YouTubeT5-01: Lognormal Value at Risk

Nicole Seaman

Director of FRM Operations
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Welcome to the first video in this new playlist that is devoted to Topic 5 in the FRM. Topic 5, Market Risk, is the first topic in Part 2. We will start here by comparing normal to lognormal VaR and, specifically, we are going to generalize to absolute VaR. Absolute VaR generalizes the relative VaR so it's the complete version of VaR. The key thing that we are going to do here is look at four different use cases so we can compare normal VaR to lognormal VaR in the single-period case. Normal is when we assume that the arithmetic returns are normally distributed and lognormal is when we assume that the geometric returns are normally distributed.

nictziak1

New Member
Hi @David Harper CFA FRM
Very useful video & explanations - as always.
My question here is whether the lognormal VaR can also be applied to bond VaR calculation and provide a similar 'solution' to the problem of time-scaling. Would the relative 10-day VaR formula (so μ=0) then look like:

%VaR = 1-exp(-(σ(YTM)*δ*a)*sqrt(10)+(0.5*c*(a*σ(YTM))^2)*sqrt(10))

where
δ=modified duration
c= convexity
σ(YTM)= the sample standard deviation of the calculated ln(YTM t/ YTM t-1)

Much appreciated

frogs

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Hi, @David Harper CFA FRM can you tell me around 17:15 in the video why you get the substitution for R' to be Mu - sigma(z)
It's very similar to normal but I can't see where you derived it.
Please clarify. Thanks!

EDIT: I kind of see how it is done. So a yes or no answer would suffice. If you want any kind of return (ie arithmetic, geometric) to be normally distributed you need Mu - sigma(z) factor?

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David Harper CFA FRM

David Harper CFA FRM
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Hi @frogs Yes that is correct! This is all based in Dowd Chapter In both the normal versus lognormal VaR there is the same assumption that returns are normally distributed! The difference is due to whether arithmetic returns are normally distributed (what we call "normal VaR" or just "VaR") or whether geometric returns are normally distributed (what we call "lognormal VaR"). Both versions of VaR assume returns are normal, the differ in the definition of the return. Consequently, both versions define the critical return, r* = μ - σ*z, which is only breached with probability typically (significance = 1 - confidence) of 5.0% or 1.0%. In the above example where μ = +1.0% and σ = 5.0%, the critical value (at 95% confidence) is 1.0% - 5.0%*1.645 = -7.22%; if the return has a normal distribution, then only 5.0% of the time do we expect it to be worse than -7.22%. But the same critical return informs both the normal and lognormal VaR. Thanks,

mreshko

New Member
Great video! Thank you. I have a question: could you please explain how in Normal VaR calculations you obtained relative % drift and relative % standard deviation for arithmetic returns. Doesn't the assumption of arithmetic returns implies that the drift and standard deviation are expressed in absolute (money) values rather than %? How do you calculate the drift and standard deviation for absolute returns such that the result is a % drift and % standard deviation? These two seem to be contradictory to me. Thank you

