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The financial crisis of 2007–2008 reopened the debate about the need for identifying a correct risk measurement in financial time series. Different risk measurements give rise to different risk-adjusted performance values. The most used and best known measurement is the Sharpe ratio, although other risk-reward ratio specifications attempt to correct for its shortcomings. This chapter describes the most popular ratios in the academic literature beyond the mean-variance approach and discusses their advantages and disadvantages. This chapter also provides an analysis of results when calculating risk ratio-based rankings.

This chapter reviews the most widely used metrics to analyze mutual fund performance. It covers tools, risk metrics, and rating criteria popular with both academics and practitioners. Measurement of relative fund performance is tested with a set of dimensionless ratios including the coefficient of variation, Sharpe ratio, and Treynor ratio. Interpreting the precise amount of relative under- or over-performance is difficult with dimensionless ratios. In response to this drawback of using such measures, several researchers developed mutual fund performance metrics that quantify risk-adjusted performance to a greater degree. For example, the M² measure, Jensen’s alpha, and the Carhart model are useful in quantifying risk-adjusted performance in percentage terms. The chapter presents recent extensions or enhancements to the Carhart model and also discusses the quantitative and qualitative risk metrics available from Morningstar.

This chapter evaluates the recent performance of REITs versus more traditional equity and debt investments. It shows that REIT returns are very closely linked to mid-cap equity returns but are only very weakly correlated to returns on bonds and returns on the underlying real estate markets. Sharpe ratios indicate that REITs have underperformed both bonds and mid-cap equities over the period 1978 to 2012. The chapter also surveys the academic literature addressing the key determinants of REIT returns: corporate governance, dividend policy, institutional ownership, and the returns on the underlying real estate market. In general, strong corporate governance and higher levels of institutional ownership positively influence REIT returns. REITs are an important class of investments for institutional investors. The data show that in the largest size decile, institutional ownership is 93.1 percent of the total.

A diverse set of measures allow investors to evaluate hedge fund portfolio managers’ performance across different dimensions. The various measures quantify the effectiveness of security selection; account for investor flows, operating risk, and worst-case investment scenarios; net out benchmark and peer-fund performance; and control for risk factors that are unique to hedge fund investment strategies. Hedge fund return information in published databases is usually self-reported, which is a conflict of interest that produces several reporting biases and inflated published average returns. After adjusting for these biases, hedge fund average returns trail equity market returns and in fact almost exactly equal U.S. Treasury bill average returns between January 1994 and March 2016. Yet, after risk adjustment, the hedge fund performance picture brightens. In the aggregate, hedge funds have higher Sharpe ratios and multifactor alphas, and lower maximum drawdown levels than equity market benchmarks.

In this chapter we study an incomplete market, but we do not look for a unique martingale measure. Instead we try to find “reasonable” bounds on arbitrage free prices. The terms “reasonable” is formalized in terms of a price rule with bounded Sharpe ratio–so-called good deal bounds. We study a factor model and show that the good deal bounds can be obtained by solving a control problem where the likelihood process acts as a state variable, and the Girsanov kernel is the control variable. The theory is then applied to concrete examples.

On the asset side, trustees must build a robust return-seeking asset portfolio in accordance with the risk and performance strategy defined in the ALM framework and the LDI strategy. This chapter provides the building blocks of an efficient investment portfolio strategy. Readers will understand the positive effect of diversification on the risk/return profile of portfolios and how to measure the skills of portfolio managers in security selection and the passive replication of risk-adjusted return of indexes. An overview is provided of the asset class universe and various management styles, the way to look at asset classes in terms of risk-adjusted returns. How to build various portfolios and undertake simulations in order to select the most appropriate portfolio to meet the objectives of performance and risk aversion are explained. Based on case studies, readers will learn how to analyse investment portfolios, simulations, build efficient frontiers and draw conclusions.

This chapter explores risk management in the Normal world. A Normal world where the mean-variance criterion can be used safely, portfolio choice is easy, the diversification principle works well, and portfolio efficiency can be measured by the Sharpe ratio. Normality assumptions imply that risk premiums are easy to compute even when markets are incomplete and are given by the Capital Asset Pricing Model (CAPM). The chapter concludes by showing the dangers of viewing the world as Normal, in spite of contrary empirical evidence.

The mean‐variance frontier is characterized with and without a risk‐free asset. The global minimum variance portfolio and tangency portfolio are defined, and two‐fund spanning is explained. The frontier is characterized in terms of the return defined from the SDF that is in the span of the assets. This is related to the Hansen‐Jagannathan bound. There is an SDF that is an affine function of a return if and only if the return is on the mean‐variance frontier. Separating distributions are defined and shown to imply two‐fund separation and mean‐variance efficiency of the market portfolio.

This chapter examines recent academic research concerning performance evaluation of equity mutual funds. Investors seeking to reach their financial goals should include investments that enhance the risk and return characteristics of their portfolios, and equities are often considered a viable asset allocation. However, studies show that many equity funds fail to produce a positive alpha. The chapter also discusses alternate means of assessing performance besides alpha, such as the Sharpe ratio and performance attribution, and problems associated with measuring alpha and other evaluation tools. Finally, it discusses the ability of managers to successfully engage in security selection for equity fund portfolios because this has a direct impact on their ability to generate positive risk-adjusted returns and manage risk.

In this chapter we study a very general multidimensional Wiener-driven model using the martingale approach. Using the Girsanov Theorem we derive the martingale equation which is used to find an equivalent martingale measure. We provide conditions for absence of arbitrage and completeness of the model, and we discuss hedging and pricing. For Markovian models we derive the relevant pricing PDE and we also provide an explicit representation formula for the stochastic discount factor. We discuss the relation between the market price of risk and the Girsanov kernel and finally we derive the Hansen–Jagannathan bounds for the Sharpe ratio.

In finance, a common way of evaluating an investment uses the investment’s expected return and the investment’s risk, in the sense of the investment’s volatility, or exposure to chance. A version of this method derives from a general mean-risk evaluation of acts, under the assumption that only money, risk, and their sources matter. Although the method does not require a measure of risk, finance investigates measures of risks to assist evaluations of risks. An investment creates possible returns, and the variance of the probability distribution of their utilities is a measure of the investment’s risk. This measure neglects some factors affecting an investment’s risk, and so is satisfactory only in special cases. Another measure of risk is known as value-at-risk, or VAR. It also neglects some factors affecting an investment’s risk, and so should be restricted to special cases.