Modified Sharpe Ratio Research Articles

Modelling the effect of macroeconomic factors on residential real estate investment returns in Abuja and Lagos, Nigeria

PurposeThis study evaluated the effect of macroeconomic factors on residential real estate (RE) investment returns in the cities of Abuja and Lagos, Nigeria, with a view to guiding RE investors and researchers.Design/methodology/approachA survey research design was employed using a questionnaire to collect RE transaction data from 2008 to 2022 from estate surveying and valuation firms in the study areas. Rental and capital value data collected were used to construct rental and capital value indices and total returns on investment. The macroeconomic data used were retrieved from the archives of the Central Bank of Nigeria (CBN). Granger causality (GC) and multiple regression models were adopted to evaluate the effect of selected macroeconomic variables on residential RE investment returns in the study areas.FindingsThe study found a progressive upward movement in rental and capital values of residential RE investment in the study areas within the study period. Total and risk-adjusted returns on investment were equally positive within the study period. Only the inflation rate, unemployment rate and real gross domestic product (GDP) per capita were found to be the major determinants of residential RE investment returns in the study areas within the study period.Research limitations/implicationsThe secrecy associated with property transaction information/data by RE practitioners in the study areas posed a challenge. Property transaction data were not adequately kept in a way for easier access and retrieval in many of the estate firms and agent offices. Consequently, there was a lack of data that spanned the study period in some of the sampled estate firms or agent offices. This data collection challenge was, however, overcome by the excess time spent retrieving the required data for this study to ensure that the findings appropriately answer the research questions. Practical implicationsInflation and GDP per capita have been found to be significant factors that influence residential RE investment performance in the study areas. Therefore, investors should pay attention to these identified macroeconomic factors for residential RE investment in the study areas whilst making investment decisions in order to mitigate a possible loss of income or return. The government should formulate and implement economic policies that would address the current high unemployment and inflation rates in Nigeria at large.Originality/valueThis study has extended and further enriched the existing body of knowledge in the field of RE investment analysis in Nigeria. To the best of the authors' knowledge, this study is the first to adopt the Cornish Fisher value-at-risk and modified Sharpe ratio models to analyse risk and risk-adjusted returns on residential RE investment, respectively, in Nigeria. It has therefore redirected the focus of RE researchers and practitioners to a more objective approach to RE investment performance analysis in Nigeria.

Assessment of Efficiency of Investment in Companies Belonging to the Rare Earth Metals Sector

Purpose – This paper aimed at assessing the efficiency of investment in companies belonging to the rare earth elements sector and examining the stability of the results. Research method – The following performance measures were used in the study: modified Sharpe and Omega, Sortino, Calmar, Sterling and Burke ratios. The investigation is based on daily quotations of selected companies whose business activities were related to extraction, processing and recycling of rare earth metals in the period: from July 2018 to June 2023 and MVIS® Global Rare Earth/Strategic Metals Index representing the global market. Results – The analysis for the entire period revealed a worse performance of investments in rare earth stocks than investment in risk-free assets. Only AREC outperformed other securities offering the best results for 3 of 6 ratios. To examine the stability of the results, the total investigation period was divided into 5 subperiods and all performance ratios were determined for each of them. A strong variation of results did not allow to recognise stable patterns. When the results for the entire investigation period were used as a benchmark, only MVREMX offered stable and attractive results of modified Sharpe ratio outperforming the five-year value in 4 out of 5 subperiods. Originality / value / implications / recommendations – This study provides a new insight into the rare earths sector regarding its growing international importance and the efficiency of investments into listed rare earth firms. The examination uses two-dimensional measures offering the advantage of simultaneously combining return and risk into a single performance number. The findings imply the necessity of improving the stock price performance to attract investors’ interest.

