Portfolio Concepts
Meanvariance analysis use of expected returns, variances, and covariances of individual investments to analyze the riskreturn tradeoff of combinations of these assets
Assumptions
 Risk aversion
 Expected returns, variances and covariances are known for all assets.
 No taxes or transaction costs
Portfolio variance = w_{1}^{2}σ_{1}^{2} + w_{2}^{2}σ_{2}^{2} + 2w_{1}w_{2} covariance
minimumvariance frontier expected returnstandard deviation combinations of the set of portfolios that have the minimum variance for every given level of expected return Steps for solving:
 Identify all possible expected returns from combining the set of assets into portfolios.
 For every expected return, determine the single portfolio with the smallest variance.
 Graph the expected return versus the variance determined in Step 2. This will yield the minimumvariance frontier.
 Global minimumvariance portfolio on the minimumvariance frontier that has the smallest standard deviation
Efficient frontier consists of portfolios that have the maximum expected return for any given standard deviation
Portfolio diversification: strategy of reducing risk by combining many different types of assets into a portfolio. Portfolio variance falls as more assets are added to the portfolio because not all asset prices move in the same direction at the same time. Therefore, portfolio diversification is affected by the:
Correlations between assets: Lower correlation means greater diversification benefits.
Variance of an equallyweighted portfolio of n stocks
Capital allocation line riskreturn line that lies tangent to the efficient frontier. The tangency point represents the best risky portfolio in the sense that it provides the best riskreturn tradeoff of any portfolio on the minimumvariance frontier (straight tradeoff line)
Capital market line(CML) capital allocation line in a world in which all investors agree on the homogeneous expectations assumption (a curve to which CAL is tangent)
Capital asset pricing model (CAPM) relationship to be expected between risk and return for individual assets. an asset’s expected return based on its level of systematic risk, as measured by the asset’s beta
 E(Ri) = R_{F} + β_{i}[E(R_{M} ? R_{F})]
Security market line (SML) the graph of the CAPM, representing the crosssectional relationship between the expected return for individual assets and portfolios and their systematic risk
Intercept = riskfree rate, R_{F} Slope = Market risk premium, E(R_{M}) ? R_{F} . Beta (systematic risk)
 β_{i} = Cov_{i,M} / σ_{M}^{2}
Sharpe ratio = slope of CML
 E(R_{M}) = R_{F} / σ_{M}
Market model regression model often used to estimate betas for common stocks
 R_{i} = α_{i} + β_{i}R_{M} + ε_{i}
Predictions:
 E(R_{i}) = α_{i} + β_{i}E(R_{M}). The expected return on Asset i depends on the market's expected return, the sensitivity of Asset i's returns to the market, and the average return to Asset i when the market return is zero.
 Var(R_{i}) = β_{i}^{2}σ^{2}_{M} + σ^{2}ε . There are two components to the variance of the returns on Asset i ? a systematic component related to the asset's beta and an unsystematic component related to firmspecific events.
 Cov_{ij} = β_{i}β_{j}σ^{2}_{M}. The covariance between two assets is the product of the betas of the two assets and the variance of the market portfolio.
Beta instability problem: Beta derived from the market model is not necessarily a good predictor of future relationships
 Adjusted beta: β_{i,t} = α_{0} + α_{1}β_{i,t1}
 where the sum α0 + α1 is set equal to 1, meanreverting level = 1
Instability in the efficient frontier
 The statistical inputs (means, variances, covariances) are unknown and forecast
 Time instability: statistical input forecasts often change over time
 Overfitting problem: small changes in the statistical inputs can cause large changes in the efficient frontier
Multifactor model timeseries regression that explains the variation over time in returns for one asset; assumes asset returns are driven by more than one factor
 Macroeconomic factor models: assume that asset returns are explained by surprises (or "shocks") in macroeconomic risk factors (e.g., GDP, interest rates, and inflation). Factor surprises are defined as the difference between the realized value of the factor and its consensus predicted value. Ad hoc.
 Fundamental factor models: assume asset returns are explained by the returns from multiple firmspecific factors (e.g., P/E ratio, market cap, leverage ratio, and earnings growth rate).
 Statistical factor models: use multivariate statistics (factor analysis or principal components) to identify multiple statistical factors that explain the covariation among asset returns. The major weakness is that the statistical factors do not lend themselves well to economic interpretation. Therefore, statistical factors are mystery factors.
