Deutsche Bank’s Handbook of Portfolio Construction gives a great introduction to two important principles for diversification and risk management of portfolios. First, tail dependence is a better guide to diversification than correlation when it really matters, i.e. in market turmoil. Second, conditional Value-at-Risk concepts (CVaR) estimates average losses one may sustain in an extreme event, and hence should be more representative for true downside risk than standard VaR. Backtests suggest that portfolio construction based on these and other risk measures produces signficant investor value.

DB Handbook of Portfolio Construction: Part 1 
DB Markets Research – Quantitative Strategy, 30 May 2013

The point

“We focus on what-so-called ‘risk-based allocation’…portfolios are constructed solely based on risk prediction. But risk, of course, can be defined in countless ways, so risk-based allocation can still be extremely complicated…Not only is managing risk becoming more paramount than outperforming a benchmark, but also risk-based allocation techniques actually do indeed outperform many (if not most) active strategies that require return prediction.”

Improving diversification

“We all know classic finance theory like Markowiz’s mean-variance optimization heavily depends on the assumption that asset returns are jointly normally distributed. We also know that empirical evidence almost universally rejects the normality assumption. The traditional statistical tools (e.g., Pearson’s correlation coefficient, portfolio volatility, etc.) are mostly based on this false assumption… One may argue that the dependence in the left tail (e.g., the chance of both assets suffering extreme losses at the same time) is more relevant in risk management and portfolio construction.”

“Rather than relying on Pearson’s correlation, we measure tail dependence between two assets using a Copula model [of the dependence between assets in a multivariate distribution]… to measure the crowdedness of plain vanilla types of low risk strategies…median tail dependence…we extend the tail dependence concept by further constructing what we called weighted portfolio tail dependence (WPTD) by taking into account asset weights. More important, we design a strategy that proactively avoids crowded trades in what we call the minimum tail dependent portfolio…[which helps] us better capture diversification benefits.”

“We attempted to measure the crowdedness of a strategy using…median pairwise correlation (MPC) and median pairwise tail dependence (MPTD). The rationale is that when investors increasingly trade a basket of stocks together, the MPC and MPTD of stocks within this basket will rise compared to the market. When the relative MPC (or MPTD) compared to the market is at certain level, it is an indication of strategy crowding. ”

“[Crowdedness also] measures the efficacy or potential diversification opportunity of a particular strategy. For example, if the weighted portfolio correlation of a risk-based strategy is approaching the average correlation of the market (i.e., a capitalization weighted benchmark index), it suggests the potential diversification benefit is shrinking and we are holding a portfolio that is increasingly similar to the index.”

Conditional value at risk (CVaR)

“Value at risk (or VaR)…is defined as a threshold value such that the probability that the loss on the portfolio over the given time horizon exceeds this value is the given probability level… [By contrast] CVaR or conditional value at risk is a statistical measure of tail risk, measured by assessing the likelihood (at a specific confidence level) that a specific loss will exceed the VaR. Mathematically speaking, CVaR is derived by taking a weighted average between the VaR and losses exceeding the VaR…CVaR can be interpreted as the average VaR when the loss is greater than VaR.”

“A portfolio that minimizes expected CVaR… attempts to manage risk better…Despite the theoretical soundness of the CVaR methodology, in practice, we have to estimate CVaR empirically and face all the usual problems with estimation errors. It is the same trade-off between model error versus estimation error, i.e., we could have a better model, but may have more estimation error.”

Backtest findings

“We empirically backtest seven risk-based allocations (equally weighted, inverse volatility/volatility parity, risk parity/equal risk contribution, global minimum variance, maximum diversification, minimum tail dependence, and minimum CVaR), compared to traditional capitalization-weighted benchmarks in different contexts from asset allocation, multi-asset (bonds, commodities, alternative betas), country/sector portfolios (MSCI ACWI, economically hedged country indices, global sectors, US sectors, European sectors, global industries, region x sector combinations), and equity portfolios (US, European, Asia ex Japan, Japan, emerging markets, and global equities). Interestingly, in almost every single context, risk-based allocations significantly outperform traditional capitalization-weighted benchmarks, with higher Sharpe ratio, lower downside risk, better diversification, and less tail dependence. “