David Harper CFA FRM

David Harper CFA FRM
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Hi @mreshko Thank you. No, the assumption of arithmetic returns does not imply that drift and standard deviation are expressed in dollar (money) units. Your question contains a few different terms, I will use Kevin Dowd's (aka, GARP's since he is their longstanding author for this) terms:
• The underlying data can be either Loss/Profit (aka, L/P) or P/L which is dollars or prices; i.e., $P0,$P1, $P2. The portfolio's value or wealth is expressed in dollars • An arithmetic return, r =$P1/$P0 - 1, is also called a simple return, while a geometric (aka, continuously compounded) return, r = ln($P1/$P0). These returns are naturally %. • Normal VaR assumes the arithmetic returns are normally distributed. Lognormal VaR assumes the geometric returns are normally distributed, and because they are conveniently lognormal, LN(.), by implication this assumes the prices are lognormal (and cannot go negative) • Absolute VaR includes the drift in the normal/lognormal function because it's the worst expected loss relative to today; i.e., "absolute" in the VaR context does not refer to the non-negative mathematical function, |x|. Relative VaR excludes the drift because it's the worst expected loss relative to the future expected value. I've found this is all teased out best with numbers, in a spreadsheet/code. Also, I'd understand your question with numbers! Hope that's helpful. mreshko New Member Hi @mreshko Thank you. No, the assumption of arithmetic returns does not imply that drift and standard deviation are expressed in dollar (money) units. Your question contains a few different terms, I will use Kevin Dowd's (aka, GARP's since he is their longstanding author for this) terms: • The underlying data can be either Loss/Profit (aka, L/P) or P/L which is dollars or prices; i.e.,$P0, $P1,$P2. The portfolio's value or wealth is expressed in dollars
• An arithmetic return, r = $P1/$P0 - 1, is also called a simple return, while a geometric (aka, continuously compounded) return, r = ln($P1/$P0). These returns are naturally %.
• Normal VaR assumes the arithmetic returns are normally distributed. Lognormal VaR assumes the geometric returns are normally distributed, and because they are conveniently lognormal, LN(.), by implication this assumes the prices are lognormal (and cannot go negative)
• Absolute VaR includes the drift in the normal/lognormal function because it's the worst expected loss relative to today; i.e., "absolute" in the VaR context does not refer to the non-negative mathematical function, |x|. Relative VaR excludes the drift because it's the worst expected loss relative to the future expected value.
I've found this is all teased out best with numbers, in a spreadsheet/code. Also, I'd understand your question with numbers! Hope that's helpful.
Thank you for your reply, but I believe there is a mistake in your second point: (a) arithmetic return is just a simple arithmetic different between values, not a ratio (hence the name); and (b) log-return is just a logarithm of a ratio of values, without 1 being deducted. From these it can be clearly seen that the arithmetic return is in units of currency.

The two stochastic models for the two types of returns are Weiner (Brownian) motion for arithmetic returns and geometric Brownian motion for log-returns. I have never seen a stochastic model for the types of returns that you are referring to.

A relative percentage returns (which is what I would call the expression you wrote for arithmetic return) can be sometimes encountered in the analysis of portfolio performance, but I have never seen it being used in market risk modelling where VaR is used to assess risk.

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David Harper CFA FRM

David Harper CFA FRM
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@mreshko In haste to reply to you (and get to my lunch), I simply misplaced the "-1", obviously. Fixed above. Simple return versus geometric (aka, continuously compounded) return, is the difference. But thank you for the dramatic "never seen"s, lol Next time I'll just refer you to the video.

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mreshko

New Member
@mreshko In haste to reply to you, I simply misplaced the "-1", obviously. Fixed above. Simple return versus geometric (aka, continuously compounded) return, is the difference. But thank you for the dramatic "never seen"s, lol Next time I'll just refer you to the video.
Yes, indeed it was a bit "dramatic", but it is nevertheless true - I have worked as a risk qaunt for over 10 years now and haven't encountered a stochastic process were the ratio of asset values is used to define a model for asset returns. If you know of one then I would be very interested in seeing it. Thank you

David Harper CFA FRM

David Harper CFA FRM
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Nah it was just an typo, like i said, an obvious typo. Back in the day, we used to call P1/P0 the "wealth ratio," then take LN(.) of the vector of "wealth ratios," when computing volatility; and another column that (. - 1) to the wealth ratio vector to give the simple returns. Or, really, wealth ratio = (P1 + D1)/P0. So the same vector of wealth ratios inputs into both return vectors. I was just trying to define the terms, including tease out the context-specific meaning of absolute VaR and lognormal VaR.

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mreshko

New Member
Nah it was just an typo, like i said, an obvious typo. Back in the day, we used to call P1/P0 the "wealth ratio," then take LN(.) of the vector of "wealth ratios," when computing volatility; and another column that (. - 1) to the wealth ratio vector to give the simple returns. Or, really, wealth ratio = (P1 + D1)/P0. So the same vector of wealth ratios inputs into both return vectors. I was just trying to define the terms, including tease out the context-specific meaning of absolute VaR and lognormal VaR.
I see. Thank you for this background information. Let me try to define precisely(-ish) what I mean. In my experience, the two basic assumptions for the distribution of asset prices are normal and lognormal, where the former means that the asset price flows a normal (aka Gaussian) distribution A(t) ~ N(m(t), s(t)), and the latter means that asset price flows a lognormal distribution A(t) ~ LN(m(t), s(t)). The normal assumption implies that a difference between asset prices on two dates (what I call arithmetic return) also follows a normal distribution, while lognormal assumptions implies that the logarithm of asset prices follows normal distribution and the logarithm of the ratio of asset prices (what I call lognormal returns) also follows a normal distribution. If I understood you correctly, you are suggesting a model where a ratio of asset prices (what you called arithmetic return) follows a normal distribution. If this is indeed what you were suggesting, then I am not sure I know what this assumption implies for the distribution of asset prices, as I am not familiar with a random variable that is distributed in such a way that taking its ratio results in a normal distribution. Hopefully this clears things up a little. Thank you

