l2 flash cards portfolio management - ss 18

Study Session 18, Reading 54

Mean Variance AnalysisMean-variance analysis - used to identify optimal or efficient

portfolios. We use the expected returns, variances, and covariance’s of individual investment returns


Assumptions underlying Mean Variance Analysis

1. All investors are risk averse (ie they prefer less risk to more for the same level of expected return)

2. Expected returns for all assets are known3. The variance and covariance of all asset returns are known4. Investors only need to know the expected returns, variances,

and covariance’s of returns to determine optimal portfolios. They can ignore skewness, kurtosis, and other attributes of a distribution.

5. There are no transaction costs or taxes


Minimum Variance Frontierminimum-variance frontier - the border of a region representing all

combinations of expected return and risk that are possible (the border of the feasible region).


Minimum Variance Frontier(cont.)minimum-variance portfolio - one that has the smallest variance among all

portfolios with identical expected return

Steps in getting minimum-variance frontier :1. Estimation step2. Optimization step

Formula:

1. The portfolio weights sum to 100%:


The Efficient Frontierefficient frontier - the portion of the minimum-variance frontier

beginning with the global minimum-variance portfolio and continuing above it

Provides the maximum expected return for a given level of variance

Represents all combinations of mean return and variance or standard deviation of return

Investor’s portfolio selection task is greatly simplified


The Efficient Frontier(cont.)

Qualities of an efficient portfolios:Minimum risk of all portfolios with the same expected return.Maximum expected return for all portfolios with the same

risk.


Instability in Minimum Variance Frontier

Challenges in the instability of the minimum variance :Greater uncertainty in the inputs leads to less reliability in the

efficient frontierStatistical input forecasts derived from historical sample often

change over time which leads to a shifting of the efficient frontierSmall changes in statistical inputs can cause large changes in the

historical frontier resulting in unreasonably large short positions and frequent rebalancing


Calculations related to the Mean Variance Frontier

Formula: Expected return on a portfolio of two assets

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

Where: E(RP) - expected return on a portfolio P

Wi - proportion (or weight) of the asset allocated to Asset i

E(Ri) - expected return on Asset i


Calculations related to the Mean Variance Frontier (cont.)

Formula: Variance of a portfolio of two assets

VARp2 = w1

2 VAR12 + w2

2 VAR22 + 2w1w2 VAR1 VAR2

Where: VARp - variance of the return on the portfolio

wi - proportion (or weight) of the asset allocated to Asset i

VARi - variance of the return on Asset i


Calculations related to the Mean Variance Frontier (cont.)

Formula: Correlation between two assets

Corr1,2 = Cov1,2 /( VAR1 * VAR2)

Where: Corr1,2 - correlation between two assets

Cov1,2 - covariance between two assets

VARi - variance of the return on Asset i


Effect of Correlation on Portfolio Diversification

Diversification - to the strategy of reducing risk by combining many different types of assets

When the correlation between the returns on two assets is less than +1, the potential exists for diversification benefits.

As the correlation between two assets decreases, the benefits of diversification When two assets have a correlation of -1, a portfolio of the two assets exists

that eliminates risk (is risk free).If the correlation between two assets declines, the efficient frontier improves.


Effect of Number of Assets on Portfolio Diversification

Diversification benefits increase as the number of assets increases.

Portfolio risk will fall at a decreasing rate, as the number of assets included in the portfolio rises.

The standard deviation of a large, well-diversified portfolio will get closer and closer to the broad market standard deviation as the number of assets in the portfolio increases.


Equally Weighted Portfolio RiskFormula: Variance of an equally-weighted portfolio

VARp2 = (1/n)* VARi

2 + {(n-1)/n}* COV

Where : VARp - variance of the return on the portfolio

n - number of assets in the portfolio

COV - average covariance of all pairings of assets in a portfolio

Portfolio variance is affected by the number of assets in a portfolio and the correlation between the assets


Capital Allocation Line (CAL)capital allocation line (CAL) - describes the combinations of expected

return and standard deviation of returns available to an investor from combining the optimal portfolio of risky assets with the risk-free asset


Capital Allocation Line EquationFormula:E(Rc) = Rf + (E(RT) – Rf)* STDEVc

STDEVT

Where: E(Rc) - expected return on an investment combination

Rf - risk free rate of return

E(RT) - expected return on the optimal risky portfolio

STDEVc - standard deviation of the combination portfolio

STDEVT - standard deviation of the optimal risky portfolio


Capital Market LineCapital Market Line (CML) - capital allocation line in a world in which all investors

agree on the expected returns, standard deviations, and correlations of all portfolio risk will fall at a decreasing rate, as the number of assets included in the portfolio rises.

