Reading 43 LOS's

Question 1

LOS 43a: Describe the implications of combining a risk-free asset with a portfolio of risky assets

Answer 1

• Risky assets can be combined into portfolios that may have a lower risk than each of the individual assets in the portfolio if the assets are not perfectly positively correlated
• An investor’s investment opportunity set includes all the individual risky assets and risk asset portfolios she can invest in
• The global minimum-variance portfolio is the portfolio of risky assets that entails the lowest level of risk among all portfolios on the MVF
• Investors aim to max return for every level of risk. Therefore, all portfolios above and to the right of the global minimum dominate those that lie below and to the right
• The section of the MVF that lies above and to the right of the global minimum is called the Markowitz efficient frontier

A risk free assest has an expected return of the RFR with a standard deviation of 0 and correlation with risky assets of 0. Once the risk-free asset is introduced into the mix:

• Any portfolio that combines a risky asset portfolio that lies on the Markowitz and the risk-free asset has a risk-return tradeoff that is linear
• The point at which a line drawn from the risk-free rate is tangent to the Markowitz efficient frontier defines the optimal risky asset portfolio. This line is known as the optimal CAL
• Each investor will choose a portfolio that contains some combination of the risk-free asset and the optimal risky portfoio

Question 2

LOS 43b: Explain the Capital allocation line (CAL) and the capital market line (CML)

Answer 2

The Capital Market Line

A capital allocation line (CAL) includes all combinations of the risk-free asset and any risky asset portfolio. The capital market line (CML) is a special case of the capital allocation line where the risky asset portfolio that is combined with the risk-free asset is the market portfolio. That is where the CAL touches the efficient frontier when it is tangent.

The risk and return characteristics of portfolios that lie on the CML can be computed using the risk and return formulas for two-asset portfolios

Expected Return on CML = E(R_p) = w_fR_f + (1 - w_f) E (R_m)

Variance on the CML = σ² = w_f²σ_f² + (1 - w_f)²σ_m² + 2w_f (1 - w_f) Cov (R_f,R_m)

Equation of CML = E (R_p) = R_f + [(E(R_m) - R_f / σ_m] x σ_p

Leverage Positions with Different Lending and Borrowing Rates

We have assumed that a investor can borrow at the RFR, which will never be the case, since they are not as realiable as the government. This makes the slope of the CML not a straight line when the investor starts to lever.

The slope to the left of the risky portfolio is [E(R_m) - R_f] / σ_m

While the slope to the right of this point where the investor is borrowing at the rate of R_b is [E (R_m) - R_b] / σ_m

The risk and return for a leveraged portfolio is higher than that of an unleveraged portfolio. Futher, given the investor’s borrowing rate is higher than the risk-free rate, for each additional unit of risk taken beyond the risky portfolio point, the investor gets a lower increase in expected return compared with portfolios to the left

Question 3

LOS 43c: Explain systematic and nonsystematic risk, including why an investor should not expect to receive additioanl return for bearing nonsystematic risk

Answer 3

The risk that disappears due to diversification in the portfolio construction process is known as unsystematic risk ( also known as unique, diversifiable, or firm-specific risk). The risk inherent in all risky assets (caused by macro-economic variables) that cannot be eliminated by diversification is known as systematic risk.

• Total Risk = Systematic risk + unsystematic risk

Once unsystematic risk has been eliminated and only systematic risk remains, a completely diversified portfolio would correlate perfectly with the market

In capital market theory, taking on a higher degree of unsystematic risk will not be compensated with a higher return because unsystematic risk can be eliminated, without additional cost, through diversification. Only if an investor takes on high risk that can not be diversified (systematic), will they be rewarded in the form of higher return.

This is an important implaction for asset pricing and expected returns. If risk is measured in terms of the standard deviation of returns of stock, the riskiest stock will not necessarily have the highest expected return. Even though a stock can have a higher level of total risk, capital market theory dictates that the market will expect a higher return on the investment that has a higher level of systematic risk, regardless of total risk. Therefore the expected return on an individual security depends only in its systematic risk.

Question 4

LOS 43d: explain return generating models (including the market model) and their uses

Answer 4

A return generating model is a model that is used to forecast the return on a security given certain parameters. A multi-factor model uses more than one variable to estimate returns:

• Macroeconomic Factor Models use economic factors that correlate with security returns to estimate returns
• Fundamental Factor models use relationships between security returns and underlying fundamentals to estimate returns
• Statistical Factor models use historical and cross-sectional returns data to identify factors that explain returns and use an asset’s sensitivity to those factors to project future returns

A general return generating model may be expressed as:

• E (R_i) - R_f = Σß_ijE(F_j) = ß_il [E(R_m) - R_f] + Σß_ilE(F_j)

The Market Model is an example of a single-index return generation model. It is used to estimate beta risk and to compute abnormal returns. Its given as:

• R_i = ¤_i + ß_iR_m + e_i ( I have used ¤ to represent greek alpha)
• The intercept ¤ and slope coefficient ß are estimated using historical asset and market returns, which are used to predict returns in the future

Question 5

LOS 43e: Calculate and Interpret beta

Answer 5

Calculation and Interpretation of Beta

Beta is a measure of the sensitivity of an asset’s return to the market’s return. It is computed as the covariance of the return on the asset and the return on the market divided by the variance of the market:

• ß_i = Cov (R_i, R_m) / σ_m² = Þ_i,mσ_iσ_m / σ_m²

Important points regarding Beta

• Beta captures an asset’s systematic or nondiversifiable risk
• A positive beta suggets that the return on the asset follows the overall trend in the market
• A negative beta indicates that the return on the asset generally follows a trend that is opposite to that of the current market trend
• A beta of zero means that the return on the asset is uncorrelated with market movements
• The market has a beta of 1. Therefore, the average beta of stocks in the market also equals 1

Estimating Beta Using Regression Analysis

We can use historical data and run regressions on them to determine their slope or beta. The more data we use, the more accurate beta becomes. When less data is used, the beta can be skewed by unsystematic events that occurred in the time frame.

