Bestia.md

Introduction

One of the primary issues in financial instability situations, like bank runs, is the privilege that first movers have. A well-designed liquidity transformation method transfers redemption costs to the redeeming investors, thereby mitigating the first-mover advantage.

Swing Pricing

Swing pricing adjusts the NAV of a fund to reflect the costs associated with redemptions. This adjustment ensures the remaining investors are equally affected by the costs incurred due to others redeeming their shares.

This is why swing pricing is the primary weapon in the management of any type of funds, designed to mitigate the adverse effects of fud-driven large scale redemptions and break the first mover advantage. There are two major variations of swing pricing:

• Liquidation-based: This method activates the fee once the fund's cash reserve is depleted and il/liquid assets need to be liquidated.
• Asset-based: With this approach, the fee is applied at the cash redemption phase, regardless of the state of the cash reserve.

Swing pricing provides two primary benefits:

1. Prevention of Panic Runs: By ensuring that redeeming investors bear the cost of liquidation, swing pricing promotes "responsible withdrawals", thereby preventing panic runs.
2. Enhanced Liquidity Provision: Due to a greater stability during bank runs, funds can hold a larger proportion of illiquid assets, improving their liquidity absorption capacity.

For this analysis we will use as sources "XXX" and "Swing Pricing for Mutual Funds". The paper by Yao Zeng et al. discusses both asset-based and liquidation-based swing pricing methods while the one by XXX is mostly focused on asset-based swing pricing methods.

Liquidation-based Swing Pricing

In the liquidation-based approach, the swing pricing fee is activated only when the fund cash reserves are empty, forcing the immediate sell of illiquid assets. This method directly connects the swing pricing mechanism to the actual process of asset liquidation, ensuring that the liquidation costs are borne by the investors who are triggering the redemptions.

Calculation

1. NAV without Swing Pricing:

   For a mutual fund holding $x$ units of cash and $y$ units of an illiquid asset with a liquidation cost $\lambda$, the NAV without swing pricing is given by:

$$\text{NAV} = \frac{x + y \phi(R)}{N}$$

where $\phi(R)$ is the value of the illiquid asset depending on the time evolution factor $R$, while $N$ is the total number of shares.

2. Swing pricing adjustment:

   When redemptions occur, the fund may need to liquidate some illiquid assets, incurring a cost. If the fund has enough cash to meet redemptions $$x \geq \phi(R) \cdot \text{redemptions}$$ no illiquid assets need to be liquidated, and no swing pricing adjustment is needed. However, if the redemptions exceed the cash holdings, the fund must liquidate illiquid assets, and swing pricing adjusts the NAV as follows:

$$\text{Adjusted NAV} = \frac{x + (y - \ell) \phi(R) - \lambda \cdot \ell}{N - \text{redemptions}}$$

where $\ell$ is the amount of illiquid asset liquidated to meet the redemptions, and $\lambda$ represents the liquidation cost per unit of the illiquid asset.

3. Determining the liquidation amount $\ell$:

   The fund first uses its cash to meet redemptions. If the redemptions exceed the available cash, the required amount of the illiquid asset $\ell$ to be liquidated is determined by:

$$\ell = \frac{\text{redemptions} \cdot \text{Initial NAV} - x}{\phi(R) \cdot (1 - \lambda)}$$

The paper's empirical analysis shows that swing pricing can improve the Liquidity Provision Index (LPI) by 6.7%, indicating a significant enhancement in liquidity provision capacity. PUT SOMEWHERE ELSE AS LIQUIDITY INDEX IS NOT DEFINED HERE

Asset-based Swing Pricing

Option1: The "Swing Pricing for Mutual Funds" paper gives us a swing pricing model that applies a fee already the cash redemption phase, spreading the liquidation costs more evenly among all investors. By doing so, it removes the first-mover advantage and reduces the overall redemption pressure in case of a bank run.

Option2: The asset-based swing pricing model adjusts the NAV based on the redemption queue, regardless of whether the fund's cash reserves are depleted or not. This proactive approach ensures that all investors bear the costs of redemptions, reducing the incentive to redeem early and avoiding the negative feedback loop of fire sales.

Therefore we can model

1. Redemptions: $$R = -\beta \Delta S$$ Where $R$ is the redemption amount, $\beta$ is the sensitivity parameter, and $\Delta S$ is the drop in NAV.

2. Price Impact: $$\Delta P = \phi(\Delta Q)$$ Where $\Delta P$ is the change in asset price, $\Delta Q$ is the quantity of asset sold and $\phi(\cdot)$ is the price impact function.

3. Fund NAV: $$S = \frac{Q \cdot P + C}{N}$$ Where $S$ is the NAV, $Q$ is the quantity of assets, $P$ is the asset price, $C$ is cash available and $N$ is the number of shares.

4. Final calculation: For $n$ funds, the aggregate redemption pressure $R_i$ on fund $i$ is influenced by the redemptions in other assets $j$: $$R_i = -\beta_i \left( \Delta S_i + \sum_{j \neq i} \alpha_{ij} \Delta S_j \right)$$ where $\alpha_{ij}$ captures the cross-fund correlation.

