Modern Portfolio Theory

 Modern Portfolio Theory

• Also called as mean-variance analysis
• Created by Harry Markowitz - 1950s (later awarded a Nobel Prize in Economic Sciences)
• It is a mathematical framework for assembling a portfolio of assets such that the expected return is maximized for a given level of risk.
•  It is a formalization and extension of diversification in investing, the idea that owning different kinds of financial assets is less risky than owning only one type. 
• Its key insight is that an asset's risk and return should not be assessed by itself, but by how it contributes to a portfolio's overall risk and return.

Assumptions

• Capital markets are perfect - 
• investors do not pay taxes, commission
• all traders have free access to all the information
• there is perfect competition between market participants
• Investors are rational and risk-averse
• Markowitz defines a rational investor as someone who seeks to maximize utility from investments. Furthermore, when presented with two investment opportunities at the same level of expected risk, rational investors always pick the investment opportunity which offers the highest expected return
• Returns are normally distributed.
• when evaluating utility, investors only consider the mean and the variance of return distributions. They ignore deviations from normality, such as skewness or kurtosis

Gist

• Investors are risk averse and wants a higher mean return( performance) and a lower variance (Risk). 
• Thus, an investor will take on increased risk only if compensated by higher expected returns. 
• An investor can reduce portfolio risk simply by holding combinations of instruments that are not perfectly positively correlated  
• While portfolio returns are calculated as weighted averages of individual asset returns, portfolio variances depend on the correlations among assets
• If all the asset pairs have correlations of 1—they are perfectly positively correlated - no diversification.
• If all the asset pairs have correlations of 0—they are perfectly uncorrelated - there is diversification. The lower the correlation, the greater the benefit becomes
• By holding a sufficiently large, diversified portfolio, investors are able to reduce, or even eliminate, the amount of company-specific (i.e., idiosyncratic) risk inherent in each individual security

Concerns regarding assumptions of MPT

• return distributions have fat tails and are assymetric
• investors ignore skewness by focusing only on mean and variance
• parameters used to apply the model- mean and variance - are estimated using historical data and it changes significantly over different timeframes.

Efficient frontier

• MPT is a mean-variance theory, and it compares the expected (mean) return of a portfolio with the standard deviation of the same portfolio

• The return - standard deviation space is sometimes called the space of 'expected return vs risk'
• The left boundary of this region is parabolic, and the upper part of the parabolic boundary is the efficient frontier in the absence of a risk-free asset (sometimes called "the Markowitz bullet").
• A portfolio lying on the efficient frontier represents the combination offering the best possible expected return for given risk level. 
• leftmost point of the efficient frontier is called as the global minimum variance portfolio
• any portfolio below the efficient frontier is, by definition, inefficient, whereas any portfolio above the efficient frontier is unattainable

 Capital market line

• Investors will combine the risk-free asset with a specific efficient portfolio that will maximize their risk-adjusted rate of return. Thus, investors obtain a line tangent to the efficient frontier whose y-intercept is the risk-free rate of return
• The tangent to the upper part of the parabolic boundary is the capital allocation line (CAL).
• The slope of the CML shows the market price of risk for efficient portfolios. slope of the CML is equal to the Sharpe measure.  Slope of the CML = (R_m – R_f) / σ_m
• One should buy assets if the Sharpe ratio is above the CML and sell if the ratio falls below the CML.
• CML shows the tradeoff between expected return and total risk.
• CML considers both systematic and unsystematic risk.

Capital Asset Pricing Model (CAPM)

• The CAPM is a model that derives the theoretical required expected return (i.e., discount rate) for an asset in a market, given the risk-free rate available to investors and the risk of the market as a whole. 
• developed by William Sharpe, Jan Mossin and John Lintner in the 1960s
• When a risk-free asset is introduced, the half-line shown in the figure is the new efficient frontier. It is tangent to the parabola at the pure risky portfolio with the highest Sharpe ratio. 
• The equation of the CML is:

• P is the sub-portfolio of risky assets at the tangency with the Markowitz bullet
• F is the risk-free asset
• C is a combination of portfolios P and F
• CAPM is usually expressed:  
• E(Rm)-Rf is the market premium, the expected excess return of the market portfolio's expected return over the risk-free rate.
• expected return, 
• The expected return, E(Ri), can be viewed as the minimum required return, or the hurdle rate, that investors demand from an investment, given its level of systematic risk.

Assumptions

• Capital markets are perfect - 
• investors do not pay taxes, commission
• all traders have free access to all the information
• there is perfect competition between market participants
• Fractional investments are possible. 
• Assets are infinitely divisible, meaning investors can take a large position as well as very small positions. 
• Market participants can borrow and lend unlimited amounts at the risk-free rate.
• Returns are normally distributed.
• when evaluating utility, investors only consider the mean and the variance of return distributions. They ignore deviations from normality, such as skewness or kurtosis
• Homogenous expectations.
•  Investors have the same forecasts of expected returns, variances, and covariances over a single period.

