VaR of Liquidity Providing to Uniswap V2 and V3

Executive Summary
This report is an application of the Value at Risk (VaR) model to Liquidity providing (LP) at Uniswap V2 and V3 platforms. VaR is a statistical measure to calculate the maximum potential loss covered by market movement. It is a standard tool used in investment banking and traditional finance. Credmark has adapted VaR to measure market risk when providing liquidity on Decentralized Exchanges.
The current Beta version of VaR can be used to monitor the risk of financial exposure when providing liquidity to a particular pool in a DEX. Especially relevant to treasury management applications, our VaR score can be used to ensure positions stay within an organization’s risk appetite.  
Credmark’s assessment indicates that Uniswap V3 holds more market risk than V2. The market risk in Uniswap V2 is comparable to HODLing tokens, while the narrow price ranges (“concentrated liquidity”) available in Uniswap V3 can create higher potential losses. This assessment did not take into account any fees that might be earned from LP and gas costs. 
Methodology
Credmark’s Value at Risk (VaR) metric uses a historical simulation model based on the past year of market activity. We apply the 99th-percentile worst-case scenario for a 10-day holding period that is consistent with Traditional Finance (TradFi) market risk capital modelling.
Decentralized exchanges might generate more loss than simply holding assets in a market because they act as automated market-makers. Market makers hold paired assets in a pool; as the market moves, the higher value asset is sold to buy more of the lower-valued asset. When price moves, the liquidity provider will not be able to extract the same amount of tokens as when he put in and what he received less is in the higher valued asset.
This type of downside is called impermanent loss [1] and is added to the standard buy and hold strategy's profit and loss.
We’ve illustrated the application of the loss valuing techniques for Uniswap V2 and Uniswap V3 protocols as of 31 December 2021. Based on these run results, as expected, Uniswap V3 showed a significantly higher level of risk compared to Uniswap V2. The narrower the liquidity range, the higher VaR. While the narrower range also entails a higher opportunity to earn money via fees, this is not included in our assessment. The position is assumed to remain the same.
Calculation
We could define the invariant at time 0 when the LP (liquidity providing) stored the tokens to a pool.
c=x0y0c=x_0y_0
c=x0​y0​
r0=x0y0r_0 = {x_0 \over y_0}r0​=y0​x0​​
x0=cr0,y0=cr0x_0 = \sqrt {c r_0} , y_0 = \sqrt {c \over r_0}x0​=cr0​​,y0​=r0​c​​
If we introduce the price, ux0u_{x0}ux0​
, they will complement with the quantity x0x_0x0​
 to r0r_0r0​
.
r0=uy0ux0r_0 = {u_{y0} \over u_{x0}}r0​=ux0​uy0​​
The initial portfolio's value is
P0=x0ux0+y0uy0=(x0+y0r0)ux0P_0 = x_0 u_{x0} + y_0 u_{y0} = (x_0 + y_0 r_0) u_{x0}P0​=x0​ux0​+y0​uy0​=(x0​+y0​r0​)ux0​
The portfolio with the price change is (HODL)
P01=x0ux1+y0uy1=(x0+y0r1)ux1P_{01} = x_0 u_{x1} + y_0 u_{y1} = (x_0 + y_0 r_1) u_{x1} P01​=x0​ux1​+y0​uy1​=(x0​+y0​r1​)ux1​
The portfolio with LP's IL is (Note quantity has change to x1x_1x1​
and y1y_1y1​
.
P1=x1ux1+y1uy1=(x1+y1r1)ux1P_1 = x_1 u_{x1} + y_1 u_{y1} = (x_1 + y_1 r_1) u_{x1}P1​=x1​ux1​+y1​uy1​=(x1​+y1​r1​)ux1​
The IL and portfolio PnL for Uniswap V2 is
IL=P01P1−1=2r1r01+r1r0−1IL = {P_{01} \over P_1 - 1} = {{2 \sqrt {r_1 \over r_0}} \over {1 + {r_1 \over r_0}}} - 1IL=P1​−1P01​​=1+r0​r1​​2r0​r1​​​​−1
PortPnL=P01P0−1=(1+r1r0)2ux0ux1−1PortPnL = {P_{01} \over P_0 - 1} = {(1 + {r_1 \over r_0}) \over 2} {u_{x0} \over u_{x1}} - 1PortPnL=P0​−1P01​​=2(1+r0​r1​​)​ux1​ux0​​−1
Uniswap V2 Protocol
Uniswap impermanent loss (IL) function is defined as:
IL(k)=2k1+k−1\text{IL}(k)={ { 2 \sqrt k} \over {1 + k} } - 1IL(k)=1+k2k​​−1
where k=1+rk=1+rk=1+r
 and rrr
 is the relative change in price [1]. In our case 10 calendar days changes since crypto market operates 24/7. Graphically IL equality could be shown as:
Uniswap v2 Impermanent Loss Function - Losses to Liquidity Providers due to Price Variation on top in Excess to Holding the Original Funds Supplied [1]
Let’s assume we have a balance of $100,000 in BTC/ETH pool distributed evenly as of 31 December 2021. Hence, we’re contributing $50,000 worth of BTC (1.06 units) and $50,000 worth of ETH (13.46 units). Based on last year’s returns, the 99th percentile, 10-day overlapping sampling, and 10-day holding period, we get the following results:
