Out of all the mathematical finance books I have seen recently, this one was by far the most pleasurable to read. This pure enjoyment is the result of the connection you have to the topics, not because of their importance, but because of their presentation in this book.

First impression: what a huge book for mathematical finance! The amount of material the author provides, both in the book and on his website, is tremendous. It really reflects time, devotion, and love of the subject.

As the title suggests, the topicof the book is the problem of asset allocation and the estimation of the corresponding risk. Asset allocation is a popular topic, with numerous applications, for an investor who seeks to optimize its investments. The author attacks the problem with lots of success.

The text is filled with graphical illustrations, which are of tremendous help in grasping the concepts with full understanding. The author provides an intuitive approach in presenting the concepts rather than just simply throwing out theorems and proofs. One of the best things I like about the book is the way each chapter is started with something like a chapter preface. Chapters begin with couple of pages explaining the problem at hand, possible ways of solving the problem and a little description on each of the subchapters (none of the chapters begin with "Let x be…."). Another vital feature of a good book are examples, and this book is swamped with them.

The author provides additional resources that complement the book:

- Technical appendix (178 pages)
- Exercise book (174 pages)
- MATLAB code (covers each chapter)

The technical appendix provides additional examples, further explanations of the concepts presented in the book, and some special cases. The "exercise book" contains computer based (MATLAB) problems.

The book is divided into fours parts. Part I provides the building blocks — the statistics of asset allocation. First 100 pages are devoted to univariate and multivariate statistics. For students it will prove worthwhile to read these two chapters even if they are familiar with the concepts. The style of writing is excellent, one which is usually not found in statistics or math books. The concepts are presented with such elegance that there is really no need for "pencil and a paper." After the first 100 pages we are ready to model the market. The concept of invariance is discussed in chapter 3. Further discussion of invariance is given for fixed income and derivatives markets. Next the author presents the ways in which to detect and obtain the distribution of the invariants. Once the distribution is obtained it is projected to the investment horizon. Finally the distribution of invariance is translated into the market prices of securities.

Part II discusses the Classical Asset Allocation. Chapter 4 provides an exploration of the estimators for the estimation of the distribution of the market invariants. First the concept of an estimator is defined. Nonparametric, maximum likelihood and shrinkage estimators are presented followed by the discussion of robust estimation. The chapter concludes with tips on how to improve the estimation of distribution of the market invariants. Chapter 5 deals with the estimation of allocations for a given investment horizon. We come back to the investor. The investor's objectives are random variables and techniques are presented for computing of the distribution of the objectives. Two approaches are presented: the distributions of the objective relative to the allocation and summarizing the distribution into a single number, an index of satisfaction. The remainder of the chapter deals with three classes of indices of satisfaction: expected utility, value at risk and expected shortfall. In chapter 6 mean-variance approach is described in detail.

Part III considers asset allocation with the inclusion of the estimation risk. Again estimation of the distribution of the invariants is considered but now the Bayesian approach to parameter estimation is applied. Chapter 8 discusses the problem of the distribution, namely, the distribution of the market is unknown and hence the estimation will contain errors. First, allocation is considered to be the outcome of a decision contingent on the realized available information. This approach is applied to the simple strategy-decision that disregards any historical information from the market. Rather, the investor's belief is considered. Further, it is applied to the sample-based allocation decision. Chapter 9 discusses allocation strategies within the allocation decision process, taking into account estimation risk. Limiting and reducing sensitivity is considered by introducing the Bayesian approach, the Black-Litterman approach and Michaud's resampling technique. The chapter ends with robust allocation decisions.

Part IV includes an appendix on linear algebra and functional analysis.

This is not an easy text to read (one reason is simply the size), but if you do decide to take on the challenge you will not end up empty-handed.

Ita Cirovic Donev is a PhD candidate at the University of Zagreb. She hold a Masters degree in statistics from Rice University. Her main research areas are in mathematical finance; more precisely, statistical mehods of credit and market risk. Apart from the academic work she does consulting work for financial institutions.