 # Risk Management

## Risk Management

Risk Management is, at last, getting our industry’s full attention. I hope that we can move forward and engage in constructive dialogue with our regulators and remain compliant. Risk Management encompasses several areas of the derivatives value chain. We’ll cover each area in turn: Price Risk, Operational Risk, etc.

### Value-At-Risk

Market Risk is calculated using Value-At-Risk or VAR. VAR calculates a probability statement such as: “I’m x% confident we won’t lose more than V dollars over the next N days”. It allows institutions to have a single number to define their market risk given the parameters used. Regulators require banks to report a 99% confidence level over the next 10 days. BIS requires the banks to have three times their 10 day VaR in capital.

• “The Bank is 99% confident we won’t lose more than \$100 million over the next 10 days.”
• The Bank is 1% confident the Bank will lose more than \$100 million over the next 10 days.

Bank Trading desks use the 99% & 95% VAR number for 1 day (tomorrow).

• 99% number because it’s consistent with bank regulators, but they care about market moves for one day.
• Bank trading desks are 1% confident they will lose more than VaR 1 day out of 100. (1/100 = 1%)
• 95% confidence figure because that’s when the hedge funds will look to scale back positions.

Hedge Funds use a one day 95% confidence figure.

• Hedge funds typically use a 95%, 1-day VaR, which they post on their websites.
• In other words, they expect to lose more than their VaR figure once a month (1/20 = 5%)

So bank traders want to know how the other side of their trade might respond if the market moves big the next day. Although most money center banks now use Monte Carlo simulation, not a normal distribution. But using volatility (the square root of variance which presumes the % returns are normally distributed) is easier to conceptualize.

### How is Value at Risk Calculated?

There are three accepted methods of calculating VaR: For the sake of brevity, we will only discuss variance-co-variance in detail.

1. Monte Carlo Simulation
2. Historical Simulation
3. Variance Co-Variance Method

### Historical Simulation

The historical simulation uses the ACTUAL returns of the assets in which the bank has open positions. They typically use the previous 200–500 days of actual price changes. EXAMPLE: using the last 200 days and calculating the daily returns they give you the actual 180th worst loss (given the Bank’s current position). Whatever loss is the 180th worst loss, while be reports as their VaR. The 180th worst loss out of 200 changes gives you the 90% confidence, one day VaR (180/200 = 90%)

### Monte Carlo Simulation

The easiest way to view Monte Carlo Simulation begins with thinking of it as similar to Historical Simulation. Except, Historical Simulation implies the actual 180th worst loss in the past 200 days can re-occur on the exact same date over the next time frame (10 days or 1 day). Monte Carlo uses a random number generator and uses 10,000 “trials” (you can view this as daily changes) to estimate the price return of the asset at the end of the analysis horizon (10 or 1 day).

### The Variance Co-Variance Method

The variance-co-variance method uses the historical volatility of the asset over the past X number of days (typically 200 – 500 days).

#### Historical Volatility & Value At Risk

Perhaps you recall the bell-shaped curve. If not, let’s run through it quickly now: Normal distribution presumes the mean of the data series is zero. The resulting volatility number you hear in the market (i.e. , VIX) is the annual volatility. We can use the annual volatility to make probability statements which are described as standard deviations from the mean:

1. STDEV = 68.3% of all occurrences, which is roughly 2 out of every 3 years
• In forecasting probable price changes in the future we say:
• The asset will be up or down 1 stdev or less 2 out of every 3 years.
• The asset will be up or down MORE THAN 1 stdev 1 out of every 3.
2. STDEV = 95.4% is roughly 19 years out of every 20
• In forecasting probable price changes in the future we say:
• The asset will be up or down 2 stdev. or less 19 out of every 20 years.
• The asset will be up or down MORE THAN 2 stdev 1 out of every 20 yrs.
3. STDEV = 99.7% which is roughly 299 years out of every 300
• In forecasting probable price changes in the future we say:
• The asset will be up or down 3 stdev or less 299 of every 300 yrs.
• The asset will be up or down MORE THAN 3 stdev 1 out of every 300 yrs.

If you look at the N= file attached here, you’ll see a 99% confidence interval on a normally distributed as is ~ 2.33 standard deviations.

#### Adjusting Annual Volatility to a Shorter Time Frame

We can adjust the annual volatility to a shorter time frame by adjusting the annual figure by the square root of time. Let’s calculate the daily VaR, using the following example:

 EXAMPLE: long a single asset valued at 100 million. <200 day historical volatility is 25%. For 10-day volatility we need to adjust our annual number by the SQRT of 365/10 as there are 365 CALENDAR days in a year. PLEASE NOTE: WE ARE NOW USING CALENDAR DAYS. 10-day Volatility = SQRT () = 6.0415 10 day Vol = .25 / 6.0415=4.303% PER 10 calendar day period. We now need to adjust our daily volatility figure to reflect 2.33 stdev’s 4.303% * 2.33 = 9.97% \$100 million * .0997 = 10- day 99% var of \$9,997,000 “I’m 99% confident we won’t lose more than \$9.997 million on this position over the next 10 calendar days. ”

#### Problems with VaR & Possible Solutions

It may be obvious that VaR focuses on the likely losses, but says little about the black swans. For a while, we performed backtesting and stress testing to ensure our VaR variables were correct. But notice, even if the volatility estimate was correct, the furthest I could go is “I’m 1% confident I’ll lose more than VaR tomorrow”. It could not tell me the size of that tail risk. For this reason, VaR has been challenged and looking to be replaced by Potential Exposure. This and other Trading Desk related risk measurements will come under the umbrella of Fundamental Review of the Trading Book (FRTB).