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CFA II: SS18 - Portfolio Concepts
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Main Assumptions of Mean Variance Analysis
1. All investors are risk averse. Investors minimize risk for any given level of expected return. Investors may differ in their degree of risk aversion, but the key is that all investors are assumed to be risk averse to some degree.
2. Expected returns, variances, and covariances are known for all assets.
3. Investors create optimal portfolios by relying solely on expected returns, variances, and covariances. No other distributional parameter is used: all returns are assumed to follow a normal distribution, in which skewness and kurtosis can be ignored.
4. Investors face no taxes or transaction costs. Therefore there is no difference between before tax-gross returns and after tax net returns
Expected Return on A Portfolio
Ex. Two asset portfolio:
E(Rp) = w₁E(R₁) + w₂E(R₂)
where:
E(Rp) = expected return on Portfolio P
wi = proportion of the portfolio allocated to asset i.
E(Ri) = expected return on asset i
Variance of a Portfolio
Ex. Two Asset Portfolio:
σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂Cov₁,₂
or
σp² = w₁²σ₁² + w₂²σ₂² + 2w₁w₂ρ₁,₂σ₁σ₂
Correlation
ρ₁,₂ = Cov₁,₂ / σ₁σ₂
Minimum Variance Frontier
A minimum variance portfolio is one that has the smallest variance among all portfolios with identical expected return.
The minimum variance frontier is a graph of the expected returns / variance combination for all minimum variance portfolios. Steps to create:
1. Estimation Step - Estimate the exp. return and variance for each individual asset and the correlation of each pair of assets
2. Optimization Step - Solve for the weights that minimize the portfolio variance subject to the following constraints:
- ∑wiE(Ri) = τ (use many diff #s here)
- ∑wi = 100%
3. Calculation Step - Calculate the expected returns and variances for all the minimum variance portfolios determined in step 2. The graph of the expected return and variance combinations is the min-var frontier.
Global Minimum Variance Portfolio
This is the min var portfolio with the smallest variance among all possible portfolios.
Efficient Portfolios
Efficient portfolios have:
- Minimum risk of all portfolios with the same expected return.
- Maximum expected return of all portfolios with the same risk
Efficient Frontier
The efficient frontier is a plot of the expected return and risk combinations of all efficient portfolios (upper part of the curve from the global min var portfolio - under ports don't make sense!).
The efficient frontier is an extremely useful portfolio management tool. Once the investor's risk tolerance is determined and quantified in terms of variance or std. dev, the optimal portfolio for the investor can be easily identified.
Equally-Weigthed Portfolio Risk
The formula for variance of an n-asset portfolio is very complicated, but the equation can be simplified dramatically if you equally weight everything:
σp² = (1/n)σi²⁻ + (n-1/n)Cov⁻
where:
σi²⁻ = avg. variance of all assets in portfolio
Cov⁻ = avg. covariance of all asset pairings in the p.
Note that as n gets larger the first time approaches 0 and the second term gets closer to the average covariance. The variance of an equally weighted portfolio approaches the average covariance as n gets large.
Equally-Weighted Port. Variance
σ² = σi²⁻ x [(1-ρ/n)+ ρ]
The main point to take away from this equation is to notice that the maximum amount of risk reduction occurs when the number of stocks is very large, such that:
σ² ≈ σi²⁻(ρ)
Inclusion of Risk-Free Assets
A risk free asset is the security that has a return known ahead of time, so the variance of the return is zero. When you add in risk free asset to a portfolio the shape of the efficient frontier changes from a curve to a line. This makes sense in the context of the exp return and variance formulas. (just plug in 0 for var and see what happens!).
Importance of Linear Relationship
The linear relationship between a portfolio and a risk free asset is a key result and is instrumental in the investor's asset allocation decisions. We use the capital allocation line to answer several important questions:
1. How should the investors choose which risky portfolio among many possible risk portfolios to combine with the risk free asset?
2. Given the investor's risk tolerance (i.e. target std. dev), what rate of return should be expected?
3. Given the investor's risk return objectives, what percentage allocation should be given to the risk free asset and the risky portfolio?
Capital Allocation Line
So we assume that the investor will choose the risky portfolio that maximizes the reward-to-risk tradeoff. This leads us to the capital allocation line which is the risk return line that lies tangent to the efficient frontier. The tangency portfolio is optimal in the sense that it has the highest possible reward to risk ratio (sharpe ratio).
