Practical Portfolio Performance Measurement and Attribution E-Book

Carl R. Bacon

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 548

Veröffentlichungsjahr: 2011

Beschreibung

Performance measurement and attribution are key tools in informing investment decisions and strategies. Performance measurement is the quality control of the investment decision process, enabling money managers to calculate return, understand the behaviour of a portfolio of assets, communicate with clients and determine how performance can be improved. Focusing on the practical use and calculation of performance returns rather than the academic background, Practical Portfolio Performance Measurement and Attribution provides a clear guide to the role and implications of these methods in today's financial environment, enabling readers to apply their knowledge with immediate effect. Fully updated from the first edition, this book covers key new developments such as fixed income attribution, attribution of derivative instruments and alternative investment strategies, leverage and short positions, risk-adjusted performance measures for hedge funds plus updates on presentation standards. The book covers the mathematical aspects of the topic in an accessible and practical way, making this book an essential reference for anyone involved in asset management.

Leseprobe

Table of Contents

Title Page

Copyright Page

Dedication

Acknowledgements

Chapter 1 - Introduction

WHY MEASURE PORTFOLIO PERFORMANCE?

THE PERFORMANCE MEASUREMENT PROCESS

THE PURPOSE OF THIS BOOK

ROLE OF PERFORMANCE MEASURERS

BOOK STRUCTURE

Chapter 2 - The Mathematics of Portfolio Return

SIMPLE RETURN

MONEY-WEIGHTED RETURNS

TIME-WEIGHTED RETURNS

TIME-WEIGHTED VERSUS MONEY-WEIGHTED RATES OF RETURN

APPROXIMATIONS TO THE TIME-WEIGHTED RETURN

HYBRID METHODOLOGIES

WHICH METHOD TO USE?

SELF-SELECTION

ANNUALISED RETURNS

CONTINUOUSLY COMPOUNDED RETURNS

GROSS- AND NET-OF-FEE CALCULATIONS

PORTFOLIO COMPONENT RETURNS

BASE CURRENCY AND LOCAL RETURNS

Chapter 3 - Benchmarks

BENCHMARKS

BENCHMARK STATISTICS

PEER GROUPS AND UNIVERSES

RANDOM PORTFOLIOS

NOTIONAL FUNDS

EXCESS RETURN

PERFORMANCE FEES

PERFORMANCE FEE STRUCTURES

Chapter 4 - Risk

DEFINITION OF RISK

RISK MEASURES

REGRESSION ANALYSIS

RELATIVE RISK

RETURN DISTRIBUTIONS

RISK-ADJUSTED PERFORMANCE MEASURES FOR HEDGE FUNDS

DRAWDOWN

DOWNSIDE RISK (OR SEMI-STANDARD DEVIATION)

VALUE AT RISK (VaR)

RETURN ADJUSTED FOR DOWNSIDE RISK

FIXED INCOME RISK

WHICH RISK MEASURES TO USE?

RISK CONTROL STRUCTURE

Chapter 5 - Performance Attribution

ARITHMETIC ATTRIBUTION

BRINSON AND FACHLER

INTERACTION

GEOMETRIC EXCESS RETURN ATTRIBUTION

SECTOR WEIGHTS

Chapter 6 - Multi-currency Attribution

ANKRIM AND HENSEL

KARNOSKY AND SINGER

GEOMETRIC MULTI-CURRENCY ATTRIBUTION

INTEREST RATE DIFFERENTIALS

Chapter 7 - Fixed Income Attribution

THE YIELD CURVE

Chapter 8 - Multi-period Attribution

SMOOTHING ALGORITHMS

Chapter 9 - Further Attribution Issues

ATTRIBUTION VARIATIONS

MULTI-LEVEL ATTRIBUTION

ATTRIBUTION STANDARDS

EVOLUTION OF PERFORMANCE ATTRIBUTION METHODOLOGIES

RISK-ADJUSTED ATTRIBUTION

Chapter 10 - Performance Measurement for Derivatives

FUTURES

FORWARD FOREIGN EXCHANGE (FFX) CONTRACT (OR CURRENCY FORWARD)

SWAPS

OPTIONS

WARRANTS

MARKET NEUTRAL ATTRIBUTION

Chapter 11 - Performance Presentation Standards

WHY DO WE NEED PERFORMANCE PRESENTATION STANDARDS?

GLOBAL INVESTMENT PERFORMANCE STANDARDS (GIPS)

ADVANTAGES FOR ASSET MANAGERS

THE STANDARDS

VERIFICATION

INTERPRETATIONS SUBCOMMITTEE

MEASURES OF DISPERSION

ACHIEVING COMPLIANCE

MAINTAINING COMPLIANCE

Appendix A - Simple Attribution

Appendix B - Multi-currency Attribution Methodology

Appendix C - EIPC Guidance for Users of Attribution Analysis

Appendix D - European Investment Performance Committee – Guidance on ...

Appendix E - The Global Investment Performance Standards

Appendix F - Guidance Statement on Composite Definition

Appendix G - Sample Global Investment Performance Standards Presentation

Appendix H - Calculation Methodology Guidance Statement

Appendix I - Definition of Firm Guidance Statement

Appendix J - Treatment of Carve-outs Guidance Statement

Appendix K - Significant Cash Flow Guidance Statement

Appendix L - Guidance Statement on Performance Record Portability

Appendix M - Guidance Statement on the Use of Supplemental Information

Appendix N - Guidance Statement on Recordkeeping Requirements of the GIPS Standards

Appendix O - Useful Websites

Bibliography

Index

For other titles in the Wiley Finance series please see www.wiley.com/finance

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Library of Congress Cataloging-in-Publication Data

Bacon, Carl R.

