Investment Risk Management 2024-2025 PDF

Investment Risk Management 2024-2025 6FNCE001C School of Business and Economics  Analysis of Active Portfolio Management School of Business and Economics 2 Today … Active Management and Value Added Comparing R...

Investment Risk Management 2024-2025 6FNCE001C School of Business and Economics  Analysis of Active Portfolio Management School of Business and Economics 2 Today … Active Management and Value Added Comparing Risk and Return The Fundamental Law of Active Management School of Business and Economics Active Management and Value Added School of Business and Economics Active Management and Value Added Measuring Value Added The value added or “active return” of an actively managed portfolio is typically calculated as the simply difference between the return on that portfolio and the return on the benchmark portfolio 𝑅𝐴 = 𝑅𝑃 − 𝑅𝐵 Hence, the value added becomes as follows: 𝑁 𝑅𝐴 = σ ∆𝑤𝑖 𝑅𝐴𝑖 𝑖=1 If ∆𝑤𝑖 is positive, it is considered an overweight compared to benchmark weight If ∆𝑤𝑖 is negative, it is considered an underweight compared to benchmark weight School of Business and Economics School of Business and Economics 6 Active Management and Value Added Choice of benchmark A benchmark or passive portfolio should have a number of qualities to serve as a relevant comparison for active management: The benchmark is a representative of the assets from which the investor will select Positions in the benchmark portfolio can actually be replicated at low cost Benchmark weights are verifiable ex ante, and return data are timely ex post School of Business and Economics Active Management and Value Added Decomposition of Value added 𝑀 𝑀 𝑅𝐴 = ෍ 𝑤𝑃,𝑗 𝑅𝑃,𝑗 − ෍ 𝑤𝐵,𝑗 𝑅𝐵,𝑗 𝑖=1 𝑖=1 𝑴 𝑴 𝑹𝑨 = ෍ ∆𝒘𝒋 𝑹𝑩,𝒋 + ෍ 𝒘𝑷,𝒋 𝑹𝑨,𝒋 𝒊=𝟏 𝒊=𝟏 We can conceptualize with just two assets classes, stocks and bonds, hence: 𝑅𝐴 = ∆𝑤𝑠𝑡𝑜𝑐𝑘𝑠 𝑅𝐵,𝑠𝑡𝑜𝑐𝑘𝑠 + ∆𝑤𝑏𝑜𝑛𝑑𝑠 𝑅𝐵,𝑏𝑜𝑛𝑑𝑠 + (𝑤𝑃,𝑠𝑡𝑜𝑐𝑘𝑠 𝑅𝐴,𝑠𝑡𝑜𝑐𝑘𝑠 + 𝑤𝑃,𝑏𝑜𝑛𝑑𝑠 𝑅𝐴,𝑏𝑜𝑛𝑑𝑠 ) Asset Value added from value added from asset allocation security selection  Comparing Risk and Return School of Business and Economics 9 Comparing Risk and Return: Sharpe Ratio 𝑅𝑝 − 𝑅𝐹 𝑆𝑅 = 𝑆𝑇𝐷(𝑅 ) An important property of the Sharpe ratio is that it is unaffected by the addition of cash or leverage in a portfolio School of Business and Economics Comparing Risk and Return: Sharpe Ratio Consider a combined portfolio with a weight of wP on the actively managed portfolio and a weight of (1 – wP) on risk-free cash. The return on the combined portfolio is RC = wPRP + (1 – wP)RF, and the volatility of the combined portfolio is just STD(RC) = wPSTD(RP) because the (1 – wP)RF portion is risk-free. Applying these two relationships gives the Sharpe ratio for the combined portfolio as 𝑅𝐶 − 𝑅𝐹 𝑤𝑃(𝑅𝑃 − 𝑅𝐹) 𝑆𝑅𝐶 = = = 𝑆𝑅 𝑃 𝑆𝑇𝐷(𝑅𝐶 ) 𝑤𝑃𝑆𝑇𝐷(𝑅𝑃) School of Business and Economics Comparing Risk and Return Constructing Optimal Portfolios An important concept from basic portfolio theory is that with a risk-free asset, the portfolio on the efficient frontier of risky assets that is tangent to a ray extended from the risk-free rate is the optimal risky asset portfolio in that it has the highest possible Sharpe Ratio 𝑆𝑅𝑃2 = 𝑆𝑅𝐵2 + 𝐼𝑅2 School of Business and Economics Comparing Risk and Return Constructing Optimal Portfolios It implies that the active portfolio with the highest (squared) information ratio will also have the highest (squared) Sharpe ratio. As a consequence, according to mean-variance theory, the expected information ratio is the single best criterion for assessing active performance among various actively managed funds with the same benchmark School of Business and Economics Comparing Risk and Return: Constructing Optimal Portfolios The preceding discussion on adjusting active risk raises the issue of determining the optimal amount of active risk, without resorting to utility functions that measure risk aversion 𝐼𝑅 