1Fixed Income InvestmentsPortfolio Management: Economic Analysis, Active Management, and Trading51. Analysis of Active Portfolio Management(Part 2)© Kaplan, Inc.Analysis of Active Portfolio ManagementThe Fundamental Law of Active ManagementThree components affecting expected information ratio:1.Information coefficient (IC): Measures manager skill Expected correlation between active returns and forecast active returns (ex-ante IC) Ex-post IC measures actual correlation between active returns and forecast active returns Typically, ex-ante IC has small positive values (< 0.2)26LOS 51.c Describe© Kaplan, Inc.Analysis of Active Portfolio ManagementThe Fundamental Law of Active Management2. Transfer coefficient (TC) Correlation between actual active weights (ΔWi) and optimal active (ΔWi*) weights. TC = 1 for unconstrained portfolios  TC < 1 w/constraints: ΔWi and ΔWi* will differ3. Breadth (BR) Number of independent bets (forecasts of active return)27LOS 51.c Describe© Kaplan, Inc.Analysis of Active Portfolio ManagementThe Fundamental Law of Active Management28ΔWi*=μiσi2×σAμi2σi2i=1N∑Optimal weight of asset iForecast active return of asset iForecast volatility of the active return of asset iActive portfolio risk (std. dev. of active return)Forecast active return of asset i squaredOptimal weights positively related to forecast active return and negatively related to forecast active riskLOS 51.c Describe2© Kaplan, Inc.Analysis of Active Portfolio ManagementGrinold (1994) Rule29iii = IC SForecast active return of asset iStandardized “scores” (with assumed variance = 1)Ex-ante ICLOS 51.c Describe© Kaplan, Inc.Analysis of Active Portfolio ManagementThe Fundamental LawExpected active portfolio return = 30i1AiE(R ) = WNiE(RA)*= IC BRσAUnconstrained portfolios TC = 1, * = optimal weightsIR* = IC BRE(RA) = (TC)IC BRσAIR = (TC)IC BRConstrained portfolios TC < 1, actual weights ≠ optimal weightsBasic Fundamental Law:Full Fundamental Law:LOS 51.c Describe© Kaplan, Inc.Analysis of Active Portfolio ManagementThe Full Fundamental Law Constrained portfolios: Actual security weights ≠ optimal weights (TC < 1) TC = risk-weighted correlation between optimal active weights and actual active weights:  or, TC 31TC = COR(ΔWi*σi,ΔWiσi)= Correlation (Δwiσi,μi/σi)LOS 51.c Describe© Kaplan, Inc.Analysis of Active Portfolio ManagementOptimal Level of Active Risk32SRP2= SRB2+ IR2Sharpe ratio and optimal active risk for unconstrainedportfoliosσ*(RA) =IRSRB×σ(RB)SRP2= SRB2+ (TC)2(IR)2σ(RA) = TCIR*SRB×σ(RB)Sharpe ratio and optimal active risk for constrainedportfoliosLOS 51.c Describe3© Kaplan, Inc.Analysis of Active Portfolio ManagementEx-post Performance Measurement Conditional expected active return:33E(RAICR) = (TC)(ICR)BRσANote that realized information coefficient (forecasting skill) over the period replaces the expected IC Actual active return = conditional active return + noise:RA= E(RAICR) + NoiseResults from constraints to optimal portfolio structureLOS 51.c Describe© Kaplan, Inc.Analysis of Active Portfolio ManagementClarke, de Silva, and Thorley (2005)Ex-post decomposition of realized active return varianceTwo components:1. Variation due to realized IC: TC22. Variation due to constraint induced noise: 1 – TC2With a TC = 0.864% of variation in performance is attributed to the variation in the realized IC, and 36% comes from constraint induced noise.34LOS 51.c Describe© Kaplan, Inc.Analysis of Active Portfolio ManagementThe Information Ratio and Manager Selection Investor chooses a combination of optimal risky portfolio and risk-free asset depending on their risk tolerance Optimal risky portfolio = portfolio with highest Sharpe ratio The actively managed portfolio with the highest information ratio will have the highest Sharpe ratio Given an information ratio and target level of active risk, expected active return can be determined:35E(RA) = IR ×σALOS 51.d Explain© Kaplan, Inc.Analysis of Active Portfolio ManagementComparison of Active Management Strategies Market timing versus security selection Information coefficient of a market timer: Note that a market timer that is only correct about the direction of the market or a sector half the time will have an IC = 036CNIC = 2 – 1NNCis the number of correct bets out of Nbets made on the direction of the market.LOS 51.e Compare/Evaluate4© Kaplan, Inc.Analysis of Active Portfolio ManagementExample: Active Management StrategiesBen Dash, a market timer, makes bets on the direction of the market and is correct 53% of the time. Ricardo Nunos, a security selector, makes monthly bets on 10 stocks with an information coefficient of 0.04.Both investors are unconstrained.Calculate the number of bets Dash needs to make to match the information ratio of Nunos.37LOS 51.e Compare/Evaluate© Kaplan, Inc.Analysis of Active Portfolio ManagementSolution: Active Management Strategies38 Nunos’s IR = Dash’s IC = 2(% correct) – 1 =IC BR= 0.04 10×12 = 0.4382(0.53) – 1= 0.06 ∴0.06 BR = 0.4380.4380.06= BR0.4380.06⎛⎝⎜⎞⎠⎟2= BR = 53.29 (approximately 53)Note this is just % correct – % incorrect betsLOS 51.e Compare/Evaluate© Kaplan, Inc.Analysis of Active Portfolio ManagementSector Rotation Sector rotation applies market timing to rotate money into sectors expected to outperform other sectors In a two-sector world (healthcare and utilities):39σA=σCBRE(RA) = IC BR ×σCσC= combined active riskσC=σhealthcare2− 2σhealthσutilitiesrhealth,utilities+σutilities2()12Annualized active riskAnnualized active returnLOS 51.e Compare/Evaluate© Kaplan, Inc.Analysis of Active Portfolio ManagementSector Rotation: ExampleJeanette Grey makes monthly bets between the information technology and energy sectors. She gives an overweighted sector a score of +1 and underweighted –1. By implication overweighting the IT sector implies underweighting the energy sector, and visa versa. Historically, the return correlation between the sectors has been 0.4 and Jeanette has been correct 65% of the time. Use the data on the next slide to compute annualized active risk of the strategy and annualized expected return. 40LOS 51.e Compare/Evaluate5© Kaplan, Inc.Analysis of Active Portfolio ManagementSector Rotation: Example41QuarterlySector E(R) σ Benchmark WeightIT 12% 5% 60%Energy 8% 3% 40%Five years of