Systematic Risk

• Specific risk is the risk associated with individual assets - within a portfolio these risks can be reduced through diversification (specific risks "cancel out"). Specific risk is also called diversifiable, unique, unsystematic, or idiosyncratic risk
•  Systematic risk (a.k.a. portfolio risk or market risk) refers to the risk common to all securities—except for selling short as noted below, systematic risk cannot be diversified away (within one market).Systematic risk is therefore equated with the risk (standard deviation) of the market portfolio.
• The systematic risk of each asset represents the sensitivity of asset returns to the market return and is referred to as the asset’s beta
• The beta of a security is a measure of its systematic risk, which cannot be eliminated by diversification.
• β, Beta, is the measure of asset sensitivity to a movement in the overall market; Beta is usually found via regression on historical data. Betas exceeding one signify more than average "riskiness" in the sense of the asset's contribution to overall portfolio risk; betas below one indicate a lower than average risk contribution

• Ri and Rm are returns on asset i and market portfolio M
•  σ i and  σM are the standard deviations.
• ρi,M is the Correlation between returns on asset i and market portfolio M
• Any security with a beta greater than 1 moves by a greater amount (has more market risk) and is referred to as cyclical (e.g., luxury goods stock). Any security with a beta below 1 is referred to as defensive (e.g., a utility stock). Cyclical stocks perform better during expansions whereas defensive stocks fare better in recessions.

Security Market Line

• SML is a visualization of the CAPM, where the x-axis of the chart represents risk (in terms of beta), and the y-axis of the chart represents expected return. 
• The security line is derived from the capital market line. CML is used to see a specific portfolio’s rate of return while the SML shows a market risk and a given time’s return. SML also shows the anticipated returns of individual assets.
• SML shows the tradeoff between the required rate of return and systematic risk.
• The slope of the SML shows the differences between the required rate of return on the market index and the risk-free rate. The slope of the SML = (R_m – R_f).

• When a security is plotted on the SML chart, if it appears above the SML, it is considered undervalued because the position on the chart indicates that the security offers a greater return against its inherent risk.

Two fund separation theorem

• All investors should allocate to 2 investments - risk free assets and market portfolio
• amount to be allocated to each investment- depends on investors risk tolerance

PERFORMANCE EVALUATION MEASURES

Sharpe performance index (SPI)

• Higher the measure, the better the risk-adjusted return.
• computes excess return (portfolio return in excess of the risk-free rate) per unit of total risk (as measured by standard deviation). = (R_m – R_f) / σ_m
• slope of the CML is the Sharpe measure of the market. A portfolio with a Sharpe measure greater than the Sharpe measure of the market offers better risk-adjusted returns compared to the market

Treynor Performance Index

• Higher the measure, the better the risk-adjusted return.
• While the Sharpe measure uses total risk as measured by standard deviation, the Treynor measure uses systematic risk as measured by beta.

• well-diversified portfolios are only exposed to market risk, having diversified away idiosyncratic risk. Beta and TPI should therefore be more relevant metrics for well-diversified portfolios
• slope of the SML can also be viewed as the Treynor measure of the market (beta =1)

Jensen’s Performance Index

• Higher the measure, the better the risk-adjusted return.
• Jensen’s Performance Index, like Treynor, assumes investors are well-diversified and, therefore, uses beta rather than standard deviation as the relevant risk metric.
• it compares the portfolio expected return to the CAPM required return. The difference between the two may be referred to as Jensen’s alpha (αP ).
• JPI = αP = E(RP ) − {RF + [E(RM) − RF ]βP }
• In equilibrium (the absence of mispricing), the portfolio expected return must equal the CAPM required return resulting in zero alpha.
• If Jensen’s alpha is positive, this implies that the portfolio is undervalued and investors would be wise to buy or hold it. Jensen’s alpha is most suitable for comparing portfolios that have the same level of systematic risk.

Tracking Error 

• Tracking error is the term used to describe the standard deviation of the difference between the portfolio return and the benchmark return. This source of variability is another source of risk to use in assessing the manager’s success
• For actively managed funds or portfolios, tracking error is also called “active manager risk.”
• it indicates how closely a portfolio follows the index to which it is benchmarked. 
• The consistency of generating excess returns is measured by the tracking error.
• Tracking Error = Standard Deviation of (P - B)

• A realized (also known as “ex post”) tracking error is calculated using historical returns. A tracking error whose calculations are based on some forecasting model is called an “ex ante” tracking error.

Information Ratio

• The information ratio (IR) divides the portfolio expected return in excess of the benchmark expected return by the tracking error
•  it is frequently used to compare the skills and abilities of fund managers with similar investment strategies.
• The information ratio and the Sharpe ratio are similar. Both ratios determine the risk-adjusted returns of a security or portfolio. However, the information ratio measures the risk-adjusted returns relative to a certain benchmark while the Sharpe ratio compares the risk-adjusted returns to the risk-free rate.

=active return/active risk (Tracking error)

• σib the standard deviation of a security or portfolio returns from the returns of a benchmark (tracking error)

Sortino Ratio

• Sortino ratio is reminiscent of the Sharpe measure except for two changes. First, we replace the risk-free rate with a minimum acceptable return, denoted RMIN
• Downside deviation is a type of semi-standard deviation. It measures the variability of only those returns that fall below the minimum acceptable return. Returns higher than RMIN are ignored from the calculation of downside deviation as they are not considered risky as far as the desired returns of our investor are concerned.
• Sortino = RP−RMIN /downside deviation