Enhancing portfolio resilience during crisis periods: Lessons from BRICS indices and multi asset strategies

This paper uses Markowitz’s mean-variance model to construct an investment portfolio incorporating multiple assets – BRICS equity indices, Gold, crude oil, bonds, and cryptocurrencies. The optimally created risky portfolios outperform alternative portfolio optimization methods – the naive portfolio and the equal risk contribution portfolio; and established indices – the S&P 500 and the MSCI Emerging Equity Index in terms of metrics – adjusted Sharpe ratio, modified Sharpe ratio, and the modified Value at Risk. The findings are validated across different periods, including the COVID-19 period and the Russian invasion of Ukraine, including various in and out of sample periods. The findings highlight the benefits of portfolio diversity, mainly using BRICS indices, Gold, and Brent Crude oil, and challenge the notion of limited diversification benefits in BRICS indices found in previous studies. This paper further suggests the potential of emerging market bonds ETF as a diversification option during turbulent economic periods and highlights the limitations of cryptocurrencies in optimizing multi asset portfolios. By adopting the recommended multi asset portfolios, investors can enhance their risk-return trade-offs and achieve superior performance compared to the S&P500 and MSCI emerging indices. Lastly, the paper recommends future research opportunities in measuring portfolio performance and hedging strategies considering risk-adjusted return measurements, transaction expenses, and dynamic rebalancing techniques.

Assessment of Residential Real Estate Investment Performance in Lafia Metropolis, Nigeria

Investors are expected to be guided through investment performance measurement and analysis in order to safeguard against potential loss of investment returns or the capital invested. Whereas residential real estate (RRE) constitutes a significant part of the portfolios of individual and institutional investors across the globe, the assessment of risk-return performance of RRE investment in Nigeria generally has not been well researched. Investors in real estate (RE) in the Lafia city of Nasarawa State, like other cities in Nigeria, still rely on mere intuition, sales comparison and the rule of thumb in real estate investment decisions-making. Consequently, these ill-informed investors often venture into poorly conceived and subsequently financially disastrous real estate investment projects thereby failing to achieve their desired investment objectives. Hence, this research was conducted to evaluate the performances of RRE investment in Lafia metropolis of north-central Nigeria with a view to guiding existing and potential real estate investors and practitioners in the region in making informed property investment decision. A survey research design was adopted to collect transacted residential property value data from estate surveying and valuation firms as well as reputable estate agent offices in the study area. Mean scores, property value index and holding period return model were used to analyse the rental/capital value indices and total return performance respectively. The modified value-at-risk model was used to assess risk while the modified Sharpe ratio was used to evaluate the risk-return performance of RRE investment performance in the study area. The research findings revealed, among others, a progressive increase in rental and capital values of RRE within the study period. RRE investment showed an average rental growth of 6.8% per annum and capital value growth of 9.4% per annum. On the basis of investment returns, it was found that Angwan Doka outperformed other selected neighbourhoods in the study area with an average mean total return and risk-adjusted return of 13.76% and 0.80% per annum respectively while Shabu underperformed other locations with average mean total return and risk-adjusted return of 12.17% and 0.55% per annum. The study recommended, among others, that potential real estate investors should consider investing in residential real estate in Angwan Doka since this location gives higher total and risk-adjusted returns on residential property investment relative to other locations in the city.

Portfolio Selection under Systemic Risk

AbstractThis paper proposes a modified Sharpe ratio to construct optimal portfolios under systemic events. The portfolio allocation problem is solved analytically under the absence of short‐selling restrictions and numerically when short‐selling restrictions are imposed. This approach is made operational by embedding it in a multivariate dynamic setting using dynamic conditional correlation and copula models. We evaluate the out‐of‐sample performance of our portfolio empirically over the period 2007 to 2020 using ex post final wealth paths and systemic risk metrics against mean–variance, equally weighted, and global minimum variance portfolios. Our portfolio outperforms all competitors under market distress and remains competitive in noncrisis periods.

Analysis of Environmental, Social, and Governance Performance in Indonesia: Role of ESG on Corporate Performance

Issues regarding environmental, social, and governance have a positive role in corporate performance, especially in developed countries. Unlike accounting performance which only considers business performance, the ESG score reflects long-term performance because it considers environmental, social, and governance factors. Firms in Indonesia cannot improve their ESG score due to the limited resource and shareholders' reactions. This study compares the investment performance between the top 9 ESG constituents and the LQ45 index. Further, this study examines the impact of the ESG score on financial performance. This research used non-financial companies listed on Indonesia Stock Exchange as a sample. Modified Sharpe ratio and panel data regression were used as a method. The result showed that the ESG portfolio has a better Sharpe ratio than LQ45. However, the regression shows a negative effect on the ESG score on financial performance. Increasing the ESG score decreases financial performance because it requires higher investment and opportunity costs. This study has several recommendations related to implementing ESG at the corporate level.

