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**FRM P1 > T1 > Intro to VaR (xls): Value at Risk**

Hello! We are uploading revised learning spreadsheets that I've been working to improve. I wanted to share some information about them. I am going to start at the beginning in this informal series.

In terms of a natural sequence, the

*first*learning workbook (

*R1-P1-T1-Intro-VaR-v2.xlsx*) is here in the Study Planner at https://www.bionicturtle.com/topic/learning-spreadsheet-intro-var/. Within this workbook, the

*first sheet*is

*and it introduces value at risk (VaR). Unlike most of our learning workbooks, this XLS is not linked directly to an assigned reading. That's because these value at risk (VaR) calculations are*

**T1.1-IntroVar****fundamental**and make several appearances in the FRM syllabus. First I want to remind of the context. There are

**three broad approaches**to estimating VaR: historical simulation, Monte Carlo simulation, and parametric (aka, analytical). This sheet is a parametric approach because it does not use raw data but instead assumes a normal probability distribution (although data would inform the inputs). So, I like to remind folks that we tend to assume the normal distribution because it is

*elegant*and

*convenient*but parametric VaR does not necessarily imply a normal distribution. In fact, it's an FRM theme that realistic returns are heavy-tailed; i.e., non-normal. Just

*today*this post is in my feed reader, "Risk management based on normality is a loser's game." At the same time, I don't mean to dismiss the justifiable use of the normal distribution when the utterly amazing central limit theorem (https://en.wikipedia.org/wiki/Central_limit_theorem) does apply. But the thing is, risk is preoccupied with loss tails, not so much the center of the distribution, alas.

The left panel (

**see below**) in this first sheet of the workbook (again that's sheet

*T1.1-IntroVar*within workbook

*R1-P1-T1-Intro-VaR-v2.xlsx*) illustrates the most basic VaR approach: we specify a

**confidence level**and a

**horizon**(e.g., one-day 95.0%) and, given the asset value and volatility, we retrieve the %aVaR with the following: %aVaR = -µ*(Δt/T) + σ*sqrt*(Δt/T)*z(α). The aVaR refers to absolute VaR which is the loss relative to the current position (relative to today), in contrast to relative VaR which would exclude the drift term and would be the loss relative to the future expected value. In market risk (aka, mVaR) when the horizon is often one day such that the drift effectively rounds to zero anyway, by convention we can exclude the drift (ie, expected return term). But it is good practice to start with absolute VaR in your thinking because basically it can't be wrong. I like to think of aVaR as

*return-adjusted risk*(compare to risk-adjusted return) because, while rVaR is just the worst-expected loss, aVaR mitigates the worst-expected loss with expected return. Further, aVaR is consistent (imo) with risk theory: in contrast to valuation (and risk-neutral pricing) which are preoccupied with

*precise present value estimation*, risk is concerned with the the approximate

*future value*in the loss tail and expected return is part of the future state; e.g., consider the drift in the Merton model.

The right panel only performs a relative VaR but on two assets (I wanted to keep the introduction as simple as possible). So we also need a correlation parameter. The VaR here relies on the traditional mean-variance framework (aka, MPT) taught in all basic finance courses, so you do want to understand this specifically. At the same time, you are probably aware that it requires assumptions which render it controversial.

Another note about terminology. The sheet refers to

**delta normal**VaR because Jorion uses this term (so historically it's been more familiar to FRM candidates). Delta normal is a bit of overkill here, because we only have a single risk factor (asset price). If we were instead computing the VaR of a call option on this asset, then the rVaR = S*σ*sqrt(Δt/T)*Δ*z(α), where Δ is the option's Greek delta and this would be "delta normal" because we are assuming the risk factor (asset price) has a

**normal**distribution and the sensitivity to the risk factor is a first partial derivative with respect to the risk factor (ie, delta). In the case of a bond, when we use duration to estimate risk, that's also a "delta normal" approach because duration is also (a function of) a first derivative; it could fairly be called duration-normal but nobody does that (in the bond case, the risk factor is the yield and we do tend to assume very short term yields are normally distributed even as their long-term evolution is decidedly non-normal). So that's a long way to explain why we use delta normal:

*delta*refers to first-derivative nature of the sensitivity to the risk factor, and

*normal*refers to the assumption about the risk factor. My XLS here does not have a delta. This approach is called

*normal linear VaR*by Carol Alexander, which makes a lot of sense in the multi-asset situation because it treats VaR as a

*linear function*of risk factor shocks. In that way, her

*linear normal*VaR is a special case of the set of

*parametric linear*VaR models. I hope that's helpful!

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