## Contents

Overview

Asset Volatility vs Asset Sensitivity to Benchmark (Beta)

Portfolio Asset Covariance

Value at Risk (VaR)

Bitcoin Futures: Basis and Proxies

Intraday Value at Risk (VaR)

Risk-Based Limits

VaR Validation (Bayesian Approach)

Scenario Analysis

Conclusion

## Overview

Crypto currencies have now become part of institutional investment strategies. According to CoinShares, assets held under management by crypto managers reached $57B at the end of Q1 2021.

Like any other financial asset, crypto investments are subject to market risk monitoring with several approaches evolving. Crypto currencies exhibit no obvious correlation to other assets classes, risk factors or economic variables. However, crypto currencies have exhibited high price volatility and have enough historical data to implement a robust market risk process.

In this paper we discuss approaches to implementing market risk analytics for a portfolio of crypto assets. We will look at betas to benchmarks, correlations, Value at Risk (VaR) and historical event scenarios.

Value at Risk allows risk managers to implement risk-based limits structures, instead of relying on traditional notional measures. The methodology we propose enables consolidation of risk for crypto assets with the rest of the portfolio. We will also discuss the use of granular time horizons for intraday limit monitoring.

## Asset Volatility vs Asset Sensitivity to Benchmark (Beta)

For exchange-traded instruments, **beta** measures the sensitivity of asset price returns relative to a benchmark. For US-listed large cap stocks, beta is generally computed relative to the S&P 500 index. For crypto currencies, several eligible benchmark indices have emerged that represent the performance of the overall crypto currency market.

We analyzed several currencies against S&P’s Bitcoin Index (SPBTC). SPBTC is designed to track the performance of the original crypto asset, Bitcoin. As market capitalization for other currencies grows, it would be more appropriate to switch to a dynamic multi-currency index such as Nasdaq’s NCI. At the time of this paper, Bitcoin constituted 62.4% of NCI.

Traditionally, beta is calculated over a variable time frame using least squares fit on a linear regression of benchmark return and asset return. One of the issues with calculating betas is the variability of the beta itself. In order to overcome that, especially given the volatility of crypto currencies, we recommend using a rolling beta.

Due to the varying levels of volatility and liquidity of various crypto currencies, a regression model may not always be a good fit. In addition to tracking fit through R-squared, it is important to track confidence level for the computed betas.

*Figure 1 History of Beta to S&P Bitcoin Index with Confidence Intervals*

The chart above shows rolling betas and confidence intervals for four crypto currencies between January 2019 and July 2021. Beta and confidence interval both vary over time and periods of high volatility (stress) cause a larger dislocation in the value of beta.

Rolling betas can be used to generate a hierarchical distribution of expected asset values.

## Portfolio Asset Covariance

Beta is a useful measure to track an asset’s volatility relative to a single benchmark. In order to numerically analyze the risk exposure (variance) of a portfolio with multiple crypto assets, we need to compute a covariance matrix. Portfolio risk is a function not only of each asset’s volatility but also of the cross-correlation among them.

*Figure 2 Correlations for 11 currencies (calculated using observations from 2021)*

The table above shows a correlation matrix across 11 crypto assets, including Bitcoin.

Like betas, correlations among assets change over time. But correlation matrices are more unwieldy to track over time than betas are. For this reason, hierarchical models provide a good, practical framework for time-varying covariance matrices.

## Value at Risk (VaR)

The VaR for a position or portfolio can be defined as some threshold Τ (in dollars) where the existing position, when faced with market conditions resembling some given historical period, will have P/L greater than Τ with probability *k*. Typically, *k* is chosen to be 99% or 95%.

**To compute this threshold **Τ**, we need to:**

- Set a significance percentile
*k*, a market observation period, and holding period*n*. - Generate a set of future market conditions (scenarios) from today to period
*n*. - Compute a P/L on the position for
*each*scenario

After computing each position’s P/L, we sum the P/L for each scenario and then rank the scenarios’ P/Ls to find the the* k *th percentile (worst) loss. This loss defines our VaR Τ at the the *k *th percentile for observation-period length *n*.

Determining what significance percentile * k* and observation length *n* to use is straightforward and often dictated by regulatory rules. For example, 99^{th} percentile 10-day VaR is used for risk-based capital under the Market Risk Rule. Generating the scenarios and computing P/L under these scenarios is open to interpretation. We cover each of these, along with the advantages and drawbacks of each, in the next two sections.

**To compute VaR, we first need to generate projective scenarios of market conditions. Broadly speaking, there are two ways to derive this set of scenarios:**

- Project future market conditions using historical (actual) changes in market conditions
- Project future market conditions using a Monte Carlo simulation framework

In this paper, we consider a historical simulation approach.

RiskSpan projects future market conditions using actual (observed) *n*-period changes in market conditions over the lookback period. For example, if we are computing 1-day VaR for regulatory capital usage under the Market Risk Rule, RiskSpan takes actual daily changes in risk factors. This approach allows our VaR scenarios to account for natural changes in correlation under extreme market moves. RiskSpan finds this to be a more natural way of capturing changing correlations without the arbitrary overlay of how to change correlations in extreme market moves. This, in turn, will more accurately capture VaR. Please note that newer crypto currencies may not have enough data to generate a meaningful set of historical scenarios. In these cases, using a benchmark adjusted by a short-term beta may be used as an alternative.