David Harper CFA FRM

David Harper CFA FRM
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HI @mreshko Oh okay, I did not realize that's what you meant originally. I don't mean to suggest that "normal VaR" comports with a good stochastic model for asset prices. (lognormal VaR matches one of your examples, of course). You probably know more than me in this regard.

We define normal absolute VaR (aVaR) = -μ + σ*z(α); where typical is 95.0% confidence such that z(0.050 or 0.950) = 1.645 normal quantile.
Just to keep it simple with a single period: If μ = 1.0% and σ = 8.0%, and because I like numerical examples, then:

The normal aVaR = -0.010 + 0.080*1.645 = .12159 or 12.159% which by convention is expressed in loss/profit format, L(+)/P(-) where losses are positives, such that mathematically, in the native P(+)/L(-) format we really mean that normal aVaR = μ - σ*z(α).

So if the current price, P(0) = $100.00, this model says the 95% VaR is 12.159% or$12.159 b/c it assumes the worst expected (with 95% confidence) return gets to a price level of P(0) + P(0) * [μ - σ*z(α)] = $100.00 +$100.00 * (0.01 - 0.080 * 1.645)
= $100.00 +$100.00 * (-12.159% or -%VaR) = $87.84. That "worst expected return" is a simple (aka, arithmetic) return: (87.84 - 100)/87.84 = 87.84/100 - 1 = 12.159%. And by defining normal aVaR = -μ + σ*z(α), we're assuming the simple (aka, arithmetic) return is normal, aren't we, by definition (not the price!)? I agree with you about the implication on prices, when we assumes the simple returns are normal. That's the (summation stability) advantage of log returns: they add over time (time additive) but the simple returns are linearly additive (cross sectionally) across the portfolio. Let me know if that's inconsistent or wrong (I hope not, it's the default approach in the FRM, and has been for >15 years!). Thanks, mreshko New Member HI @mreshko Oh okay, I did not realize that's what you meant originally. I don't mean to suggest that "normal VaR" comports with a good stochastic model for asset prices. (lognormal VaR matches one of your examples, of course). You probably know more than me in this regard. We define normal absolute VaR (aVaR) = -μ + σ*z(α); where typical is 95.0% confidence such that z(0.050 or 0.950) = 1.645 normal quantile. Just to keep it simple with a single period: If μ = 1.0% and σ = 8.0%, and because I like numerical examples, then: The normal aVaR = -0.010 + 0.080*1.645 = .12159 or 12.159% which by convention is expressed in loss/profit format, L(+)/P(-) where losses are positives, such that mathematically, in the native P(+)/L(-) format we really mean that normal aVaR = μ - σ*z(α). So if the current price, P(0) =$100.00, this model says the 95% VaR is 12.159% or $12.159 b/c it assumes the worst expected (with 95% confidence) return gets to a price level of P(0) + P(0) * [μ - σ*z(α)] =$100.00 + $100.00 * (0.01 - 0.080 * 1.645) =$100.00 + $100.00 * (-12.159% or -%VaR) =$87.84.

That "worst expected return" is a simple (aka, arithmetic) return: (87.84 - 100)/87.84 = 87.84/100 - 1 = 12.159%.
And by defining normal aVaR = -μ + σ*z(α), we're assuming the simple (aka, arithmetic) return is normal, aren't we, by definition (not the price!)?

I agree with you about the implication on prices, when we assumes the simple returns are normal. That's the (summation stability) advantage of log returns: they add over time (time additive) but the simple returns are linearly additive (cross sectionally) across the portfolio. Let me know if that's inconsistent or wrong (I hope not, it's the default approach in the FRM, and has been for >15 years!). Thanks,
I am in agreement with you on how VaR should be calculated in the two cases: normal asset prices and log-normal asset prices. What I am not 100% comfortable with is the use of the relative change of asset prices to define "normal returns" - this does not seem consistent with the two common assumptions made for asset prices. If "normal returns" were defined as a simple difference of normally distributed asset prices, then the whole thing would be fine... expect for the point that I raised at the start about the drift and standard deviation quoted in percent (%).

