Portfolio Management - Cfa Level Ii

Front 1

Mean-variance analysis

Back 1

based on the idea that the value of investment opportunities can be meaningfully measured in terms of mean return and variance of return

Front 2

Mean-variance analysis assumptions

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All investors are risk averse; they prefer less risk to more for the same level of expected return (not all investors have the same level of tolerance though)

Expected returns for all assets are known

The variances and covariances of all asset returns are known

Investors need only know the expected returns, variances, and covariances of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes of distribution

There are no transaction costs or taxes

Front 3

Efficient portfolio

Back 3

one offering the highest expected return for a given level of risk as measured by variance or standard deviation of return

Front 4

Minimum-variance portfolios

Back 4

portfolios that have minimum variance for each given level of expected returns; the set of efficient portfolios is a subset of the set of minimum variance portfolios

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Minimum-variance frontier

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a curve that represents the minimum variance (risk) that can be achieved for a given level of expected return
i. Better than portfolio possibilities curve, can be used for portfolios with more than two assets
ii. As you shift to the right on the frontier what you gain in return, you gain in risk
iii. All the points below the minimum-variance portfolio should be avoided, as they will have a higher variance (risk) with less expected return
iv. Efficient frontier – is only the positively sloped part of the minimum-variance frontier, from the global minimum variance portfolio and up and to the right

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To solve for the Minimum-variance frontier

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Calculated the range of possible expected returns – minimum and maximum

Calculate the proportion of each of the two assets (asset weights) in the minimum-variance portfolio for each possible level of expected return

Calculate the variance for each possible level of expected return to find the optimal combination: lowest variance for highest return

Solve for the weights to minimize variance with the only restriction being all weights sum to 1 (this means you can short an asset, would have to add wj>0 to eliminate)

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Impact of increasing the # of assets within a portfolio

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1) you can improve the risk-return trade-off by expanding the set of assets in which we can invest

2) the composition of the minimum-variance portfolio for any particular level of expected returns depends on the expected returns, the variances and correlations of those returns, and the number of assets

Front 8

Benefits of diversification and how the correlation in a two-asset portfolio and the number of assets in a multi-asset portfolio affect the diversification benefits

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Correlation has a significant impact on the risk-return trade-off among assets

While the end points and starting points are all the same for each correlation level, the path between those points varies dramatically – to “more curvy” near -1 and straight lines at -1 and +1

At +/-1 correlation, the return on one asset is an exact positive/negative linear function of the return on the other asset, because of this the returns on one asset cannot dampen or smooth out fluctuations of another – at +/-1 correlation, diversification has no potential benefits

In between +/-1 correlation, diversification is achieved – as correlation approaches -1, the potential benefits of diversification increase

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Capital allocation line (CAL)

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describes the combinations of expected return and standard deviation of return available by combining an optimal portfolio of risky assets with the risk-free asset; the graph of this starts at the intersection of the RFR return and is tangent to the efficient frontier of risky assets – the line itself represents an optimal portfolio of risky assets

An individual investor’s risk tolerance will determine where on the CAL they will prefer to be

CAL is the maximum slope from the minimum variance portfolio to 100% risky portfolio

If we can invest in a RF asset, CAL represents the best risk-return trade-off achievable

The CAL has a y-intercept = to the RFR

The CAL is tangent to the efficient frontier of risky assets

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Capital market line (CML)

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when investors share identical expectations about mean returns, variance of returns, and correlations of risky assets; when the CAL is the same for all investors

CML equation = E(Rp) = RFR + ((E(market return) – RFR) * sd of port) / sd of the market

The slope of this line equation = the market price of risk, because it indicates the market risk premium for each unit of market risk

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Relationship between CAL and CML

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CML is when the CAL is the same for all investors

Front 12

CAPM underlying assumptions

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Investors need only know the expected returns the variances, and the covariances of returns to determine which portfolio are optimal for them (proven by mean-variance theory)

Investors have identical views about risky assets’ mean returns, variances of returns and correlations

Investors can buy and sell assets in any quantity without affecting price, and all assets are marketable (can be traded)

Investors can borrow and lend at the RFR without limit, and they can sell short any asset in any quantity

Investors pay no taxes on returns and pay no transaction costs on trades

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CML and CAPM

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CML represents the efficient frontier when the assumptions of the CAPM hold