​
Strategy
HODL
LP (based on total portfolio return)
LP (based on
 IL)
Observation Date
22 May 2021
22 May 2021
4 May 2021
VaR
$49,105
$49,515
-$17,856
Impermanent Loss
0
$410
$1,552
Unrealized Loss
$49,105
$49,105
-$19,688
In the case of LP strategy to Uniswap v2, the impact of IL for the worst-case scenario is minimal and the worst-case scenario is the same as in the case for bu y and hold strategy.
The 99th worst-case IL is equal to $1,552 and is associated with an unrealized gain during the same period. Note that negative loss is a profit.
Uniswap V3 Protocol
In the case of Uniswap V3, in contrast to V2, IL is also driven by the LP price range and initial price of the deal, which for our calculations is assumed to be the price at the reporting date.
The function defining IL is then defined as:
When Pa≤P≤PbP_a \le P \le P_bPa​≤P≤Pb​
​
ILa,b(k)=2k−1−k1+k−paP−kPpb=IL(k)×(11−paP+kPpb1+k)\text{IL}_{a,b}(k) = { { 2 \sqrt k - 1 - k} \over { 1 + k - \sqrt {p_a \over P} - k \sqrt{P \over p_b} } } = \text{IL}(k) \times \left ( { 1 \over { 1 - { \sqrt {p_a \over P} + k \sqrt { P \over p_b } \over {1 + k} } } } \right )ILa,b​(k)=1+k−Ppa​​​−kpb​P​​2k​−1−k​=IL(k)×⎝⎛​1−1+kPpa​​​+kpb​P​​​1​⎠⎞​
When P≤PaP \le P_aP≤Pa​
​
1Pa+1PbPb−P0P0−Pa+(P0−Pa)P1{{1 \over \sqrt{P_a}} + {1 \over \sqrt{P_b}}} \over
{
{{\sqrt{P_b} - \sqrt{P_0}} \over { \sqrt{P_0} - \sqrt{P_a} }} +
( {\sqrt{P_0} - \sqrt{P_a}} ) P_1
}P0​​−Pa​​Pb​​−P0​​​+(P0​​−Pa​​)P1​Pa​​1​+Pb​​1​​
​When Pb≤PP_b \le PPb​≤P
​
Pb−PaPb−P0P0−Pa1P1+P0−Pa{ {\sqrt{P_b} - \sqrt{P_a}} }
\over
{
  { {\sqrt{P_b} - \sqrt{P_0}} \over {\sqrt{P_0} - \sqrt{P_a}} } { 1 \over P_1} + { \sqrt{P_0} - \sqrt{P_a}}
}P0​​−Pa​​Pb​​−P0​​​P1​1​+P0​​−Pa​​Pb​​−Pa​​​
where pap_apa​
 and pbp_bpb​
 are the lower and the higher bound of the set liquidity providing (LP) price range [2], and PPP
 is the current price of the associated pair. 
Based on the same parameters:
Reporting Date
31-Dec-21
LP Pool
BTC/ETH
Position in $
                         100,000
Price
                         12.70
The VaR estimates for different price ranges are:
Note that the range with the lower bound equal to 0 and high upper bound (x10) price corresponds to the Uniswap V2 that doesn’t have any LP range setting option.
Implementation
The model is implemented within the Credmark Model Framework (CMF) with model slug finance.var-dex-lp, more details on the implementation can be found here.
Model Output
Testing result is shown below for end of date 2021-12-31 with 270-day horizon, 10-day return and 99% confidence level. The result differs from earlier calculation slightly due to different prices used.
Pools: 
Uniswap V2: 0xbb2b8038a1640196fbe3e38816f3e67cba72d940 WETH/WBTC
Uniswap V3: 0xcbcdf9626bc03e24f779434178a73a0b4bad62ed WETH/WBTC
Pool Type
Range for upper/lower
VaR
VaR without IL
VaR for IL
Uniswap V2
100%
-37%
-37%
-2%
Uniswap V3
1%
-100%
-37%
-100%
Uniswap V3
5%
-64%
-37%
-70%
Uniswap V3
10%
-41%
-37%
-35%
Uniswap V3
20%
-38%
-37%
-18%
Uniswap V3
40%
-37%
-37%
-9%
Uniswap V3
60%
-37%
-37%
-6%
Uniswap V3
80%
-37%
-37%
-4%
Uniswap V3
100%
-37%
-37%
-2%
Conclusion
Uniswap V2 does not introduce significant additional market risk to HODLing while Uniswap V3 can introduce impermanent loss exceeding 40% of the notional. However, inthe  case of VaR the primary driver is Unrealized loss from HODLing position hence observation date remains the same for most of the wider ranges.
It is assumed that no fees are received from Liquidity Providing (LPing) in both cases. Uniswap V3 does have the potential to amplify fees earned by providing concentrated liquidity (not covered in this report), but Credmark’s suggestion is not to provide liquidity without a deep understanding of Uniswap V3 and specialized tools.
Contributors
Discord Handle
ETH Address
Reward
Contribution
atulemis#0983
0x5fb7584838fB467e90bb8a1df3a278482e34E856
0 CMK (internal)
Original version
kunlun#8324
x109B3C39d675A2FF16354E116d080B94d238a7c9
0 CMK (internal)
Update
​
References
1.Uniswap V2 - Understanding Returns, Uniswap.org (Accessed on 17 November 2021), https://docs.uniswap.org/protocol/V2/concepts/advanced-topics/understanding-returns​
2.Impermanent Loss in Uniswap V3, Auditless (Accessed on 17 November 2021), https://medium.com/auditless/how-to-calculate-impermanent-loss-full-derivation-803e8b2497b7 
3.Atis Elts, 30 September 2021 (Accessed on 1 February 2022), http://atiselsts.github.io/pdfs/uniswap-v3-liquidity-math.pdf​

