All about that VaR.

In this paper, we are going to derive how the VaR came to be, not just a statement of what VaR is but how can anyone use logic and intuition to get to a mathematically correct way of calculating the losses in a portfolio and perhaps a little less known fact about it.

To begin, let’s imagine this setup, a trader thinks to himself:

“I need to know how much I could lose, theoretically , given my current portfolio positioning”

At first this seems simple, the mental map is, risk is equivalent to dispersion or deviation from ‘expected behavior’. Let’s calculate the standard deviation and call it a day? But, the standard deviation only cares about variability, the trader is not worried about the upside deviation (profits), rather the downside risk and both do not carry the same weight (behaviorally and mathematically) to the trader.

What If we add probabilistic weights to each observation and effectively transform the return/loss observations into a return/loss distribution. In sophisticated terms, what if we model losses/returns probabilistically, treating them as realizations of a random variable with an associated distribution.

Now that we have the idea of a loss distribution, we think,

I need to find the the smallest number ℓ such that losses are less than or equal to ℓ with probability at least P.

In simpler terms, “What is the maximum I can lose at a time t, with a probability of P?”, all these are nothing but a way to describe a ‘quantile’, which in our case turns out to be the Value at Risk. Value at risk is the a quantile of the loss distribution at a specific level of confidence.

This in fancy math is,

(Inverse CDF)

Now that we have intuitively grasped the concept of VaR, let’s head towards deriving the formula for ‘VaR’.

Let’s assume the following things,

1. Return is a random variable represented by X, which makes loss L = -X.
2. Let f(l) be the PDF of the loss distribution (in full generality).
3. Let F(l) be the CDF of the loss distribution.
4. Let ‘alpha’ be the confidence level.

We want to find the exact loss amount, VaR(alpha), where the probability of losing less than or equal to that amount is exactly our confidence level ‘alpha’.

The probability of a loss being less than or equal to VaR(alpha) is the area under the PDF curve (for loss, which is the flipped return distribution, easier for Reimann integrals as we integrate left to right) from -inf to VaR(alpha), which is,

Now by definition, this is simply the CDF of the loss distribution. So we can write that integral as,

To solve for the VaR, we take the inverse here,

And thus we arrive at the same intuitive formulation mathematically.

In a special case, we assume that the return distribution is normal with (mu,sigma²) rather the return random variable is normally distributed and hence the loss is normally distributed with a negative mean compared to the return distribution,

Loss variable

If we further standardize the loss distribution (for ease, not wanting to solve complex integrals, rather use pre-computed area under curve or the inverse thereof, that’s right the ‘Z table’) to a standard normal distribution (and our concerned loss variable to make it fair) we get Z,

Now that the CDF is of a standardized normal distribution and can be compared to Z (the standardization helps in a way that we already know the inverse CDF values from the Z table and don't have to solve for anything),

For an arbitrary loss L(alpha)

More specifically, the L(alpha) we are looking for is the VaR(alpha),

The inverse of a standard normal distribution for alpha resolves to simply the Z score (alpha), and solving for VaR(alpha) we get,

The Delta Normal VaR

This brings us to a very important mathematical result, the formulation for VaR assuming the loss generating distribution in normally distributed.

Practitioners at major AM firms, IBs tend to have many complex models for calculating the VaR, which could range from simple Historical simulation VaR to Monte-Carlo VaR to weighted VaR schemes (we shall touch upon these in further papers and their limitations), for now we build intuition. But, there’s a big flaw to VaR, it is not axiomatically sound, it’s not a coherent measure of Risk. Moreover, even if its non-parametric or parametric, there is a ‘tail blind spot’. If the VaR is 1000$ for a given alpha, if the loss breaches the VaR threshold, is it going to be 1001$ or 10,000$? nobody knows. This naturally motivates an alternative risk measure called the ‘Conditional Value at Risk, CVaR or as I like to call it, the ‘Cooler VaR’. We shall tackle these in subsequent papers.

As always, don’t hesitate to reach out to me on LinkedIn, adios!