What is VaR (Value-at-Risk)?

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Introduction

Among all the market risk measures, Value-at-Risk (“VaR”) is the champion in terms of popularity and usage. Pioneered by J.P. Morgan in 1995 as part of their “RiskMetrics” software, it had swiftly become the risk measure of choice for both banks and regulators alike. Even today, VaR continues to be a key market risk measure reported by financial institutions.

What this means for you as an interviewee is this – you will be asked about it and so, read on and get prepared.

In today’s post, I will broadly cover the following about VaR:

• What is it and why is it important?
• How is it computed?
• What are its shortcomings?

Definition

Let’s first focus on the key idea and intuition.

VaR, in essence, represents how much loss you could incur on your investment portfolio.

However, this is not precise enough. What’s the scenario or context? For example, is it potential loss due to a certain event happening? Or is it potential loss over a specific time period? Further, how do we quantify and agree on what we mean by “potential” loss? So clearly, there is a need to be more specific on definition.

The 3 key components

So here’s the more precise definition of VaR: a measure of potential loss over a given time horizon and confidence level.

Let’s examine the key terms and (units):

• Potential loss (dollars) : Amount of money your portfolio could lose. Note that this is uncertain.
  
  
• Time horizon (days): Time period over which this potential loss could occur. For example, 1 day or 10 days.
  
  
• Confidence level (%): How conservative this loss estimate is; a higher number translates to a larger loss i.e. more conservative. A common number for this is 99%, which means if I have 100 predictions of this loss figure, I’ll pick the 99th highest one.

Let’s put in some concrete numbers that occur in real-life.

For example, banks often compute the 1-day 99th percentile VaR. For illustration, suppose this figure turns out to be 5 million dollars. Then, the three parameters are: potential loss of 5mio, time horizon of 1 day, and confidence level of 99%.

The easiest way to think about this is as follows: our portfolio 1-day loss should NOT exceed 5mio for 99% of the time. That is, with roughly 250 trading days in a year i.e. 250 observations of portfolio PnL, we would expect our portfolio to lose > 5mio for ~3 days ( 1% of 250 = 2.5 days).

Why is VaR important?

VaR is important because it’s an industry-standard market risk measure. This facilitates comparison across different industry players and in turn, enables monitoring and regulation.

It also gives one an intuitive way to describe the risk of a portfolio in dollar terms via the 3 key components. It recognizes that future loss on a portfolio is uncertain and tries to describe such loss with just enough precision for it to be useful yet, still understandable.

From your point of view as someone who wants to land a market risk role, VaR will inevitably be asked in the interview. Thus, it’s imperative that you prepare to answer this.

How is it computed?

There are two main types of VaR that differ in terms of how you compute it: Historical VaR and Parametric VaR. Most financial institutions use Historical rather than Parametric VaR; the latter is typically studied in schools / FRM by GARP and remains there mostly.

Let’s have a look at them one-by-one.

1) Historical VaR

Here, history is directly used as the main source of information to predict the future – hence, the term “historical”.

The main idea is as follows: value the portfolio across actual historical market data and take a percentile of the PnL figures obtained to obtain VaR.

Many financial institutions use this since it’s straightforward to compute. No assumption is required for the distribution of market risk factors and portfolio PnL – market moves and resultant PnL are taken as-is.

However, while simple, the key drawback of Historical VaR is the absolute reliance on historical data to predict future PnL. This assumes the past fully predicts the future, which is certainly unrealistic.

2) Parametric VaR

Here, as the name suggests, is one where once you have the required parameters, you can compute it.

The main idea is as follows: fit a statistical distribution to historical market data and take a percentile of the distribution to obtain VaR.

This is similar to Historical VaR in that it also makes use of historical market data, albeit indirectly. Nonetheless, the key difference between the two is: Parametric assumes a statistical distribution for portfolio PnL while Historical simply takes it as observed in the market (sometimes called the “empirical distribution”).

The key drawback of Parametric VaR similar to that of Historical VaR where there exists an unrealistic assumption. In this case, it’s that portfolio PnL follows the assumed statistical distribution.

Historical vs Parametric – which should I use?

As with many things in life, the answer is: it depends.

To see this, Table 1 below shows a side-by-side comparison of Historical vs Parametric VaR. Depending on your objectives, constraints, and beliefs about the veracity of the assumptions, one of the two would be a better choice.

For instance, if you don’t have strong views about the statistical distribution of the portfolio PnL or underlying market risk factors, Historical VaR may be the preferred choice as no distribution has to be assumed.

What are the shortcomings of VaR?

Despite the popularity of VaR as a risk measure in the industry, it too has deficiencies. In particular, there are two main ones.

1) No information about losses beyond it

The first deficiency is that VaR is really just a single point on the loss distribution. Hence, it does not give any further information about the extent of losses that could occur beyond it.

For example, if the 99th percentile 1-day VaR is \$100, we only know that the 1-day loss will be less than \$100 and with a probability of 99%. However, what happens at the 99.9% confidence level? Could the loss be way higher than \$100?

Hypothetically, if the 99.9th percentile loss increases sharply to \$1,000, this suggests a fat-tailed loss distribution.

2) It is not subadditive

Suppose we have a portfolio comprising two stocks, Apple and Google.

Now, if VaR is subadditive, it means that the VaR of the portfolio is less than or equal to the sum across each stock’s VaR. Mathematically, it is:

V(Portfolio) $\leq$ V(Apple) + V(Google)

However, it has been proven that VaR is NOT subadditive in general (see here for more information on the same) so the above property doesn’t hold most of the time. That is, the risk of a portfolio is possibly larger than the sum of risks of across each asset. This violates the common wisdom of diversification in finance where adding assets to a portfolio should make it less risky.

Conclusion

Although a related risk measure – Expected Shortfall (ES) – is catching on, VaR remains an ever popular risk measure that’s used extensively across financial institutions.

Therefore, if you can demonstrate some fundamental knowledge about it, in addition to giving the interviewer a great first impression, you will definitely be one step closer to landing your market risk role!

Ray Yeo

Risk Manager by Profession, Mentor and Coach by Passion.