Über die Autorin bzw. den Autor

Gregory Connor is professor of finance at the National University of Ireland, Maynooth, and senior research associate at the London School of Economics and Political Science. Lisa R. Goldberg is executive director of analytic initiatives at MSCI Barra and adjunct professor of statistics at the University of California, Berkeley. Robert A. Korajczyk is professor of finance at Northwestern University.

Von der hinteren Coverseite

"Thorough and well-cited, this is a comprehensive treatment of techniques for portfolio risk management. It provides a unique perspective, from the fundamentals to practical applications. There are few books that cover this material in this particular way."--Christopher L. Culp, author of Structured Finance and Insurance

"The range of topics is wide and the coverage is deep. An impressive book."--Peter Christoffersen, McGill University

"The conceptual framework of this book is presented in a lucid and clear manner. The treatment is mathematically rigorous where it matters, without ever becoming pedantic and without cutting corners."--Riccardo Rebonato, Royal Bank of Scotland

"This book takes major steps forward in the crucially important area of portfolio risk measurement, making significant strides toward incorporating industry and country risk, as well as macroeconomic, FX, credit, transactions cost, and liquidity risks. It will be an essential reference text for academics, central bankers, and others in the financial services industry."--Francis X. Diebold, University of Pennsylvania

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Portfolio Risk Analysis

Princeton University Press

Contents

Acknowledgments...........................................................xiIntroduction..............................................................xiiiKey Notation..............................................................xix1 Measures of Risk and Return.............................................12 Unstructured Covariance Matrices........................................363 Industry and Country Risk...............................................614 Statistical Factor Analysis.............................................795 The Macroeconomy and Portfolio Risk.....................................1016 Security Characteristics and Pervasive Risk Factors.....................1177 Measuring and Hedging Foreign Exchange Risk.............................1348 Integrated Risk Models..................................................1559 Dynamic Volatilities and Correlations...................................16710 Portfolio Return Distributions.........................................19111 Credit Risk............................................................21212 Transaction Costs and Liquidity Risk...................................24113 Alternative Asset Classes..............................................27114 Performance Measurement................................................29915 Conclusion.............................................................319References................................................................323Index.....................................................................345

Chapter One

Section 1.1 gives definitions of portfolio return. Section 1.2 introduces some key portfolio risk measures used throughout the book. Section 1.3 discusses risk–return preferences and portfolio optimization. Section 1.4 discusses the capital asset pricing model (CAPM). In that section we relate the CAPM to the more general state-space pricing model, and critically assess its applications to portfolio risk management. Section 1.5 takes a broader perspective, discussing the institutional environment and overall objectives of portfolio risk management, and its fundamental limitations.

1.1 Measuring Return

Except where noted, throughout the book we work in terms of a unit investment, that is, an investment with an initial value of $1. The analysis can be scaled up to cover an investment of any size.

1.1.1 Arithmetic and Logarithmic Return

We define the investment universe as the set of assets that are current or potential holdings in the portfolio. In many applications, the investment universe is very large since it includes individual stocks, bonds, commodities, and other assets from around the world. We use n to denote the number of assets in the investment universe.

We usually measure the random per-period payoff on an asset in terms of its arithmetic return, denoted by r. This is defined as the end-of-period price plus any end-of-period cash flow, divided by the beginning-of-period price, minus one. Intermediate cash flows can be artificially shifted to the end of the period using an appropriate reinvestment rate. The arithmetic return on a portfolio is defined analogously as the end-of-period value plus cash flow divided by the beginning-of-period value, minus one. Note that return is a random variable; its expected value is called the expected return.

The arithmetic portfolio return equals the weighted sum of constituent asset returns. We use r to denote the n-vector of asset returns on all the assets in the investment universe, and w the n-vector of portfolio weights. These weights give the proportion invested in each asset, and therefore sum to one. (In a typical application, the vast majority of portfolio weights are zero, since the number of assets in the investment universe tends to be much larger than the number held in nonzero amounts in the portfolio.) The portfolio return can be expressed as

r_w = w' r. (1.1)

Arithmetic return supports subportfolio analysis: the arithmetic return on the portfolio equals the value-weighted sum of the returns on any complete collection of nonoverlapping subportfolios. We can apply (1.1) to nested subsets, first using it to find the returns to subportfolios, where w is the vector of asset weights within a subportfolio and r is the vector of assets in the subportfolio, and then to the aggregate portfolio, where w is the vector of weights in the subportfolios and r is the vector of subportfolio returns. At each step, the investment universe is defined appropriately. So, for example, consider a large diverse portfolio of common stocks, government bonds, corporate bonds, and real estate. Defining the investment universe as the collection of all individual assets in all the categories, the total portfolio return is the weighted sum of all the individual returns. Alternatively, if desired, we can first calculate the return on the common stock portfolio using the universe of common stocks as the set of assets and the total amount invested in common stocks as total value. From this we get a common stock portfolio return. Repeating this procedure for government bonds, then corporate bonds, and then real estate gives analogously a government bond portfolio return, a corporate bond portfolio return, and a real estate portfolio return. Now we redefine the investment universe as consisting of these four "assets" (our computed subportfolio returns) so that in the second step n = 4, and we can compute the total portfolio return as the weighted sum of these four asset returns.