Pool Type	Range for upper/lower	VaR	VaR without IL	VaR for IL
Uniswap V2	100%	-37%	-37%	-2%
Uniswap V3	1%	-100%	-37%	-100%
Uniswap V3	5%	-64%	-37%	-70%
Uniswap V3	10%	-41%	-37%	-35%
Uniswap V3	20%	-38%	-37%	-18%
Uniswap V3	40%	-37%	-37%	-9%
Uniswap V3	60%	-37%	-37%	-6%
Uniswap V3	80%	-37%	-37%	-4%
Uniswap V3	100%	-37%	-37%	-2%

Discord Handle	ETH Address	Reward	Contribution
atulemis#0983	0x5fb7584838fB467e90bb8a1df3a278482e34E856	0 CMK (internal)	Original version
kunlun#8324	x109B3C39d675A2FF16354E116d080B94d238a7c9	0 CMK (internal)	Update

Last modified 1yr ago

Reporting Date	31-Dec-21
LP Pool	BTC/ETH
Position in $	100,000
Price	12.70

Strategy	HODL	LP (based on total portfolio return)	LP (based on IL)
Observation Date	22 May 2021	22 May 2021	4 May 2021
VaR	$49,105	$49,515	-$17,856
Impermanent Loss	0	$410	$1,552
Unrealized Loss	$49,105	$49,105	-$19,688