficient portfolios are combinations of the market portfolio and the risk free asset

• The market portfolio is the tangency portfolio

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Example 1.

• Consider a levered portfolio that consists of $1,200 invested in the market portfolio, with $200 of that $1,200 borrowed at the riskless rate of 5%. If the expected return on the market is 12%, what is the expected return on your portfolio?

• Recall:

• Return:

Example 2

• Aoife Kavanagh has invested 60% of her money in stock A with a beta of 1.5. The rest is in stock B with a beta of 2.1. The risk free rate is 5% and the expected market return is 12%.

• Based on the CAPM, what is the expected return of each stock? Risk premium is 7%.

• What is the beta of her portfolio?

• Based on the CAPM and given her portfolio beta, what is the expected return on her portfolio?

Example 3

• A stock is expected to pay no dividends for the first three years, i.e., D1 = $0, D2 = $0, and D3 = $0. The dividend for Year 4 is expected to be $5.00 (i.e., D4 = $5.00), and it is anticipated that the dividend will grow at a constant rate of 8% a year thereafter. The risk-free rate is 4%, the market risk premium is 6%, and the stock's beta is 1.5. Assuming the stock is fairly priced, what is the current price of the stock?

Let’s Take a Step Back

• The total risk of a stock can be measured by its variance, or standard deviation.

• By combining stocks into a portfolio, we can come up with better risk/return combinations through diversification. Some of the risk of each stock in a portfolio is diversified away.

• This is a critical idea. We do not care about all the risk, but only about systematic, undiversifiable risk!

• => How do we measure the common source of risk?

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Beta: Intuition

• Conceptually, the relative risk of an asset in the portfolio is:

βi=cov(Ri,Rp)/σ2(Rp)

• The beta of the stock is a measure of its relative risk in the market. For a given stock market index, beta measures the asset’s relative risk contribution to that index.

• The beta of the market is 1. Beta measures a stock’s relative sensitivity to market movements. β ><1?

• Important in determining discount rates:

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Measuring Beta

21

beta

MarketReturn

StockReturn

1. Total risk = diversifiable risk + market risk2. Market risk is measured by beta, the sensitivity to market changes.

What is the Market?

• What is a market return?

– The return on the market portfolio of all assets in the economy. In practice, a broad stock market index, such as the S&P 500 Composite, is used to represent the market.

• Why use the market return?

– It reflects the movement of the broader economy and we care about broad economic risks, not the individual risk of any firm

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Market Risk

• How do we get a measure for market risk?

– First, compute the stock’s excess return over the risk free rate and then regress that on the market return over the risk free rate!

• Let’s see how this works. The following page has a brief introduction to the idea behind regression!

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An Example

Year S&P 500 IBM Return T-Bill Rate S&P 500 IBM's RiskReturn Risk Premium Premium

1991 26.0% -17.5% 9.7% 16.3% -27.2%1992 4.4% -40.0% 5.6% -1.2% -45.6%1993 7.1% 16.0% 4.1% 3.0% 11.9%1994 -1.5% 32.2% 2.3% -3.8% 29.9%1995 34.1% 25.7% 8.4% 25.7% 17.3%1996 20.0% 67.6% 5.5% 14.5% 62.1%1997 31.0% 39.3% 6.2% 24.8% 33.1%1998 27.0% 77.5% 6.1% 20.9% 71.4%1999 19.5% 17.5% 4.3% 15.2% 13.2%2000 -10.0% -20.8% 7.0% -17.0% -27.8%

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• A regression is simply an estimation of a linear relationship between 2 or more variables. Let’s review it by using a relevant example. Find the realized RISK PREMIUM for both IBM and the S&P 500.

• Note: Risk Premium = Return – Risk Free Rate

A Linear Relationship

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-60%

-40%

-20%

0%

20%

40%

60%

80%

-20% -10% 0% 10% 20% 30%

IBM

Ris

k P

rem

ium

Market Risk Premium

Risk Premia

Can you notice a relationship between IBM’s return and the index return? Draw a line through it! What does it tell you?

Calculating Beta

Year S&P 500 IBM Return T-Bill Rate S&P 500 IBM's RiskReturn Risk Premium Premium

1991 26.0% -17.5% 9.7% 16.3% -27.2%1992 4.4% -40.0% 5.6% -1.2% -45.6%1993 7.1% 16.0% 4.1% 3.0% 11.9%1994 -1.5% 32.2% 2.3% -3.8% 29.9%1995 34.1% 25.7% 8.4% 25.7% 17.3%1996 20.0% 67.6% 5.5% 14.5% 62.1%1997 31.0% 39.3% 6.2% 24.8% 33.1%1998 27.0% 77.5% 6.1% 20.9% 71.4%1999 19.5% 17.5% 4.3% 15.2% 13.2%2000 -10.0% -20.8% 7.0% -17.0% -27.8%

Getting a Regression Estimate: (Excel: = SLOPE(rangeofy ; rangeofx))Slope 1.42

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Again: Beta - Intuition

• The relative risk of an asset in the portfolio is:

β=cov(Ri,Rm)/σ2m

• The beta of the stock is a measure of its relative risk in the market. For a given stock market index, beta measures the asset’s relative risk contribution to that index.

