📊Portfolio Optimization

Using knowledge-based and computational techniques, the optimal DeFi management can be separated into 2 parts including the reliable DeFi asset selection and the optimal DeFi asset allocation. The total value locked (TVL) and its changes are considered as the reliability of the DeFi assets that can be mathematically expressed as follows:

Step 1: Filtering DeFi assets based on the reliability score calculated by:

$$Reliability\_Score_n = \frac{TVL^n_t}{Max TVL} + \frac{TVL^n_t - TVL^n_{t-20}}{TVL^n_{t-20}}$$

Where:

$Reliability\_Score_n$ is the reliability score of the DeFi asset n .

$TVL^n_t$ is the total value locked of DeFi asset $n$ at the present.

$TVL^n_{t-20}$ is the total value locked of the DeFi asset n in the last 20 days.

$MaxTVL$ is the maximum TVL value of the selected DeFi assets.

Note that selected DeFi assets need to be audited by at least 1 outstanding smart contract audit company. And the timeframe to consider the TVL is set to 20 days. In the next step, the expected return and the return’s volatility (risk) of the selected DeFi assets calculated from historical data of rewards, governance token prices, and liquidity provider (LP) token prices are used in the optimal DeFi asset allocation model as:

Step 2: Providing the optimal DeFi asset allocation by:

max_{w \in (0,1)} \sum_{n=1}^{N} ( w_n \cdot \text{exp_return}_n)

$$\text{subject to: } \big( \sum_{n=1}^N \big(w_n \times SD\_Return_n\big) \big) \leq SD\_limit$$

$$\big( \sum_{n=1}^N w_n \big) = 1$$

Where:

$w_n$ is the allocation weight of DeFi asset n.

$\mu_n$ is the expected value of returns of DeFi asset $n$.

$\sigma_n$ is the standard deviation (volatility) of returns of DeFi asset n.

$\hat\sigma$ is the standard deviation (volatility) limit selected by users.

$N$ is the number of considered DeFi assets.

The Markowitz Portfolio Optimization model mentioned above considers the expected return maximization for different values of volatility so-called "efficient frontier". The optimal solutions at different volatility acceptance levels are shown in Figure 1 (blue line):

Figure 1 Optimal solutions of DeFi portfolio allocation using quadratic programming and Monte Carlo Simulation

In Figure 1, the blue line is the efficient frontier used to select the optimal asset allocation for the DeFi investment at given volatility tolerance levels. Axes X and Y represent the risk level (standard deviation in %) and the expected return of the DeFi asset portfolio in %, respectively.

The efficient frontier solves for a weighted portfolio of assets that maximizes the reward-risk ratio - named the Sharpe Ratio. Given a risk-free reward, one can find a tangent line that stretches from the Y-axis to the efficient frontier that achieves the highest slope (reward per unit risk). In the diagram above, it is the red star on the efficient frontier. Hence, the weights to achieve this portfolio reward-risk tradeoff is our solution to the efficient frontier optimization.

We are currently researching the increasing complexity of the problem. This posits the use of a more advanced, multi-objective function that requires a more advanced numerical method such as the Particle Swarm Optimization.

References:

Quadratic programmingWikipedia

Monte Carlo methodWikipedia

Particle swarm optimizationWikipedia