Arbitrage pricing theory (APT) crosssectional equilibrium pricing model that explains the variation across assets’ expected returns
 Returns are derived from a multifactor model
 Unsystematic risk can be completely diversified away
 No arbitrage opportunities exist
 lack of clarity
 E(RP) = RF + βP,1(λ1) + βP,2(λ2) + … + βP,k(λk)
 λ:expected risk premium associated with each risk factor
 β:sensitivity of the Portfolio P to each risk factor
Active return(tracking error) = differences in returns between a managed portfolio and its benchmark
 active return = returns from managed portfolio  benchmark returns
Active risk (tracking risk) = standard deviation of the active return:
Active risk^{2} = active factor risk + active specific risk
 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.
 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, after controlling for differences in factor sensitivities of the portfolio versus the benchmark.
Information ratio = portfolio’s average active return / portfolio’s tracking risk
 Pure factor portfolio constructed to have sensitivity equal to 1.0 to only one risk factor, and sensitivities of zero to the remaining factors; particularly useful for speculation or hedging purposes.
 Tracking portfolios constructed to have the same set of factor exposures to match a predetermined benchmark; usually outperforming
 CAPM: a singlefactor asset pricing model, in which only risk relative to the broad market is priced; suggests some combination of the market portfolio and the riskfree asset
 APT captures multiple dimensions of risk besides the overall market risk, and suggests that investors make decisions relative to multiple sources of risk
Key Assumptions of the CAPM:
 Investors can borrow and lend at the riskfree rate.
 Unlimited shortselling is allowed with full access to short sale proceeds.
Implications of the CAPM:
 market portfolio lies on the efficient frontier (i.e., the market portfolio is efficient).
 linear relationship between an asset’s expected returns and its beta: E(Ri) = R_{F} + βi[E(R_{M}) ? R_{F}]
 intercept = riskfree rate
 slope = market risk premium (E(RM) ? RF).
If Key Assumptions are Violated:
 The market portfolio might lie below the efficient frontier (i.e., it might be inefficient).
 The relationship between expected return and beta might not be linear
Practical consequences of restrictions on borrowing at the riskfree rate and on short selling are:
 Expected return/beta relationship still might not be linear
 No portfolios of risky assets is likely to be the market portfolio:
 CAPM riskadjustments may not be appropriate for security performance evaluation
International Asset Pricing
International market integration capital can flow freely across borders Impediments: Psychological barriers, Legal restrictions, Transactions costs, Discriminatory taxation, Political risks, Foreign currency risk
International capital mobility increase over the past 20 years:
 many private and institutional investors who are internationally active
 all major corporations have multinational operations
 corporations and governments borrow&lend on an international scale
Necessary conditions for CAPM:
 Riskaverse investors
 Homogeneous expectations
 Concerned with nominal returns home currency
 Ability to borrow & lend unlimited amounts at the riskfree rate
 No taxes nor transaction costs
Assumptions for extended CAPM
 Investors throughout the world have identical consumption baskets
 Purchasing power parity holds exactly at any point in time
Forward currency risk premium
 FCRP = {[E(S_{1})  F] / S_{0}} = [{E(S_{1})  S_{0}} / S_{0}]  (r_{DC}  r_{FC})
ICAPM riskpricing relationship specifies expected return as function of investor’s domestic riskfree rate, world market risk premium, and sensitivity of the asset to changes in all foreign currencies.
 E(R) = RF +(β_{g} × MRP_{g}) + sum(γ_{k} × FCRP_{k})
ICAPM applies only in a world with integrated capital markets. If markets are segmented, risk is not priced the same in all markets, and the ICAPM does not accurately capture the relationship between risk and return across all capital markets.
Currency exposure can be measured in the local currency. Correlation between local currency asset returns and real exchange rate movements dictates the size and sign of the foreign currency risk premium.
 Domestic currency sensitivity: γ = γ(LC) + 1(accounts for exposure of a currency to itself)
 local currency appreciation hurts exporters(stock value of domestic firms goes up > GDP growth), benefits importers
 Money demand model: positive correlation between changes in domestic currency and stock return
 Free markets theory: real interest rate ↑ > domestic currency ↑ > bonds exposure to currency risk 
 Government intervention theory: domestic governments frequently defend declines in their currency’s value, bonds exposure to currency risk +
Jcurve effect(traditional approach) decline in a currency’s real exchange rate will improve the country’s competitiveness
 in shortrun, cost of imports ↑ > domestic inflation > real income ↓ > domestic demand and production ↓ > real GDP ↓
 in longrun, improved international competitiveness, export demand ↑
Money demand model real economic activity > domestic currency demand ↑  > currency value ↑ > GDP ↑; positive longrun correlation between domestic currency and stock returns
Active Portfolio Management
Need for Active portfolio management
 Develop capital market forecasts for the major asset classes.
