The Tangent or Efficient Portfolio

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Identifying the Tangent Portfolio• Sharpe Ratio: Measures the ratio of reward-to-volatility

provided by a portfolio

• The portfolio with the highest Sharpe ratio is the portfolio where the line with the risk-free investment is tangent to the efficient frontier of risky investments. The portfolio that generates this tangent line is known as the tangent portfolio.

[ ] Portfolio Excess ReturnSharpe Ratio

Portfolio Volatility ( )

−= = P f

P

E R r

SD R

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Identifying the Tangent Portfolio• Combinations of the risk-free asset and the tangent portfolio provide the best risk and

return tradeoff available to an investor.

• This means that the tangent portfolio is efficient and that all efficient portfolios are combinations of the risk-free investment and the tangent portfolio. Every investor should invest in the tangent portfolio independent of his or her taste for risk.

• An investor’s preferences will determine only how muchto invest in the tangent portfolio versus the risk-free investment.

• Conservative investors will invest a small amount in the tangent portfolio.

• Aggressive investors will invest more in the tangent portfolio.

• Both types of investors will choose to hold the same portfolio of risky assets, the tangent portfolio, which is the efficient portfolio .

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LECTURE 06

Cost of Capital

Berk, De Marzo

Chapter 12 and 13

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The Capital Asset Pricing Model• The Capital Asset Pricing Model (CAPM) allows us to identify the efficient

portfolio of risky assets without having any knowledge of the expected return of each security.

• Instead, the CAPM uses the optimal choices investors make to identify the efficient portfolio as the market portfolio, the portfolio of all stocks and securities in the market.

• Investors hold only efficient portfolios of traded securities—portfolios that yield the maximum expected return for a given level of volatility.

• Investors have homogeneous expectationsregarding the volatilities, correlations, and expected returns of securities.

• Homogeneous Expectations

• All investors have the same estimates concerning future investments and returns.

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AssumptionCAPM• NO transcation costs and No personal income taxes

• All investor have the same level of information

• Investor’s trade securities at competitive market price

• Investor choose the efficcient portfolios

• Investors have the homogenous expectations reagarding the volatilitiescorrelations and expected return of a securities

• Unlimited short sale allowed

• Unlimited lending and borrowing allowed at riskless rate.

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CAPM• In equilibrium, every asset must be priced so that its risk-adjusted required

rate of return falls exactly on the security market line.

• • Total Risk = Systematic Risk + Unsystematic Risk.

• – Systematic Risk – a measure of how the asset co-varies with the entire economy (e.g., interest rate, business cycle).

• – Unsystematic Risk – idiosyncratic shocks specific to asset i, (e.g., loss of key contract, death of CEO).

• CAPM quantifies the systematic risk of any asset as its beta• Beta measures the sensitivity of a security to market-wide risk factors

• Expected return of any risky asset depends linearly on its exposure to the market (systematic) risk, measured by beta .

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Beta• We can use the beta of the investment to determine the scale of the investment in the

market portfolio that has equivalent systematic risk.

• The Capital Asset Pricing Model (CAPM) is a practical way to estimate. The cost of capital of any investment opportunity equals the expected return of available investments with the same beta. The estimate is provided by the Security Market Line equation:

• The equation implies there is a linear relationship between a stock’s beta and its expected return. The line going from the risk-free to the market portfolio is called security market line (SML).

r

i=r

f+β

i× (E[R

Mkt]-r

f)

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The Security Market Line

The SML shows the expected return for each security as a function of its beta with the market. According to the CAPM, the market portfolio is efficient, so all stocks and portfolios should lie on the SML.

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The Security Market Line

• Beta of a Portfolio: The beta of a portfolio is the weighted average beta of the securities in the portfolio.

( ) , ( , ) ( , )

( ) ( ) ( )= = = =

∑∑ ∑

i i MktiP Mkt i MktP i i ii i

Mkt Mkt Mkt

Cov x R RCov R R Cov R Rx x

Var R Var R Var Rβ β

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CAPM• Under CAPM assumptions it is the line along which all the securities should

lie when plotted according to their expected return and betas. Expected return of a portfolio depends on portfolio’s beta.

• The cost of capital of any investment opportunity equals the expected return of available investments with the same beta. This estimate is provided by the Security.

• Investors will require a risk premium comparable to what they would earn taking the same market risk through an investment in the market portfolio.

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Application • Linear Regression: The statistical technique that identifies the best-

fitting line through a set of points.

αi is the intercept term of the regression.

βi(RMkt – rf) represents the sensitivity of the stock to market risk. When the market’s return increases by 1%, the security’s return increases by βi%.

εi is the error term and represents the deviation from the best-fitting line and is zero on average.

(R

i − r

f) = α

i + β

i(R

Mkt − r

f) + ε

i

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Using Linear Regression

• Linear Regression• Since E[εi] = 0:

• αi represents a risk-adjusted performance measure for the historical returns. If αi is positive, the stock has performed better than predicted by the CAPM. If αi is negative, the stock’s historical return is below the SML.

• When the market portfolio is efficient, all stocks are on the security market line and have an alpha of zero.

