“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 port-folio 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.


INTRODUCTION
Markowitz said in his seminal 1952 paper, "The only free lunch in finance is diversification".Informed investors are keen to use portfolio diversification to reduce the risk of their investments.They are ready to go beyond the national boundaries in their pursuit of diversification to minimize the risk of their investments.Since the 1980s, world capital markets have become increasingly deregulated and opened up opportunities for international investment.Opening up the global financial markets has increased the preference of investors for cross-border investments as a way to diversify their portfolios.
The unprecedented COVID-19 pandemic plunged humanity into crisis, catching the world off guard and disrupting the global landscape in previously unimaginable ways.Beyond its devastating impact on public health, the pandemic reverberated across the global economy and sent shockwaves through the financial markets.As the world adjusted to the "new normal" imposed by the pandemic, the Russian invasion of Ukraine in February 2022 further destabilized the world stage and crude oil prices.In such turbulent times, the task of portfolio managers became exceptionally challenging.Balancing portfolios across various asset classes while consistently delivering positive risk-adjusted returns becomes formidable.Consequently, it has become imperative for portfolio managers to explore different asset classes and returns, outperforming popular benchmarks such as the S&P500, the broad-based Index for developed markets, and the MSCI market index for emerging markets.
For several compelling reasons, BRICS equities continue to shine in the contemporary financial landscape.The brilliant performance of BRICS equities in contemporary times can be attributed to their favorable growth prospects, expanding global significance, improved business environments, enhanced accessibility, interconnectedness with global markets, and regional economic cooperation.BRICS countries have exhibited considerable momentum in their financial market, creating a prospective hub for international investors.This paper recommends the creation of a multi asset portfolio for international investors comprising emerging market equities indices of BRICS countries along with other select asset classes, which generate better risk-adjusted returns and have exhibited resilience during crisis periods.

