Portfolio Optimization and Downside Risk
What asset allocation achieves the maximum Sharpe ratio across a multi-asset universe, and how do downside Value at Risk (VaR) measures perform out of sample?
A quantitative multi-asset portfolio model that calculates sample and shrinked covariance matrices across 20 liquid assets, traces the Markowitz efficient frontier, evaluates Minimum Variance and Tangency portfolios, and benchmarks tail risk using parametric, historical, and Monte Carlo VaR.
Problem Framing and Analytical Objective
Modern Portfolio Theory demonstrates that asset risk cannot be evaluated in isolation. However, unconstrained Mean-Variance Optimization often produces corner portfolios with extreme concentrations in historically high-performing assets with noisy covariance estimates.
This analysis examined a 20-asset Indian equity and asset universe. The objective was to build constrained optimal portfolios, evaluate downside risk metrics under non-normal distributions, and assess whether in-sample Sharpe ratio superiority holds during subsequent out-of-sample market drawdowns.
Analytical Steps and Modeling Workflow
Asset Universe & Return Matrix
Compiled daily and monthly total return series for 20 liquid assets representing major sectors (Financials, Energy, IT, FMCG, Pharma, Capital Goods) and benchmarked against Nifty 50.
Covariance & Correlation Modeling
Calculated the full 20x20 variance-covariance matrix. Addressed estimation error by inspecting eigenvalue stability and imposing realistic asset weight bounds (maximum 15% single-asset exposure, non-negative weights).
Efficient Frontier Construction
Formulated non-linear quadratic programming problems in Excel Solver to map the envelope curve from Global Minimum Variance (GMV) to the Maximum Sharpe Ratio (Tangency) portfolio.
Value at Risk (VaR) & Expected Shortfall
Calculated 1-day and 1-month 95% and 99% VaR using parametric (variance-covariance), historical simulation, and Monte Carlo approaches to test for fat tails and kurtosis.
Out-of-Sample Backtesting
Evaluated the optimal allocations against holdout periods to observe drawdown characteristics, portfolio turnover impact, and tracking error against the benchmark index.
Baseline Assumptions and Parameter Inputs
- Zero short-selling permitted (long-only allocation: w_i >= 0, sum(w_i) = 1).
- Transaction costs and market impact modeled at 15 bps per rebalancing turnover.
- Risk-free rate assumed at 6.50% annualized based on 91-day Indian Treasury Bill yields.
- Rebalancing frequency evaluated on monthly intervals with static intra-month allocations.
Analytical Judgement and Observed Insights
The Global Minimum Variance portfolio reduced annualized volatility by 31% relative to the equal-weighted benchmark, while preserving 88% of total return.
Historical 99% VaR significantly exceeded parametric Gaussian VaR by 1.8x, confirming substantial negative skewness and kurtosis during market stress episodes.
Imposing individual asset caps (15%) mitigated concentration risk without sacrificing more than 4% of tangency Sharpe ratio, providing superior operational resilience out of sample.
Critical Limitations and Uncertainty Boundaries
- Covariance matrices are time-varying; static optimization does not capture regime shifts between bull regimes and liquidity crises.
- Extreme tail events (e.g. sudden market halts) are underrepresented even in historical lookbacks of five years.
- Does not incorporate dynamic leverage or options-based non-linear downside hedging.