Risk Aversion Adjusted Volatility

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In finance and risk management, Risk Aversion Adjusted Volatility (also known as RAAV, or HSBC RAAV), measures the potential net volatility of a risk profile. Unlike Value at Risk (VaR), which depends only on one particular outcome (typically the 5th percentile of historical returns or simulated returns), RAAV takes into account the entire risk distribution and can more accurately account for non-linear payoffs.

It was first proposed by Adco Leung from HSBC in 2018.

Definition

Mathematical formula

RAAV is defined as:

$RAAV=Upside\,Volatility-Downside\,Volatility\times Weight$

where $Weight$ is the "loss aversion" coefficient.

Choice of weight

Leung proposes to use $Weight=2$ as the default, given behavioural science research has found that people generally find losses to be twice as painful as the pleasure of having an equal-size gain. See loss aversion.

If $Weight=1$, it suggests a risk neutral preference whereby the investor or risk manager would be indifferent so long as one unit increase in downside volatility can be compensated by an equal unit increase in upside volatility (as is usually the case with increasing the delta of an investment with linear payoff).

If $Weight>1$, it suggests risk aversion, or loss aversion, is at play as investor or risk manager would demand more than 1 unit of upside volatility to compensate for 1 unit increase in downside volatility.

Calculation of upside and downside volatilities

Mathematical formulae

Upside (downside) volatilities can be intuitively viewed as the standard deviation of returns above (below) the target return. Mathematically,

$Upside\,Volatility={\sqrt {\frac {\sum (r-T)_{+}^{2}}{n}}}$,

$Downside\,Volatility={\sqrt {\frac {\sum (r-T)_{-}^{2}}{n}}}$

Here,

• $r$ denotes the net return of an investment or any position, $T$ denotes the target return of an investment (which may be the risk free rate return, or simply zero in the context of hedging), and $n$ denotes the total number of observations ($n-1$ may also be used instead for sample standard deviations).
• $\sum (r-T)_{+}^{2}$ denotes the summation of squared excess returns where the observations of $(r-T)>0$.
• $\sum (r-T)_{-}^{2}$ denotes the summation of squared excess returns where the observations of $(r-T)<0$.

As decomposition of total volatility

A special case occurs when $T=0$,

$RMS\,Return={\sqrt {\frac {\sum r^{2}}{n}}}={\sqrt {{\frac {\sum r_{+}^{2}}{n}}+{\frac {\sum r_{-}^{2}}{n}}}}={\sqrt {(Upside\,Volatility)^{2}+(Downside\,Volatility)^{2}}}$

In other words, upside volatility and downside volatility are two orthogonal principal components of the root-mean-square return when $T=0$.

Two investments with the same overall volatility (8.06% p.a.) yet very different upside and downside component volatilities.

Examples

Consider two investments with the same overall volatility:

• An investment has a total volatility of 8.06% p.a., whose components are 7% p.a. of upside volatility and 4% p.a. of downside volatility ($0.0806^{2}=0.07^{2}+0.04^{2}$). This investment has an RAAV of -1% p.a. ($0.07-0.04\times 2=-0.01$).
• An investment has a total volatility of 8.06% p.a., whose components are 1% p.a. of upside volatility and 8% p.a. of downside volatility ($0.0806^{2}=0.01^{2}+0.08^{2}$). This investment has an RAAV of -15% p.a. ($0.01-0.08\times 2=-0.15$).

RAAV conveniently captures the weighted difference between upside and downside volatilities.

Treatment of cost of hedging or income

Any fixed payoff, whether positive or negative, would simultaneously impact both upside and downside volatilities. Consider a 10% p.a. cost of hedging, it would lead to a higher downside volatility due to a) more observations falling below the target return, and b) larger magnitudes of downside returns. Similarly, it would also lead to a lower upside volatility due to a) fewer observations that remain above the target return, and b) smaller magnitudes of upside returns.

Use in risk management

Leung argues that for hedging purposes, risk managers lack a flexible metric that can accurately reflect their risk profile which may be overlaid with options. For example, in order to hedge the FX risk of a particular cash flow exposure, risk managers may want more flexibility than to lock in the FX forward rate using a FX forward or currency swap, and would tend to deploy options such as call spreads, out of the money options and collar etc.

Value-at-Risk (VaR) may be a useful indicator for financial institutions whose portfolio exposure can be approximated by Gaussian distribution due to their small individual exposures and the central limit theorem.

However, for many corporates and individuals whose portfolios typically consist of one or a few dominant positions, VaR may not be a good representation of the actual risk profile and would fail in evaluating non-linear hedging strategies.

RAAV fills this gap by acknowledging the entire risk distribution as is and requires no assumption of the underlying distribution, i.e. the underlying distribution may or may not be Gaussian.

Use in hedging

In the hedging context, RAAV is a more practical metric to use than Sharpe or Sortino ratios which are more suited for investment purposes.

Consider two risk positions:

• A has an upside volatility of 5 and downside volatility of 0.1; and
• B has an upside volatility of 0.1 and downside volatility of 0.

Clearly, as a risk manager, position A would be more preferable given its much larger upside than B, and their downsides are similar in absolute magnitudes.

Using RAAV, A would have an RAAV of +4.8 and B would have an RAAV of +0.1, and A should be favoured.

Using Sortino ratio (which we roughly define as the ratio of upside to downside volatility), A would have a Sortino ratio of 5 and B would have a Sortino ratio of infinity, and B should be favoured based on this metric.

Intuition

RAAV captures the decision function that would be used by most human risk managers.

Risk managers do not generally set out to eliminate volatility entirely. In fact, eliminating volatility entirely can inflict large psychological toll in terms of regret risk. Consider the airline industry whose fuel hedging programmes may be justified by reducing fuel price volatility. However, when oil prices plummeted, many airline companies faced huge losses in their fuel swap contracts.

Ideally, risk managers would prefer to eliminate only the downside volatility and retain as much as possible any upside volatility.

However, in practice, linear instruments can only remove one part of downside volatility for a roughly equal part of upside volatility, assuming the cost of hedging is negligible.

Because of the loss aversion coefficient (>1) embedded in the RAAV formula, linear hedging instrument can still improve RAAV because the "downside volatility times coefficient" would reduce faster than "upside volatility times one."

Risk managers can apply backtesting or apply future-looking Monte Carlo simulations to evaluate RAAV for various hedging strategies, which may be linear or non-linear instruments, on a consistent basis.

External links

• Rethinking Volatility, 1 December 2018, Adco Leung, https://www.gbm.hsbc.com/insights/markets/rethinking-volatility
• Three Scientific Reasons Why Companies Hedge, 26 February 2018, Adco Leung, https://www.gbm.hsbc.com/insights/markets/rethinking-treasury-three-scientific-reasons

See also

• Decision making
• Risk metric
• Risk measure
• Volatility
• Value at risk
• Sharpe ratio
• Sortino ratio
• Upside potential ratio
• Behavioural economics
• Loss aversion
• Risk aversion
• Regret (decision theory)
• Hedge (finance)
• Backtesting

References

Category:Risk management

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