CAL Equation
To determine the rate of return commensurate with the investor's risk tolerance, we can use the mathematical equation for the CAL.
E(Rc) = Rf + [ (E(Rt) - Rf / δt ]δc
where:
E(Rc) = expected return on the investment combo
δc = standard dev. of investment combo
Rf = risk free rate (also the y intercept)
Sharpe Ratio = Slope! indicates the change in return for every 1unit change in investor's portfolio risk (δc).
What you need to remember about CAL
- If a risk free investment is available, investors can combine it with a risky portfolio to increase their return at all levels of risk.
- The CAL is the straight line that intersects the y-axis at the risk-free rate and lies tangent to the efficient frontier.
- The intercept of the CAL equals the risk-free rate and the slope equals the maximum portfolio reward-to-risk ratio (defined as the sharpe ratio).
- The tangency portfolio is the optimal risky portfolio because it has the highest possible expected reward to risk tradeoff
- The CAL can be used to determine the risk associated with any desired target return, or the expected return associated with any desired target std. dev.
- The intercept and slope of the CAL depend on the asset expectations of the investor. Therefore, the investors with different asset expectations face different CALs.
The Capital Market Line
The CML is the capital allocation line in a world in which all investors agree on the expected returns, std. devs, and correlations of all assets (known as the homogenous expectations assumption). Assuming identical expectations, there will only be one capital allocation line, and it is the CML.
Under these assumptions all investors agree on the optimal risk portfolio. This is called the market portfolio, M, defined as the portfolio of all marketable assets, weighted in proportion to their relative market values.
The key conclusion is that all investors will make optimal investment decisions by allocation between the risk free asset and the market portfolio.
Differences Between the CAL and CML
- There is only one CML. Homogenous expectations.
- There are an unlimited number of CALs because each is developed uniquely for the investor.
- The tangency portfolio for the CML is the market portfolio, and there is only one market portfolio.
- The tangency portfolio for the CAL can differ across investors depending on differences in their expectations.
- The CML is a special case of the CAL.
Capital Asset Pricing Model
The model describes the relationship we should expect to see between risk and return for individual assets. Specifically the CAPM provides a way to calculate an asset's expected return (or required return) based on its level of systematic risk as measured by the asset's beta.
Underlying Assumptions:
- Investors only need to know expected returns, variances, and covariances in order to create optimal portfolios.
- All investors have the same forecasts of risky asset's returns, vars and covars.
- All assets are marketable, and the market is perfectly competitive.
- Investors are price takers, whose individual buy and sell decisions have no effect on asset prices.
- Investors can borrow and lend at the risk free rate, and unlimited short selling is allowed.
- There are no frictions to trading, such as taxes or transaction costs.
Implication of CAPM Assumptions
Given the previous assumptions, there are four important implications for CAPM:
1. Due to homogenous expectations and all investors using minimum variance analysis, all investors identify the same risky tangency portfolio and combine that risky portfolio with the risk free asset when creating their portfolios.
2. Because all investors hold the same risky portfolio the weight on each asset must be equal to the proportion of its market value to the MV of the entire port.
3. The Security Market Line (SML), which is the graph of the CAPM, describes the relationship between the expected return and systematic risk for all assets.
4. Systematic risk as measured by beta is the only risk priced by the market.
Security Market Line
The SML is the graph of the CAPM, representing the cross sectional relationship between an asset's expected return and its systematic risk.
E(Ri) = Rf + β[E(Rm - Rf)]
Calculating Beta Coefficients
Systematic risk is estimated by the asset's beta, which is a standardized measure of an asset's systematic risk. The formula for the beta (systematic risk) for security i is:
βi = Covi,m / δm² = ρi,mδiδm / δm² = ρi,m(δi / δm)
where:
i = security
m = market
SML v. CML
Key differences between CML and SML:
1. Measures of risk: SML - Systematic. CML - Std Dev. (total risk).
2. Application: SML - Tool used to determine the appropriate exp. returns for securities. CML - Tool used to determine the appropriate asset allocation for the investor.
3. Definition: SML - Graph of the CAPM. CML - Graph of the efficient frontier
4. Slope: SML - MRP. CML - Market Portfolio Sharpe Ratio
Mean Variance Model - Historical Estimates
The efficient frontier is derived by creating optimal portfolios that consider all assets in the market. However, the forecast process needed to derive the efficient frontier is daunting.