 Allocate funds across the major risky asset classes to form the optimal risky portfolio.
 Allocate funds between the riskfree asset and the optimal risky portfolio.
 Rebalance the portfolio as capital market forecasts and the investor’s risk aversion changes.
TreynorBlack model portfolio optimization framework that combines modern portfolio theory and market inefficiency
 markets are nearly efficient
 number of mispriced assets is limited
 balances the need for diversification with the need for superior performance
 The specific steps in the model are:
 Develop capital market expectations for the passively managed market index portfolio.
 Identify a limited number of mispriced securities using security analysis.
 Determine weightings across the mispriced securities to form an actively managed portfolio.
 Determine weightings to the actively managed portfolio and to the passively managed market index to form an optimal portfolio combination of the two.
 Allocate funds to the riskfree asset and to the optimal portfolio, which maximizes the client’s utility.
Steps to adjusted alphas (higher the more accurate) for analyst’s forecasting accuracy:
 Collect the timeseries alpha forecasts for the analyst.
 Calculate the correlation between the alpha forecasts and the realized alphas.
 Square the correlation to derive the R^{2}.
 Adjust (shrink) the forecast alpha by multiplying it by the analyst’s R^{2}.
Taxation
Global taxation regimes
Regime 
Ordinary Income Tax Structure 
Favorable Treatment for: 
Interest Income? 
Dividend Income? 
Capital Gains? 
Common Progressive 
Progressive 
Yes 
Yes 
Yes 
Light Capital Gain Tax 
Progressive 
No 
No 
Yes 
Heavy Dividend Tax 
Progressive 
Yes 
No 
Yes 
Heavy Capital Gain Tax 
Progressive 
Yes 
Yes 
No 
Heavy Interest Tax 
Progressive 
No 
Yes 
Yes 
Flat and Light 
Flat 
Yes 
Yes 
Yes 
Flat and Heavy 
Flat 
Yes 
No 
No 
Investment income tax: FVIF_{IT} = [1 + R(1 ?T_{1})]^{N}
Deferred capital gains tax (MV = cost basis): FVIF_{CGT} = [(1 + R)^{N}(1 ?T_{CG}) + T_{CG}]
Deferred capital gains tax (MV ≠ cost basis): FVIF_{CGBT} = [(1 + R)^{N}(1 ?T_{CG})] + T_{CG}B
Wealthbased tax: FVIF_{WT} = [(1 + R)(1 ? T_{W})]^{N}
Return after realized taxes: R_{ART} = R(1 ? P_{1}T_{1} ? P_{D}T_{D} ? P_{CG}T_{CG})
For the annual taxes already paid
 Effective capital gains tax rate: T_{ECG} = T_{CG}(1 ? P_{1} ? P_{D} ? P_{CG}) / (1 ? P_{1}T_{1} ? P_{D}T_{D} ? P_{CG}T_{CG})
 Future accumulation factor: FVIF_{T} = [(1 + R_{ART})N (1 ? T_{ECG})] + T_{ECG} ? (1 ? B)T_{CG}
Accrual equivalent aftertax return: annual return that produces the same terminal value as the taxable portfolio
 moves closer to the pretax return as the time horizon increases and as more of the portfolio return is deferred
Accrual equivalent tax rate: T_{AE} = 1 ? (R_{AE}/R)
 lower the accrual equivalent tax rate, the more tax efficient the investment is. Higher portfolio allocations to tax disadvantaged assets will result in less tax efficiency and higher accrual equivalent tax rates.
Tax drag: percent of the investment gain lost to taxes

 Tax Drag > Tax Rate.
 As the Investment Horizon increases → the Tax Drag increases.
 As the Investment Return increases → the Tax Drag increases.
 Considering deferred capital gains tax independent of other types of taxation:
 Tax Drag = Tax Rate.
 As the Investment Horizon increases → the Tax Drag is unchanged.
 As the Investment Return increases → the Tax Drag is unchanged.
 Considering wealthbased taxes independent of other types of taxation:
 Tax Drag > Tax Rate.