Formula:E(Rc) = Rf + (E(RM) – Rf)* STDEVc

STDEVM

Where: E(Rc) - expected return on an investment combination

Rf - risk free rate of return

E(RM) - expected return on the market portfolio

STDEVc - standard deviation of the combination portfolio

STDEVM - standard deviation of the market portfolio


Capital Asset Pricing Model (CAPM)

Describes the expected relationship between risk and return for individual assets.

Expresses returns as a function of beta, thus simplifying risk return calculations

Provides a way to calculate an asset’s expected based on its level of systematic risk, as measured by the asset’s beta.


Security Market Line (SML)Security Market Line (SML) - graph of the CAPM representing the

cross-sectional relationship between the expected return for individual assets and portfolios and their systematic risk. The intercept equals the risk free rate and the slope equals the market risk premium.


Security Market Line (SML) (cont.)

Security Market Line (SML) Equation:

E(Ri) = RF + βi[E(RM – RF)]

Where: E(Ri) - expected return on the asset

RF - risk free rate of return

βi - beta of the asset

E(RM – RF)]- expected risk premium


CAPM equation

The beta for a stock is the ratio of its standard deviation to the standard deviation of the market multiplied by its correlation with the market


CAPM equation (cont.)

Market risk premium equals the expected difference in returns between the market portfolio and the risk-free asset.


Differences between the SML and CML

The SML uses systematic (non diversifiable risk) as a measure of risk while the CML uses standard deviation (total risk)

SML is a tool used to determine the appropriate expected (benchmark) returns for securities while the CML is a tool used to determine the appropriate asset allocation (percentages allocated to the risk-free asset and to the market portfolio) for the investor.

Then SML is a graph of the capital asset pricing model while the CML is a graph of the efficient frontier.

The slope of the SML represents the market risk premium while the slope of CML represents market portfolio Sharpe ratio.


The Market Model market model - regression model used to estimate betas. It assumes two types of risk:

macroeconomic (systematic) or firm specific (unsystematic) risksFormula:

Ri = αi + βi*RM + εi

Where: Ri - return on Asset i

RM - return on the market Portfolio M

αi - intercept (the value of Ri when RM equals zero)

βi - slope (estimate of the systematic risk for Asset i)

εi - regression error with expected value equal to zero (firm-specific surprises)


Underlying Assumptions of the Market Model

The expected value of the error term is zero.The errors are uncorrelated with the market return.The firm-specific surprises are uncorrelated across assets.


Market Model PredictionsThe expected return on Asset i depends only on the expected return

on the market portfolio, E(RM), the sensitivity of the returns on Asset i to movements in the market, βi, and the average return to Asset i when the market return is zero, αi.

The variance of the returns on Asset i consists of two components: a systematic component related to the asset’s beta, βi σM , and an unsystematic component related to firm-specific events.

The covariance between any two stocks is calculated as the product of their betas and the variance of the market portfolio.


Application of the Market Model

Simplify the calculation for estimating the covariances To trace out the minimum-variance frontier with n assetsCorrelation between the returns on two assets


Calculation of Adjusted and Historical Beta

Historical beta is calculated by the use of the historical regression estimate derived from the market model.

Often some adjustments are made to the historical beta to improve its ability to forecast the future beta.

Adjusted beta is a historical beta adjusted to reflect the tendency of beta to mean revert (towards one).

An adjusted beta tends to predict future beta better than historical beta does.


Multifactor ModelsDescribe the return of an asset in terms of the risk of the asset

with respect to a set of factors.Include systematic factors, which explain the average returns

of a large number of risky assets.

Categories: macroeconomic factor models fundamental factor models statistical factor models


Macroeconomic factor modelsIt assume that asset returns are explained by surprises in

macroeconomic risk factorsThe main features are systematic and priced risk factors and

factor sensitivities.Investors will be compensated for bearing priced risk factors. Different assets have different factor sensitivities to the

priced risk factors defined above.


Macroeconomic factor models (cont.)