CAPM = E(Ri) = RFR + Beta * (E(R of Market) – RFR)

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Security market line (SML)

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is the graph of the CAPM model, or the CAPM equation

is a linear function of beta

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Beta definition as it relates to the market

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Beta is a measure of the asset’s sensitivity to movements in the market

Beta = Cov(Ri, Rm) / Var(Rm)

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Market risk premium

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E(Rm) - RFR

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Sharpe Ratio

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the ratio of mean return in excess of the RFR to the standard deviation of return

Sharpe Ratio = (E(Ri) – RFR) / sd of Asset i

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Adding assets to the portfolio and the Sharpe Ratio

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Adding a new asset to your portfolio is optimal if the following condition holds:

1) E(Rnew) – RFR / sd of new > (E(Rport) - RFR / sd of port) * Corr (Rnew, Rport)

2) As long as sharpe of new asset is greater than sharpe of portfolio, (i.e. Corr = 1), should add

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Market Model

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describes a regression relationship between the returns on an asset and the returns on the market portfolio

Ri = alpha’i’ + beta’i’ * Rm + error’i’

Where alpha is the average return on asset’i’ unrelated to the market return

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Market Model assumptions

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The expected value of the error term is 0

The market return is uncorrelated with the error term, Cov(Rm, error) = 0

The error terms are uncorrelated among different assets

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Market Model predictions

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Expected return for asset ‘i’ depends on the expected return to the market, E(Rm), the beta, and the alpha

Variance of the return to asset ‘i’ depends on variance of the return to the market, variance of the error term for asset ‘i’, and beta

Covariance of the return to assets ‘i’ and ‘j’ depends on the variance of the return to the market, and the sensitivities of each asset

Correlation of returns for assets ‘i’ and ‘j’ = Cov(Ri, Rj) / square root of Var(Ri) * square root of Var(Rj)

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Adjusted beta

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if historical beta is not deemed to be the best predictor, can use adjusted beta

Adjusted beta uses instead a first order autoregression: Bt+1 = alpha initial + alpha 1 * Bt + error

To simply, given mean reverting tendencies, alpha initial = 1/3 and alpha 1 = 2/3

Adjusted Beta = (1/3) + (2/3)*(Beta t)

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Historical beta

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assumes that beta for each stock is a random walk from one period to the next, and the error term mean is “0” – so Beta t+1 = Beta t + error (or 0)

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Reasons for and problems related to instability in the minimum-variance frontier

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Small changes in input assumptions can lead to large changes in the minimum-variance (and efficient) frontier, because uncertainty exists about the expected returns, variances, and covariances used in tracing out the minimum-variance frontier

To avoid/respond to instability:
1) Add constraints against short sales
2) Improve the statistical quality of inputs to optimization
3) Reflect the fact that the inputs to optimization are random variables rather than constants

If is unstable when calculated using historical data for difference time periods – time instability exists

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Multi-factor model

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factoring in additional factors other than market returns to explain asset returns – market model is market return only, multi-factor models could also address: interest rate movements, inflation, or industry-specific returns

Multi-factor models are used in portfolio management, risk analysis, and the evaluation of portfolio performance; have gained importance as they explain asset returns better than the market model, and they provide a more detailed analysis of risk than does a single factor model

Ri = ai + bi1 * F1 +…+ bik * Fk + error

Where ai is the expected return on Asset i, bi’s are the sensitivities of each factor, and F’s are the surprise in each factor

Factor sensitivity = a measure of the response of return to each unit of increase in a factor, holding all other factors constant

Error is the part that is unexplained by the expected return and factor surprises, is therefore defined as: asset-specific risk

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Macroeconomic factor models

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the factors are surprise in macroeconomic variables that significantly explain equity returns; can affect either the expected future cash flows of companies or the interest rate used to discount these cash flows back to the present

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Fundamental factor models

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factors are attributes of stocks or companies that are important to explaining cross-sectional differences in stock prices (factors that have been used: P/B, Market Cap, P/E, and financial leverage)

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Statistical factor models

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statistical methods are applied to a set of historical returns to determine portfolios that explain historical returns in one or two senses (less used)

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Factor analysis models

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the factors are the portfolios that best explain (reproduce) historical return covariances

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Principal –components models

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the factors are portfolios that best explain (reproduce) the historical return variances

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Surprise in multi-factor models

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the actual value minus the predicted (or expected) value