Practical portfolio performance : measurement and attribution / Carl R Bacon. – 2nd ed.

p. cm. – (Wiley finance series)

Includes bibliographical references and index.

ISBN 978-0-470-05928-9 (cloth/cd)

1. Investment analysis. 2. Portfolio management. I. Title.

HG4529.B33 2008

332.6 — dc22

2008007637

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

This book is dedicated to Alex for her continued love and support

Acknowledgements

This book is based on a series of performance measurement training courses I have had the pleasure of running around the world over the last decade. I have learnt so much and continue to learn from the questions and observations of the participants over the years, all of whom must be thanked.

I should also like to thank the many individuals I’ve had the pleasure to work with at various institutions, those I’ve met at conferences and at numerous GIPS committee meetings that have influenced my views over the years.

Naturally from the practitioner’s perspective I’ve favoured certain methodologies over others. My strong preferences are difficult to disguise, nevertheless I’ve attempted to present each methodology as fairly as possible – apologies to those who may feel their methods have been unfairly treated.

Of course all errors and omissions are my own.

Carl R. Bacon CIPM Deeping St James [email protected]

1

Introduction

The more precisely the position is determined, the less precisely the momentum is known in this instant, and vice versa.

Heisenberg (1901-1976) The Uncertainty Principle (1927)

Learn as much by writing as by reading.

Lord Acton (1834-1902)

WHY MEASURE PORTFOLIO PERFORMANCE?

Whether we manage our own investment assets or choose to hire others to manage the assets on our behalf we are keen to know “how well” our collection or portfolio of assets is performing.

The process of adding value via benchmarking, asset allocation, security analysis, portfolio construction, and executing transactions is collectively described as the investment decision process. The measurement of portfolio performance should be part of the investment decision process, not external to it.

Clearly, there are many stakeholders in the investment decision process; this book focuses on the investors or owners of capital and the firms managing their assets (asset managers or individual portfolio managers). Other stakeholders in the investment decision process include independent consultants tasked with providing advice to clients, custodians, independent performance measurers and audit firms.

Portfolio performance measurement answers the three basic questions central to the relationship between asset managers and the owners of capital:

1. What is the return on their assets?

2. Why has the portfolio performed that way?

3. How can we improve performance?

Portfolio performance measurement is the quality control of the investment decision process providing the necessary information to enable asset managers and clients to assess exactly how the money has been invested and the results of the process. The US Bank Administration Institute (BAI, 1968) laid down the foundations of the performance measurement process as early as 1968. The main conclusions of their study hold true today:

1. Performance measurement returns should be based on asset values measured at market value not at cost.

2. Returns should be “total” returns, that is, they should include both income and changes in market value (realised and unrealised capital appreciation).

3. Returns should be time-weighted.

4. Measurement should include risk as well as return.

THE PERFORMANCE MEASUREMENT PROCESS

Performance measurement is essentially a three-stage process:

1. Measurement:

Calculation of returns, benchmarks and peer groups Distribution of information

2. Attribution:

Return attribution

Risk analysis (ex post and ex ante)

3. Evaluation:

Feedback

Control

THE PURPOSE OF THIS BOOK

The writing of any book is inevitably a selfish activity, denying precious time from family who suffer in silence but also work colleagues and friends driven by the belief that you have something to contribute in your chosen subject.

The motivation to write the first edition was simply to provide the book I most wanted to read as a performance analyst which did not exist at the time.

The vocabulary and methodologies used by performance analysts worldwide are extremely varied and complex. Despite the development and global success of performance measurement standards there are considerable differences in terminology, methodology and attitude to performance measurement throughout the world.

The main aims of the first edition were:

1. Provide a reference of the available methodologies and to hopefully provide some consistency in their definition.

2. Promote the role of performance measurers.

3. Provide some insights into the tools available to performance measurers.

4. Share my practical experience.

Since the first edition I’m pleased to say the CFA Institute have launched the CIPM designation which further reinforces the role of performance measurement and is a major step in developing performance measurement as a professional activity. I can certainly recommend the CIPM course of study and I’m delighted to have successfully achieved the CIPM designation. The CIPM curriculum has also to some degree influenced the content of this second edition.

With practical examples this book should meet the needs of performance analysts, portfolio managers, senior management within asset management firms, custodians, verifiers and the ultimate clients. I’m particularly pleased that this second edition includes a CD including many of the practical examples used throughout the book.

ROLE OF PERFORMANCE MEASURERS

Performance measurement is a key function in an asset management firm, it deserves better than to be grouped with the back office. Performance measurers provide real added value, with feedback into the investment decision process and analysis of structural issues. Since their role is to understand in full, make transparent and communicate the sources of return within portfolios they are often the only independent source equipped to understand the performance of all the portfolios and strategies operating within the asset management firm.

Performance measurers are in effect alternative risk controllers able to protect the firm from rogue managers and the unfortunate impact of failing to meet client expectations.

BOOK STRUCTURE

The chapters of this book are structured in the same order as the performance measurement process itself, namely:

Chapter 2 Calculation of portfolio returns

Chapter 3 Comparison against an appropriate benchmark

Chapter 4 Proper assessment of the reward received for the risk taken

Chapters 5 to 10 Attribution of the sources of excess return

Chapter 5 Fundamentals of attribution

Chapter 6 Multi-currency attribution

Chapter 7 Fixed income attribution

Chapter 8 Multi-period attribution

Chapter 9 Attribution issues

Chapter 10 Attribution for derivatives

Chapter 11 Presentation and communicating the results

Inevitably, due to time constraints the first edition was not complete; the second edition affords me the opportunity to make substantial additions and improvements.