LITERATURE REVIEW
Diversifying investments across asset classes and nations reduce portfolio risks.Multi asset portfolios deliver greater liquidity, stepped-forward variety, and decreased volatility (Vo & Tran, 2020) and can match effectively alongside numerous investment procedures and asset categories (Peskin, 2018).Foreign investors look to diversify their portfolios in emerging markets, like BRICS, which also benefits those firms (Vo & Chu, 2019).With significant foreign investment flows, the volatility of oil prices will be very significant for emerging economies such as BRICS countries (Naeem et al., 2022).Advanced economies are getting integrated with BRICS economies (Majumder, 2012).The study done by Buchanan et al. (2021) found that including BRICS equity indices in a multiasset portfolio increases risk-adjusted returns.
Apart from BRICS equities, commodities also add dimensions to a multi asset portfolio.Gold is one such commodity.Gold is an effective hedge for debt and equity markets with an investment horizon of one year (Bredin et al., 2015).Gold has also emerged as an option for a safe investment for equity investors interested in long-term investments, particularly during times of financial turmoil.Aboura (2016) found Gold essential in a multi asset portfolio constructed with commodities as an asset class across all economic regimes.Dong Yoon (2021) found that Gold's hedging role concerning stock differs in everyday and extreme market situations.In 2008, with the onset of the global financial crisis, Gold proved to be a safe bet in a multi asset portfolio.The safe haven properties of Gold were again witnessed during the COVID-19 period.However, the diversification properties of Gold weakened as the COVID-19 pandemic intensified, as noted by Ren et al. (2021).
Crude oil futures are another commodity that adds to the diversification benefits of a multi asset portfolio.The relationship of oil with the economic markets is a well-researched topic (Alquist & Gervais, 2013;Zhang & Wang, 2015;Ma et al., 2019).Wang et al. (2020) study found that the equity markets of BRICS strongly correlate to the WTI Crude under severe circumstances.Consequently, mentioning WTI provides no diversification advantages to a BRICS portfolio.Nunez-Mora and Sachchez-Ruenes (2020) covered oil mixes and BRICS equity indices to assemble a multi asset portfolio.Their research observed that India, China, and Brazil's equity indices and oil mixes of China and Brazil contribute to portfolio performance.However, the Russian equity index and oil mix contributed significantly to the portfolio's efficient frontier.Wang et al. (2022) discovered that portfolio diversification in a multi asset portfolio using energy futures provided higher risk-adjusted returns for all energy futures markets except for WTI crude oil futures for the study, a period between 2011-2020.In maximum allocation strategies, using energy futures facilitates and enhances the overall performance of a bond-stock portfolio (Bessler & Wolff, 2015).
Apart from equities and commodities, bonds as an asset have gained popularity during the past decade, and institutional investors have primarily embraced this asset as a new asset class (Sterge & Van der Stichele, 2016).Brinson et al. (1991) used stock and bond market assets to analyze performance in a multi asset portfolio created for longterm investors.The structure of bonds segregates investors from market-related risk to event-related risk, which is why they have become an essential source of portfolio diversification (Litzenberger et al., 1996).A study by Kish (2016) found that bonds could provide benefits of diversification when added to an investor's portfolio comprising stocks, commodities, and real estate.The empirical research works of other researchers reveal that tail risk and drawdown measures under various market regimes can be brought down by using bonds (Clark et al., 2016;Sterge & van der Stichele, 2016).Bonds can act as an effective diversifier; however, in some cases, they may be a poor hedge but can emerge as a good haven when there is a downward movement in stock prices during the post-crisis period (Drobetz et al., 2020).
A substantial body of research has focused on the correlation between cryptocurrencies and other financial assets.In a study conducted in U.S. markets from 2010-2013, Briere et al. (2015) found that investment portfolios with crypto assets have a better risk-return ratio than portfolios without it.Wu and Pandey (2014) found that incorporating Bitcoins in a portfolio enhances portfolio returns and mitigates the risk of losses.Corbet et al. (2018) for the period covering 2013-2017, found crypto assets to have a low correlation with traditional financial markets.Therefore, including crypto assets in a multi asset portfolio would benefit investors.Bouri et al. (2017) found evidence of Bitcoin being a good diversification asset in a multi asset portfolio.Studies by Koziuk (2022) (for countries with worsening inflation and institutional performance) and Akyildirim et al. (2021) (during pandemic times) found the inclusion of cryptocurrencies in multi asset portfolios to improve risk-return performance.
Based on the review of the past studies, the authors find a gap in the literature, as very few studies have taken the joint inclusion of these multi assets in a global portfolio, especially during times of stress.The paper attempts to achieve the following research questions: 1. Construct multi asset portfolios for international investors in a way that delivers better risk-adjusted returns and shows less vulnerability to incidences of extreme fall in portfolio value in times of stress.
2. Suggest the best portfolio construction method among the popular portfolio construction methods, which provides better risk-adjusted returns over popular indices-S&P 500 and MSCI Emerging market indices.

METHOD
The The Optimal risky, Naïve and Equal Risk contributions methods create multi asset portfolios.
Microsoft Excel Solver has been used to create the portfolios using the GRG Nonlinear method with non-negativity constraints.Hence, short selling was prohibited while calculating asset weights under each method.The multiple ways of portfolio evaluation using risk-adjusted methods -Adjusted Sharpe ratio, Modified Sharpe ratio, and the Value of Risk method -have added to the robustness of the study.

Optimal Risky Portfolio:
The mean-variance model based on the portfolio selection theory introduced by the economist Harry Markowitz (1952Markowitz ( , 1959) ) utilizes quantitative techniques, such as mean-variance optimization or other advanced optimization algorithms, to determine the allocation of assets that provide the highest Sharpe ratio or another risk-adjusted performance measure.The return on security i is given by: ( ) where P 1 and P 0 are the share prices at time t and t -1, respectively.
The investor attains the expected return by adopting a variance of return and reducing the variance in expected return through diversified asset allocation in a portfolio.The asset allocation process is done by allotting the optimal weights of investments to risky assets.The weights become optimal when the portfolio attains expected utility maximization and risk minimization for risk-averse investors (Moulya et al., 2019).

