Min-CDaR for Indian Equity Portfolios

Concept Overview

Minimum Conditional Drawdown at Risk (Min-CDaR) is a portfolio optimization approach designed specifically to minimize the expected severity of drawdowns at a chosen confidence level. Unlike traditional variance-based methods that penalize both upside and downside movements, Min-CDaR directly targets portfolio drawdowns—the peak-to-trough declines experienced during market downturns. This makes it particularly valuable for investors who are sensitive to temporary capital depreciation and recovery periods, rather than short-term volatility.

Understanding Drawdowns

Drawdown measures the decline from a historical peak to a subsequent trough in the value of a portfolio. It captures both the magnitude and duration of downward movements, making it an intuitive risk metric for practitioners.

Drawdown Definition

For a portfolio with value process V(t), the drawdown at time t is defined as:

D(t) = \max_{0 \leq \tau \leq t} \frac{V(\tau) - V(t)}{V(\tau)}

This represents the percentage decline from the highest previous peak to the current value.

Maximum Drawdown

Maximum Drawdown (MDD) is the largest drawdown observed over a specific time period:

\text{MDD} = \max_{0 \leq t \leq T} D(t)

While MDD focuses on the single worst event, CDaR considers the expected severity of all drawdowns exceeding a threshold.

From Drawdown to CDaR

Conditional Drawdown at Risk (CDaR) extends the drawdown concept by focusing on the expected value of drawdowns that exceed a threshold at a specified confidence level. This parallels the relationship between Value at Risk (VaR) and Conditional Value at Risk (CVaR), but in drawdown space.

Drawdown at Risk (DaR) vs. Conditional Drawdown at Risk (CDaR)

Drawdown at Risk (DaR):

A threshold value that drawdowns will not exceed with a certain probability α (e.g., 95%). Comparable to VaR but for drawdowns.

Conditional Drawdown at Risk (CDaR):

The expected value of drawdowns that exceed the DaR threshold. This represents the average severity of the worst drawdowns in the distribution.

\text{CDaR}_\alpha = \mathbb{E}[D(t) | D(t) \geq \text{DaR}_\alpha]

Mathematical Formulation

The Min-CDaR optimization problem can be stated as follows:

\begin{aligned} \min_{\mathbf{w}} \quad & \text{CDaR}_\alpha(\mathbf{w}) \\ \text{s.t.} \quad & \mathbf{w}^\top \mathbf{1} = 1 \\ & \mathbf{w} \geq \mathbf{0} \quad \text{(optional non-negativity constraint)} \end{aligned}

Where:

$\mathbf{w}$ is the vector of portfolio weights
$\alpha$ is the confidence level (typically 0.90, 0.95, or 0.99)

Linear Programming Formulation

Similar to CVaR optimization, CDaR can be reformulated as a linear programming problem. For a discrete set of time points with observed portfolio values, the optimization becomes:

\begin{aligned} \min_{\mathbf{w}, \gamma, \mathbf{u}, \mathbf{z}} \quad & \gamma + \frac{1}{(1-\alpha)T}\sum_{t=1}^T z_t \\ \text{s.t.} \quad & u_t \geq V(\tau, \mathbf{w}) - V(t, \mathbf{w}), \quad \forall \tau \leq t \\ & z_t \geq u_t - \gamma, \quad t = 1,2,...,T \\ & z_t \geq 0, \quad t = 1,2,...,T \\ & \mathbf{w}^\top \mathbf{1} = 1 \\ & \mathbf{w} \geq \mathbf{0} \quad \text{(optional)} \end{aligned}

Where:

$V(t, \mathbf{w})$ is the portfolio value at time t given weights w
$u_t$ are auxiliary variables representing the drawdown at time t
$z_t$ are auxiliary variables capturing excess drawdowns beyond the threshold
$\gamma$ is the DaR at confidence level α
$T$ is the number of time points in the historical sample

Our Implementation

In our portfolio optimization system, Min-CDaR optimization is implemented using the following approach:

Historical data processing: Calculate a time series of portfolio values based on historical asset returns and candidate portfolio weights.
Drawdown calculation: For each time point, compute the drawdown from the previous peak.
Linear programming formulation: Convert the CDaR minimization into a linear program that can be efficiently solved.
Additional constraints: The basic formulation can be enhanced with sector constraints, individual asset limits, or target return thresholds.