In Chapter 2 the “what” of performance measurement is introduced describing the many forms of return calculation including the relative merits of each method together with calculation examples.

Performance returns in isolation add little value; we must compare these returns against a suitable benchmark. Chapter 3 discusses the merits of good and bad benchmarks and examines the detailed calculation of commercial and customised indexes. I’ve added a section on random portfolios, some additional remarks, a few benchmark statistics and extended the section on performance fees.

Chapter 4 is substantially enhanced to include risk measures for hedge funds in an attempt to catalogue all the available risk measures used by performance analysts including suggestions for consistent definition were such definitions are lacking.

The original Chapter 5 in the first edition was perhaps too long. Attribution is a broad subject; Chapter 5 is now focused on the fundamentals of attribution, principally the Brinson model and its adaptation to arithmetic and geometric approaches.

Chapter 6 focuses on multi-currency attribution, including the important work of Karnosky and Singer plus a detailed description of geometric multi-currency attribution.

Chapter 7 is largely new material focusing on fixed income attribution. Since the investment decision process of fixed income managers is fundamentally different the unadjusted Brinson model is not appropriate.

Attribution analysis is useful for analysing performance not only for the most recent period but also for the longer-term requiring the linking of multi-period attribution results. The issues of multi-period attribution are discussed in Chapter 8.

Chapter 9 is new material covering a variety of technical attribution issues including security-level analysis, off benchmark investments, and balanced and multi-level attribution.

Chapter 10 is also new material covering the measurement and attribution of derivative instruments and attribution for various alternative asset strategies such as market neutral and 130:30 funds.

Finally, in Chapter 11 we turn to the presentation of performance and consider the global development of performance presentation standards. The second edition is updated for the latest version of the GIPS standards published in 2006.

2

The Mathematics of Portfolio Return

Mathematics is the gate and key of the sciences.... Neglect of mathematics works injury to all knowledge, since he who is ignorant of it cannot know the other sciences or the things of this world.

Roger Bacon, Doctor Mirabilis, Opus Majus (1214-1294)

Mathematics has given economics rigour, alas also mortis.

Robert Helibroner (1919-2005)

SIMPLE RETURN

In measuring the performance of a “portfolio” or collection of investment assets we are concerned with the increase or decrease in the value of those assets over a specific time period, in other words the change in “wealth”.

This change in wealth can be expressed either as a “wealth ratio” or a “rate of return”.

A wealth ratio greater than 1.0 indicates an increase in value, a ratio less than 1.0 a decrease in value.

Starting with a simple example, take a portfolio valued at £100m initially and valued at £112m at the end of the period. The wealth ratio is calculated as follows:

Exhibit 2.1 Wealth ratio

The value of a portfolio of assets is not always easy to obtain, but should represent a reasonable estimate of the current economic value of the assets. Firms should ensure internal valuation policies are in place and consistently applied over time. A change in valuation policy may generate spurious performance over a specific time period.

Economic value implies that the traded market value, rather than the settlement value of the portfolio, should be used. For example, if an individual security has been bought but the trade has not been settled (i.e. paid for) then the portfolio is economically exposed to any change in price of that security. Similarly, any dividend declared and not yet paid or interest accrued on a fixed income asset is an entitlement of the portfolio and should be included in the valuation. Since it is a potential asset of the portfolio any reclaimable withholding tax should also be accrued in the market value. Although it may take some time before any withholding tax is recovered it should remain in the market value until either it is recovered or written off thus capturing performance in the appropriate period.

The rate of return denoted r describes the gain (or loss) in value of the portfolio relative to the starting value, mathematically:

(2.2)

Rewriting Equation (2.2):

(2.3)

Using the previous example the rate of return is:

Exhibit 2.2 Rate of return

Equation (2.3) can be conveniently rewritten as:

(2.4)

Hence, the wealth ratio is actually the rate of return plus one.

Where there are no “external cash flows” it is easy to show that the rate of return for the entire period is the “compounded return” over multiple subperiods.

Let Vt equal the value of the portfolio after the end of period t then:

(2.5)

External cash flow is defined as any new money added to or taken from the portfolio, whether in the form of cash or other assets. Dividend and coupon payments, purchases and sales and corporate transactions funded from within the portfolio are not considered external cash flows. Income from a security or stock lending programme initiated by the portfolio manager is not considered to be external cash flow, while if initiated by the client such income should be treated as external cash flow; the performance does not belong to the portfolio manager.

Substituting Equation (2.4) into Equation (2.5) we establish Equation (2.6):

(2.6)

Exhibit 2.3 Chain linking

This process (demonstrated in Exhibit 2.3) of compounding a series of subperiod returns to calculate the entire period return is called “geometric” or “chain” linking.

MONEY-WEIGHTED RETURNS

Unfortunately, in the event of external cash flows we cannot continue to use the ratio of market values to calculate wealth ratios and hence rates of return. The cash flow itself will make a contribution to the valuation. Therefore we must develop alternative methodologies that adjust for external cash flow.

Internal rate of return (IRR)

To make allowance for external cash flow we can borrow a methodology used throughout finance, the “internal rate of return” or IRR.

The internal rate of return has been used for many decades to assess the value of capital investment or other business ventures over the future lifetime of a project. Normally, the initial outlay, estimated costs and expected returns are well known and the internal rate of return of the project can be calculated to determine if the investment is worth undertaking. IRR is also used to calculate the future rate of return on a bond and called the yield to redemption.

