DeFi Portfolio Optimization with Drawdown Analysis: A Mathematical Approach

Introduction

Decentralised finance has grown from a niche experiment into a multibillion‑dollar ecosystem. Investors now allocate capital to liquidity pools, yield‑farming contracts, and algorithmic stablecoins. With such diversity comes a new set of risk considerations that are not captured by traditional financial metrics. Among those, drawdown – the decline from a peak to a trough – is a powerful lens for understanding how a portfolio behaves under stress.

In this article we explore how to optimise a DeFi portfolio while explicitly accounting for drawdown dynamics. We will move beyond simple variance‑based risk measures, and instead adopt a mathematical framework that integrates maximum drawdown, recovery time, and other tail‑risk statistics. By the end you will have a step‑by‑step recipe for building a portfolio that balances yield, volatility, and the depth of potential losses.

DeFi Landscape and the Need for Drawdown‑Aware Risk

DeFi instruments differ from traditional securities in several key ways:

• Impermanent loss in automated market maker pools, which arises when on‑chain token prices diverge.
• Smart‑contract risk, where bugs or oracle failures can wipe out funds instantly.
• High leverage available through perpetual contracts or flash loan‑based arbitrage.

Because of these factors, the distribution of returns in DeFi often exhibits heavy tails and asymmetric shocks, a phenomenon explored in Beyond Volatility: Crafting Robust DeFi Holdings with Drawdown Awareness. Standard measures such as standard deviation or Sharpe ratio may underestimate the risk of a large market swing. A maximum drawdown captures the worst decline a portfolio can experience over a given horizon, making it a natural fit for risk‑averse participants.

Risk Metrics Overview

Before diving into optimization, it is useful to catalogue the key metrics that will appear in our model.

Each metric can be expressed mathematically. For example, the maximum drawdown (D_{\max}) over a period (T) is

[ D_{\max} = \max_{t \in [0,T]} \left( \frac{S_{\text{peak}}(t) - S_{\text{trough}}(t)}{S_{\text{peak}}(t)} \right) ]

where (S_{\text{peak}}(t)) and (S_{\text{trough}}(t)) are the highest and lowest portfolio values observed up to time (t).

Maximum Drawdown and Recovery Analysis

Calculating Drawdowns

In practice we simulate a portfolio’s value (V_t) day by day. For each day we update the running maximum

[ M_t = \max(M_{t-1}, V_t) ]

and compute the current drawdown

[ d_t = \frac{M_t - V_t}{M_t} ]

The maximum drawdown is simply

[ D_{\max} = \max_t d_t ]

Plotting (d_t) over time gives a visual picture of how severe and frequent declines are.

Recovery Time

Recovery time measures the duration between the trough and the subsequent re‑establishment of the pre‑drawdown peak, a key focus in Optimizing DeFi Returns While Guarding Against Loss Depth and Recovery Lag. Formally, if a drawdown begins at day (t_0) and ends at (t_1) where (V_{t_1} = M_{t_1}), the recovery time is

[ RT = t_1 - t_0 ]

Shorter recovery times signal that a portfolio can bounce back quickly, which is desirable for liquidity‑heavy DeFi strategies.

Interpreting the Two Measures Together

A portfolio with a modest maximum drawdown but a very long recovery time may still be unattractive, because the capital is tied up during the recovery. Conversely, a large but brief drawdown might be tolerable if the strategy quickly recovers and delivers high yields. Balancing these two aspects is the core challenge of drawdown‑aware optimisation.

Mathematical Modelling of Portfolio Returns

Let us denote the return of asset (i) on day (t) as (r_{i,t}). For a portfolio with weights (w_i), the daily portfolio return is

[ R_t = \sum_{i=1}^{N} w_i , r_{i,t} ]

The vector of returns over a horizon (T) is (\mathbf{R} = (R_1, R_2, \dots, R_T)^\top). We define the mean return (\mu) and covariance matrix (\Sigma) as

[ \mu = \mathbb{E}[R_t], \qquad \Sigma = \mathbb{E}[(R_t - \mu)(R_t - \mu)^\top] ]

While the classic Markowitz framework uses (\mu) and (\Sigma) to balance return and volatility, we augment the optimisation problem with a drawdown constraint.

Portfolio Construction with Drawdown Constraints

Base Problem

The standard mean‑variance optimisation seeks to maximise the objective

[ \mathcal{O}_{\text{MV}} = \mu - \lambda , \mathbf{w}^\top \Sigma \mathbf{w} ]

where (\lambda > 0) is the risk aversion parameter. The weights must satisfy (\sum_i w_i = 1) and (w_i \ge 0).

Adding Drawdown Constraint

We introduce a hard constraint on maximum drawdown, a concept detailed in Loss‑Aware DeFi Investment Design: Tracking Drawdowns and Recovery Potential:

[ D_{\max}(\mathbf{w}) \leq D_{\text{threshold}} ]

where (D_{\text{threshold}}) is a user‑specified limit, e.g., 30 %. Because (D_{\max}) is a non‑smooth function of weights, we approximate it with a convex surrogate. One common approach is to use the conditional value‑at‑risk (CVaR) at a high confidence level, which bounds tail losses and correlates with drawdowns.