Front 32

Arbitrage pricing theory (APT)

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is an alternative to CAPM and describes expected return on an asset (or portfolio) as a linear function of the risk of the asset (or portfolio) with respect to a set of factors. Like CAPM, APT describes a financial market equilibrium, but makes less-strong assumptions than CAPM

States that the expected return on a well-diversified portfolio is linearly related to the factor sensitivities of that portfolio

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Assumptions of Arbitrage pricing theory (APT)

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A factor model describes asset returns

There are many assets, so investors can form well-diversified portfolios that eliminate asset-specific risk

No arbitrage opportunities exist among well-diversified portfolios

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Factor risk premium (or factor price)

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the expected return in excess of the RFR for a portfolio with a sensitivity of 1 to that factor and 0 to all other factors

Termed a pure factor portfolio

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APT compared to multi-factor model

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APT relates to multi-factor models, in that its intercept is the expected return of an asset in equilibrium vs. a multi-factor model intercept as just the expected return of an asset

If all variables are not given, can solve for RFR and set two equations equal to one another to find the other variable, and then plug that answer in to find RFR

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APT compared to Fundamental factor model

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uses the same equation structure, but factors are stated as returns rather than surprises, the expected return intercept has a different value/meaning, and the sensitivities are specific to the asset and standardized

Standardized beta = (asset i’s attribute value – avg attribute value) / sd of attribute values

Front 37

Active return

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return on portfolio – return on the benchmark (comparable to the portfolio)

split into two components: active factor sensitivities (sector weightings) and asset selection (stocks per sector)

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Active risk

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the standard deviation of active returns

Front 39

Active factor risk

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the contribution to active risk squared resulting from the portfolio’s different-than-benchmark exposures relative to factors specified in the risk model

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Active selection risk or Active specific risk

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the contribution to active risk square resulting from the portfolio’s active weights on individual assets as those weights interact with assets’ residual risk = sum of weight differences and variances of the asset’s returns unexplained by factors

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Tracking error

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synonym with active risk, but the term “error” is confusing as it is meant to represent “difference” here

Front 42

Tracking risk

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also a synonym of active risk =
sd * (Rportfolio – Rbenchmark)

Make sure the same time periods are used for each return

Can vary from 1% with a passive investment to 6-9% for very active investment management

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Information ratio

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a tool for evaluating mean active returns per unit of active risk

IR = (mean Rportfolio – mean Rbenchmark) / standard deviation * (Rportfolio – Rbenchmark)

Or IR = sample mean active return / tracking risk

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Factor portfolio

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has a sensitivity of 1 for a factor and 0 for all other factors within a multi-factor model; a portfolio with exposure to only one risk factor, exactly representing that risk

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Tracking portfolio

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a portfolio with factor sensitivities that are matched to those of a benchmark or other portfolio, “tracking the benchmark” to control the risk relative to the benchmark

To construct will need to determine the weights of each factor/sensitivity to match the benchmark sensitivity or the desire combination of portfolios to track the benchmark

1) All weights sum to 1
2) Sum of Weights * sensitivities of portfolios = benchmark sensitivity
3) Do this again (for additional sensitivities needed to be tracked), and solve for weights

Front 46

Why an investor can possibly earn a substantial premium for exposure to dimensions of risk unrelated to market movements?

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CAPM provides an incomplete description of risk compared to multifactor models with greater transparency/visibility into the drivers of return

Investors should instead look towards multifactor models to tilt towards the appropriate risks that they can take to improve and individualize portfolio selection –cyclical risk, recession risk…

Front 47

Efficiency of the market portfolio in the CAPM and the relation between the expected return and beta of an asset when restrictions on borrowing at the risk-free rate and on short selling exist

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Since the linear relation between betas and expected returns does not necessarily hold when borrowing is limited and short selling is restricted or not possible, the CAPM risk adjustment is questionable (two CAPM assumptions conflict with one another)

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Practical consequences that follow when restrictions on borrowing at the risk-free rate and on short selling exist

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The relationship between expected return and beta is not linear and that the market portfolio may not be efficient

High risk tolerance investors hold portfolios of risky assets that differ from those held by cautious investors

Risk adjustments using beta may be misleading

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International market integration

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Integrated world financial market would achieve international efficiency, in that capital flows across markets would instantaneously take advantage of any new information throughout the world

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International market segmentation

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Impediments to capital mobility – legal restrictions or other forms of constraints that segment one national market from others
a. Psychological barriers, legal restrictions, transaction costs, discriminatory taxation, political risks, foreign currency risks

International asset pricing: are “similar” securities priced in the same manner on different national markets?