Simple internal rate of return

Modified internal rate of return

The standard internal rate of return method in Equation (2.8) is often described by performance measurers as the modified internal rate of return method to differentiate it from the simple rate of return method described in Equation (2.7) which assumes midpoint cash flows.

This method assumes a single, constant force of return throughout the period of measurement, an assumption we know not to be true since the returns of investment assets are rarely constant. This assumption also means we cannot disaggregate the IRR into different asset categories since we cannot continue to use the single constant rate.

For project appraisal or calculating the redemption yield of a bond this assumption is not a problem since we are calculating a future return for which we must make some assumptions.

IRR is an example of a money-weighted return methodology; each amount or dollar invested is assumed to achieve the same effective rate of return irrespective of when it was invested. In the US the term “dollar-weighted” rather than “money-weighted” is more often used.

The weight of money invested at any point of time will ultimately impact the final return calculation. Therefore if using this methodology it is important to perform well when the amount of money invested is largest.

For example, assume cash flow occurs on the 236th day of the 3rd year for a total measurement period of 5 years. Then:

Although today spreadsheets offer easy solutions to IRR calculations, historically solutions were more difficult to obtain, even taxing Sir Isaac Newton in the 17th century. Controversy surrounds the authorship of various iterative methods, in particular the Newton-Raphson method which perhaps should be attributed to Thomas Simpson (1740), (Kollerstrom 1992).

Simple Dietz

Even in its simple form the internal rate of return is not a particularly practical calculation, especially over longer periods with multiple cash flows. Peter Dietz (1966) suggested as an alternative the following simple adaptation to Equation (2.2) to adjust for external cash flow, let’s call this the simple (or original) Dietz method:

(2.13)

where:

C represents external cash flow.

The numerator of Equation (2.13) represents the investment gain in the portfolio. In the denominator replacing the initial market value we now use the average capital invested represented by the initial market value plus half the external cash flow. An assumption has been made that the external cash flow is invested midway through the period of analysis and has been weighted accordingly. The average capital invested is absolutely not the average of the start and end values which would factor in an element of portfolio performance into the denominator.

This method is also a money-(or dollar-) weighted return, and is in fact the first-order approximation of the internal rate of return method.

To calculate a simple Dietz return, like the simple IRR, only the start market value, end market value and total external cash flow are required.

Exhibit 2.7 Simple Dietz

Using the existing example data:Market start value$74.2 mMarket end value$104.4 mExternal cash flow$37.1 mThe simple Dietz rate of return is:

Dietz originally described his method as assuming one half of the net contributions are made at the beginning of the time interval and one half at the end of the time interval:

(2.14)

This simplifies to the more common description:

The Dietz method is easier to calculate, easier to visualise than the IRR method and can also be disaggregated, that is to say the total return is the sum of the individual parts.

ICAA method

Extending our previous example in Exhibit 2.8:

Exhibit 2.8 ICAA method

Market start value$74.2 mMarket end value$104.2 mExternal cash flow$37.1 mTotal income$0.4 mIncome reinvested$0.2 m

In this method income (equity dividends, interest or coupon payments) is not automatically assumed to be available for reinvestment. The gain in the numerator is appropriately adjusted for any reinvested income included in the final value by including reinvested income in the definition of external cash flow.

Interestingly, although the average capital is increased for any reinvested income in the denominator there is no negative adjustment for any income not reinvested. This is perhaps not unreasonable from the perspective of the client if the income is retained and not paid until the end of period.

However, from the portfolio manger’s viewpoint if this income is not available for reinvestment it should be treated as a negative cash flow as follows:

(2.16)

Extending our previous example again in Exhibit 2.9:

Exhibit 2.9 Income unavailable

Market start value$74.2 mMarket end value$104.0 mExternal cash flow$37.1 mTotal Income$0.4 m

In Equation (2.16) any income received by the portfolio is assumed to be unavailable for investment by the portfolio manager and transferred to a separate income account for later payment or alternatively paid directly to the client.

Obviously, income paid or transferred is no longer included in the final value VE of the portfolio. In effect, in this methodology income is treated as negative cash flow. Since income is normally always positive, this method has the effect of reducing the average capital employed, decreasing the size of the denominator, and thus leveraging (or gearing) the final rate of return.

Consequently, this method should only be used if portfolio income is genuinely unavailable to the portfolio manager for further investment. Typically, this method is used to calculate the return of an asset category (sector or component) within a portfolio.

Modified Dietz

In determining Dt the performance analyst must establish if the cash flow is received at the beginning or end of the day. If the cash flow is received at the start of the day then it is reasonable to assume that the portfolio manager is aware of the cash flow and able to respond to it, therefore it is reasonable to include this day in the weighting calculation. On the other hand, if the cash flow is received at the end of the day the portfolio manager is unable to take any action at that point and therefore it is unreasonable to include the current day in the weighting calculation.

For example, take a cash flow received on the 14th day of a 31-day month. If the cash flow is at the start of the day, then there are 18 full days including the 14th day available for investment and the weighting factor for this cash flow should be (31 − 13)/31. Alternatively, if the cash flow is at the end of the day then there are 17 full days remaining and the weighting factors should be, (31 − 14)/31.

Performance analysts should determine a company policy and apply this consistently to all cash flows.