One key consideration for the historical simulation approach is the selection of the observation window or lookback period. Most regulatory guidelines require at least a one-year window. However, practitioners also recommend a shorter lookback period for highly volatile assets. In the chart below we illustrate how VaR for our portfolio of crypto currencies changes for a range of lookback periods and confidence intervals. Please note that VaR is expressed as a percentage of portfolio market value.

Use of an exponentially weighted moving average methodology can be used to overcome the challenges associated with using a shorter lookback period. This approach emphasizes recent observations by using exponentially weighted moving averages of squared deviations. In contrast to equally weighted approaches, these approaches attach different weights to the past observations contained in the observation period. Because the weights decline exponentially, the most recent observations receive much more weight than earlier observations.

*Figure 3 Daily VaR as % of Market Value calculated using various historical observation periods*

VaR as a single number does not represent the distribution of P/L outcomes. In addition to computing VaR under various confidence intervals, we also compute expected shortfall, worst loss, and standard deviation of simulated P/L vectors. Worst loss and standard deviation are self-explanatory while the calculation of expected shortfall is described below.

Expected shortfall is the average of all the P/L figures to the left of the VaR figure. If we have 1,000 simulated P/L vectors, and the VaR is the 950th worst case observation, the expected shortfall is the average of P/Ls from 951 to 1000.

The table below presents VaR-related metrics as a percentage of portfolio market value under various lookback periods.

*Figure 4 VaR for a portfolio of crypto assets computed for various lookback periods and confidence intervals*

## Bitcoin Futures: Basis and Proxies

One of the most popular trades for commodity futures is the basis trade. This is when traders build a strategy around the difference between the spot price and futures contract price of a commodity. This exists in corn, soybean, oil and of course Bitcoin. For the purpose of calculating VaR, specific contracts may not provide enough history and risk systems use continuous contracts. Continuous contracts introduce additional basis as seen in the chart below. Risk managers need to work with the front office to align risk factor selection with trading strategies, without compromising independence of the risk process.

*Figure 5 BTC/Futures basis difference between generic and active contracts*

## Intraday Value

The highly volatile nature of crypto currencies requires another consideration for VaR calculations. A typical risk process is run at the end of the day and VaR is calculated for a one-day or longer forecasting period. But Crypto currencies, especially Bitcoin, can also show significant *intraday* price movements.

We obtained intraday prices for Bitcoin (BTC) from Gemini, which is ranked third by volume. This data was normalized to create time series to generate historical simulations. The chart below shows VaR as a percentage of market value for Bitcoin (BTC) for one-minute, one-hour and one-day forecasting periods. Our analysis shows that a Bitcoin position can lose as much as 3.5% of its value in one hour (99^{th} percentile VaR).

## Risk-Based Limits

Right from the inception of Value at Risk as a concept it has been used by companies to manage limits for a trading unit. VaR serves as a single risk-based limit metric with several advantages and a few challenges:

Pros of using VaR for risk-based limit:

- VaR can be applied across all levels of portfolio aggregation.
- Aggregations can be applied across varying exposures and strategies.
- Today’s cloud scale makes it easy to calculate VaR using granular risk factor data.

VaR can be subject to model risk and manipulation. Transparency and use of market risk factors can avoid this pitfall.

Ability to calculate intra-day VaR is key for a risk-based limit implementation for crypto assets. Risk managers should consider at least an hourly VaR limit in addition to the traditional daily limits.

## VaR Validation (Bayesian Approach)

Standard approaches for back-testing VaR are applicable to portfolios of crypto assets as well.

Given the volatile nature of this asset class, we also explored an approach to validating the confidence interval and percentiles implied from historical simulations. Although this is a topic that deserves its own document, we present a high-level explanation and results of our analysis.

Building an approach first proposed in the Pyfolio library, we generated a posterior distribution using the Pymc3 package from our historically observed VaR simulations.

Sampling routines from Pymc3 were used to generate 10,000 simulations of the 3-year lookback case. A distribution of percentiles (VaR) was then computed across these simulations.

The distribution shows that the mean 95^{th} percentile VaR would be 7.3% vs 8.9% calculated using the historical simulation approach. However, the tail of the distribution indicates a VaR closer to the historical simulation approach. One could conclude that the test indicates that the original calculation still represents the extreme case, which is the motivation behind VaR.

*Figure 6 Distribution of percentiles generated from posterior simulations*

## Scenario Analysis

In addition to standard shock scenarios, we recommend using the volatility of Bitcoin to construct a simulation of outcomes. The chart below shows the change in Bitcoin (BTC) volatility for select events in the last two years. Outside of standard macro events, crypto assets respond to cyber security events and media effects, including social media.

*Figure 7 Weekly observed volatility for Bitcoin*

## Conclusion

Given the volatility of crypto assets, we recommend, to the extent possible, a probability distribution approach. At the very least, risk managers should monitor changes in relationship (beta) of assets.

For most financial institutions, crypto assets are part of portfolios that include other traditional asset classes. A standard approach must be used across all asset classes, which may make it challenging to apply shorter lookback windows for computing VaR. Use of the exponentially weighted moving approach (described above) may be considered.

Intraday VaR for this asset class can be significant and risk managers should set appropriate limits to manage downward risk.

Idiosyncratic risks associated with this asset class have created a need for monitoring scenarios not necessarily applicable to other asset classes. For this reason, more scenarios pertaining to cyber risk are beginning to be applied across other asset classes.