Containment of drawdowns and optimization of performance ratios for multi-asset portfolios is critical for trading strategies. Alas, short data series or structural changes often render estimates of covariance matrices unreliable. A popular solution is risk-parity with volatility targeting. An alternative is ‘MinMax’ drawdown control, which builds on a broad interpretation of drawdowns as maximum actual or opportunity losses from not adjusting a benchmark portfolio to a specific underlying asset. In the case of one risky and one safe asset, this boils down to managing simultaneously the risks of conventional PnL drawdowns and foregone risk returns. Optimal asset allocation depends only on aversion to different types of drawdowns. Averaging over a plausible range of aversion parameters gives a model portfolio. Empirical evidence for the case of cryptocurrencies suggests that in an environment of uncertain returns MinMax delivers better PnL return-to-drawdown ratios than conventional volatility control.

The post ties in with SRSV’s summary lecture on risk management.

The below are excerpts from the paper. Emphasis and cursive text have been added.

## The portfolio construction problem

“A central challenge for portfolio allocation…[is] the implementation of mean-variance optimal portfolios…Using historical data to estimate returns and correlations, frequently lead to unattractive, leveraged, and highly unstable portfolios…The limitations of historical data as a determinant of portfolio allocation seem particularly salient in the case of innovative asset classes with limited track record… it is difficult to form reliable estimates of expected returns and covariance matrices needed as inputs for standard portfolio optimization. Even if such estimates are available, they may be useless to investors if the behavior of underlying assets changes over time.”

*N.B.: This issue would apply to almost all financial contracts in the case of a significant structural change of the macro environment.*

“Under the standard Bayesian framework, the investor is equipped with a prior…This prior captures the investor’s beliefs over the possible evolution of returns. This may include the presence of positive auto-correlation in returns, the possibility that a bubble may be underway, and so on…The difficulty we confront…is that forming beliefs is difficult, especially when there is little data to inform the decision-maker. An indicator of this difficulty is that different well-informed investors will frequently disagree about expected market behavior.”

## The risk parity approach

“One response to this issue [*of predicting return covariance matrices*] has been…risk-parity [*which*] allocates portfolio weights on the basis of volatility estimates alone…Risk-parity is a particular implementation of modern portfolio theory. It assigns weights to risky assets that are inversely proportional to each asset’s volatility…[*i.e based on the inverse of*] a volatility estimate for asset i at time t, and [*proportionate to*] a scaling parameter used to adjust overall portfolio volatility.”

“In the case of a single risky asset, risk parity boils down to a volatility-control strategy.”

## The MinMax drawdown control framework

“[*The MinMax framework*] tackles the problem of portfolio allocation over novel or changing assets and argues the merits of worst case drawdown guarantees as a benchmark objective for dynamic asset allocation: a good dynamic asset allocation framework should guarantee low drawdowns relative to both the safe asset, and underlying risky assets. MinMax drawdown control helps an agnostic investor achieve low drawdowns….if the investor has correct beliefs over the possible distribution of returns, her portfolio returns should experience low drawdowns with very high probability…If a decision-maker experiences high drawdowns, then, her beliefs are likely misspecified.”

“Drawdown corresponds to the usual drawdown against the safe asset, i.e. peak-to-trough losses against the safe asset. More generally drawdowns can be interpreted as a sample version of optimality conditions: [*drawdowns*] correspond to the maximum potential gains from adjusting a benchmark portfolio towards a particular underlying asset.”

“MinMax drawdown control…[*refers to*] asset allocation strategies which guarantee the best possible drawdown…against arbitrary sequences or returns…Setting maximum drawdown targets against the safe asset will pin down parameters [of] the relative importance of different drawdowns for the investor…[*Under certain assumptions*] worst-case drawdowns under MinMax drawdown strategies are of order [*equal to the square root of the periods over which it is optimized*].”

“[*For example*] in the case of two reference assets a risk-free asset and a risky asset, MinMax drawdowns map out an intuitive two-dimensional frontier: [*one dimension*] captures losses against the safe asset; [*the second dimension*] captures foregone performance relative to the risky asset. For every guaranteed maximum drawdown against the safe asset, the frontier associates the best possible drawdown guarantee for drawdowns against the risky asset… This lets the decision-maker express preferences over risk without referring to a prior. A particularly risk-averse investor will prefer very low drawdowns with respect to the safe asset at the expense of higher drawdowns against the risky asset. Inversely, an aggressive investor will prefer low drawdowns against the risky asset…. It turns out that the optimal asset allocation strategy…depends only on regrets [*over specific types of drawdowns*].”

“A natural portfolio aggregation strategy is simply to average out the MinMax drawdown portfolios of active investors.”

“Averaged-out low-drawdown strategies are well suited to form the basis of portfolio indices suitable for a broad class of investors.”

## Empirical evidence

“Crypto-currencies are an interesting case-study for the MinMax drawdown approach…crypto-currencies have a short track-record…and will mostly likely continue to experience, considerable changes, reflecting evolutions in the set of investors, regulation, and technological use.”

As [*the below table and figure*] illustrate, both volatility control and MinMax drawdown control considerably improve the risk-reward trade-off of investing in bitcoin as captured by the Sharpe ratio. However, in order to match the realized drawdowns of MinMax, volatility control must considerably reduce its exposure throughout the investment period, which significantly reduces the corresponding annualized returns. As a result, MinMax achieves a much better performance-to-drawdown ratio than volatility control.”

“[*For*] the case of multiple risky assets…[*we*] consider a portfolio allocation problem in which the investor seeks to optimize over cash, bitcoin, ethereum and ripple. Both risk-parity and MinMax improve on the Sharpe ratio achieved by a regularly rebalanced equal-weight portfolio. However, MinMax achieves much higher annualized returns than risk-parity for the same worst-case drawdown. In fact, by successfully adjusting its exposure toward the best performing underlying crypto-currency, MinMax achieves better raw performance than the much riskier equal-weight portfolio.”