Portfolio Optimization for Extreme Risks with Maximum Diversification: An Empirical Analysis

Heavy tailedness and interconnectedness widely exist in stock returns and large insurance claims, which contributes to huge losses for financial institutions. Diversification ratio (DR) measures the degree of diversification using the Value-at-Risk, which is known to capture extreme risks better than variance. The portfolio optimization strategy based on DR maximizes the effect of diversification for extreme risks. In this paper, we empirically examine the DR strategy by using more than 350 S&P 500 stocks under the assumption that the stock losses are modeled with a flexible multivariate heavy-tailed model. This assumption is verified empirically. The performance of DR strategy is compared with four benchmark strategies: equally weighted portfolio, minimum-variance portfolio, extreme risk index portfolio, and most diversified portfolio. The performance of comparison includes annualized portfolio return, modified Sharpe ratio, maximum drawdown, portfolio concentration, portfolio turnover, and the degree of diversification. DR outperforms other strategies. In particular, DR shows the highest return and maintains the highest level of diversification during the global financial crisis of 2007–2009.

Measuring the risk-adjusted performance of selected soft agricultural commodities

In this paper, we used several elaborate return-to-risk methods to investigate the risk-adjusted performances of five soft commodities. Regarding only the level of risk, we found that cocoa had the highest risk of losses, followed by orange juice. Cotton and coffee had the lowest risk of losses. However, according to the return-to-risk output, cotton was the worst asset in which to invest because it had negative average returns. In contradistinction, sugar had a relatively high risk of losses but also the highest average returns, which put it in the first place according to the Sharpe, Sortino and modified Sharpe ratios. Although orange juice had the second-worst downside risk performance, it came in second place according to the return-to-risk ratio because it had relatively high average returns.

Green Bonds for the Transition to a Low-Carbon Economy

The green bond market is emerging as an impactful financing mechanism in climate change mitigation efforts. The effectiveness of the financial market for this transition to a low-carbon economy depends on attracting investors and removing financial market roadblocks. This paper investigates the differential bond performance of green vs non-green bonds with (1) a dynamic portfolio model that integrates negative as well as positive externality effects and via (2) econometric analyses of aggregate green bond and corporate energy time-series indices; as well as a cross-sectional set of individual bonds issued between 1 January 2017, and 1 October 2020. The asset pricing model demonstrates that, in the long-run, the positive externalities of green bonds benefit the economy through positive social returns. We use a deterministic and a stochastic version of the dynamic portfolio approach to obtain model-driven results and evaluate those through our empirical evidence using harmonic estimations. The econometric analysis of this study focuses on volatility and the risk–return performance (Sharpe ratio) of green and non-green bonds, and extends recent econometric studies that focused on yield differentials of green and non-green bonds. A modified Sharpe ratio analysis, cross-sectional methods, harmonic estimations, bond pairing estimations, as well as regression tree methodology, indicate that green bonds tend to show lower volatility and deliver superior Sharpe ratios (while the evidence for green premia is mixed). As a result, green bond investment can protect investors and portfolios from oil price and business cycle fluctuations, and stabilize portfolio returns and volatility. Policymakers are encouraged to make use of the financial benefits of green instruments and increase the financial flows towards sustainable economic activities to accelerate a low-carbon transition.

Analisis Kinerja Saham Syariah Indonesia dan Malaysia Periode 2011-2018

Indonesia and Malaysia are two countries in Southeast Asia that are always included in the top ten countries in the world that have the health and development of Islamic Finance according to the State of The Global Islamic Economy (GEI) Report. This study analyzes the performance of Indonesian sharia stocks that are incorporated in the Jakarta Islamic Index (JII) and Malaysian Islamic stocks that are incorporated in the FTSE Bursa Malaysia Hijrah Shariah Index (FHSI) for the period 2011-2018 with a technical analysis approach using 4 indicators, namely: return, volatility, Value at Risk (VaR), and Modified Sharpe Ratio (MSR). The results obtained from this study are that there have no significant differences in return and MSR performance between Indonesian and Malaysian Islamic stocks, but there have differences in volatility and VaR, where Indonesian Islamic stocks have higher volatility and VaR than Malaysia at a 95% confidence level.