Formula: Return for stocks using macroeconomic model

Formula: Return on portfolio using two-factor macroeconomic factor mode


Fundamental factor modelsIt assume that asset returns are explained by multiple firm

specific factors.Sensitivities are not regression slopes. Instead, the

sensitivities are standardized attributesThe fundamental factors are rates of return associated with

each factor


Statistical factor modelsApplied to a set of historical returns to determine factors that

explain historical returns.

Two primary statistical factor models: factor analysis models - the factors are the portfolios that best

explain (reproduce) historical return covariances. principal-components models - the factors are portfolios that best

explain (reproduce) the historical return variances.


Arbitrage Pricing Theory (APT)An equilibrium asset-pricing k-factor model which assumes

no arbitrage opportunities exist. Describes the expected return on an asset (or portfolio) as a

linear function of the risk of the asset with respect to a set of factors.

Makes less-strong assumptions.


Assumptions of APTReturns are derived from a multifactor model.Unsystematic risk can be completely diversified away.No arbitrage opportunities exist

arbitrage opportunity - an investment opportunity that bears no risk, no cost, and yet provides a profit


APT Equation


Differences between APT and Multifactor Models

Arbitrage Pricing Theory (APT) models look similar to multifactor models

While APT models are equilibrium models, multifactor models are statistical regressions

APT models explain the results over a single time period as functions of different factors, while multifactor models are based on data from multiple time periods


Active Risk and Return, Information Ration

Active return - return in excess of the return of the benchmarkFormula:

Active Return = RP – RB

Active risk - the standard deviation of active returns.Components: Active factor risk Active specific risk

Information ratio standardizes the return achieved by a portfolio manager by dividing the return with the standard deviation of the return.


Factor and Tracking Portfolios

pure factor portfolio (or simply a factor portfolio) - a portfolio that has been constructed to have a sensitivity equal to 1.0 to only one risk factor, and sensitivities of zero to the remaining factors.

tracking portfolios - have a deliberately designed set of factor exposures. That is, a tracking portfolio is deliberately constructed to have the same set of factor exposures to match (“track”) a predetermined benchmark.


Implications of CAPM assumptionsTwo key assumptions necessary to derive the CAPM:

Investors can borrow and lend at the risk-free rate.Unlimited short selling is allowed with full access to short sale proceeds.

Two major implications of the CAPM:The market portfolio lies on the efficient frontier.There is a linear relationship between an asset’s expected returns and

its beta.

If these assumptions don’t hold, then:The market portfolio might lie below the efficient frontier.The relationship between expected return and beta might not be linear.


Impediments to Market IntegrationPsychological barriersLegal restrictionsTransaction costsDiscriminatory taxationPolitical risksForeign currency risk


Factors favouring Market Integration

There are many private and institutional investors who are internationally active.

Many major corporations have multinational operations.Corporations and governments borrow and lend on an

international scale.


Extended CAPMextended CAPM - domestic CAPM extended to the international

environment is called theThe risk-free rate (Rf) is the investor’s domestic risk-free rate, and the

market portfolio is the market capitalization-weighted portfolio of all risky assets in the world

Assumptions needed to extend CAPMInvestors throughout the world have identical consumption baskets. Purchasing power parity holds exactly at any point in time.


ICAPM EquationE(r)= Rf +(βg×MrPg )+(g1×FcrP1)+(g2×FcrP2 )+...........+(gk ×FcrPk )

Where: E(r) - asset’s expected return

Rrf - domestic currency risk-free rate

βg - sensitivity of the asset’s domestic currency returns to changes in the global market portfolio

MrPg - world market risk premium [E(rm ) - r ]

E(r m) - expected return on world market portfolio

g1 to gk - sensitivities of asset’s domestic currency returns to changes in the values of currencies 1 through k FcrP1 to FcrPk - foreign currency risk premiums on currencies 1 through k


Change in the Real Exchange RateThe real exchange rate is the spot exchange rate, S, multiplied by the ratio of the consumption basket price levels Formula:

X = S × (PFC /PDC)

The expected foreign currency appreciation or depreciation should be approximately equal to the interest rate differential

Formula: E(s) = rDC - rFC,

where: s - percentage change in the price of foreign currency (direct exchange rate)


Foreign Currency Risk Premium (FCRP)Foreign Currency Risk Premium (FCRP) i- s the expected

exchange rate movement minus the (risk free) interest rate differential between the domestic currency and the foreign currency


Expected Return on Foreign Investments

Formula: Expected return on an unhedged foreign investment

E(R) =E(RFC) + E(s)

Where: E(R) - Expected domestic currency return on the investment

E(RFC) - Expected foreign investment return

E(s) - Expected percentage currency movement


Expected Return on Foreign Investments (cont.)