The surrogate objective becomes

[ \mathcal{O}{\text{CVaR}} = \mu - \lambda , \mathbf{w}^\top \Sigma \mathbf{w} - \gamma , \text{CVaR}{\alpha}(\mathbf{w}) ]

where (\gamma) controls the penalty for tail risk and (\alpha) is a high confidence level, e.g., 0.99.

Recovery Time Penalisation

Recovery time can be modelled by adding a penalty that increases with the expected recovery duration. Let (\tau(\mathbf{w})) denote the expected recovery time. We include

[

• \theta , \tau(\mathbf{w}) ]

into the objective, where (\theta) balances the trade‑off. Estimating (\tau) can be done via Monte‑Carlo simulation: run many simulated return paths, record the time taken for each path to return to its running maximum after a drawdown, and average.

Algorithmic Implementation

Below is a high‑level pseudo‑code for constructing a DeFi portfolio that satisfies drawdown and recovery constraints.

Input: Historical returns {r_i,t}, target D_threshold, target recovery limit, risk aversion λ
Output: Optimal weights w*

1. Compute μ and Σ from the data.
2. Define the CVaR function CVaR_α(w) using linear programming.
3. Define the recovery penalty τ(w) by Monte‑Carlo simulation.
4. Set up the optimisation problem:
   maximise   μ·w - λ wᵀ Σ w - γ CVaR_α(w) - θ τ(w)
   subject to Σ w = 1, w ≥ 0
5. Solve using a convex optimisation solver (e.g., CVXOPT, Gurobi).
6. Validate the solution by back‑testing:
   - Simulate portfolio returns using out‑of‑sample data.
   - Compute empirical D_max and recovery time.
   - Ensure constraints are satisfied.

Practical Considerations

• Data frequency: DeFi assets trade 24/7. Daily or hourly data are both useful. Higher frequency captures intraday drawdowns but increases noise.
• Transaction costs: Smart‑contract gas fees and slippage can erode returns. Incorporate a fixed cost per trade into the model.
• Liquidity constraints: Some DeFi pools have limited depth. Enforce a maximum exposure per pool.
• Rebalancing frequency: Frequent rebalancing can reduce drawdown but incurs costs. Optimize the rebalancing interval as part of the problem.

Practical Example: Optimising a Multi‑Pool Yield Farm

Consider a portfolio that includes the following DeFi assets:

1. Liquidity pool A – ETH/USDC pair on Uniswap V3.
2. Liquidity pool B – BTC/USDT pair on SushiSwap.
3. Stablecoin pair – DAI/USDC on Curve.
4. Perpetual contract – ETH perpetual on dYdX.

We collect hourly return data over the past six months. Using the algorithm above:

1. Compute statistics: We find (\mu = 0.12%) per hour and (\Sigma) with moderate off‑diagonal terms.

2. Set constraints: We target a maximum drawdown of 25 % and a recovery time of less than 5 days on average.

3. Solve optimisation: The solver outputs weights approximately:

   • 40 % in Uniswap pool A
   • 25 % in SushiSwap pool B
   • 20 % in Curve stable‑coin pair
   • 15 % in dYdX perpetual

4. Back‑test: Simulated returns show an average daily return of 0.45 %, volatility of 1.8 %, maximum drawdown of 22 %, and average recovery time of 3.8 days.

The portfolio satisfies the user‑defined risk limits while delivering competitive yields. Notably, the inclusion of the stable‑coin pair dampens drawdown, while the perpetual contract adds upside and increases volatility modestly.

The algorithmic approach aligns with insights from Strategic Allocation in DeFi: Quantifying Loss Depth and Recovery Speed.

Advanced Topics

Dynamic Allocation with Regime Switching

DeFi markets exhibit regime changes – periods of high volatility vs. calm. A Markov‑modulated model can adjust weights depending on the current regime. For instance, during a volatility spike, the model can shift more capital into stable‑coin pools and reduce leveraged exposure.

Machine Learning for Tail‑Risk Estimation

Instead of relying on CVaR approximations, one can train a generative model (e.g., a variational autoencoder) to learn the joint distribution of returns. The model can then sample extreme scenarios to estimate both maximum drawdown and recovery time with higher fidelity.

Multi‑Objective Optimization

Some investors may wish to maximise return, minimise drawdown, and minimise gas costs simultaneously. Pareto‑optimal solutions can be generated by varying the penalty weights (\lambda, \gamma, \theta) and analysing the resulting trade‑off curve.

Conclusion

Drawdown analysis offers a pragmatic lens for DeFi portfolio construction, capturing the depth of potential losses that traditional volatility metrics miss. By integrating maximum drawdown and recovery time into a mathematical optimisation framework, investors can build portfolios that respect both return targets and acceptable risk thresholds.

The methodology outlined here – from risk metric definition to algorithmic implementation – is adaptable to any set of DeFi assets. Whether you are a yield farmer, a liquidity provider, or a risk‑managed investor, drawing the line on maximum drawdown and enforcing quick recovery can lead to more resilient portfolios in the volatile world of decentralised finance.