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## Portfolio Concepts

Mean-variance analysis use of expected returns, variances, and covariances of individual investments to analyze the risk-return 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 = w12σ12 + w22σ22 + 2w1w2 covariance

minimum-variance frontier expected return-standard deviation combinations of the set of portfolios that have the minimum variance for every given level of expected return Steps for solving:

1. Identify all possible expected returns from combining the set of assets into portfolios.
2. For every expected return, determine the single portfolio with the smallest variance.
3. Graph the expected return versus the variance determined in Step 2. This will yield the minimum-variance frontier.
• Global minimum-variance portfolio on the minimum-variance 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 equally-weighted portfolio of n stocks Capital allocation line risk-return line that lies tangent to the efficient frontier. The tangency point represents the best risky portfolio in the sense that it provides the best risk-return trade-off of any portfolio on the minimum-variance frontier (straight trade-off 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) = RF + βi[E(RM ? RF)]

Security market line (SML) the graph of the CAPM, representing the cross-sectional relationship between the expected return for individual assets and portfolios and their systematic risk

Intercept = risk-free rate, RF Slope = Market risk premium, E(RM) ? RF . Beta (systematic risk)

βi = Covi,M / σM2

Sharpe ratio = slope of CML

E(RM) = RF / σM

Market model regression model often used to estimate betas for common stocks

Ri = αi + βiRM + εi

Predictions:

1. E(Ri) = αi + βiE(RM). 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.
2. Var(Ri) = βi2σ2M + σ2ε . 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 firm-specific events.
3. Covij = βiβjσ2M. 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: βi,t = α0 + α1βi,t-1
where the sum α0 + α1 is set equal to 1, mean-reverting 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 time-series 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 firm-specific 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) cross-sectional 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 risk2 = 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 single-factor asset pricing model, in which only risk relative to the broad market is priced; suggests some combination of the market portfolio and the risk-free 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 risk-free rate.
• Unlimited short-selling 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) = RF + βi[E(RM) ? RF]
• intercept = risk-free 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 risk-free 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 risk-adjustments 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:

• Risk-averse investors
• Homogeneous expectations
• Concerned with nominal returns home currency
• Ability to borrow & lend unlimited amounts at the risk-free 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

FCRP = {[E(S1) - F] / S0} = [{E(S1) - S0} / S0] - (rDC - rFC)

ICAPM risk-pricing relationship specifies expected return as function of investor’s domestic risk-free rate, world market risk premium, and sensitivity of the asset to changes in all foreign currencies.

E(R) = RF +(βg × MRPg) + sum(γk × FCRPk)

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 +

J-curve effect(traditional approach) decline in a currency’s real exchange rate will improve the country’s competitiveness

• in short-run, cost of imports ↑ -> domestic inflation -> real income ↓ -> domestic demand and production ↓ -> real GDP ↓
• in long-run, improved international competitiveness, export demand ↑

Money demand model real economic activity -> domestic currency demand ↑ - > currency value ↑ -> GDP ↑; positive long-run 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 risk-free asset and the optimal risky portfolio.
• Rebalance the portfolio as capital market forecasts and the investor’s risk aversion changes.

Treynor-Black 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:
1. Develop capital market expectations for the passively managed market index portfolio.
2. Identify a limited number of mispriced securities using security analysis.
3. Determine weightings across the mispriced securities to form an actively managed portfolio.
4. Determine weightings to the actively managed portfolio and to the passively managed market index to form an optimal portfolio combination of the two.
5. Allocate funds to the risk-free 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:

1. Collect the time-series alpha forecasts for the analyst.
2. Calculate the correlation between the alpha forecasts and the realized alphas.
3. Square the correlation to derive the R2.
4. Adjust (shrink) the forecast alpha by multiplying it by the analyst’s R2.

## Taxation

 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: FVIFIT = [1 + R(1 ?T1)]N

Deferred capital gains tax (MV = cost basis): FVIFCGT = [(1 + R)N(1 ?TCG) + TCG]

Deferred capital gains tax (MV ≠ cost basis): FVIFCGBT = [(1 + R)N(1 ?TCG)] + TCGB

Wealth-based tax: FVIFWT = [(1 + R)(1 ? TW)]N

Return after realized taxes: RART = R(1 ? P1T1 ? PDTD ? PCGTCG)

For the annual taxes already paid

Effective capital gains tax rate: TECG = TCG(1 ? P1 ? PD ? PCG) / (1 ? P1T1 ? PDTD ? PCGTCG)
Future accumulation factor: FVIFT = [(1 + RART)N (1 ? TECG)] + TECG ? (1 ? B)TCG

Accrual equivalent after-tax return: annual return that produces the same terminal value as the taxable portfolio moves closer to the pre-tax return as the time horizon increases and as more of the portfolio return is deferred

Accrual equivalent tax rate: TAE = 1 ? (RAE/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 wealth-based 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 (1-t) portion of the investment risk, taxable portion assumed by the government. Investor bears all the investment risk for a TDA and tax-exempt 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.

• Traders: short-term 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 long-term and taxed at preferential rates when realized
• Exempt investors: all of stock held in tax-exempt 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 highest-in, first-out (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 after-tax basis
• The mean-variance optimization should optimally allocate assets and determine the optimal asset location for each asset.
• Accrual equivalent after-tax returns would be substituted for before-tax returns and after-tax risk would be substituted for before-tax risk.

## Portfolio Management Planning

Portfolio perspective portfolio managers should analyze the risk-return 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, long-term 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 longer-term, less liquid, higher-risk 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:

• Promote long-term discipline for portfolio decisions
• Help insure against short-term 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 long-term 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 buy-and-hold 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
• Semi-active, risk-controlled 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 risk-return characteristics to be included so that the portfolio’s expected risk-return profile is well-understood

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