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The notion that asset diversification reduces risk is ancient and can be traced as far back as the Talmud which states, “A man should always keep his wealth in three forms: one-third in real estate, another in merchandise, and the remainder in liquid assets” (Baba Metzia, verse 42a). Somewhat more recently, in 1738, Daniel Bernoulli observed, “it is advisable to divide goods which are exposed to some small danger into several small portions rather than to risk them all together” (1738/1954, p. 30). Arguably, however, it was not until 1935 that the future Nobel laureate J. R. Hicks offered some early direction for modern portfolio theory. Although his research was more concerned with explaining the demand for money, he points out two important considerations for modeling risk. Hicks writes, “The risk factor comes into our problem in two ways: First, as affecting the expected period of investment, and second, as affecting the expected net yield of investment” (Hicks, 1935, p. 7). Regarding Hicks' first point, both Markowitz (1952) and Roy (1952) emplace their analyses in a one-period investment horizon. Second, and even more relevant to modern portfolio theory, is Hicks' suggestion of using an expected value calculated with subjective probabilities. Hicks continues, “It is convenient to represent these probabilities to oneself, in statistical fashion, by a mean value, and some measure of dispersion” (1935, p. 8). Clearly, Hicks comes very close to articulating a mean–variance solution. Crucially, and unlike Roy or Markowitz, Hicks does not develop this line of reasoning nor does he suggest the particular use of variance or standard deviation as that measure of risk. Nonetheless, Hicks' suggestion anticipates the work of Markowitz and Roy.¹

Beginning from the onset of the Covid-19 pandemic, crypto assets have intensely been in the spotlight and have attracted significant investor attention. By being the first blockchain product, Bitcoin is the first crypto asset and still dominates the entire crypto market capitalization. In this study, we shed light on whether this energy-hungry crypto asset is an effective tool for portfolio volatility reduction from the perspective of the Modern Portfolio Theory. Based on a two-year period from April 2019 to April 2021, which includes the extreme impacts (crash and rally) of the pandemic on markets, we conclude that Bitcoin is not a beneficial instrument for volatility reduction if short-selling is not allowed. After removing this restriction, Bitcoin has very small negative investment weights in minimum variance portfolios. In other words, short-selling Bitcoin slightly reduces portfolio volatility.

Since the inception of modern portfolio theory, traditional asset classes have been the standard investment products for portfolio construction. With the introduction of cryptoassets such as cryptocurrencies, tokenized securities, smart contracts, and blockchain-based token assets, the crypto “revolution” has created a new asset class for consideration in a modern portfolio. This chapter explains cryptoassets in a portfolio, including their limitations and parameters as an asset class in a diversified portfolio. Finally, it reports an improvement in a new portfolio’s reward/risk ratio using the Sharpe ratio when adding cryptoassets to simulated equity, bonds, and real estate portfolios. A caveat is that a cryptoasset’s contribution to a well-diversified traditional portfolio differs from the performance of investing in a single and isolated cryptoasset.

Purpose: The present study aims to test the Quadratic Programming model for Optimal Portfolio selection empirically.

Need for the Study: All the investors who buy financial products are motivated to obtain higher profits or, in other words, to maximise their returns. However, the high returns are often accompanied by higher risks, and avoiding such risks has become the primary concern for all investors. There is a great need for such a model to maximise profits and minimise risk, which can help design an investment portfolio with minimum risk and maximum return. The Quadratic Programming model is one such model which can be applied for selected shares to build an optimised portfolio.

Methodology: This study optimises the stock samples using a two-level screening of correlation coefficient and coefficient of variation. The monthly closing prices of the NSE-listed Indian pharmaceutical stocks from December 2019 to January 2022 have been used as sample data. The Lagrange Multiplier method is used to apply the model to achieve the optimal portfolio solution. Based on the market reality, the transaction costs have also been considered. The Quadratic programming model is further optimised to achieve the optimal portfolio for the select stocks.

Findings: The traditional portfolio theory and the modified quadratic model gives similar and consistent results. In other words, the modified quadratic model asserts the accuracy of the conventional portfolio model. The portfolio constructed in the present study gives a return much higher than the return of the benchmark portfolio of Nifty Fifty, indicating the usefulness of applying the Quadratic Programming model.

Practical Implications: The construction of an optimal portfolio using the traditional or modified Quadratic model can help investors make rational investment decisions for better returns with lower risks.

Portfolio selection has been extensively studied in field of business and economics. Many methods have been developed to construct a well-diversified portfolio that is expected to result in higher investment return with minimum risk. One of the most foundational works contributing to modern portfolio selection is the Markowitz mean variance optimization approach. The Markowitz approach heavily relies on past stock price performance, both in term of correlation structure and the return, to predict the future outcome. We constructed both Markowitz portfolio and the Fundamental Indexing portfolio independently, then using Buffet ratio to weight, combined both portfolio into a newly blended portfolio, test out-of-sample the new portfolio in term of return and then compare it to the Indonesian LQ45 benchmark index. The result shows that the new combined portfolio returns annually on average 43.89% higher than the benchmark index.

In this study, we analyze the power of the individual return-to-volatility security performance heuristic (ri/stdi) to simplify the identification of securities to buy and, consequently, to form the optimal no short sales mean–variance portfolios. The heuristic ri/stdi is powerful enough to identify the long and shorts sets. This is due to the positive definiteness of the variance–covariance matrix – the key is to use the heuristic sequentially. At the investor level, the heuristic helps investors to decide what securities to consider first. At the portfolio level, the heuristic may help us find out whether it is a good idea to invest in equity to begin with. Our research may also help to integrate individual security analysis into portfolio optimization through improved security rankings.