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Factors that favor international market integration

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Private and institutional investors are extensively invested abroad

All major corporations have truly multinational operations – shares listed on multiple indices

Large corporations and governments borrow internationally and take advantage of relative bond mispricing between countries (improving market efficiency)

Flow of foreign investments has grown rapidly, recently– segmentation has mitigated

Whether similar firms are priced identically in their respective national markets…

Front 52

Assumptions of the domestic capital asset pricing model (CAPM)

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Investors care about risk & return; are risk-averse and prefer less risk and more expected return

Consensus among all investors holds; everyone agrees about the expected return & risk of assets

Investors care about nominal returns in their domestic currency

Risk-free interest rate exists, with unlimited borrowing or lending capacity at this rate

There are no transaction costs or taxes

Separation theorem – everyone holds the same portfolio of risky assets and individual investor’s determine the weight of that portfolio with their domestic RFR “separately”

Average beta of all securities is equal to one (the beta of the market portfolio)

Front 53

Separation theorem

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everyone holds the same portfolio of risky assets and individual investor’s determine the weight of that portfolio with their domestic RFR “separately”

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Why an extension of domestic CAPM is needed to fit an International context

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Domestic CAPM in an international context would require the domestic RFR + the market cap weighted portfolio of all risky assets in the world for the market portfolio. This can only be justified when:
i. Investors throughout the world have identical consumption baskets
ii. Real prices of consumption goods are identical in every country; purchasing power parity holds exactly at any point in time

These assumption would suggest that exchange rates would simply mirror inflation differentials between two countries; and that exchange rate uncertainty would not technically exist

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Real exchange rate movements

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are defined as movements in the exchange rates that are not explained by the inflation differential between the two countries

% chg real exchange rate = % chg nominal exchange rate + foreign inflation – domestic inflation

For extended CAPM to hold, there can be no real exchange rate movement; x = 0; appreciation or depreciation must be fully offset by the inflation differential between the two countries

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Foreign currency risk

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is the risk that real prices of consumption goods might not be identical in every country; also called real exchange rate risk, purchasing power risk

With regard to the International CAPM, a US domestic RFR may be risk-free in the US, but bears foreign currency risk in other countries

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Foreign currency risk premium working in concert with interest rate parity

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E(R) – RFR, or the expected movement in the exchange rate less the interest rate differential (domestic RFR – foreign RFR), and after factoring in appreciation/depreciation for the period

Linear approx says that (F – S) / S is approximately equal to RFRdc – RFRfc (interest rate differential) – the best predictor of exchange rates is the interest rate differential

The expected return may be less with currency hedging, as it bears less risk –> difference between expected return no hedging – hedging = foreign currency risk premium

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Risk pricing relation

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when the expected return on any asset is simply a function of its covariance with the domestic market portfolio

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The effect of market segmentation on the ICAPM

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Segmentation occurs when securities that have the same risk characteristics and are listed in two different markets, have different expected returns

Segmentation skews ICAPM for a given security based on biases that cannot be parsed out of the broader market portfolio precisely – may relate to the “safety” of international investments, the tax implications of each country, whether FX hedging is available (physically and/or legally), etc.

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Currency exposure

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defined as the sensitivity of the asset return, measured in the investor’s domestic currency, to a movement in the exchange rate – an international investor measures total return as the sum of returns on assets, in local currencies, plus any currency movements (bearing both market and foreign currency risks)

Local currency exposure =1 + the price return on the FC / % movements in the exchange rate

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Currency exposure in terms of correlations

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Zero correlation btw local currency stock returns and exchange rate movements would mean no systematic reaction to exchange rate adjustments

Negative correlation: means the local stock price benefits from a depreciation of the local currency – so a loss on Swiss currency would be offset partly by a capital gain on the stock price

Positive correlation: means the local stock price drops in reaction to a depreciation of the local currency – foreign asset prices would negatively compound the currency effect; a “bad hedge”

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Likely exchange rate exposure of a company based on the company’s activities, and explain the impact of both real and nominal exchange rate changes on the valuation of the company