Return is typically measured at daily, weekly, monthly, quarterly, and annual frequencies. However, the T single-period arithmetic returns do not add up to the arithmetic return over T periods. Instead, the T-period arithmetic return on an asset or portfolio is the compound product of the constituent one-period returns:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.2)

When written in this form, the T-period return is called the compound return to highlight the fact that it is a product of returns at higher frequency. If the intervening single-period returns are uncorrelated, then the expected compound return equals the product of the returns (each augmented by one) and then minus one:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

If, in addition, the single-period returns are independent through time with constant mean and variance, then the variance of the compound return can be expressed in terms of the single-period mean and variance:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.3)

However, this relatively simple formula for variance (1.3) does not apply if the single-period mean or variance is time dependent.

Temporal analysis of return is simpler in the continuously compounded or logarithmic framework. The log return of an asset or portfolio is defined as the log of the end-of-period value plus cash flow minus the log of the beginning-of-period value. Mathematically, this is written

r_l = log(1 + r).

Log returns add up over time. Letting rl0,T denote the log return over T periods,

r_l0,T = r_l0,1 + r_l1,2 + ··· + r_lT-1,T. (1.4)

Formula (1.4) facilitates temporal aggregation of risk. For example, if returns are uncorrelated, then the variance of the T-period log return is the sum of the variances of the one-period returns. Consequently, log returns often provide a better foundation for multiperiod and long-horizon risk–return analysis than arithmetic returns since the additive formula (1.4) is much more tractable than the multiplicative formula (1.2).

A significant drawback to log returns is that they do not satisfy the portfolio linearity property (1.1). In other words, the log return of a portfolio is not the value-weighted sum of the log returns of the constituent assets. Another drawback is that investment objectives are less naturally stated in terms of log return than arithmetic return. Thus, log and arithmetic returns have complementary strengths and weaknesses. In this book we will use both definitions; the term return without modifier will refer to arithmetic return.

1.1.2 Approximate Relationships between Arithmetic and Log Returns

A second-order Taylor expansion of log(1 + r) evaluated at r = 0 gives a useful relationship between arithmetic and log returns:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.5)

If r is small enough for r²/2(1 + z)² to be small relative to r, then the log and arithmetic returns are approximately equal. For highfrequency returns, this condition is often satisfied so there is little difference between the values of the two types of returns, relative to their size. Figure 1.1 compares log and arithmetic returns over the interval r = (-0.99, 1.0); figure 1.2 shows the difference between them as a proportion of arithmetic return.

If the log return of a particular asset or portfolio has a normal distribution, then the moments of the arithmetic and log returns have exact analytical relationships. For the mean and variance,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

However, the assumption of normality for log asset returns implies that the log portfolio return is nonnormal. Formally, this limits the applicability of normal log returns in portfolio analysis. Campbell and Viceira (1999, 2001) argue that the log return of a portfolio can be treated as approximately normal, based on the multivariate normality of the log returns of the constituent assets.

1.1.3 Excess Return, Active Return, and Long–Short Return

The excess return of an asset or portfolio is defined as the arithmetic return minus the riskless return. It is typically preferable in risk analysis to use excess return rather than total return since it removes the riskless component of the return. Let x = r - 1ⁿr₀ denote the vector of asset excess returns, where 1ⁿ is an n-vector of ones and r₀ is the riskless return. There are several equivalent ways to express the excess return of a portfolio w in terms of its components:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.6)

Note that in the final expression in (1.6) the n + 1 vector of portfolio weights (including the -1 weight in the riskless return) sums to zero rather than one. Excess log return is usually defined as r_l - r_l0, which is equal to log((1 + r)/(1 + r₀)).

Large institutional investors typically allocate funds to more than one asset class and there may be a different manager for every class. Each manager is typically benchmarked to an index portfolio appropriate to his asset class, and for that manager risk and return are measured relative to the assigned benchmark. The difference between the return to the manager's portfolio and the benchmark return is called the active return.