 As the Investment Horizon increases → the Tax Drag increases.
 As the Investment Return increases → the Tax Drag decreases.
Taxable account bears (1t) portion of the investment risk, taxable portion assumed by the government. Investor bears all the investment risk for a TDA and taxexempt account.
Tax management investor would locate heavily taxed assets in tax advantaged accounts and hold lightly taxed assets in taxable accounts. The value created by the effective tax management of investment securities is referred to as the tax alpha.
Trading behavior
 Traders: shortterm gains taxed on an annual basis; forgo tax advantages of equity due to frequent trading
 Active investors: trade less frequently so that gains are taxed at lower rates
 Passive investors: buy and hold equity so that gains are deferred for the longterm and taxed at preferential rates when realized
 Exempt investors: all of stock held in taxexempt accounts, avoiding taxation altogether
Tax loss harvesting uses investment losses to offset investment gains or income, resulting in a tax savings; saves on current taxes, but reduces the cost basis for future taxes, thereby resulting in higher taxes in the future
 When highestin, firstout (HIFO) tax lot accounting is allowed by a government, an investor liquidates the portion of a position with the highest cost basis first, thereby minimizing current taxes; allows tax savings to be reinvested earlier, creating a tax alpha that compounds through time
 If tax rates are expected to be higher, it may be optimal for the investor to postpone tax loss harvesting and to liquidate low cost basis stock now
 Efficient frontier of portfolios should be viewed on an aftertax basis
 The meanvariance optimization should optimally allocate assets and determine the optimal asset location for each asset.
 Accrual equivalent aftertax returns would be substituted for beforetax returns and aftertax risk would be substituted for beforetax risk.
Portfolio Management Planning
Portfolio perspective portfolio managers should analyze the riskreturn tradeoff of the portfolio as a whole, not the risk return tradeoff of the individual investments in the portfolio, because some risk (unsystematic risk) can be diversified away by combining the investments into a portfolio
Ongoing portfolio management process
 Planning > Execution > Feedback
1. Analyzing objectives and constraints
Investment objectives
 Risk objectives investor’s willingness and ability to take risk (risk tolerance vs. risk aversion)
 Affecting ability to accept risk: required spending needs, longterm wealth targets, financial strength, liabilities
 Return objectives desired or a required return
Investment constraints internally/externally restrictions or limitations on available investment choices.
 Liquidity constraints: either expected or unexpected cash outflows that will be needed at some specified time
 Time horizon constraints: time periods during which a portfolio is expected to generate returns to meet major life events
 Longer time horizons indicate a greater ability to take risk and influence the appropriate asset allocation by allowing the investor to invest in longerterm, less liquid, higherrisk securities
 Legal and regulatory factors: externally generated; mainly affect institutional investors; specifcations on investment classes not allowed or limited
 Taxation factors
 Unique circumstances: internally generated, individual preference
2. Investment Policy Statement
Main role:
 Be readily implemented by current or future investment advisers
 Promote longterm discipline for portfolio decisions
 Help insure against shortterm shifts in strategy when either market environments or portfolio performance cause panic or overconfidence
Typical elements:
 detailed client description that allows a common understanding of the client’s situation
 purpose with respect to policies, objectives, goals, restrictions, and limitations
 duties and responsibilities
 objectives and constraints
 calendar schedule for both portfolio performance and IPS review
 asset allocation ranges and statements regarding flexibility and rigidity
 guidelines for portfolio adjustments and rebalancing
3. Strategic asset allocation
 formulate longterm target weightings for the asset classes to be included in the portfolio
 need for flexibility in the asset allocation to allow for temporary shifts
 common approaches for implementation
 Passive investment: not very responsive to changes in expectations; e.g. indexing and buyandhold investment
 Active investment: responsive to changing expectations; attempt to capitalize on differences between a portfolio manager’s beliefs concerning security valuations and marketplace representation; e.g. alpha generation and accordance to a particular investment style
 Semiactive, riskcontrolled active, or enhanced index strategies: somewhat of a hybrid of passive and active decisions; e.g. Index tilting(:track an underlying index but attempt to outperform by differetly weighted index compositions)
 forecasts of riskreturn characteristics to be included so that the portfolio’s expected riskreturn profile is wellunderstood
Ethical standards requirement: portfolio managers should be in a position of trust and must meet the highest standards of ethical conduct in order to truly serve his clients