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The importance of the exchange rate for an individual firm depends on the currency structure of its exports, imports, and financing – may be beneficial in terms of costs from one country to another, but limited in pricing potential for another portion/geography of the business

Local stock market should react favorably to a depreciation of its currency given the rapid internationalization of many corporations, who gain demand for product/services with a weaker currency; but on a “real” basis, should not greatly influence the “real” value of the company

Front 63

Currency exposures of national economies

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currency changes may mean different things for different countries, in emerging markets correlation is positive, as a drop in the currency signals distress, and vice versa

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Currency exposures of Equity markets

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economic activity is a major driver of stock market returns, so a decline in the currency’s real exchange rate tends to improve competitiveness of that developed country abroad, but also increases the cost of imports, which creates additional domestic inflation, reduces real income and hence, reduces domestic demand and production

However, the initial destruction of GNP should eventually be offset by improved international competitiveness and export demand until purchasing power parity is restored

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Currency exposures of Bond markets

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related to the change in long-term interest rates and exchange rates – bond returns are negative when bond yields rise

A rise in the domestic real interest rate causes appreciation of the domestic currency, as international investment is attracted to the higher rate, resulting in lower bond prices because of the risk in yields; but is important to determine whether the change is rate driven, or simply a change in inflation, as increased inflation could also lead to higher interest rates

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J-curve effect

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The reaction of the deficit to the depreciation is called the j-curve effect, because of the j-shape of trade balance curve as a function of time – if the economy, however, is slow to improve, could result in a vicious cycle: immediate economy activity and trade balance worsen, leading to more depreciation, which in turn may worsen domestic economic conditions, and so on…

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Traditional trade approach

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suggests that a real exchange rate appreciation tends to reduce the competitiveness of the domestic economy, and therefore, reduces domestic activity

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Money-demand model

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proposes that real growth in the domestic economy leads to increased demand for the domestic currency through a traditional money-demand equation; an increase in currency demand induces a rise in the relative value of the domestic currency; therefore domestic stock prices are strongly influenced by real growth

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Justify active portfolio management when security markets are nearly efficient

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The less active management, the less efficient a market becomes – as prices will no longer reflect sophisticated forecasts, thus exposing profit opportunities/outsized return potential, once again luring active management back to the market

Empirical evidence – some managers have produced streaks of abnormal returns that would be hard to describe as lucky; “noise” in realized rates, and suggests managers have beaten the passive strategy by statistically small but still significant margin; persistence in returns related to the timing of buying and selling by some managers have beaten passive strategies

Decent profits to the better analysts should be the rule rather than the exception

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Theory behind the Treynor-Black model for security selection

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TB model represents a portfolio management theory that assumes security markets are nearly efficient – the essence of the model is this:

1) Security analysts in an active investment management organization can analyze in depth only a limited # of stocks, making the assumption that those stocks unanalyzed are fairly priced
2) The market index portfolio is the baseline portfolio, which is dubbed the passive portfolio
3) The macro forecasting unit of the investment management firm provides forecasts of the expected rate of return and variance of the passive (market-index) portfolio
4) Objective of security analysis is to form an active portfolio of a necessarily limited # of securities – perceived mispricing is what guidance these selections
5) Macroeconomic forecasts for the passive and active portfolios are used to determine the optimal risky portfolio, which will be a combination of the passive and the active portfolios

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Steps towards an active portfolio include

Back 71

Estimate the beta of each security and its residual risk, from E(Rm) – RFR determine the securities’ required rate of return

Determine securities’ expected return and its expected abnormal or alpha return

Determine nonsystematic risk of the mispriced stock, the variance of the stock’s residual, which offsets the benefit (alpha) of specializing in an underpriced security

Use estimates for alpha, beta, and variance (error) to determine the optimal weights

Compute alpha, beta, and variance (error) of the active portfolio from the weights of the securities in the portfolio

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Measuring analyst accuracy with Treynor-Black

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Under TB approach, a market portfolio on the CML is deemed inefficient, and instead an active portfolio generating alpha will reside on the CAL, which should “by design” always lie above CML

CAL expected return on A = E(RA) = alphaA + RFR + betaA *(E(Rm) – RFR)

Meanwhile, market portfolio M, is the point of tangency for the CML

Combining the ultimately efficient A with M gets portfolio P, the point of tangency for the CAL