Combining Upside and Downside Volatility in Investment Decision

This paper deals with asset allocation decisions when the considered risk measure is directly related to the investor’s level of risk aversion. It is well known that the optimal portfolio weights are considerably sensitive to how assets are ranked on the basis of their risk-return profile. We propose a procedure to construct optimal portfolios that adapt quickly to changes in risk using a time varying asset allocation model based on a modified Sharpe Ratio measure related to downside and upside risk weighted using an aversion parameter. The model is applied, as an illustrative example, to six stock markets located in Western Europe basing the analysis on monthly data covering the period January 2000-December 2020.

Measuring Islamic Stock Performance in Indonesia with A Modified Sharpe Ratio

Since the late 1960s, one of the stock performance analysis tools commonly used is Sharpe Ratio. The Sharpe Ratio consists of three components, namely stock return, risk-free returns, and stock risk. Many studies approach risk-free returns with interest rates, including when measuring the performance of Islamic stocks, while interest rates are prohibited in the concept of Islamic finance. Moreover, the stock risk is measured by a standard deviation which assumes returns are normally distributed, while many stock returns are non-normally distributed. This paper intends to measure the performance of Islamic stocks listed on the Indonesian Stock Exchange (IDX) for the period of January 2011 to July 2018 using a modified Sharpe Ratio. The ratio is modified by replacing the interest rate with four approaches: eliminating the interest rate, changing with zakah rates, changing with inflation, changing with the nominal gross domestic product, and replacing the risk measurement from Standard Deviation to Value at Risk (VaR). The findings provide almost the same results as the original measurement and thus, show very high suitability for using these models in other circumstances. Therefore, on the concept of Islamic finance, risk-free returns can be measured using these four approaches, especially inflation and GDP. This study also recommends inflation and GDP to measure risk-free returns in the Sharia's Compliant Asset Pricing Model (SCAPM) or Islamic Capital Asset Pricing Model (ICAPM).====================================================================================================ABSTRAK – Pengukuran Kinerja Saham Syariah di Indonesia menggunakan Sharpe Ratio Modifikasi. Sejak akhir 1960-an, salah satu alat mengukur kinerja saham yang biasa digunakan adalah Sharpe Ratio. Model Sharpe Ratio terdiri atas tiga komponen, yaitu return saham, return bebas risiko, dan risiko saham. Return bebas risiko diukur mengunakan variabel suku bunga yang digolongkan riba dan dilarang dalam konsep keuangan islam. Sedangkan risiko saham diukur dengan standar deviasi yang mengasumsikan data berdistribusi normal. Paper ini bertujuan untuk mengukur kinerja saham syariah yang terdaftar pada Bursa Efek Indonesia (BEI) untuk periode Januari 2011 sampai Juli 2018 dengan menggunakan Sharpe Ratio modifikasi. Kajian akan memodifikasi model Sharpe Ratio dengan mencari variabel alternatif penganti suku bunga dengan empat pendekatan, yaitu: menghilangkan variabel suku bunga tersebut, mengganti dengan zakat rate, mengganti dengan inflasi, dan mengganti dengan produk domestik bruto, serta mengganti standar deviasi dengan Value at Risk (VaR) sebagai pengukur risiko saham yang selanjutnya diimplementasikan pada pasar modal syariah di Indonesia periode Januari 2011 - Juli 2018. Hasil kajian menunjukkan kesesuaian yang sangat tinggi untuk hasil pengukuran kelima model tersebut. Dilihat dari kedekatan hasil pengukuran kinerja, kelima model tersebut dapat dikelompokkan menjadi dua, yaitu model dengan tingkat suku bunga, inflasi, dan PDB sebagai kelompok pertama, sedangkan model tanpa suku bunga dan tingkat zakat sebagai kelompok kedua

Proper use of the modified Sharpe ratios in performance measurement: rearranging the Cornish Fisher expansion