𝑆𝑇𝐷 𝑅𝐴 = 𝑆𝑇𝐷(𝑅𝐵 ) 𝑆𝑅 𝐵 By definition, the total risk of the actively managed portfolio is the sum of the benchmark return variance and active return variance 𝑆𝑇𝐷 𝑅𝑃 2 = 𝑆𝑇𝐷 𝑅𝐵 2 + 𝑆𝑇𝐷 𝑅𝐴 2 School of Business and Economics Comparing Risk and Return: Constructing Optimal Portfolios School of Business and Economics Comparing Risk and Return Constructing Optimal Portfolios School of Business and Economics  The Fundamental Law of Active Management School of Business and Economics 17 The Fundamental Law: Active security returns we assume that the investor is concerned about maximizing the managed portfolio’s active return subject to a limit on active risk (also called “benchmark tracking risk”). To this end, the investor uses forecasts for each security of the active return, RAi, or thus the benchmark relative return is 𝑅𝐴,𝑖 = 𝑅𝑖 − 𝑅𝐵 Our notation for the investor’s forecasts of the active security returns is μi (Greek letter mu). The term μi can be thought of as the security’s expected active return, μi = E(RAi), referring to the investor’s subjective expectation, in contrast to an expectation based on a formal equilibrium model The individual security active return can be defined as the residual return in a multi-factor statistical model 𝐾 𝑅𝐴,𝑖 = 𝑅𝑖 − ෍ 𝛽𝑗,𝑖 𝑅𝑗 𝑗=1 with K market-wide factor returns, Rj, and security sensitivities, βj,i, to those factors School of Business and Economics The Fundamental Law: Active security returns The correlation between any set Signal quality is measured by the of active weights, Δwi, in the correlation between the left corner, and forecasted forecasted active returns, μi, at active returns, μi, at the top of the top of the triangle, and the the triangle, measures the realized active returns, RAi, at the degree to which the investor’s right corner, commonly called the forecasts are translated into information coefficient (IC) active weights, called the transfer coefficient (TC) School of Business and Economics The Fundamental Law: Active security returns The Transfer Coefficient is the cross-sectional correlation between the forecasted active security returns and the actual active weights, adjusted for risk 𝜇 𝑇𝐶 = 𝐶𝑂𝑅( ; ∆𝑤𝑖 𝜎 ) 𝜎𝑖 The Information Coefficient is found by calculating the correlations between each manager’s forecasts and the realized risk-weighted returns 𝑅𝐴𝑖 𝜇𝑖 𝐼𝐶 = 𝐶𝑂𝑅( ; ) 𝜎𝑖 𝜎𝑖 School of Business and Economics The Fundamental Law The Basic Fundamental Law The anticipated value added for an actively managed portfolio, or expected active portfolio return, is the sum product of active security weights and forecasted active security returns 𝑁 𝐸 𝑅𝐴 = ෍ ∆𝑤𝑖 𝜇𝑖 𝑖=1 𝑬 𝑹𝑨 ∗ = 𝑰𝑪 𝑩𝑹𝝈𝑨 where * indicates that the actively managed portfolio is constructed from optimal security weights. Breadth is the number of securities: BR=N. Bear in mind that information ratio of the unconstrained optimal portfolio can be expressed as 𝐸(𝑅 ) ∗ 𝐴 𝐼𝑅∗ = = 𝐼𝐶 𝐵𝑅 𝜎𝐴 School of Business and Economics The Fundamental Law The Full Fundamental Law Including the impact of the transfer coefficient, the full fundamental law is expressed in the following equation 𝑬 𝑹𝑨 = (𝑻𝑪)(𝑰𝑪) 𝑩𝑹𝝈𝑨 where an * is not used because the managed portfolio is constructed from constrained active security weights School of Business and Economics The Fundamental Law The Full Fundamental Law The full fundamental law of active management, which we will refer to simply as the fundamental law, hence forward, states that the expected active return, E(RA), is the product of four key parameters: the transfer coefficient, TC; the assumed information coefficient, IC; the square root of breadth, BR; and portfolio active risk, σA Bear in mind that the information ratio of the unconstrained optimal portfolio can be expressed as 𝐸(𝑅𝐴) 𝐼𝑅 = = (𝑇𝐶)(𝐼𝐶) 𝐵𝑅 𝜎𝐴 School of Business and Economics  Thank you School of Business and Economics 24