Extending our standard example in Exhibit 2.10:

Exhibit 2.10 Modified Dietz

Market start value31 December$74.2 mMarket end value31 January$ 104.4 mExternal cash flow14 January$37.1 mAssuming the cash flow is at the end of day 14:Assuming the cash flow is at the beginning of day 14 with 18 full days in the month left:

TIME-WEIGHTED RETURNS

True time-weighted

Time-weighted rate of returns provide a popular alternative to money-weighted returns in which each time period is given equal weight regardless of the amount invested, hence the name “time-weighted”.

In the “true or classical time-weighted” methodology performance is calculated for each subperiod between cash flows using simple wealth ratios. The subperiod returns are then chain linked as follows:

(2.18)

where: Vt is the valuation immediately after the cash flow Ct at the end of period t.

In Equation (2.18) we have made the assumption that any cash flow is only available for the portfolio manager to invest at the end of the day. If we make the assumption that the cash flow is available from the beginning of the day we must change Equation (2.18) to:

(2.19)

Alternatively, we may wish to make the assumption that the cash flow is available for investment midday and use a half weight assumption as follows:

(2.20)

Note from Equation (2.13):

Equation (2.20) is really a hybrid methodology combining both time weighting and a money-weighted return for each individual day and therefore ceases to be a true time-weighted rate of return.

Using our standard example data we now need to know the value of the portfolio immediately after the cash flow as shown in Exhibits 2.11, 2.12 and 2.13:

Exhibit 2.11 True time-weighted end of day cash flow

End of day cash flow assumption:Market start value31 December$74.2 mMarket end value31 January$104.4 mExternal cash flow14 January$37.1 mMarket value end of14 January$103.1 m

Unit price method

The “unit price” or “unitised” method is a useful variant of the true time-weighted methodology. Rather than use the ratio of market values between cash flows, a standardised unit price or “net asset value” price is calculated immediately before each external cash flow by dividing the market value by the number of units previously allocated. Units are then added or subtracted

Exhibit 2.12 True time-weighted start of day cash flow

Start of day cash flow assumption:Market start value31 December$74.2 mMarket end value31 January$104.4 mExternal cash flow14 January$37.1 mMarket value start of14 January$67.0 m

Exhibit 2.13 Time-weighted midday cash flow

Midday cash flow assumption:Market start value31 December$74.2 mMarket end value31 January$104.4 mExternal cash flow14 January$37.1 mMarket value start of14 January$67.0 mMarket value end of14 January$103.1 m

(bought or sold) in the portfolio at the unit price corresponding to the time of the cash flow – the unit price is in effect a normalised market value.

The starting value of the portfolio is also allocated to units, often using a notional, starting unit price of say 1 or 100.

The main advantage of the unit price method is that, the ratio of the end of period unit price with the start of period unit price always provides the rate of return irrespective of the change of value in the portfolio due to cash flow. Therefore to calculate the rate of return between any two points the only information you need to know is the start and end unit prices.

Let NAVi equal the net asst value unit price of the portfolio after the end of period i. Then:

(2.21)

The unitised method is so convenient for quickly calculating performance that returns calculated using other methodologies are often converted to unit prices for ease of use, particularly over longer time periods.

The unitised method is a variant of the true or classical time-weighted return and will always give the same answer as can be seen in Exhibit 2.14.

Typically, the performance of mutual funds is calculated using net asset value prices for external presentation and with a true time-weighted methodology for internal analysis. It can be a challenge to reconcile both sets of returns; the performance analyst must ensure that

Exhibit 2.14 Unit price method

the internal valuations are aligned with valuations produced by the unit pricing accountant including the timing of external cash flows. Unless there are no external cash flows it will be impossible to reconcile the time-weighted unit price method with a money-weighted return.

TIME-WEIGHTED VERSUS MONEY-WEIGHTED RATES OF RETURN

Time-weighted returns measure the returns of the assets irrespective of the amount invested. This can generate counterintuitive results as shown in Exhibit 2.15:

Exhibit 2.15 (Time-weighted returns versus money-weighted returns)

Start period 1 Market value£100End period 1 Market value£200Cash flow£1000Start period 2 Market value£1200End period 2 Market value£700Time-weighted return:Money-weighted return:

In Exhibit 2.15 the client has lost £400 over the entire period, yet the time-weighted return is calculated as a positive 16.67%. The money-weighted return reflects this loss, −66.67% of the average capital employed. It is important to perform well in the second period when the majority of client money is invested.

If the client had invested all the money at the beginning of the period of measurement then a 16.67% return would have been achieved. The difference in return calculated is due to the timing of cash flow. Over a single period of measurement the money-weighted rate of return will always reflect the cash gain and loss over the period.

The time-weighted rate of return adjusts for cash flow and weights each time period equally, measuring the performance that would have been achieved had there been no cash flows. Clearly, this return is most appropriate for comparing the performance of different portfolio managers with different patterns of cash flows and with benchmark indexes, which for the most part are calculated using a time-weighted approach.

In effect, the time-weighted rate of return measures the portfolio manager’s performance adjusting for cash flows and the money-weighted rate of return measures the performance of the client’s invested assets including the impact of cash flows.

With such large potential differences between methodologies, which method should be used and in what circumstances?

Most performance analysts would prefer time-weighted returns. By definition time-weighted returns weight each time period equally, irrespective of the amount invested, therefore the timing of external cash flows or the amount of money invested does not affect the calculation of return. In the majority of cases portfolio managers do not determine the timing of external cash flows, nor does the amount of money invested normally change the investment decision process, therefore it is desirable to use a methodology that is not impacted by the timing of cash flow.

A few performance analysts argue that from a presentational perspective, particularly when dealing with private clients, money-weighted returns are preferred since over a single period a loss always results in a negative return and a gain in a positive return. While this is indeed true it may not be a true reflection of the portfolio manager’s performance, and we are certain that the central assumption of money-weighted returns, that of a constant force of return, is most unlikely to be correct.