The swing pricing adjustment for multiple funds requires a coordinated approach: $$\Delta S_{sw,i} = -\phi_i \left( R_i + \sum_{j \neq i} \alpha_{ij} R_j \right)$$ ensuring that the first-mover advantage is eliminated across all participating funds.

The swing pricing adjustment can be expressed as: $$\Delta S_{sw} = -\phi\left( R + F \right)$$ Where $R$ is the immediate redemption and $F$ is the net asset outflow.

Derivation Steps

1. Initial condition: $$|\Delta Z| > 0$$ This is the initial drop in asset prices

2. Redemptions: $$R_{fm} = -\beta \Delta S_{tot}$$ Where $\Delta S_{tot}$ includes the initial price drop and the impact of subsequent redemptions.

3. Price impact function: $$\Delta P = \gamma (\Delta Q)$$ Assuming a linear price impact function where $\gamma$ is the price impact coefficient.

4. NAV adjustments: $$S_{sw} = S_0 + \Delta Z + \Delta S_{sw}$$ Adjusting the NAV based on the swing pricing formula.

Bestia own swing pricing

The provided mathematical expressions seem to be inconsistently labeled and misaligned, leading to potential confusion in applying the swing pricing model. Here's a corrected and coherent presentation of the formulas:

Definition of Swing Pricing Function

The swing pricing function aims to transition smoothly from linear to exponential behavior based on deviations from a target threshold. We start by defining a basic function for this behavior:

$$f(x) = a \cdot x \cdot e^{c \cdot x}$$

Where:

• x: Deviation from the target percentage.
• a: Scales the magnitude of the function, aligning with the fee range.
• c: Controls the growth rate from linear to exponential.

This function remains near-linear for small x as $e^{c \cdot x} \approx 1$ when $c \cdot x$ is close to zero. For larger x, exponential growth dominates due to the exponential term.

Incorporating Market Dynamics

Next, we account for additional market dynamics such as net flows and asset price velocity:

$$Adjusted Swing Factor = \left( a \cdot |F| \right) \cdot \left( 1 + b \cdot \sigma \right)$$

Where:

• $F = \frac{\text{assetsIn/Outgoing}}{\text{time}}$: Net flow as a percentage of the AUM.
• $\sigma = \frac{\partial \text{price}}{\partial t}$: Represents the asset price velocity.
• a and b: Scaling constants to fine-tune the response.

Final Swing Pricing Model

Finally, we integrate the transient term to smoothen the swing pricing transition:

$$Final Swing Factor = Adjusted Swing Factor \cdot \left( 1 + d \cdot f(Deviation) \right)$$

Where:

• $Deviation = |T_{ideal} - T_{now}|$
• $f(Deviation) = e^{c \cdot Deviation}$: The smoothing function from the initial definition.
• d: A scaling constant to adjust the impact of the deviation.

Leading to:

$$\left( a \cdot |F| \right) \cdot \left( 1 + b \cdot \sigma \right) \cdot \left( 1 + d \cdot e^{c \cdot |T_{ideal} - T_{now}|} \right)$$

We now need to normalise the function such that it ranges from 0% (no fee applied) to 100% (entire amount paid as fee). This can be done by introducing a sigmoid function which naturally bounds its output between 0 and 1:

$$f(x) = \frac{1}{1 + e^{x} - k)}}$$

Where:

• x: Deviation from the target percentage.
• k: Adjusted constant to scale and shift the response curve appropriately.

The sigmoid function compresses the entire range of real numbers into the interval [0, 1], making it an ideal choice for normalisation.

Liquidity Provision Index (LPI)

The Liquidity Provision Index (LPI) is a measure developed to quantify the liquidity provided by financial intermediaries, both issuing debt (such as banks) and issuing equity (such as mutual funds).

"The LPI captures the extent to which an intermediary enhances the liquidity available to investors by allowing them to redeem their investments at a value higher than what they would obtain from directly liquidating the underlying assets. " COPIED FROM PAPER, REPHRASE

Calculation of the LPI

The LPI for a financial intermediary $k$ is defined as:

$$\text{LPI}_k = \frac{c^_k(R)}{c(x^_k, y^*_k)} - 1$$

where:

• $c^*_k(R)$ represents the contract payment to investors under intermediary $k$, given economic fundamentals $R$.
• $c(x^_k, y^_k)$ is the liquidation value of the intermediary's portfolio, composed of cash $x^_k$ and illiquid assets $y^_k$, without the intermediary's involvement.
• $x^_k$ and $y^_k$ are the optimal portfolio allocations chosen by the intermediary.

In the case of mutual funds without swing pricing, the LPI is: $$ \text{LPI}_f = \begin{cases} \frac{x + y \phi(R)}{x + y (1 - \lambda) \phi(R)} - 1 & \text{if } \text{redemptions} \leq \text{liquidation value}, \ 0 & \text{if } \text{redemptions} > \text{liquidation value}. \end{cases} $$

For mutual funds with swing pricing, the LPI is adjusted to account for the liquidation costs incurred during redemptions.