Naive Portfolio:
The naive portfolio construction method, an equally weighted portfolio, assigns equal weights to all assets without considering their individual risk or return characteristics.The framework of mean-variance, as proposed by Markowitz (1952), is well accepted in all modern portfolio theories where the investors follow the naïve (1/N) diversification strategy for allocating their investments across various assets (Berbartzi & Thaler, 2001;Brown et al., 2007).The main attraction of this approach is its simplicity, where a fraction of 1/N of wealth is allocated to each of the N assets available for investment.Generally, the equation for the variance of the portfolio with n as the size of its distribution in the following way: Here w i and w j are the proportion of investment of assets (i and j), and r i and r j are considered as return of the assets (i and j), respectively.The covariance of the returns between the assets is represented by cov (r i , r j ).When an equally weighted portfolio is formed using the naïve diversification, we get w i = w j = 1 / n, and the equation ( 1) can be represented as: Let the risk of portfolio x be measured by R(x) and the risk contribution of asset i be represented as Ci(x).Then, as per the definition, it is held that If the variance of its return measures the portfolio's risk, then R(x) = x T Qx and Ci(x) = xi(Qx)i, where ( ) In this equation T denotes the transpose and Qx denotes the estimated variance covariance matrix of the asset return.Qij is the covariance between assets and xj is the acceptable weight of the portfolio that will make it feasible.
Similarly, if we use the standard deviation of the return as the risk measure yields ).
In an ERC portfolio, one considers only the variance risk measure, recognizing that all results will be applied equally to the standard deviation (Mausser & Romanko, 2014).
In the current study, as the asset returns exhibited non-normal properties, as measured by the Jarque Bera test (see Table 2), the Adjusted Sharpe Ratio (ASR) is used to evaluate portfolio performance.
There are various ways to calculate an adjusted Sharpe ratio (Maillard, 2018).One of the methods is presented here: where S is the skewness of the distribution of return and kurtosis is represented as K, which can be computed by the third and fourth moment of the distribution, have represented the skewness.When two assets have the same return, the asset with the lower risk will be chosen, so the worse asset should be measured when there is more significant risk involved and this will resolve any inconsistency that may occur due to inadequate treatment of risk in the original Sharpe ratio for certain circumstances.
In mathematical terms, the partial derivatives of an efficiency index for a given risk parame- .µr SR σ = (10) Now, one can find the partial derivative of the alternative ratio compared to its risk, and it becomes no longer necessary that fund returns exceed the risk-free asset to avoid an inappropriate risk treatment.

.
p SRm µ r Since the Modified Sharpe ratio can make a more precise adjustment of profitability with the volatility assumed, it becomes a better measure of profitability.The current study uses a modified Sharpe ratio and an adjusted Sharpe ratio to measure portfolio performance.
Value at Risk (VaR) is the process of quantifying the risk of adverse price movements.It provides the maximum potential portfolio loss amount that will not exceed over a given time horizon with a small probability (Jorion, 2007).Thus, if the random variable X describes potential portfolio profits and losses with the related quantile x A , then A will represent the percentage of worst cases considered, i.e.

( )
In this case, the supreme of the worst cases percentage α will be considered the VaR.
It is neither sub-additive nor coherent as a measure of risk (Artzner et al., 1999).The mandate of the expected shortfall has been introduced and mandated by the regulatory authorities (Burdorf & Van Vuuren, 2018).So, the value of maximum shortfall has emerged as an essential risk metric.Modified VaR estimates the tail conditional expectation and calculates the maximum shortfall expected shortfall.(i.e., the average of losses greater than the VaR at a significance level).
When the value at risk adjusts, the risk is measured by volatility alone, along with skewness and kurtosis of the distribution.This helps to measure the risk of a multi asset portfolio with non-normally distributed assets and solve the portfolio optimization process by minimizing the modified value at risk at a given confidence level (Favre & Galeano, 2002).In the current study, the portfolio performance of the Optimal Risky Portfolio, Naïve Portfolio, and Equal Risk Contribution Portfolio is measured by the adjusted Sharpe ratio, modified Sharpe ratio, and measurement of extreme shortfall in a portfolio using the modified value at risk method.The authors compare the performance of the three portfolios created using the metrics with the MSCI Index and S&P500.