A major drawback of true time-weighted returns is that accurate valuations are required at the date of each cash flow. This is an onerous and expensive requirement for some asset managers. The manager must make an assessment of the benefits of increased accuracy against the costs of frequent valuations for each external cash flow and the potential for error. Asset management firms must have a daily valuation mindset to succeed with daily performance calculations. Exhibit 2.16 demonstrates the impact of a valuation error on the return calculation:

Exhibit 2.16 Valuation error

Market start value31 December$74.2 mMarket end value31 January$104.4 mExternal cash flow14 January$37.1 mErroneous market value14 January$101.1 mA significant and permanent difference from the accurate time-weighted return of −9.93% calculated in Exhibit 2.11.

Not unreasonably, institutional clients such as large pension funds paying significant fees might expect that the asset manager has sufficient quality information on a daily basis to manage their portfolio accurately. Most large managers will also have mutual or other pooled funds in their stable which in most cases will already require daily valuations (not just at the date of each external cash flow). The industry, driven by performance presentation standards and the demand for more accurate analysis, is gradually moving to daily calculations as standard.

In terms of statistical analysis daily calculation adds more noise than information; however, in terms of return analysis, daily calculation (or at the least valuation at each external cash flow which practically amounts to the same thing) is essential to ensure the accuracy of long-term returns.

I do not believe in the daily analysis of performance, which is far too short term for long-term investment portfolios, but I do believe in accurate returns, which require daily calculation. It is also useful for the portfolio manager or performance measurer to analyse performance between any two dates other than the standard calendar period ends.

APPROXIMATIONS TO THE TIME-WEIGHTED RETURN

Asset managers without the capability or unwilling to pay the cost of achieving accurate valuations on the date of each cash flow may still wish to use a time-weighted methodology and can use methodologies that approximate to the “true” time-weighted return by estimating portfolio values on the date of cash flow, such as the methodologies outlined in the next three subsections.

Index substitution

Assuming an accurate valuation is not available, an index return may be used to estimate the valuation on the date of the cash flow thus approximating the “true” time-weighted return as demonstrated in Exhibit 2.17:

Exhibit 2.17 Index substitution

In Exhibit 2.17 the index is a good estimate of the portfolio value and therefore the resultant return is a good estimate of the true time-weighted rate of return. However, if the index is a poor estimate of the portfolio value, see Exhibit 2.18, then the resultant return may be inaccurate, although in this case a better estimate of underlying return than say the modified Dietz or IRR.

Exhibit 2.18 Index substitution

Regression method (orβmethod)

The regression method is an extension of the index substitution method. A theoretically more accurate estimation of portfolio value can be calculated adjusting for the systematic risk (as represented by the portfolio’s beta) normally taken by the portfolio manager.

Exhibit 2.19 Regression method

The index substitution method is only as good as the resultant estimate of portfolio value; making further assumptions about portfolio beta need not improve accuracy.

Analyst’s test

Rearranging Equation (2.22):

(2.23)

or

(2.24)

In other words, the time-weighted return of the portfolio can be approximated by the ratio of the money-weighted return of the portfolio divided by the money-weighted return of notional fund and then multiplied by the notional fund time-weighted rate of return. Since all commercial indexes are time weighted (they don’t suffer cash flows and are therefore useful for comparative purposes) we can use an index return for the time-weighted notional fund.

The advantage of these three approximate methods is that a time-weighted return may be estimated even without sufficient data to calculate an accurate valuation and hence an accurate time-weighted return. The disadvantages are clear: if the index, regression and notional fund assumptions respectively are incorrect or inappropriate the resultant return calculated will also be incorrect. Additionally, the actual portfolio return appears to change if a different index is applied which is counterintuitive (surely the portfolio return ought to be unique) and is very difficult to explain to the lay trustee.

HYBRID METHODOLOGIES

In practice, many managers use neither true time-weighted nor money-weighted calculations exclusively but rather a hybrid combination of both.

If the standard period of measurement is monthly, it is far easier and quicker to calculate the modified (or even simple) Dietz return for the month and then chain link the resulting monthly returns. This approach treats each monthly return with equal weight and is therefore a version of time weighting. All of the methods mentioned previously can be calculated for a specific time period and then chain linked to create a time-weighted type of return for that time period.

Linked modified Dietz

Currently, the standard approach for institutional asset managers is to chain link monthly modified Dietz returns. Often described as a time-weighted methodology, in fact it is a hybrid chain-linked combination of monthly money-weighted returns. Each monthly time period is given equal weight and therefore time weighted, but within the month the return is money weighted.

BAI method (or linked IRR)

The US Bank Administration Institute (BAI, 1968) proposed an alternative hybrid approach first proposed by Fisher, (1966), which essentially links simple internal rates of return rather than linking modified Dietz returns.

Because of the difficulties in calculating internal rates of return this is not a popular method and is virtually unknown outside of the US.

For clarification both the BAI method and the linked modified Dietz methods can be described as a type of time-weighted methodology because each standard period (normally monthly) is given equal weight. True time weighting requires the calculation of performance between each cash flow.

The index substitution, regression and analyst tests methods are approximations to the true time-weighted rate of return. The simple Dietz, modified Dietz and ICAA methods are approximations of the internal rate of return and are therefore money weighted.

WHICH METHOD TO USE?

Determining which methodology to use will ultimately depend on the requirements of the client, the degree of accuracy required, the type and liquidity of assets, availability of accurate valuations and cost and convenience factors.