The first problem in using historical estimates is that we have to estimate a very large number of inputs (n exp returns, + n individual asset std. dev, + n(n-1)/2 covariances).
Obviously we need more practical methods for computing inputs to mean-variance model.
Mean Variance Model - Market Model
A more practical and useful model is the market model. Premise is that there are two sources of risk:
1. Unanticipated macroeconomic events (systematic)
2. Firm specific events (unsystematic)
The market model is the regression model often used to estimate betas for common stocks:
Ri = αi + βiRm + εi
where:
Ri = Return on asset i
Rm = Return on the market portfolio
αi = intercept ( value Ri when Rm = 0)
βi = slope (estimate for systematic risk of i)
εi = regression error with expected values equal to 0.
Market Model Assumptions
1. The expected value of the error term is 0.
2. The errors are uncorrelated with the market return.
3. The firm specific surprises are uncorrelated across assets.
Market Model Predictions
The market model offers a simple way to derive forecasts of expected returns, variances, and covariances for individual assets. Make a few simplifying adjustments and walla:
exp return: E(Ri) = αi + βiE(Rm)
var of asset i: δi² = βi²δm² + δε²
cov of i & j: Covij = βiβjδm²
Market Model thus makes 3 predictions:
1. The exp return on Asset i depends only on the exp ret or the market and the sensitivity of the returns on asset i to movements in the market, βi.
2. The variance of the returns consists of two components: systematic risk βi²δm², and firm specific δε²
3. The covariance between any two stocks is calculated as the product of their betas and the variance of the market portfolio.
Instability of the Minimum Variance Frontier
The instability of the min var frontier and ∴ the efficient frontier is a concern for number of reasons:
1. The statistical inputs are unknown and must be forecast; greater uncertainty in the inputs leads to less reliability in the efficient frontier.
2. Statistical input forecasts derived from historical sample estimates often change over time, causing the estimated efficient frontier to change over time (this is called time instability).
3. Small changes in the statistical inputs can cause large changes in the efficient frontier resulting in unreasonable large short positions and overly frequent rebalancing.
Multifactor Models
The market model could be described as a single factor model, because it assumes asset returns are explained by a single factor: the return on the market portfolio. A multi factor model assumes asset returns are driven by more than one factor. There are three general classifications of multi factor models:
1. Macroeconomic Factor Models
2. Fundamental Factor Models
3. Statistical Factor Models
1. Macroeconomic factor models
Macroeconomic factor models assume that asset returns are explained by surprises in macroeconomic risk factors (e.g. GPD, Interest Rates, Inflation). Factor surprises are defined as the difference between the realized value of the factor and its consensus predicted value.
2. Fundamental Factor Models
Fundamental factor models assume asset returns are explained by the returns from multiple firm-specific factors (e.g. P/E Ratio, Market Cap, Leverage Ratio and Earnings Growth Rate).
3. Statistical Factor Models
Statistical Factor Models use statistical methods to explain asset returns. Two primary types of stat. factor models are used: factor analysis and principal component models.
In factor analysis, factors are portfolios that explain covariance in asset returns.
In principal component models, factors are portfolios that explain the variance in asset returns. The major weakness is that the statistical factors do not lend themselves well to economic interpretation. Therefore stat factors are mystery factors.
Macro Economic Factor Models An Example
Ri = E(Ri) + bi₁F GDP + bi₂F QS + εi
The stocks expected return comes from CAPM
The F's are factor surprises
The b's are sensitivities of stock to the surprises
εi is the part of the return that can't be explained by the model.
Macro Economic Factor Models - Priced Risk
A risk that does not affect many assets (i.e. unsystematic) can usually be diversified away in a portfolio and will not be priced by the market; i.e. you cannot be expected to be compensated for taking on this type of rid.
GPD and Credit Quality Spread shocks are systematic risk factors, meaning they can affect even well diversified portfolios. Since they cannot be avoided, systematic factors represent priced risk; risk for which investors can expect compensation.
Macro Economic Factor Models - Factor Sensitivities
In a macro economic factor model, asset returns are a function of unexpected surprises to systematic factors, and different assets have different factor sensitivities. The sensitivities can be estimated by regressing historical asset returns on the corresponding historical macroeconomic factors.
