Portfolio Management - Cfa Level Ii

Front 1

Mean-variance analysis

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

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

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Correlation formula

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p1,2 = Cov1,2

-------------

σ1 σ2

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Variance of a two asset portfolio

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with covariance:

σ2 = w1^2 *σ1^2 + w2^2*σ2^2 + 2*w1*w2*covariance(1,2)

with correlation:

σ2 = w1^2 *σ1^2 + w2^2*σ2^2 + 2*w1*w2*p1,2σ1σ2

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Expected return on a 2 asset portfolio

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E(Rp) = w1E(R1) + w2E(R2)

where E(Rp) is the expected return on portfolio P

w = weighting of that asset

E(R) expected return on that asset

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Efficient portfolio

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

Front 6

Minimum-variance portfolios

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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
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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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

Y = a +bX

E(Rc) =

[E(Rt) - Rfr]

RfR + --------------------- x σc

σt

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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(RA) =

[E(RM) - Rfr] x σA

RfR + ---------------

σM

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

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Equally weighted portfolio risk

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Front 13

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

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

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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 =

Covi,m pi,mσiσm σi

---------- = ------------- = pi,m x --------

σm^2 σm^2 σm

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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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multi-factor models could also address: interest rate movements, inflation, or industry-specific returns

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

Front 31

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 36

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

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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 41

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

Front 42

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

Front 43

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

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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…

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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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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 50

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”

Front 51

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 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

Front 55

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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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 = annualized residual return 𝛂

--------------------------------------- = ------

annualized residual risk w

IR = IC x √BR

Front 57

Information Coefficient

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measures managers forecasting accuracy

if a manager makes N bets on the direction of the market and Nc are correct, the IC is the covariance between forecast and actual direction of the market

IC = Ncorrect

2x ( ------- ) -1

N

when we add another source of information that is correlated, the skill (IC) of the manager does not increase proportionately. ICcom represents the new info.

ICcom = ICorig x √(2/1+r)

where r = correlation

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ex-post information ratio

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related to the t-stat one obtains for alpha in the regression of portfolio excess returns against benchmark excess returns:

tα t statistic of alpha

-----

√n number of years of data

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Value added

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Objective of active management is to maximize value added

VA = α - (λ x ω^2)

λ = risk aversion

ω = residual risk

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Highest achievable value added *

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function of optimal level of residual risk and the portfolio managers IR

VA* = ω* x IR

---------- or

2

VA* = IR^2

---------- or

4λ

VA* = IC^2 x√BR

-------------

4λ

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Breadth

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# of forecasts made in a year

IR = IC x √BR

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Optimal level of residual risk

ω*

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ω* = IR IC x √BR

----- = ------------

2λ 2λ

λ = risk aversion

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Steps towards an active portfolio include

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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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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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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

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Systematic Risk

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reflects factors that have general effect on the securities market as a whole and cannot be diversified away

for example macroeconomic risk

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Unsystematic risk

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can be reduced through diversification

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The single factor market model covariance calculation

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One of the predictions of the single-factor market model is that Cov(Ri,Rj) = bibjsM2. In other words, the covariance between two assets is related to the betas of the two assets and the variance of the market portfolio.