Parametric VaR - When Taylor Series meets Normal Distribution

Introduction

Just as all assets and liabilities are quantified in monetary terms in the balance sheet to make it amenable for basic arithmetic such as addition and subtraction, Value-at-Risk (VaR) quantifies diverse sources of risk in monetary terms making it one unified risk measure. When we say $1 million is the 10-day 95% Value-at-Risk(VaR), it essentially means “we are 95% confident that the loss in 10 days will NOT exceed $1 million”. There are three broad methods for calculating VaR i.e. Historical, Parametric and Monte Carlo. Parametric VaR enjoys an advantage in terms of analytical tractability. Of course, it suffers from significant model risk as one must fit a known distribution on returns and model correlations. However, it remains as a popular tool because of its simplicity.

While calculating parametric VaR for a single instrument with one risk factor (e.g. stock) is quite straightforward, things become real dirty real quick when the number of instruments and/or risk factors increase. When we throw non-linear instruments in the mix, it gives rise to sub-atomic particles such as delta, vega (for Options), duration(for Bonds) and we end up calculating delta-normal VaR etc. Often students are bombarded with these isolated analytical formulae for VaR for different portfolios and they end up memorizing them. In my experience, memorization does not scale with humans.

In this article, I will introduce a simple 3 step procedure that requires the knowledge of ‘Taylor Series’, ‘Expectation and Variance Algebra’ and ‘Normal Distribution’ to be able to systematically arrive at VaR of a portfolio.?

Expectation & Variance Algebra

When dealing with random variables X, Y , there are a couple of super useful equations one needs to know.

Expectation is a linear operator

?, ??(α,β) are scalars

We can say Expectation of a linear combination is the linear combination of expectations

Variance is NOT a linear operator

Taylor Series

Taylor series is fundamentally a very important tool in calculus and has profound use in Quant finance. It approximates change in a function from the change in input variables

Univariate Function

Taylor series approximation till second order

The above is also called delta-gamma approximation. ?The gamma term is non-linear with respect to X which compromises the analytical elegance of the parametric VaR infrastructure. We will therefore conveniently avoid the second order term and pretend life is good. So just the delta approximation?for now

Taylor Series for a multivariate function

Again keeping only the delta terms,

Notice that the ordinary derivatives have been replaced with partial derivatives (differentiation w.r.t one variable keeping others constant)

Normal Distribution

And finally, one needs to be able familiar with quantiles and CDF of a normal distribution. CDF at x, gives the probability to the left of x which is the same as the CDF of the z transformation of x. If I want to calculate x at which the CDF is α, I will proceed like this.

In risk management, we typically assume μ=0. It’s the volatility σ that gets the spot light. So that gets us

Now, in our case, x would be some risk factor (e.g. stock price, interest rate, commodity price, FX rate etc..). And our value of portfolio V = (sum of value of all instruments) eventually becomes a function of those risk factors. So, V is a multivariate function of risk factors. Our target for VaR is a worst possible loss i.e. a worst possible ?V. Obtaining a worst possible ?V is a piece of cake if we have the distribution of ?V

I put a negative sign to report a positive loss. But how to go from distribution of x (risk factor) to the distribution of ?V ? That is where ‘Taylor Series’ and ‘Variance Algebra’ come together. And that is it! Here, I outline the systematic process to do that.?

The 3-step procedure to calculate delta based VaR

If you look at following schematic, it sort of looks like a neural network with risk factors being the input layer, change in instrument values being the hidden layer and finally the change in value of portfolio as the output layer. Change in risk factors drive the change in value of instruments through sensitivities. And change in instrument values simply add up at the portfolio level. Remember, our goal is to establish a link between distribution of risk factors and the distribution of portfolio PnL (?V)

Step 1 –

Use Taylor Series to write the change in each instrument value as the delta approximation linked to the risk factor change.

Step 2 –

Portfolio PnL (?V) is the sum of each instrument’s PnL

Write the portfolio PnL equation

Under delta approximations, you will find that the above equation for Portfolio PnL is a linear combination normal randoms. And we know that a linear combination of normal randoms produces another normal random. Mean of every PnL is conveniently 0. So, it finally boils down to calculating the standard deviation of Portfolio PnL . And this is a simple application of the Variance Algebra that we previously looked at. And yes, here we will also need the correlations to compute the covariance terms.

Step 3 –

That is all.

Let us look at some examples.

Examples

( A single stock )

r = arithmetic return on the stock?

S acts as a scalar

( two stocks )

IF ?r1 , r2 and ?are two normal randoms, ?will also be a normal random as it is a linear combination of two random normals.?

(1 bond (B)?)

D is the modified duration and ?y is the change in interest rates?

Note that,

so the above is Taylor Series delta approximation?

notice we had volatility of stock price relative returns, here we have volatility of change in interest rates?

( A Call Option (C) with underlier stock (S) and a Foreign Bond (B))

If the spot FX rate = X, then value of our portfolio is?

There are 3 risk factors in total ( S, X and interest rate y )

Taylor Series first order approximation?

Note that X naturally pops up as a multiplier in the bond’s sensitivity w.r.t interest rate because reported PnL is in domestic currency. Also for FX risk, the sensitivity is the market value of the Bond in domestic currency.

If we want to model them jointly, we can. It’s simply a linear combination of 3 normal randoms i.e. return on stock , change in interest rate ?and return on FX rate ?, but we need cross correlations across these 3 asset classes. As per FRTB delta risk charge, we need to capitalize these 3 terms separately under Equity, GIRR (General Interest rate risk) and FX respectively.?

It was not that difficult after all. With this one can systematically derive analytical VaR for any portfolio. Can we derive the FRTB equations for delta risk charge based on the above premise. I have and I will share the derivation in my next article. Stay tuned.

Cheers!

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