**Leverage is a powerful tool to amplify the gains (and losses) of a strategy – in this article we explain how it works.**

In finance, leverage refers to the concept of borrowing funds to buy assets and thereby have more exposure than the original funds would allow. Of course, the risk of insolvency is amplified as well!

## What are leverage indices?

Leverage indices such as **Scalara’s Flexible Leverage Indices** (FLIs) are designed to seek returns that are a **multiple of an underlying asset’s return**. For example, the ETH 2x Flexible Leverage Index (ETH2x-FLI) aims to return two times the return of the price of ETH. This also means that if the ETH price falls, the leverage index falls even more.

Intuitively, the leverage index performance can be formulated as:

\[LI_{T+1}=LI_{T} \cdot (1 + L \cdot r_{UL, T+1})\]

where *L* is the leverage ratio and *r _{UL}, T+1* the return from time

*T*to

*T+1*of the underlying asset.

An alternative representation, that allows for greater insight, tracks the value of the **collateralized debt position** (CDP) that is entered into to generate leverage. The index value is the **net asset value** (*NAV*) of this CDP.

\[LI_{T+1}=NAV_{CDP, T+1} = Collateral_{T+1} – Debt_{USD, T+1} = Collateral_{T} \cdot (1 + r_{UL, T+1}) – Debt_{USD, T}\]

whereby we assume there is no cost to borrow (i.e. the debt balance is unchanged from *T* to *T+1*).

This second formula is helpful to understand how the performance of the underlying asset affects the **leverage ratio** (*LR*) of the CDP over time. The *LR* of the CDP is the value of the **collateral divided by its NAV**.

After one time period, if the underlying asset’s price increases, the leverage ratio decreases and vice versa:

\[LR_{T+1} = {Collateral_{T+1} \over NAV_{T+1}} = {Collateral_{T} \over Collateral_{T} – {Debt_{T} \over 1 + r_{UL, T+1}}}\]

which is less than *LR _{T}* if

*r*and greater than

_{UL,T+1}> 0*LR*if

_{T}*r*.

_{UL,T+1}< 0## Why rebalance?

As long as this collateralized debt position is left unchanged, the return over the holding period will be as expected: the underlying return multiplied by the leverage ratio at inception of the CDP.

There are two reasons why leverage indices are rebalanced:

Firstly, to **avoid insolvency and liquidation**. Rebalancing to a lower leverage ratio increases the health of the debt position. As we have seen above the leverage ratio increases when the underlying asset’s price falls.

Secondly, to ensure that the **leverage ratio remains close to the target** leverage. Otherwise, the leverage ratio of the index could deviate significantly from the target leverage depending on the time passed since entering the CDP. Rebalancing will set the LR back to the target leverage so it is providing the expected return (over short holding periods) no matter when a user starts tracking the index.

As we will later see, the tradeoff of this decision to frequently rebalance is that **over longer time periods the leverage index return will not necessarily be close to the the underlying return** **multiplied with the “advertised” leverage**.

The following chart shows the performance of a leverage index over **two time periods with rebalancing**. In this example, the index is rebalanced “fully”, i.e. the CDP is adjusted to a leverage ratio of 2 again after the first time period. As a result, during the second time period the leverage index returns two times the underlying return again. The two period return is simply the compounded return *(1+r _{1})(1+r_{2})-1* (which is clearly not equal to

*r*!).

_{1}+ r_{2}In practice this resetting of the leverage ratio is done by either selling collateral assets and repaying debts (if the underlying asset’s price decreased) or borrowing more and adding to the collateral (if the underlying increased).

We see that while each single period alone returns twice the underlying return of the corresponding period, over both periods together this holds not true. We will answer why in the next section.

## What is volatility decay?

Redrawing the above chart with a smaller return difference between the first and second time period, i.e. **lower volatility** in the underlying asset, shows the impact of volatility on the multiperiod performance of a rebalanced leverage strategy.

By comparing the two charts, we can see that in the first more volatile case the leverage index performed worse in relative terms (*8 *compared to “naive” *2*4.5=9 *if not rebalanced). In the second less volatile case the leverage index actually does better than “expected” (*32* compared to *2*15.5=31*).This **underperformance compared to the un-rebalanced strategy** due to volatility is often referred to as **volatility decay or volatility tax**.

**There is of course no money stolen or taxed!** As seen above, introducing rebalancing changes the strategy’s leverage “path” compared to keeping the CDP unchanged. Every period in the example starts at *LR=2* if fully rebalanced. On the other hand if the index is not rebalanced the LR is not equal to 2 at beginning of each period.

A more mathematical explanation can be derived from the difference between arithmetic and geometric averages but is beyond the scope of this article.

## Flexible is better

So far we assumed rebalancing fully back to the target leverage after every time period. This is true for most daily leverage ETFs in traditional finance.

**Scalara’s Flexible Leverage Indices** do not fully rebalance to the target leverage at each rebalance though. Instead, as long **as the leverage ratio remains in a band around the target the CDP is only partially adjusted** (e.g. in the case of **ETH2x-FLI**, at each rebalance the CDP is rebalanced to move the LR 5% of the “distance” from its current *LR* to the target *LR* of 2).

Since every rebalance requires a swap between borrow asset and collateral asset this flexible algorithm **reduces the overall turnover** (and hence transaction fees).

As a side effect, because the leverage is not set back to its target every time, FLIs may also **closer align with an unrebalanced position** that returns the targeted return multiple over a longer holding period.

This is achieved **without sacrificing safety** since if the *LR* moves beyond the acceptable band the index is rebalanced back to that upper or lower band level fully.

**Flexible Leverage Indices** are implemented as single tokens by the **Index Coop** and are currently available for ETH, BTC and MATIC. In addition, **inverse versions** that provide -1x exposure, i.e. increase when markets fall, to these assets are available.

*Scalara is dedicated to creating and maintaining indices for a decentralized world.*