Performance analysis is a key process in finance to evaluate or compare investment opportunities, allocations, or management. The classical method is to compute the market or sub-market returns and volatilities, and then calculate the standard performance measure, namely, the Sharpe ratio. This measure is based on the first two moments of a return distribution. Therefore, a significant weakness of this method is that it implicitly assumes that the distribution is Gaussian (if it is not Gaussian, the approach may lead to a bad fit). In fact, risk comes from not only volatility, but also from higher moments of distribution such as skewness and kurtosis. The standard method to resolve this issue is to use the modified Sharpe ratio; this method replaces the classical Sharpe ratio volatility with the value at risk. The latter is computed using the Cornish Fisher expansion, a tool based on the first four moments of return distribution. This methodology, however, may present a major pitfall: in some cases, quantile functions do not stay monotone. In this paper, we show how this tool can be used effectively through a specific procedure, rearrangement. We compare various metrics using rank correlation, and demonstrate how and in which cases the proposed procedure delivers ranking different from the standard Sharpe ratio ranking. Furthermore, we show how our technique offers better distribution approximations and is therefore a more useful performance metric. Institutional investors may find the technique proposed here useful in that it allows for considering non-normality in performance analysis.

Realistic energy commitments in peer-to-peer transactive market with risk adjusted prosumer welfare maximization

As the local energy sources are mostly uncertain and fluctuating in nature, the ‘energy risk’ due to discrepancies between committed energy transactions and metered measurements is prominent in peer to peer (P2P) markets. We propose a P2P market settlement mechanism which lowers this risk and maximizes the welfare of buyers and sellers. The risk in energy production is modelled using Markowitz portfolio theory and the best point where energy return per unit risk is maximum is obtained from the efficient frontier by using the modified Sharpe ratio. The energy portfolio thus obtained is used as a constraint while optimizing the conflicting prosumer benefits using multi-objective stochastic weight trade-off chaotic non-dominated sorting particle swarm optimization (SWTC-NSPSO). In effect, only a reliable proportion of total energy demand submitted in the bid is cleared in the market, foreseeing the real-time fluctuations. The proposed market settlement mechanism also gives room to the existing distribution system operators by assigning them the duty of 1) optimally allocating energy among buyers and sellers in accordance with their competitive bids 2) providing the infrastructure, managing the market and charging for the service and 3) checking the technical feasibility by performing load flow and monitoring power transfer sensitivities to encourage short distance transactions. The energy allocation is done in CIGRE LV benchmark microgrid with ten peers having solar and wind generation. The allocated energy is found to be closer to the metered measurements and hence the reserve cost is observed to be less.

Alpha Alchemy: Diversifying Without Diluting Alpha

Managers are often evaluated and selected on the basis of their portfolio’s stand-alone risk and return properties. This is too narrowly focused. The basics of mean-variance optimization tells us that covariance and the relationship of a security’s returns within the context of the overall portfolio is important. We illustrate how the basics of portfolio construction with individual securities also applies to building a portfolio of managers. First, we show the statistical properties of different asset classes, as more and more managers are used to get exposure to the asset class. Depending on the asset class, tracking error, relative skewness, and relative excess kurtosis improve at different rates. This is important for considering how many managers may be necessary to get exposure to an asset class. We then illustrate an optimization method of combining the managers. Although the managers, on their own, may not have statistically positive alphas, they can be combined in a way to improve the likelihood that the portfolio of managers has a statistically positive alpha. The optimization method is not a substitute for traditional due diligence. It is just one additional tool useful for manager selection and portfolio construction. TOPICS:Wealth management, portfolio construction, manager selection Key Findings • Combining managers can reduce tracking error, but the relationship between tracking error reduction and manager addition depends on the asset class. Specifically, it depends on the average tracking error and average correlation of excess returns within an asset class. • A slightly modified Sharpe ratio is equivalent to a t-value. An information ratio (with Jensen’s alpha in the numerator) can also be interpreted as t-value. Maximizing these ratios is equivalent to maximizing confidence that the excess return or alpha will be positive. • Individual managers with no or even negative alpha can be combined to give positive alpha.

Portfolio Selection Under Systemic Risk

This paper proposes a novel methodology to construct optimal portfolios that explicitly incorporates the occurrence of systemic events. Investors maximize a modified Sharpe ratio that is conditional on a systemic event, with the latter interpreted as a low market return environment. We solve the portfolio allocation problem analytically under the absence of short-selling restrictions and numerically when short-selling restrictions are imposed. This approach for obtaining an optimal portfolio allocation is made operational by embedding it in a multivariate dynamic setting using dynamic conditional correlation and copula models. We evaluate the out-of-sample performance of our portfolio empirically on the US stock market over the period 2007 to 2020 using ex-post wealth paths and systemic risk metrics against mean-variance, equally-weighted, and global minimum variance portfolios. Our portfolio maximizing a modified Sharpe ratio outperforms all competitors under market distress and remains competitive in non-crisis periods.