Fundamental Factor Models - Example
Ri = ai + bi₁ F P/E + bi₂ F SIZE + εi
F's are risk associated with P/E or SIZE factors
Bs are standardized sensitivity measures
Standardized sensitivity measures in most models are not slopes. Instead, the fundamental factor sensitivities are standardized attributes:
bi₁ = (P/E)i - P/E⁻ / δP/E
i.e. P/E ratio for stock i minus average P/E across all stocks divided by the std. dev of P/E across all stocks.
By standardizing we are able to use different fundamental factors measured in different units in the same factor model.
Macro v. Fundamental
The key differences between the macro-economic and fundamental factor model are such:
- Sensitivities - Standardized sensitivities in fundamental are not estimated.
- Interpretation Factors - Macro factors are surprises in the macro variables. Fundamental are rates of return associated with each factor and are estimated using multiple regression.
- Number of factors - Macro are usually small in number as they are intended to represent systematic risk. Opp for fundamental.
- Intercept Term - Intercept for macro equals the stocks expected return. Intercept of fundamental has no economic interpretation; it is simply a regression intercept necessary to make the unsystematic risk of the asset = 0.
Arbitrage Pricing Theory (APT)
APT Assumes:
- Returns are derived from a multi factor model. Not a lot of practical guidance here for the identification of risk factors.
- Unsystematic risk can be completely diversified away.
- No arbitrage opportunities exist. Remember an art opp is defined as an investment opportunity that bears no risk, no cost and yet provides a profit.
The APT Equation
The APT describes the equilibrium relationship between expected returns for well-diversified portfolios and their multiple sources of systematic risk.
E(Rp) = Rf + βp,₁(λ₁) + βp,₂(λ₂) ..... etc
Each λ stands for the expected risk premium associated with each risk factor. Each λj equals the risk premium for a portfolio with factor sensitivity equal to 1 to factor j and factor sensitivities equal to 0 for the remaining factors.
Unlike CAPM the APT does not require that one of the risk factors is the market portfolio. THis is a major advantage of the arbitrage pricing model. The APT assumes there are no market imperfections preventing investors from exploiting arbitrage opportunities. As a result, extreme long and short positions are permitted and misplacing will disappear immediately.
APT v. Multi-Factor Models
- The APT is a cross sectional equilibrium pricing model that explains the variation across assets' expected returns during a single time period. The multi factor model is a time-series regression that explains the variation over time in returns for one asset.
- The APT is an equilibrium pricing model that assumes no arbitrage pops. The macro econ multi factor models are ad hoc.
- The intercept term in a macro econ is the asset's expected return, and is derived from the APT equation. The APT intercept is the Rf rate.
Active Return & Risk
Active return equals the differences in returns between a managed portfolio and its benchmark:
active return = Rp - Rb
Active Risk (also known as tracking error or tracking risk) is defined as the standard deviation of the active return:
active risk = S(rp-rb) = √(∑(Rp-Rb)²/(n-1)
Active risk can be split into two components:
1. Active Factor Risk - risk from active factor tilts attributable to deviations of the portfolio's factor sensitivities versus the benchmark's sensitivities to the same set of factors.
2. Active Specific Risk - Risk from active asset selection attributable to deviations of the portfolio's individual asset weightings versus the benchmark's individual asset weightings
Information Ratio
Active return alone is not enough to measure a manager's performance over a time series. To demonstrate a manager's consistency in generating active return, we utilize the information ratio in which we standardize average active returns by dividing it by its standard deviation. In other words, the info ratio equals the portfolio's average active return divided by the portfolio's tracking risk:
IR = (Rp⁻-Rb⁻) / S(Rp-Rb)
Pure Factor Portfolio
A pure factor portfolio is a portfolio that has been constructed to have sensitivity equal to 1.0 to only one risk factor, and sensitivities of zero to the remaining risk factors. They are really useful for speculation or hedging purposes. Think of a manager that believes GDP growth will be stronger than expected but wishes to hedge against all other factor risks. The manager can take a long position in the GDP factor portfolio. It is exposed to the GDP risk factor but has zero sensitivity or is hedged against all other factors.
Tracking Portfolios
Tracking portfolios have a deliberately designed set of factor exposures. namely they are designed to track a predetermined benchmark. Usually this is an active bet on asset selection. The manager constructs the portfolio to have the same factor exposures as the benchmark but then selects what they believe to be superior securities, thus hopefully outperforming w/out taking on more systematic risk.
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