Formula: Expected return on an hedged foreign investment

E(R) =E(RFC) + (F-S)/S

Where: E(R) - Expected domestic currency return on the investment

E(RFC) - Expected foreign investment return

F - Forward rate in direct quotes

S - Spot rate in direct quotes


Currency Exposurelocal currency exposure – the sensitivity of the returns in the stock denominated

in the local currency to changes in the value of the local currency

domestic currency exposure - because the exposure of a currency to itself is 1, domestic currency exposure is equal to local currency exposure plus 1.


Exchange Rate Exposure

exchange rate exposure – the way the value of an individual company changes in response to a change in the real value of the local currency

We can estimate the currency exposure of a particular firm by regressing the firm’s stock return on local currency changes.


Economic activity and exchange rate movements

Two theories to explain the relationship between economic activity and exchange rate:

1. traditional model - predicts that depreciation in the value of the domestic currency will cause an increase in the competitiveness of the domestic industry and, thus, an increase in the stock value of domestic firms

2. money demand model - an increase in real economic activity leads to an increase in the demand for the domestic currency


Active portfolio management

active portfolio management - refers to decisions of the portfolio manager to actively manage and monitor the broad asset allocation and security selection of the portfolio. Equilibrium is the desirable end result of active portfolio management.


Justification of active portfolio management

Develop capital market forecasts for major asset classesAllocate funds across the major risky asset classes to form the optimal risky

portfolio that maximizes the reward-to-risk ratio.Allocate funds between the risk-free asset and the optimal risky portfolio in

order to satisfy the investor’s risk aversion.Rebalance the portfolio as capital market forecasts and investor’s risk

aversion changes (also known as market timing)


Treynor Black ModelTreynor-Black model - a portfolio optimization framework that combines

market inefficiency and modern portfolio theory. The model is based on the premise that markets are nearly efficient.

Objective: To create an optimal risky portfolio that is allocated to both a passively managed (indexed) portfolio and to an actively managed portfolio

Formula:


Adjustments in Treynor Black Model

Collect the time-series alpha forecasts for the analystCalculate the correlation between the alpha forecasts and the

realized alphasSquare the correlation to derive the R2

Adjust (shrink) the forecast alpha by multiplying it by the analyst’s R2


The Portfolio Management ProcessImportant features:1. The process is ongoing and dynamic (there are no end points, only feedback to previous steps).2. Investments should be evaluated as to how they affect portfolio risk and return characteristics.

Phases:1. Planning2. Execution3. Feedback


Investment Constraints1. Liquidity constraints - relate to expected cash outflows that will be needed at some

specified time and are in excess of available income2. Time horizon constraints - associated with the time period(s) over which a portfolio is

expected to generate returns to meet specific future needs 3. Tax constraints - depend on how, when, and if portfolio returns of various types are taxed 4. Legal and regulatory factors - usually associated with specifying which investment classes

are not allowed or dictating any limitations placed on allocations to particular investment classes.

5. Unique circumstances - internally generated and represent special concerns of the investor


Investment Policy Statement (IPS)investment policy statement (IPS) - a formal document that governs

investment decision making, taking into account objectives and constraints.

Main role of the IPS: Be readily implemented by current or future investment advisers Promote long-term discipline for portfolio decisions. Help protect against short-term shifts in strategy


Elements of the IPSA client description Identification of duties and responsibilities of parties involved.The formal statement of objectives and constraints.A calendar schedule for both portfolio performance and IPS review.Asset allocation ranges and statements regarding flexibility and rigidity

when formulating or modifying the strategic asset allocation.Guidelines for portfolio adjustments and rebalancing.


Strategic Asset AllocationStrategic asset allocation is the final step in the planning stage.

Common Approaches to Strategic Asset Allocation1. Passive investment strategies – represent strategies that are not responsive

to changes in expectations2. Active investment strategies - attempt to capitalize on differences between

a portfolio manager’s beliefs concerning security valuations and those in the marketplace.

3. Semi-active, risk-controlled active, or enhanced index strategies - hybrids of passive and active strategies


Factors affecting Strategic Asset Allocation

1. Risk-return2. Capital market expectations3. The length of the time horizon

Affect of Time Horizon:The longer the investment time horizon, the more risk an

investor can take on

l2 flash cards portfolio management - ss 18

Business