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**FIXED INCOME PERFORMANCE ATTRIBUTION**

A PRESENTATION TO THE EUROPEAN BOND COMMISSION Wolfgang Marty Portfolio Analytics CSAM Zürich

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1. INTRODUCTION 2. DECOMPOSING THE RETURN 3. FACTOR ANALYSIS 4. FIXED INCOME RISK MODELS 5. THE RISK MODELS OF WILSHIRE 6. EXAMPLES

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1. INTRODUCTION

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**INTRODUCTION THE INVESTMENT PROCESS**

Target asset allocation (Exante) Forecasts (active) Optimization Re-balancing Market movements Performance evaluation (Expost)

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**INTRODUCTION PERFORMANCE MONITORING PROCESS**

Production / Reporting Portfolio Analytics / Risk Control (quantitative aspect) (qualitative aspect) Considering of output (ex-post) and input (ex ante) Performance Performance Performance Performance Performance Portfolio Performance measurement accounting Reporting Analysis Watch List Analytics Review Feed Forward and Feed Back Efficient monitoring of of Investment process

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**INTRODUCTION DEFINITION PORTFOLIO ANALYTICS**

Portfolio analytics is concerned with quantifying the sources of the return and assessing the risk of a portfolio. It not only measures the evolution of the wealth over a certain time period but it provides a comprehensive discussion of the performance of specific portfolios.

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**INTRODUCTION RETURN MEASUREMENT**

Different source of return (Currency return, local market return, return from high yield investment, etc.) Computation ideally daily Input data are critical Return is measured either absolute or relative to a reference portfolio (Benchmark) Return measurement is conceptually easier to understand than risk measurement

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**INTRODUCTION RISK ASSESSMENT**

Different source of risk (Exchange rate risk, Interest rate risk, Credit risk, Sector risk, etc.) Computation based on historical data (time series) Updating once a month is sufficient In portfolio analytics we mostly consider variance or covariance as risk measures Differentiation between forward and backward looking risk Computation of absolute and relative risk measures Tracking Error is relative Risk Measure

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**DEFINITION PERFORMANCE ATTRIBUTION**

INTRODUCTION DEFINITION PERFORMANCE ATTRIBUTION Return attribution mathematically: Decompose a real number into a sum Risk attribution mathematically: Consider generalisation of theorem of Pythagoras b . a c

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**DEFINITION PERFORMANCE ATTRIBUTION**

INTRODUCTION DEFINITION PERFORMANCE ATTRIBUTION More precisely: Analyse the portfolio performance and the relative performance in terms of the decisions that generate returns. Conclusion: Need mapping from the decision making process to a performance attribution model.

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**2. DECOMPOSING THE RETURN**

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**THE WILSHIRE APPLICATION AXIOM RETURN OF A PORTFOLIO**

(relative) return rp of a portfolio The arithmetic (relative return) of a portfolio is the (relative) weighted average (wi), (wi - bi), of the arithmetic return (ri) of the individual securities. On the right hand side and the left hand side is the same (no model)

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**THE WILSHIRE APPLICATION AXIOM ABSOLUTE DECOMPOSITION**

Consider portfolio of 3 securities Mac Donalds IBM CS Group USA Switzerland Country Portfolio return w: weight r: return rP = w1 r w2 r w3 r3 Sector Food TMT Bank

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**THE WILSHIRE APPLICATION AXIOM SLICING THE INVESTMENT UNIVERSE**

Country (Subgroup) Sector (Subgroup) Multi step decision Universe

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**THE WILSHIRE APPLICATION AXIOM CROSS PRODUCT**

Interaction 4 2 Portfolio Ending point y Benchmark portfolio Starting point 3 5

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**THE WILSHIRE APPLICATION AXIOM PROGRAM GLOBAL PERFORMANCE ATTRIBUTION**

Universe: Bond Markets of JP Morgan Global Bond Index The Application computes 3 different Model Returns (‘ Format ’) for 3 different return attribution allows the measurement of 3 different investment processes Model 1 and 2 are based on a form of the capital asset pricing model. leverage factor is ratio of Bond duration and Benchmark Duration Model 1 uses short term rate, Model 2 uses yield of a bond

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**THE WILSHIRE APPLICATION AXIOM PROGRAM GLOBAL PERFORMANCE ATTRIBUTION**

Model computes the Model return of a Bond The Application computes Buy and Hold return in USA Dollars: Example: U.S. Treasury Bond Price 07/01/2002: % Price 07/31/2002: % local return = ((end_price + end_accrued)/(begin_price + begin_accrued) -1)*100 = 3.29%

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**THE WILSHIRE APPLICATION AXIOM PROGRAM GLOBAL PERFORMANCE ATTRIBUTION**

The difference between Model Return and Buy and Hold return is the selection effect. Illustration: Return due to Market Movement versus Return calculated by duration times yield change. Common in all Formats: Currency effect The currency effect examines the impact of active currency exposure of the portfolio versus the benchmark Return in Swiss Francs with currency return -0.47%: base currency return = ((1 + local_return/100) * ( currency_return/100) - 1)*100 = 2.80%

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**THE WILSHIRE APPLICATION AXIOM FORMAT 1: DURATION/COUNTRY**

This format measures the following decisions: Duration effect Value added by being longer than the benchmark in a country where interest rate fell and shorter than benchmark in a country where interest rose. It is only not zero if you have duration exposure in the benchmark and the portfolio.