Time-weighted returns neutralise the impact of cash flow. If the purpose of the return calculation is to measure and compare the portfolio manager’s performance against other managers and commercially published indexes then time weighting is the most appropriate. On the other hand, if there is no requirement for comparison and only the performance of the client’s assets are to be analysed then money weighting may be more appropriate.

As demonstrated in Exhibit 2.15, a time-weighted return, which does not depend on the amount of money invested, may lead to a positive rate of return over the period in which the client may have lost money. This may be difficult to present to the ultimate client although in truth the absolute loss of money in this example is due to the client giving the portfolio manager more money to manage prior to a period of poor performance in the markets. If there had been no cash flows the client would have made money.

Confidence in the accuracy of asset valuation is crucial in determining which method to use. If accurate valuations are available only on a monthly basis, then a linked monthly modified Dietz methodology may well be the most appropriate. The liquidity of assets is also a key determinant of methodology. If securities are illiquid it may be difficult to establish an accurate valuation at the point of cash flow. For illiquid assets such as private equity any perceived accuracy in the true time-weighted return could be quite spurious.

Internal rates of return are traditionally using for venture capital and private equity asset categories for a number of reasons:

1. The initial investment appraisal for non-quoted investments often uses an IRR approach.

2. Assets are difficult to value accurately and are illiquid.

3. The venture capital manager often controls the timing of cash flow.

It is interesting to note that like private equity, real estate performance has been traditionally calculated using a money-weighted methodology; however, with increased frequency of property valuations (albeit not independent) real estate has strived to be treated with equivalence to other assets and more recently time-weighted approaches have become more common. It is often said that private equity uses a money-weighted methodology because the asset manager has more control of the timing of cash flows; that may well be true but I believe the more likely reason is that historically accurate valuations at the time of cash flows are simply not available, and therefore a time-weighted methodology cannot be applied.

Money-weighted rates of return are often used for private clients to avoid the difficulty of explaining why a loss could possibly lead to a positive time-weighted rate of return. Perhaps the most significant advantage of money-weighted returns is that if the portfolio generates a profit, the rate of return with be positive (and vice versa) regardless of the pattern of cash flows.

Because the recognised advantage of time-weighted returns is that they neutralise the impacts of cash flows they are clearly favoured when the portfolio manager does not influence the timing of cash flows. This does not imply that if the portfolio manager influences the timing of cash flows then money-weighted returns should be preferred, the resultant return is simply the constant force of return required when applied to the pattern of cash flow that results in the end market value of the portfolio. This return is unique and not really comparable with other portfolios enjoying a different pattern of cash flow or indeed benchmark indexes; there are other ways of measuring the impact of timing, notably attribution analysis.

Mutual funds suffer a particular performance problem caused by using backdating unit prices as illustrated in Exhibit 2.21.

This is in effect what happened in the “late trading and market timing” scandal in US mutual funds revealed in 2003. Privileged investors were allowed to buy or sell units in international funds at slightly out-of-date prices with the knowledge that overseas markets had risen or fallen significantly already, resulting in small but persistent dilution of performance for existing unit holders.

SELF-SELECTION

With the choice of so many different, acceptable calculation methodologies, managers should establish an internal policy to avoid both intentional and unintentional abuse.

Table 2.1 illustrates the range of different returns calculated for our standard example in just the one period.

The fundamental reason for the difference in all of these returns is the assumptions relating to external cash flow. Without cash flow all these methods – money-weighted, time-weighted and approximations to time-weighted – will give the same rate of return.

Exhibit 2.21 Late trading

The reason for the difference effectively lies in the denominator (or average capital invested) of the return calculation. Each of these methods makes different assumptions about the impact of external cash flow on the denominator: the greater the cash flow the greater the impact.

The differences in the simple Dietz and the modified Dietz returns in Table 2.1 are so significant in this example because the cash flow is large relative to the starting value. If the cash flow is not large then the assumptions used to weight the cash flow will not have a measurable effect.

Because this effect is often not significant it is not always worth revaluing the portfolio for each cash flow. Many institutional asset managers employ a standard modified Dietz method and only revalue for a large external cash flow above a set percentage limit (10% is common). Asset management firms should set a limit and apply it rigorously. The limit may be defined to apply to a single cash flow during the period or the total cash flow during the period.

Table 2.1 Return variations due to methodology

MethodReturn (%)Simple Dietz−7.44Modified Dietz (end of day)−7.30Modified Dietz (beginning of day)−7.21Simple IRR−7.41Modified IRR−7.27True time-weighted (end of day)−9.93True time-weighted (beginning of day)−9.63Time-weighted midday cash flow−9.40Index substitution−9.79Regression−9.99Analyst’s test−10.14

If multiple returns are routinely calculated for each methodology and the best return chosen for each period, even poor performing portfolios could appear to be performing quite well. Clearly, it is unethical to calculate performance using multiple methods and then choose the best return.

Intentional self-selection of the best methodology is easy to avoid but unintentional abuse can occur. Portfolio managers are well aware that cash flow can impact performance and often they have a good feel for the performance of their own portfolios. If they have underperformed their expectations by say 0.2% they may require the performance measurer to investigate the return. The measurer identifying that a cash flow has occurred (but less than the normal limit) may conclude that the return has been adversely impacted by the cash flow. It would be entirely inappropriate for the performance measurer to adjust the return (even though it is theoretically more accurate) because the portfolio manager is unlikely to require the same analysis if the return is 0.2% above expectations, resulting in only positive adjustments taking place.