• Beta measures a stock’s relative sensitivity to market movements.

• β >1 => above average risk

• β <1 => below average risk

• Will be important in determining discount rates.

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Key Takeaways

• Use variance to measure the total risk.

• BUT diversifiable risk doesn’t matter because when you hold a portfolio, it doesn’t exist any more!

• Use beta to figure out its systematic (or non-diversifiable) risk. This is the risk that we will use to produce the discount rate (or expected/required return).

• Key ideas: risk-return tradeoff; what is an efficient portfolio; how diversification works. Beta.

For risky cash flows: r = risk-free + risk premium. Risk premium will be a function of beta. Estimate it.

Alphas and betas for investment funds / companies

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Application of CAPM

• Using the SML equation, we can estimate the expected rate of return on equity.

• Step 1. Risk-free rate, Rf.– Use the current yield on one-year T-bills.

• Step 2. Excess return of market, E(RM)-Rf.– Use the historical data

• Step 3. Beta of stock j, βj.– Use the regression method or Beta books (e.g. Merrill Lynch)

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Alternatives to CAPM: the APT

• Arbitrage Pricing Theory was pioneered by Steve Ross at MIT as an Arbitrage Pricing Theory was pioneered by Steve Ross at MIT as an alternative to CAPMalternative to CAPM

• Does not ask which portfolios are efficient. Instead, assumes that each stock’s return depends in part on macro factors and in part on idiosyncratic events unique to each firm.

• Imposes the following structure on expected return / risk premium

• E(Risk Premium) = r - rf

= βf1(rfactor1 - rf) + βf2(rf2 - rf) + …

• E(Return) = a + bf1(rfactor1) + bf2(rf2) + …

In the matter of APT vs. CAPM

• Like in the CAPM, in APT expected return depends on market-wide factors and is unaffected by unique risk.

• If the expected risk premium on each of the portfolios is proportional to the portfolio’s market beta, then the APT and CAPM will give the same answer. Otherwise, will not.

• APT: need not measure the market portfolio. Can test on small datasets. BUT

• APT does not tell us what the factors are!

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Does the CAPM hold up?

High-minus-low book-to-market

Return vs. Book-to-MarketDollars(log scale)

Small-minus-big

http://mba.tuck.dartmouth.edu/pages/faculty/ken.french/data_library.html

2003

CAPM: The Three Factor Model

• Take the regular CAPM and add to the market factor on the right hand side.

• Work by Gene Fama & Ken French: historically, small firms and high BTM (value firms) have provided above-average returns.

• Pick up these additional risk factors.

• Include these 2 factors on the RHS of the CAPM.

• E(R) = rf+bmkt(rm-rf)+bsize(rsmall-rlrg)+bmtb(rhibtm-rlobtm)

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CAPM: The Three Factor Model

• How to apply the Fama-French 3 factor CAPM?

1. Identify the factors. Market, Size, BTM

2. Estimate the risk premium for each using historical data. ~7% market; ~3.7% size, ~5.2% BTM (’26-’06)

3. Estimate the betas for each stock as we have done with the regular CAPM (slope coefficient).

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Applying the Three Factor CAPM

• Same as we did with the CAPM

• Mastercard’s market beta is 0.92. It also has a size beta of -0.17 and the BTM beta of 0.13.

• What is its expected return if the T-bill rate is 3.5%?

– E(R) = Rf+βmkt (Rm-Rf)+βsize(Rsm-Rlrg)+ βbtm (Rhi-Rlo) =

= 3.5% + 0.92*7%-0.17*3.7%+0.13*5.2%

= 9.99%

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Takeaways

• Intuition about risk and portfolios.

• What do you need to make the most efficient portfolios with the CAPM? How would you do it?

• You only need the tangency portfolio (which is the market portfolio) plus a risk free asset to create any efficient portfolio.

• CAPM & the enhanced 3 factor CAPM model.

• Determine expected return (or discount rate) of assets if based on the market risk premium and the risk free rate.

• => Discount the cash flows.

Think about it

• Draw the Security Market Line (SML) and plot asset C such that it has less risk than the market but plots above the SML, and asset D such that it has more risk than the market and plots below the SML. (Be sure to indicate where the market portfolio is on your graph, and label axes.)

• Which of the stocks C or D is overpriced? Explain why you reached this conclusion.

• Explain why such pricing cannot persist in a market that is in equilibrium.