Modified Sharpe Ratio Application in Calculation of Mutual Fund Star Ranking

Purpose of this study is to apply to modify Sharpe Ratio to calculate Star Ranking of Equity-based mutual funds registered in Mutual Fund Association of Pakistan, further, the idea was to recalibrate locally developed models being used in Pakistan by autonomous professional bodies who professionally assigns star ranking of mutual funds, equity market exhibited negative returns from July 2017 onwards this research which brought the problem to assign star ranking due to model structure, model relies on risk-adjusted return (Sharpe Ratio), therefore Sharpe Ratio has a limitation in negative excess return. Two developed models were simultaneously compared to witness the predictive power of these models, (1) modified Sharpe and (2) VIS Credit Rating Company (Explaining the Stars) Model. Data was collected from March 2013 to March 2018 quarterly and the exercise was done quarterly. Findings revealed a magnificent piece of work, (1) there is no difference between model 1 and model 2 by both way results exhibited same mutual fund star rankings, (2) both methods have a different way of calculating final score with same results, and (3) modified Sharpe ratio is quite well when excess return is negative but when there is a mix of negative and positive better to use VIS model as well as in positive excess returns. A research paper could not calibrate other models developed by rating companies (Pakistan Credit Rating Company) which is a future research gap.

The SOCIMIs in Spain. An investment opportunity in the real estate sector

The purpose of the present work is to analyze the SOCIMIs (Real estate listed investment trusts in Spain), its performance and its potential as a diversification asset between June 2014 and December 2017. The tax advantages they offer, the main incentive for the creation of these types of companies, are bringing about an increase in development of real estate investment in Spain, which is behind other developed markets where they are generally called REITs (Real Estate Investment Trusts). First, the characteristics, operation, evolution in the market, portfolios and shareholders’ composition are studied. Next, the analysis shows high performance levels measured by the modified Sharpe Ratio and the Jensen’s Alpha for the main SOCIMIs on the market as well as a high portfolio diversification potential demonstrated by the Markowitz Efficient Frontier Model. SOCIMIs are characterized by being less risky investments than the market in general. Compared to traditional Real estate listed companies on the continuous market, SOCIMIs show better performance and lower levels of risk and, in general, all the companies studied –SOCIMIs and traditional Real estate listed companies- can be defined as defensive investments, less risky than the market.

Joint tails impact in stochastic volatility portfolio selection models

This paper examines the impact of the joints tails of the portfolio return and its empirical volatility on the optimal portfolio choices. We assume that the portfolio return and its volatility dynamic is approximated by a bivariate Markov chain constructed on its historical distribution. This allows the introduction of a non parametric stochastic volatility portfolio model without the explicit use of a GARCH type or other parametric stochastic volatility models. We describe the bi-dimensional tree structure of the Markov chain and discuss alternative portfolio strategies based on the maximization of the Sharpe ratio and of a modified Sharpe ratio that takes into account the behaviour of a market benchmark. Finally, we empirically evaluate the impact of the portfolio and its stochastic volatility joint tails on optimal portfolio choices. In particular, we examine and compare the out of sample wealth obtained optimizing the portfolio performances conditioned on the joint tails of the proposed stochastic volatility model.

Global minimum variance portfolios under uncertainty: a robust optimization approach

This paper presents new models which seek to optimize the first and second moments of asset returns without estimating expected returns. Motivated by the stability of optimal solutions computed by optimizing only the second moment and applying the robust optimization methodology which allows to incorporate the uncertainty in the optimization model itself, we extend and combine existing methodologies in order to define a method for computing relative-robust and absolute-robust minimum variance portfolios. For the relative robust strategy, where the maximum regret is minimized, regret is defined as the increase in the investment risk resulting from investing in a given portfolio instead of choosing the optimal portfolio of the realized scenario. The absolute robust strategy which minimizes the maximum risk was applied assuming the worst-case scenario over the whole uncertainty set. Across alternate time windows, results provide new evidence that the proposed robust minimum variance portfolios outperform non-robust portfolios. Whether portfolio measurement is based on return, risk, regret or modified Sharpe ratio, results suggest that the robust methodologies are able to optimize the first and second moments without the need to estimate expected returns.