Table 2.2 lists the advantages and disadvantages of each return methodology available to the performance measurer together with my personal preference from the asset manager’s perspective. My preferences are consistent with the evolution of performance returns methodologies shown in Figure 2.1.

Different asset categories and different product types have progressed through this evolution of return methodologies at different rates-I would suggest driven by the availability of accurate valuations. Mutual funds evolved to true time-weighted return first because an accurate

Table 2.2 Calculation methodologies

valuation must be derived to allow investors to enter and exit the mutual fund at a fair price. Pension funds are perhaps one step behind in evolutionary development seeking time-weighted returns to neutralise the impact of cash flows and allow fair comparison of their managers which each other and benchmarks. In fact, pension funds are still in the process of evolving to the final stage with many institutional asset managers still using linked modified Dietz; they are not able, or unwilling because of cost, to calculate daily valuations. In reality, the GIPS standards mandate from 2010 a half-way house between linked modified Dietz and true time weighting which requires assets managers to value for large cash flows only.

Figure 2.1 Evolution of performance returns

Internal rate of return is clearly the starting point followed by the simple and modified Dietz approximations to the internal rate of return. Private equity, because of the difficulty of obtaining accurate valuations, is unable to evolve further. Interestingly, real estate has evolved from money-weighted to time-weighted more recently facilitated by more frequent property valuations.

ANNUALISED RETURNS

The geometric average or annualised return is the return which, if compounded with itself for the cumulative period, will result in the cumulative return.

It is poor performance practice to annualise returns for short periods less than one year. It is inappropriate to assume that the rate of return achieved in the year to date will continue for the remainder of the year.

The terms “arithmetic” and “geometric” are common in the field of performance measurement: arithmetic reflects additive relationships and geometric reflects multiplicative or compounding relationships.

Investment returns compound. When assessing historic performance it is essential to use the constant rate of return that will compound to the same value as the historic series of returns as shown in Exhibit 2.23:

Exhibit 2.23 Positive bias

Arithmetic averages are positively biased; if returns are not constant the annualised return will always be less than the arithmetic average. Note that the annualised return when compounded with itself over the entire period will reconcile to the original cumulative return; the arithmetic average return will not. Hence, the annualised return provides a better indicator of wealth at the end of the period than the arithmetic average. Performance analysts should use annualised rather than average returns.

Return hiatus

If there is any hiatus or gap in the performance track record, however short, it is impossible to bridge the gap and link performance returns to produce cumulative and hence annualised returns.

Some analysts might argue that it is possible to substitute an index return for the gap but I would argue that it is not best practice.

CONTINUOUSLY COMPOUNDED RETURNS

While simple returns are positively biased, continuously compounded returns are not.

We observe from the operation of our bank accounts that interest paid into our accounts compounds over time, in other words we receive interest on our interest payments. The more frequent the payments the higher the compounded return at the end of the year.

The nominal rate of return in each monthly period required to achieve an effective rate of return of 12% is 11.39%.

If we continue to break down the periods into smaller and smaller periods eventually we find the continuously compounded return or in effect the “force of return”:

(2.28)

(2.29)

The main advantage of continuously compounded returns is that they are additive. The total return can be calculated as follows:

(2.30)

Continuously compounded returns should be used in statistical analysis because unlike simple returns they are not positively biased.

GROSS- AND NET-OF-FEE CALCULATIONS

A key component in long-term investment performance is the fee charged by the asset manager. Fees are charged in many different ways by several different parties, in evaluating and comparing the performance of a portfolio manager it is essential that the impact of fees be appropriately assessed.

There are three basic types of fees or costs incurred in the management of an investment portfolio:

1. Transaction fees – the costs directly related to buying and selling assets including broker’s commission, bid/offer spread, and transaction-related regulatory charges and taxes (stamp duty, etc.), But excluding transaction-related custody charges.

2. Portfolio management fee – the fees charged by the asset manager for the management of the account.

3. Custody and other administrative fees including audit fees, performance measurement fees, legal fees and any other fee.

Portfolio managers should be evaluated against those factors that are under their control. Clearly, the portfolio manager has a choice whether or not to buy or sell securities; therefore performance should always be calculated after (or net of) transaction costs. This is naturally reflected in the valuations used in the methods described previously and no more action need be taken.

Portfolio management fees are traditionally taken directly from the account, but need not be; the portfolio manager may invoice the client directly, thereby receiving payment from another source.

If payments were not taken directly from the portfolio then any return calculated would be before or “gross” of fees.

The gross-of-fee effect can be replicated if the fee is deducted from the account by treating the management fee as a negative external cash flow. If the fee is not treated as an external cash flow, then the return calculated is after or “net” of fees.

The gross return is the investment return achieved by the portfolio manager and normally the most appropriate return to use for comparison purposes since institutional clients are normally able to negotiate fees.

Custody and other administration fees are not normally in the control of the portfolio manager and hence should not be reflected in the calculation of performance return for evaluation purposes. It should be noted, however, that the “client return” after administration fees is the real return delivered to the client.

Portfolio managers may bundle all of their services together and charge a “bundled fee”. If the bundled fee includes transaction costs that cannot be separated, then the entire fee must be subtracted to obtain the investment return.

In most countries local regulators will require mutual funds to report and advertise their performance net of all fees.

If calculating performance net of fees then to reflect the correct economic value of the portfolio, fees (including performance fees) should be accrued negatively in the valuation.

Estimating gross- and net-of-fee returns

The most accurate way to calculate the gross and net series of returns for a portfolio would be to calculate each set of returns separately, treating the fee as an external cash flow for the gross return but making no adjustment for the net return.