RESULTS
From January 2018 to March 2020 (Pre Covid-19), most assets realized daily negative mean returns.Gold was the least volatile asset class, with a 0.83 % standard deviation.(Table 3).Gold diversifies portfolios but is not a core investment.Gold provided the best risk-adjusted returns when the MSCI emerging Index and S&P 500 stock market had negative daily means but a greater standard deviation.The kurtosis of all assets showed that Gold had the lowest skewness at 0.57 and the minimal tail risk event.Before COVID-19, when the 10-year U.S. Treasury yield averaged 2.42% p.a. (Table 4), the optimal risky portfolios had the highest Adjusted Sharpe ratio (ASR) at 0.364 (Table 7) and modified Sharpe ratio (MSR) at 0.09 ( The optimal risky portfolio had the best ASR of 4.849 (Table 7) and MSR of 2.18 (Table 8).The MSCI Index had the second-best ASR, with 3.508.The Equal Risk portfolio has the second-highest adjusted Sharpe ratio, behind the ideal risky portfolio at 1.25 (Table 8).Conversely, the Naïve portfolio creation approach had the lowest ASR, and the S&P500 had the lowest MSR.The 100 million Portfolio, closely followed by the S&P 500 at 95.69, saw a significant deficiency (Table 9).With the lowest 100 million portfolio gap, the Optimal The optimal portfolio creation approach again had the greatest ASR of 2.459 (Table 7), followed by the S&P 500 at 1.59.The former had the greatest MSR at 0.77 (Table 8), slightly higher than the S&P 500's 0.61 modified Sharpe ratio, and the S&P Index followed suit.
In the Post-Covid Out-of Sample Period from February 1, 2022, to August 25, 2022, Brent Crude delivered the highest mean return, averaging 0.08% and a standard deviation of 3.33%.(Table 3).Crude oil, with a kurtosis of 9.94, indicated a substantial tail risk.Cryptocurrencies had the best returns at 0.14% but the highest standard deviation at 24.61%.The 10-year Treasury yield reached 2.66% (Table 4).All three portfolio creation metrics increased Sharpe ratios throughout this era.
After the Russian invasion of Ukraine, the world accepted COVID-19 as normal.The optimal portfolio creation strategy has the highest Sharpe ratio.However, the ASR was 0.354, while the other two portfolio techniques exhibited negative ASR and MSR (Tables 7 and 8).MSR included tail risk and was a meagre 0.10 for the optimal risky portfolio.
The MSCI emerging equities index had the lowest adjusted and modified Sharpe ratios at -2.183 and -0.74, followed by the S&P Index.
The Russian invasion of Ukraine caused asset price volatility, especially in crude oil and Moex.From 01/02/2022 -28/02/2023, Moex delivered daily mean returns of -0.19% with a standard deviation of 3.51% (Table 3) and a very high tail risk (kurtosis at 74.30).Brent Crude oil prices were tracked with daily mean returns of -0.02%, a standard deviation of 2.93%, and a kurtosis of 8.70.This volatility in Crude oil prices affected the Sharpe Ratios negatively.Thus, the optimal risky portfolio, which suggested 100% crude oil exposure in the past, now recommends 100% South African Index -STOP 40.STOP 40 had daily mean returns of 0.02%, a low standard deviation of 1.32%, and a low kurtosis of 1.41, unlike Crude, Moex, and the Chinese Index (SSE).Zero risk weightage  in an equal risk contribution portfolio included the South African Index STOP 40's risk.The Optimal risky portfolio has the greatest risk-adjusted Sharpe ratio of 0.084 (Table 7).The Naïve Portfolio and Equal Risk Contribution Portfolio had negative Sharpe Ratios throughout the same time.The MSCI index had the worst ASR at -1.415.S&P500 had a -0.730ASR.The MSR and risk-adjusted returns ranked similarly.
In the post-Russian invasion of Ukraine Out-of-Sample Period: 01/03/2023 -14/06/2023, the U.S. 10-year Treasury yield peaked at 3.6% a year after the invasion of Ukraine by Russia owing to rising bond rates.All assets except Brent Crude and Cryptocurrencies had a standard deviation of 0.06% to 1.16% (Table 3).The ideal risky portfolio weighted Moex 39%, Bovespa 22%, Sensex 16%, and Gold 23% (Table 5).Gold's glitter returned to an optimal risky portfolio, as seen in the pre-COVID period.The equal risk contribution portfolio's only risky assets are Bovespa at 62% (see Table 6) and Brent Crude at 34%.The optimal risky portfolio design strategy again produced the greatest ASR of 5.778 (Table 7), compared to the S&P 500's -2.830 and the MSCI emerging market index's 0.620.Similar outcomes were seen with MSR-based portfolios.The ideal portfolio to the worst MSR had a ratio of 2.96 (Table 8), followed by the MSCI Emerging Equity Index and S&P 500 at -1.20.