• Investors can improve the performance of their portfolios by buying stocks with positive alphas and by selling stocks with negative alphas

{Distance above / below the SMLExpected return for from the SML

[ ] ( [ ] ) = + − +14444244443i f i Mkt f i

i

E R r E R rβ α

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The Debt Cost of Capital• Consider a one-year bond with YTM of y. For each $1 invested in the bond today,

the issuer promises to pay $(1+y) in one year.

• Suppose the bond will default with probability p, in which case bond holders receive only $(1+y-L), where L is the expected loss per $1 of debt in the event of default.

• So the expected return of the bond is:

rd = (1-p)y + p(y-L) = y – pL

= Yield to Maturity – Prob(default) X Expected Loss Rate

• The importance of the adjustment depends on the riskiness of the bond.

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Annual Default Rates by Debt Rating (1983–2011)

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Average Debt Betas by Rating and Maturity

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Example- Estimating the Debt Cost of Capital

In early 2013, auto parts retailer Autozone had outstanding10-year bondswith a yield to maturity of 3% and a BBB rating. If corresponding risk-freerates were 1.5%, and the market risk premium is 8%, estimate the expectedreturn of Autozone’s debt, where the expected loss rate is 60%.

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Example-1: Estimating the Debt Cost of Capital

• Using the average estimates in Table 12.2 and an expected loss rate of 60%, we have:

rd = Y-PL= 3% - 0.5%(0.60) = 2.7%

• Alternatively, we can estimate the bond’s expected return using the CAPM and an estimated beta of 0.10 from Table 12.3. In that case,

rd = 1.5% + 0.10(8%) = 2.3%

• Both estimates are rough approximations and they bothsuggest that the expected return of Autozone’s debt is belowits yield-to-maturity of 3%.

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Risk-freeRate• In order to determine the appropriate risk-free rate, it is necessary to look at

the risk-free assets traded in the financial market.

• • Long term government rate (even on a coupon bond) as the risk free rate on all of the cash flows in a long term analysis.

• For short term analysis, short term government security rate as the risk free rate.

• • The risk free rate that you use in an analysis should be in the same currency that your cash flows are estimated.

• The conventional practice of estimating risk free rates is the government bond rate.

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Beta for non TradedAssets• In the case of a firm´s equity or debt, we estimate the cost of capital based

on the historical risk of these securities. Because a new project is not a traded security, this approach is not possible.

• • Instead the most common method for estimating a project´s beta is to identify comparable firms in the same line of business as the project. If we can estimate the cost of capital of assets of comparable firms, we can use that estimate as a proxy for the project´s cost of capital.

• • The simplest setting is one in which we can find an all-equity financed firm in a single line of business that is comparable to the project. Because the firm is all equity, holding the firm´s stock is equivalent to owning the portfolio of its underlying assets If the firm has similar market risk of project, we can use comparable firm´s equity beta and cost of capital as estimates for the project

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A Project’s Cost of Capital

• Asset (unlevered) cost of capital: Expected return required by investors to hold the firm’s underlying assets.• Weighted average of the firm’s equity and debt costs of capital

• Asset (unlevered) beta

U E D

E Dr = r + r

E+D E+D

U E D

E Dβ = β + β

E+D E+D

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Financing and the Weighted Average Cost of Capital

• How might the project’s cost of capital change if the firm uses leverage to finance the project?

• Perfect capital markets• In perfect capital markets, choice of financing does not affect cost of

capital or project NPV

• Taxes – A Big Imperfection• When interest payments on debt are tax deductible, the net cost to the

firm is given by: Effective after-tax interest rate = r(1-τC)

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The Weighted Average Cost of Capital

• Weighted Average Cost of Capital (WACC)

• Given a target leverage ratio:

1wacc E D C

E Er = r + r ( - τ )

E+D E+D

wacc U C D

Dr =r - τ r

E+D

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Example:2-Cost of Capital

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Ch-13Information and Rational Expectations

• Rational Expectations

• All investors correctly interpret and use their own information, as well as information that can be inferred from market prices or the trades of others.

• Regardless of how much information an investor has access to, he can guarantee himself an alpha of zero by holding the market portfolio.

• Because the average portfolio of all investors is the market portfolio, the average alpha of all investors is zero.

• If no investor earns a negative alpha, then no investor can earn a positive alpha, and the market portfolio must be efficient.

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Information and Rational Expectations

• The market portfolio can be inefficient only if a significant number of investors either:

• Misinterpret information and believe they are earning a positive alpha when they are actually earning a negative alpha, or

• Care about aspects of their portfolios other than expected return and volatility, and so are willing to hold inefficient portfolios of securities.

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The Behavior of Individual Investors

• Underdiversification and Portfolio Biases• There is much evidence that individual investors fail to diversify their portfolios

adequately.

• Familiarity Bias

• Investors favor investments in companies they are familiar with

• Relative Wealth Concerns

• Investors care more about the performance of their portfolios relative to their peers.

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The Behavior of Individual Investors

• Excessive Trading and Overconfidence

• According to the CAPM, investors should hold risk-free assets in combination with the market portfolio of all risky securities.

• In reality, a tremendous amount of trading occurs each day.

• Overconfidence Bias

• Investors believe they can pick winners and losers when, in fact, they cannot; this leads them to trade too much.

• Sensation Seeking

• An individual’s desire for novel and intense risk-taking experiences.

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