Finding the ultimate weight* in portfolio A =
i. alphaA / (alpha*(1 – betaA) + Rm * (variance of A / variance of market, M))
ii. based on this equation, weight* increases when betaA increases, because the greater systematic risk of the active portfolio A, the smaller is the benefit from diversifying it with the index, M, and the more beneficial it is to take advantage of the mispriced securities

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The importance of the portfolio perspective

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Portfolio perspective underlies the portfolio management process and the investment policy statement – it is the focus on the aggregate of all the investor’s holdings: the portfolio

Developed over time into modern portfolio theory – the analysis of rational portfolio choices based on the efficient use of risk – and is now widely accepted as a how to achieve investment objectives

Helped to spread the use and knowledge of quantitative methods in portfolio management

Today, quantitative and qualitative concepts complement each other in investment management practice

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Steps of the portfolio management process

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Planning: identify and specify the investor’s objectives and constraints (IPS development); strategic asset allocation decisions + forecasts of capital market expectations

Execution: portfolio construction and revision – taking into account IPS + capital market expectations to start the selection of assets; interacting constantly with the feedback step

Feedback: two components: monitoring and rebalancing (investor-related factors or investor circumstances, and economic and market input factors), and performance evaluation (calculate the portfolio’s rate of return, performance appraisal (sources of good and bad)

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Portfolio management process

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is an integrated set of activities that combine in a logical, orderly manner to produce a desired product – the process is dynamic and flexible, meant to be a one-size fits all, for any investment type; it should be a continuous and systematic complete with feedback loops for monitoring and rebalancing

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Investor Objectives

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desired investment outcomes, pertaining to return (relative is suggested) and risk

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Investor Constraints

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limitations on the investor’s ability to take full or partial advantage of particular investments, are either internal (client’s liquidity needs, time horizon, and unique circumstances) or external (tax issues and legal and regulatory requirements)

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Risk Tolerance

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the capacity to accept risk, is both the investor’s willingness and ability to do so

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Purpose of a Investment policy statement (IPS)

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serves as the governing document for all investment decision-making; may cover additional issues/guidelines beyond objectives and constraints

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Elements of IPS

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i. Brief client description
ii. Purpose of establishing policies and guidelines
iii. Duties and investment responsibilities or parties involved (related to fiduciary duties, communication, operational efficiency, and accountability); parties involved include the client, any investment committee, the investment manager, and the bank custodian
iv. Statement of investment goals, objectives, and constraints
v. Schedule for review of investment performance as well as the IPS itself
vi. Performance measures and benchmarks to be used in performance evaluation
vii. Any considerations to be taken into account in developing the strategic asset allocation
viii. Investment strategies and investment style(s) – must clearly state the basis for investment decisions and guides those decisions toward achieving investment objectives
ix. Guidelines for rebalancing the portfolio based on feedback

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Asset allocation included in the IPS

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requires the examination of the interaction of objectives and constraints with long-run capital market expectations; the planning process involves concrete elaboration of an investment strategy either: passive (or indexing or strict buy and hold), active (holdings differ from the benchmark; looking to produce alpha), or semi-active (risk controlled and somewhat indexed)

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Capital market expectations

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include long-run risk and return forecasts for various asset classes

form the basis for choosing portfolios that maximize expected return for a given level of risk

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Strategic allocation

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combines the IPS and capital market expectations to determine target asset class weights – in single and/or multi-period perspectives; single period having the advantage of simplicity and multi-periods addressing liquidity and tax considerations that arise from rebalancing portfolios over time, as well as serial correlations (long- and short-term dependencies) in returns, but is more costly to implement

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Short-term investment horizon

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decreased risk-taking availability; typically when the investor faces unanticipated short-term liquidity needs

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Long-term investment horizon

Back 85

10 or more years

the longer the time horizon the more risk the investor can take on, and the greater the investor’s ability to replenish investment resources by increasing savings

Should average results over several markets and business cycles

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Multi-period investment horizon

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a combo of short-term and long-term

“funding children’s education shorter term and the investor’s retirement longer term”

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Professional standards (two types) for managing investment portfolios

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standards of competence and standards of conduct; merely drawing a livelihood from managing or advising on the investment of client monies is insufficient in itself to make an investment professional

Portfolio manager must keep foremost in mind that he or she is in a position of trust, requiring ethical conduct towards the public, client, prospects, employers, employees, and fellow workers