DISCUSSION
Rajput and Sufiya (2022) found cointegration in the BRICS indices, implying overseas investors had few diversification benefits.On the contrary, the current research found evidence supporting the inclusion of BRICS indices in multi asset portfolios in the different sub-periods investigated in the study.
In the Pre-COVID Period: 3/01/2018 -18/03/2020, the Optimal Portfolio had the best ASR and MSR and the lowest deficit in a 100 million portfolio, as   9).This conclusion supports Wang et al. ( 2019), who found a strong link between BRICS equity markets and oil markets under severe situations.The optimal risky portfolio showed that crude oil optimizes the Sharpe Ratio the most.
In the Russian invasion of Ukraine war Period from 01/02/2022 to 28/02/2023, the optimal risky portfolio had the greatest MSR of 0.03, as seen in Table 8.The emerging Index had the worst Minimum Sharpe Ratio (MSR) of -0.52, while the equal risk portfolio had -0.30.The S&P 500 and Naïve portfolio had MSR values of -0.28 and -0.20, respectively.At a 99% confidence level, most portfolios, the MSCI emerging stocks and the S&P 500 had a large decline in portfolio value over this period.

CONCLUSION
This research paper constructs and investigates multi asset portfolios using three portfolio construction strategies: BRICS equities, Gold, Crude oil, bonds, and cryptocurrencies from January 2018 to June 2023, outperforming the S&P500 and MSCI Index for three periods.The authors validated their findings using various optimization methodologies, including the Optimal Risky Portfolio, Naïve Portfolio, and Equal Risk Contribution Portfolio.The validations occurred during the pre-COVID-19 crisis, post-COVID-19, and the Russian invasion of Ukraine and subsequent wars led by the former.The portfolios outperformed the Naïve portfolio, an equal risk contribution portfolio, the MSCI Index, and the S&P 500 in terms of adjusted Sharpe ratios and modified ratios.The researchers also evaluated the results using the adjusted Sharpe ratio, modified Sharpe ratio, and extreme portfolio value shortfall.This study aims to demonstrate the benefits of portfolio diversity using BRICS indices, Gold, Brent crude oil futures, and the supremacy of the traditional optimal risky portfolio technique in creating international multi asset portfolios.
The optimal risky portfolio outperforms the portfolio's expected shortfall in value throughout the analyzed periods, except for two out-of-sample periods: the COVID-19 period from March 2020 to February 2021 (used as the pre-COVID-19 period) and the post-sample period from February 2022 to August 2022.
Based on the ASR evaluation metric, the MSCI emerging stock index performed worse during the pre-COVID-19, post-COVID-19 out-of-sample, and Russian invasion of Ukraine and the subsequent war, the S&P 500 performed worst in the year after Russia invaded Ukraine.This phenomenon can also be attributed to the 10-year government bond yield rising to 3.60% annually during the research, and during and after COVID-19, the Naïve portfolio had the lowest Average Sharpe Ratio (ASR).
In the entire research period, the portfolio that used an equal risk contribution method did not perform optimally according to risk-adjusted metrics -ASR and MSR.The MSCI developing index had the worst Modified Sharpe Ratio (MSR).Underperformances were notable in the weeks before the COVID-19 epidemic.The MSCI emerging market Index had the least MSR post-COVID-19 and during the Russian invasion of Ukraine.After the COVID-19 pandemic, the Naïve portfolio had the MSR.The COVID-19 pandemic and the year after the Russia-led war against Ukraine caused the S&P 500 index's biggest market stress reaction.The S&P 500 underperformed in all three stages, while the Naïve portfolio underperformed throughout the COVID-19 and post-COVID-19 eras.