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**THE WILSHIRE APPLICATION AXIOM FORMAT 1: DURATION/COUNTRY**

Country effect The country effect quantifies the effect of the managers active country bets on management performance by taking the difference between the portfolio weights and the benchmark weights for each country and then multiplying this bet by the relative return to that country (relative to the average local benchmark). Asset allocation approach as it incorporates cross - country decisions

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**THE WILSHIRE APPLICATION AXIOM FORMAT 2: YIELD/EXPOSURE**

This format measures the following decisions: Yield effect The yield component measures the return contribution in the portfolio relative to the benchmark by being invested in higher yielding securities (e.g. corporate bonds).

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**THE WILSHIRE APPLICATION AXIOM FORMAT 2: YIELD/EXPOSURE**

Market effect Is a combination of country-weighting and duration within country decisions. Thus a large country weighting offset by a short duration might result in a neutral or even negative net market exposure. In the duration/country format we would have a large country bet and a large negative duration bet. Suited for investment process that centres around yield curve shifts

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3. FACTOR ANALYSIS

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**THE WILSHIRE APPLICATION AXIOM FORMAT 3: FACTOR EXPOSURE**

The Format measures the decomposition the returns in multi-currency portfolios in terms of different types of yield curve movements and currency changes. Characteristics: Measures investment process that makes bets on different sections of the yield curve Consistent with Risk model Best explanatory power of the three formats It is regression based and represents a detailed description of returns in terms of a common set of risk factors

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**THE WILSHIRE APPLICATION AXIOM EFFECTIVE DURATION**

shift(d1) term structure 100bps spot rate shift(d1) time

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**THE WILSHIRE APPLICATION AXIOM EFFECTIVE DURATION**

Extension of flat yield concept In Wilshire the spot rates are held in cubic spline forms Takes shape of the spot rate curve into account Effective duration and option adjusted duration is the same Takes into account change of the bonds with different cash flows (callable bonds) Defined as price sensitivity of bond to shift in actual yield curve

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**CHANGES OF THE YIELD CURVE WILSHIRE SHIFT SLOPE AND CURVATURE**

term structure spot rate slope (d2) curvature (d3) shift(d1) time

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**4. FIXED INCOME RISK MODEL**

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**FIXED INCOME RISK MODEL FACTOR ANALYSIS**

Historical Data Specify factors Principal components analysis Regression analysis Problem: Interpretation of factors Problem: Goodness of the fit

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**FIXED INCOME RISK MODEL COMPARISON**

Barra Shift Twist Butterfly Wilshire Shift Slope Curvature Prespecified by yield curve shapes with few parameters Estimated by historical data

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**FIXED INCOME RISK MODEL BARRA SHIFT TWIST AND BUTTERFLY**

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**FIXED INCOME RISK MODEL BENCHMARK BOND EXPOSURES**

Step 1: Compute by regression analysis the Risk Matrix given the risk factors d1,d2,d3, country and currencies. Different length of time periods (90, 180, 360 days, exponential weightings) Independent of Portfolio Step 2: Compute the breakdown of the relative Risk (Tracking error) given the Portfolio Holdings and the Benchmark holdings.

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**5. THE RISK MODELS OF WILSHIRE**

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**THE RISK MODELS OF WILSHIRE DIFFERENT MODELS**

Wilshire Global Risk Model Wilshire Global Credit Risk Model

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**THE RISK MODELS OF WILSHIRE SOME CHARACTERISTICS**

Factors in Global Risk Model (based on 13 Countries of J.P. Morgan GBI, Total: 47 factors) d1, d2, d3 Currency

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**THE RISK MODELS OF WILSHIRE CHARACTERISTICS**

Factors in Credit Risk Model Total (16 countries, approx factor) d1, d2, d3 Currency Sectors Quality Rating (Three Groups: Moody Aa, A, Baa) Other spreads Euro country spread special Instrument in USA (e.g. GNMA prepay)

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6. EXAMPLES

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**EXAMPLE 1 ILLUSTRATION: BUCKETING APPROACH**

Benchmark: JPM EMU Index Portfolio: All Bonds with Duration > 8 Years (subset of Benchmark)

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**EXAMPLE 1 ILLUSTRATION: BUCKETING APPROACH**

Benchmark: JPM EMU Index Portfolio: All Bonds with Duration > 8 Years (subset of Benchmark)

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**EXAMPLE 1 ILLUSTRATION: DURATION /COUNTRY FORMAT**

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**EXAMPLE 1 ILLUSTRATION: YIELD/EXPOSURE FORMAT**

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**EXAMPLE 1 ILLUSTRATION: FACTOR FORMAT**

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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EXAMPLE 2 SAMPLE REPORT

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Copyright © 2007 by The McGraw-Hill Companies, Inc. All rights reserved. Derivatives Appendix A.

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