Risk Analysis of Passive Portfolios

Paper: arXiv 2407.08332

Abstract

In this work, we present an alternative passive investment strategy. The passive investment philosophy comes from the Efficient Market Hypothesis (EMH), and its adoption is widespread. If EMH is true, one cannot outperform market by actively managing their portfolio for a long time. Also, it requires little to no intervention. People can buy an exchange-traded fund (ETF) with a long-term perspective. As the economy grows over time, one expects the ETF to grow. For example, in India, one can invest in NETF, which suppose to mimic the Nifty50 return. However, the weights of the Nifty 50 index are based on market capitalisation. These weights are not necessarily optimal for the investor. In this work, we present that volatility risk and extreme risk measures of the Nifty50 portfolio are uniformly larger than Markowitz’s optimal portfolio. However, common people can’t create an optimised portfolio. So we proposed an alternative passive investment strategy of an equal-weight portfolio. We show that if one pushes the maximum weight of the portfolio towards equal weight, the idiosyncratic risk of the portfolio would be minimal. The empirical evidence indicates that the risk profile of an equal-weight portfolio is similar to that of Markowitz’s optimal portfolio. Hence instead of buying Nifty50 ETFs, one should equally invest in the stocks of Nifty50 to achieve a uniformly better risk profile than the Nifty 50 ETF portfolio. We also present an analysis of how portfolios perform to idiosyncratic events like the Russian invasion of Ukraine. We found that the equal weight portfolio has a uniformly lower risk than the Nifty 50 portfolio before and during the Russia-Ukraine war. All codes are available on GitHub (\url{“https://github.com/sourish-cmi/quant/tree/main/Chap_Risk_Anal_of_Passive_Portfolio"}).

Complexity vs Empirical Score

• Math Complexity: 5.0/10
• Empirical Rigor: 6.0/10 (capped from 7.5: transaction costs not modelled; no out-of-sample test; see penalty flags)
• Quadrant: Street Traders — practical and empirical, lighter on theory

Why this score: The paper contains standard financial mathematics and statistical theory suitable for an undergraduate or advanced undergraduate level, not heavy advanced proofs or dense abstract mathematics. However, it is highly data-driven, with explicit empirical comparisons between real-world indices (Nifty50) and proposed portfolios, includes event analysis (Russia-Ukraine war), and provides reproducible code on GitHub, indicating strong backtest readiness.

Research Flowchart

  flowchart TD
  A["Research Goal: Improve Passive Investing"] --> B["Input: Nifty 50 Data"]
  B --> C["Methodology: Compare Portfolios"]
  C --> D["Computational Process: Risk Analysis"]
  D --> E{"Findings"}
  E --> F["Key Outcome: Equal Weight Strategy"]
  
  subgraph C ["Methodology: Compare Portfolios"]
      C1["Market Cap Weighted<br>Nifty 50 ETF"]
      C2["Markowitz Optimal<br>Portfolio"]
      C3["Proposed Equal Weight<br>Portfolio"]
  end
  
  subgraph D ["Computational Process: Risk Analysis"]
      D1["Volatility Risk"]
      D2["Idiosyncratic Risk<br>Minimization"]
      D3["Extreme Event Analysis<br>Russia-Ukraine War"]
  end
  
  subgraph E ["Findings"]
      F1["Uniformly Lower Risk<br>than Nifty 50 ETF"]
      F2["Risk Profile Similar to<br>Markowitz Optimization"]
      F3["Robust Performance<br>During Crisis"]
  end

Cited References (24)

1. Sparse Portfolio Selection via Bayesian Multiple Testing (2017) — link
2. X-Sigma-Rho and Market Efficiency (2014) — link
3. Regularizing Portfolio Risk Analysis: A Bayesian Approach (2014) — link
4. The Art of R Programming: A Tour of Statistical Software Design (2011)
5. Risk Contribution Is Exposure Times Volatility Times Correlation: Decomposing Risk Using the X-Sigma-Rho Formula (2011) — link
6. On Bayesian Inference for Generalized Multivariate Gamma Distribution (2010) — link
7. Stochastic Calculus for Finance : The Binomial Asset Pricing Model (2007)
8. Improved estimation of the covariance matrix of stock returns with an application to portfolio selection (2003) — link

0 of 24 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.