The value deficit of a $100 million portfolio must be estimated using modified Value at Risk (mVaR) ideas.The S&P 500 performed poorly on this criterion before the COVID-19 pandemic.Scarcity increased during the COVID-19 epidemic and the Russian invasion of Ukraine.The Naïve portfolio had the lowest mVAR performance during four phases: COVID-19, post-COVID-19, Russian invasion of Ukraine, and Out-of-sample Russo-Ukraine war.The minimum value at risk (mVaR) measure showed that the equal risk portfolio and MSCI Emerging Markets Index performed below ideal levels, except during the Russia-led war on Ukraine and the post-COVID-19 out-of-sample period.
In conclusion, the naïve and equal risk portfolios performed poorly during economic downturns, as measured by performance criteria such as adjusted risk-return and value at risk.The academic literature has focused on Optimal Risky portfolio construction.The optimal risky portfolio's supremacy over the MSCI emerging market Index and S&P 500 portfolios is crucial.
The discovery in this work advances the knowledge frontier.Gold outperformed other investments in the year before COVID-19.BRICS equities performed well during the COVID-19 pandemic.After the COVID-19 pandemic and the Russian-led war against Ukraine, Brent crude oil helped boost risk-adjusted profits.Gold's influence on India, Russia, and Brazil's equity indices shows it is again a viable asset for diversifying international investment portfolios after months of the Russo-Ukraine war and crisis.
In the turbulent COVID-19 era, this analysis shows that Emerging Market Bonds ETF provided diversification.Cryptocurrencies failed to optimize multi asset portfolios during the period of study.One exciting area of future opportunity research could be measuring portfolio performance and hedging approaches using risk-adjusted return measurements, including transaction expenses and dynamic rebalancing techniques.

Table 1 .
(Maillard et al., 2010)tudy A mean-free strategy suggests that it will result in equal risk from each asset in the portfolio from a risk perspective(Maillard et al., 2010).It allocates the weights based on the assets' volatilities and correlations, ensuring that each asset contributes equally to the overall portfolio risk.It provides a systematic way to balance risk across assets and can help manage downside risk and improve portfolio stability.

Table 8
pre-COVID-19 period, the MSCI emerging market stock index had the weakest performance and the worst negative adjusted and modified Sharpe ratio.The adjusted value at risk and severe deficit Investment Management and Financial Innovations, Volume 20, Issue 4, 2023 http://dx.doi.org/10.21511/imfi.20(4).2023.09As COVID-19 spread, global financial markets were unsettled, and the 10-year Treasury rate hovered around 0.79% (Table 4).

Table 3 .
Extract of descriptive statistics (4)p://dx.doi.org/10.21511/imfi.20(4).2023.09high kurtosis, indicating a 13.11 tail risk.Thus, it did not qualify for the ideal risky portfolio due to its low risk-adjusted return.The U.S. 10-year Treasury rate rose to 1.35% p.a. during the second and third waves and the appearance of many COVID variations as the global economy recovered.

Table 5 .
Recommended optimal risky portfolios

Table 7 .
Portfolio evaluation as per Adjusted Sharpe Ratio

Table 8 .
Portfolio evaluation as per Modified Sharpe Ratio

Table 9 .
Portfolio evaluation as expected shortfall using modified Value at Risk (MVaR)