Key Implementation Details:

Aspect	Detail
Data preparation	Historical returns are converted to cumulative portfolio values to calculate drawdowns.
Optimization method	Linear programming using auxiliary variables to handle the path-dependent drawdown calculation.
Default confidence level	95% (focusing on the worst 5% of drawdowns).
Results reporting	Both portfolio weights and risk metrics (DaR, CDaR, expected maximum drawdown) are calculated and displayed.

Advantages and Limitations

Advantages

Directly Targets Investor Experience: Focuses on drawdowns, which directly affect investor psychology and often trigger emotional decisions.
Path-Dependent Risk Measure: Unlike variance, CDaR accounts for the sequence and persistence of losses over time.
Recovery Periods: Indirectly addresses the time to recover from losses, which is critical for investors with specific time horizons.
No Distribution Assumptions: Works with empirical return distributions without assuming normality or other specific distributions.

Limitations

Computational Complexity: More computationally intensive than traditional MVO due to path-dependency and the need to calculate drawdowns over time.
Data Requirements: Needs substantial historical data to capture meaningful drawdown patterns and rare events.
May Sacrifice Returns: Like other risk-minimization approaches, can result in portfolios with lower expected returns if not constrained.
Time Period Sensitivity: Results can vary significantly based on the historical time period used for optimization.

Ideal Use Cases

Long-Term Investors

Well-suited for pension funds, endowments, and individual retirement portfolios where large drawdowns can seriously impact long-term objectives.

Target-Date Investments

Excellent for investments with specific time horizons where recovery time from drawdowns becomes increasingly important as the target date approaches.

Drawdown-Sensitive Products

Useful for financial products with explicit guarantees against drawdowns or those marketed as "drawdown-controlled" investments.

Comparison with Other Optimization Methods

Optimization Method	Risk Measure	Key Differences from Min-CDaR
Mean-Variance (MVO)	Variance (σ²)	Penalizes both upside and downside movements; ignores sequential losses
Minimum Volatility	Standard deviation (σ)	Similar to MVO but without explicit return targets; ignores path dependency
Minimum CVaR	Conditional Value at Risk	Focuses on tail losses in the return distribution rather than drawdowns over time
Maximum Diversification	Diversification Ratio	Focuses on correlation structure rather than explicit risk minimization

Real-world Applications and Examples

Example: Traditional vs. Min-CDaR Portfolio

Consider two portfolios during the 2008-2009 financial crisis:

Traditional MVO Portfolio:

Expected Return: 7.5% annually
Volatility (σ): 14%
Maximum Drawdown: -48%
Time to Recovery: 37 months

Min-CDaR Portfolio:

Expected Return: 6.2% annually
Volatility (σ): 11%
Maximum Drawdown: -28%
Time to Recovery: 19 months

The Min-CDaR portfolio sacrificed 1.3% in expected annual return but significantly reduced the maximum drawdown and recovery time during a severe market downturn.

Industry Applications

Target-Date Funds: Use drawdown control to reduce sequence-of-returns risk as investors approach retirement age.
Absolute Return Funds: Employ Min-CDaR as part of their toolkit to provide more stable return patterns with controlled drawdowns.
Risk Parity Strategies: Some risk parity implementations incorporate drawdown management as a secondary objective.

References

Chekhlov, A., Uryasev, S., & Zabarankin, M. (2005). "Drawdown measure in portfolio optimization." International Journal of Theoretical and Applied Finance, 8(01), 13-58.
Goldberg, L. R., & Mahmoud, O. (2017). "Drawdown: From practice to theory and back again." Mathematics and Financial Economics, 11(3), 275-297.
Zabarankin, M., Pavlikov, K., & Uryasev, S. (2014). "Capital asset pricing model (CAPM) with drawdown measure." European Journal of Operational Research, 234(2), 508-517.
Alexandrovich, C., Stanislav, U., & Michael, Z. (2003). "Portfolio optimization with drawdown constraints." Asset and Liability Management, 263-278.

Minimum Conditional Drawdown at Risk (CDaR)

Portfolio optimization focusing on minimizing severe drawdowns