Question 6

LOS 43f: Explain the capital asset pricing model (CAPM), including its assumptions and the security market line (SML)

LOS 43g: Calculate and interpret the expected return of an asset using the CAPM

Answer 6

The capital asset pricing model (CAPM) is a single-index model that is widely used to estimate returns given security betas. The CAPM is expressed as:

• E (R_i) = R_f + ß_i [E ( R_m) - R_f]

ASsumptions of the CAPM

• Investors are utility-maximizing, risk-averse, rational individuals
• Markets are frictionless and there are no transaction costs and taxes
• All investors have the same single-period investment horizon
• Investors have homogenous expectations and therefore arrive at the same valuation for any given asset
• All investments are infinitely divisible
• Investors are price-takers. No investor is large enough to influence security prices

The Security Market Line

The SML illustrates the CAPM equation. Its y-intercept equals the risk-free rate and its slope equals the market risk premium, (R_m - R_f)

Unlike the CAL and CML that consider only efficient portfolios, the SML and CAPM apply to any security of portfolio, regardless of whether it is efficient. This is because they are based only on a security’s systematic risk, not total risk.

The CAPM equation tells use that the expected (required) rate of return for a risky asset is determined by the RFR with some risk premium, which is determine by the systematic risk of the investment (ß) and the prevailing market risk premium (R_m-R_f)

Portfolio Beta

The CAPM can also be applied to portfolios of assets

• The beta of a portfolio equals the weighted average of the betas of the securites in the portfolio
• The portfolio’s expected return can be computed using the CAPM

Question 7

LOS 43h: Describe and demonstrate applications of the CAPM and the SML

Answer 7

Applications of the CAPM

Estimate of expected return: The expected rate of return computed from the CAPM is used by investors to value stocks, bonds, real estaet, and other assets. In capital budgeting, where NPV is used to make investing decisions, the CAPM is used to compute the required rate of return, which is then used to discount expected future cash flows.

Portfolio Performance Evaluation

The Sharpe ratio is used to compute excess returns per unit of total risk and is calculated as so:

• Sharpe Ratio = (R_p - R_f) / σ_p

A portfolio with a higher Sharpe ratio is preferred to one with a lower Sharpe ratio given that the numerator of the portfolios being compared is positive.

Two drawbacks of the Sharpe ratio are that is uses total risk as a measure of risk even though only systematic risk is priced, and that the ratio itself is not informative

The Treynor ratio basically replaces total risk in the Sharpe ratio with systematic risk (beta). It is calculated as:

• Treynor Ratio = (R_p-R_f) / ß_p

For this ratio to offer meaninful results, both the numerator and denominator must be positive. Neither the Treynor or Sharpe ratio offer any info about the significance of the differences between ratios for portfolios

M-squared (M²) is also based on total risk, not beta risk. It is calculated as:

• M²= (R_p-R_f) (σ_m/σ_p) - (R_m-R_f)

M² offers identical ratings as the Sharpe, but these are easier to interpret because they are in percentage terms. A portfolio that has matches the market’s performance will have a M² of zero, while one that outperforms will be positive.

Jensen’s alpha is based on systematic risk. It first eliminates a portfolio’s beta risk using the market model, and then used the CAPM to determine the required return from the investment. The difference between the portfolio’s actual return and the required return is called Jensen’s alpha and is calculated as so

• ¤_p= R_p - [R_f + ß_p (R_m -R_f)]

Jensen’s alpha for the market equals zero. The higher the Jensen’s alpha for a portfolio, the better its risk-adjusted performance

Security Characteristic Line

The SCL plots the excess returns of a security against the excess returns on the market. The equation of the SCL is given as:

• R_i-R_f = ¤_i + ß_i (R_m- R_f)
• Note that Jensen’s alpha is the y-intecept and beta is the slope of the SCL

Security Selection: Identifying Mispriced Securities

To determine whether a security is undervalued or overvalued, we compare the return that the security offers to the return it should offer to compensate investors for systematic risk (beta). For example a stock that offers 9% but based on CAPM is required to compensate 8%, is undervalued and investors should buy it.

• If the expected return using price and dividend forecasts is higher than the investor’s required return given systematic risk in the security, the security is undervalued.
• If the expected return using price and dividend forecasts is lower than the investor’s required return given the systematic risk in the security, the security is overvalued, so the investor should sell or short it.

Constructing a Portfolio

The CAPM tells us that investors should hold a portfolio that combines the risk-free asset with the market portfolio. We want to add securities to our portfolio that have positive alphas and have low correlation with other securities.

The weight of each nonmarket security in the portfolio should be proportional to:

• ¤_i / σ_ei²

where:

• ¤_i= Jensen’s alpha
• σ_ei²= nonsystematic variance of the security

Beyond the CAPM

Limitations of The CAPM

Theoretical limitations

• The CAPM is a single-factor model; only systematic risk is priced in the CAPM
• it is only a single period model

Practical Limitations

• A true market portfolio is unobservable as it would also inlcude assets that are not investable (ex, human capital)
• In the absence of a true market portfolio, the proxy for the market portfolio used varies across analysts, which leads to different return estimates for the same asset

Extensions of the CAPM

Theoretical Models like the apitrage pricing theory (APT) expand the number of risk factors

Practical Models use extensive research to uncover risk factors that explain returns