Letting w_B denote the benchmark portfolio weights, we define the active portfolio weights by

w_A = w - w_B. (1.7)

In the active framework, a zero-cost constraint w'_A1ⁿ = 0 replaces the unit-cost constraint w'1ⁿ = 1. Despite the fact that an active portfolio has zero cost, its return is well-defined due to formula (1.7), which implies that active return can be expressed as the difference between the return on two unit-cost portfolios.

The long–short return is a variation on active return. Many hedge fund investment portfolios consist of a collection of long positions and short positions split equally, supported by a cash account earning a riskless return (not included in the portfolio weights). Assume that the long and short positions have equal total value and scale the portfolio weights so that the sum of the long positions equals one and the sum of the short positions equals one. The difference between these long and short portfolio returns is called the long–short return.

Note that the total value of a long–short portfolio equals zero by construction. Typically, the cash position of the hedge fund will be significantly smaller than the total position, giving rise to a leverage effect. The leverage ratio (the ratio of the fund's position size to its cash value) has a big effect on the riskiness of the fund.

An increasingly popular product design in institutional portfolio management is "relaxed constraint" funds, sometimes called "130/30" funds. For this type of fund, the manager invests the value of the fund in securities and then short-sells securities with a total value equal to 30% of the fund value and invests the short-sale receipts in more security purchases. The return to the fund is measured as a proportion of the initial fund value. See Jacobs and Levy (2007) for a detailed overview of relaxed constraint funds.

1.2 The Key Portfolio Risk Measures

In this section we introduce some key risk measures used throughout the book.

1.2.1 Portfolio Risk Components

Jorion (2007) decomposes portfolio risk into market risk, liquidity risk, credit risk, operational risk, and legal risk. This book will cover the first three of these but not operational risks, which Jorion defines as "inadequate systems, management failure, faulty controls, fraud, or human error." In practical applications, systems for the measurement and control of operational risk are extremely important. However, they lie outside the scope of this book. We will only consider legal risks to the extent that they are related to the structure and maintenance of quantitative risk models. We also consider model risk, which we define as risk due to misspecification or estimation error in the risk analysis model.

In this first chapter we focus on market risk, which is risk stemming from the variation in the market prices of the assets in the portfolio.

1.2.2 Return Distribution and Moments

The most general notion of quantitative portfolio risk is the return distribution of r, which gives the cumulative probability cum(x) that an asset or portfolio return r is no larger than x for every fixed value of x:

cum(x) = Pr(r ≤ x),

where Pr(x) denotes the probability of a given set of events. A related concept is the portfolio return density den(ω), which satisfies

cum(x) = ∫^x_-∞ den(ω)dω for all x.

In principle, all risk measures can be derived from the return distribution. However, as is often the case, generality comes at a cost. Without additional assumptions, the return distribution is difficult to estimate.

An alternative characterization of the return distribution cum(x) is given by the moments of the distribution function. The first moment of a distribution is the mean. We denote mean return by µ and demeaned return by [??] = r - µ. The kth central moment of the return distribution is the expected value of demeaned return raised to the power k. One widely used risk measure is variance, which is the second central moment:

ω² = E][??]²].

It is often convenient to take the square root of variance, which we will call volatility, denoting it by ω = E][??]²])1/2.

To separate out the effect of variance on the distribution, it is standard practice to scale demeaned return by volatility in the definition of moments for k greater than two. This scaled version of return, [??]/ω, is called standardized return. Skewness is the third central moment of the standardized return,

E]([??]/ω)³],

and kurtosis is the fourth central moment of the standardized return:

E]([??]/ω)⁴].

If return is normally distributed, then return variance completely determines the distribution of demeaned return; the higher moments provide no additional information. In particular, in the case of normality all the moments equal zero for odd values of k, and the fourth moment (kurtosis) has a value of 3. Since the normal distribution serves as the benchmark case, it is useful to define excess kurtosis as raw kurtosis minus 3.

The plausibility of assuming that a return distribution is normal depends on the portfolio in question and the return horizon. Table 1.1 shows the variance, skewness, and excess kurtosis of four return series for three different return intervals (1-day, 5-day, and 20-day). In addition to a stock market index, a long-term bond index, and an exchange rate return, we show the return to an option-based portfolio that involves writing out-of-the-money put options on the stock market index combined with a cash account. We call this the market insurance provision portfolio, since the portfolio holder is essentially "selling insurance" against market declines, underwritten by his cash account. Lo (2001) uses this type of portfolio strategy to illustrate the dangers of using variance as a risk measure for derivatives-based investment strategies. Note the extreme negative skewness and positive excess kurtosis of the market insurance provision portfolio return.

(Continues...)

Excerpted from Portfolio Risk Analysisby Gregory Connor Lisa R. Goldberg Robert A. Korajczyk Copyright © 2010 by Princeton University Press. Excerpted by permission of Princeton University Press. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
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