Not sure I fully understood the points you were making in your last paragraph about "cross sectional" aditivity of simple returns across the portfolio. I prefer the precise language of mathematics when discussing this sort of things, as words can be easily misinterpreted or ones per-conseptions projected onto them. Kind regards.

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David Harper CFA FRM

David Harper CFA FRM
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Thanks, but the words about mathematics are imprecise, and is why I often use a numerical illustration (e.g., what is the mathematical meaning "absolute"? Well, it depends on the context). And mathematics is not the only domain. I'm trained in data science which relies on statistics and econometrics which has a language about time series (And this is a forum, with norms, another contextual fact). It's my opinion that there exists no universal semantic(s) in mathematics (i.e., textual language about mathematics) but rather that mathematics is logic, symbols and syntax ... but I believe it is not grammar. (Am I allowed to invoke my semiotics training, or must everything be math?). Put simply, words about math are always susceptible to misinterpretation because they are interpretations. But, then again, I have >17,000 posts in this forum and >15 years learning that lesson the hard way, by exchanging ideas with practitioners, many of who are/were much better mathematicians/quants/data scientists/econometricians/statisticians/financial analysts/etc than me. Maybe you just don't recognize cross-sectional because you don't know its familiar domain. A recurring pattern is different people using different words, each believing the words are correct, and the beauty of math is how the actual math tends to resolve misinterpretations.

Your first post (if you want my honest opinion) wasn't precise at all. If you wanted to be precise, you'd have used a numerical illustration (i.e., actual math). That's an opinion I've generated over a decade+ hosting this forum. You used the word "absolute" in a way that wasn't immediately familiar to me, or what I really mean, is that I was instantaneously weighing different interpretations, based on experience answering questions.

The delicious irony in what you just wrote is that my response to your original question (to itemize some basic definitions) was precisely because I perceived your question to be imprecise . Instinctively, I answered with definitions because I was not settled in my interpretation of your question.

In any case, cross-sectional (https://en.wikipedia.org/wiki/Cross-sectional_data), which is a valid term is statistics and econometrics, refers to point-in-time. So if you have a typical dataframe (aka, matrix or tibble is a cute word for that simply because I happen to prefer R's tidyverse to python. But "tibble" would be a valid but teasing example of how contextual can be language. If a quant hadn't heard of tibble because they'd only used python and never used the tidyverse, is tibble imprecise? No. It's just unfamiliar) of portfolio returns where the columns are (eg) daily returns and the rows are position components (eg., assets), then the can add the components cross-sectionally (ie., the portfolio's return on a given day is the sum of its components or positions) or we can add them over time. In a typical portfolio matrix setup, a cross-sectional operation is column-wise. Log returns are time-additive, simple returns are cross-sectionally additive. But it's all just words, the ultimate language (imo) is in the numbers and/or code.

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mreshko

New Member
Thanks, but the words about mathematics are imprecise, and is why I often use a numerical illustration (e.g., what is the mathematical meaning "absolute"? Well, it depends on the context). And mathematics is not the only domain. I'm trained in data science which relies on statistics and econometrics which has a language about time series (And this is a forum, with norms, another contextual fact). It's my opinion that there exists no universal semantic(s) in mathematics (i.e., textual language about mathematics) but rather that mathematics is logic, symbols and syntax ... but I believe it is not grammar. (Am I allowed to invoke my semiotics training, or must everything be math?). Put simply, words about math are always susceptible to misinterpretation because they are interpretations. But, then again, I have >17,000 posts in this forum and >15 years learning that lesson the hard way, by exchanging ideas with practitioners, many of who are/were much better mathematicians/quants/data scientists/econometricians/statisticians/financial analysts/etc than me. Maybe you just don't recognize cross-sectional because you don't know its familiar domain. A recurring pattern is different people using different words, each believing the words are correct, and the beauty of math is how the actual math tends to resolve misinterpretations.

Your first post (if you want my honest opinion) wasn't precise at all. If you wanted to be precise, you'd have used a numerical illustration (i.e., actual math). That's an opinion I've generated over a decade+ hosting this forum. You used the word "absolute" in a way that wasn't immediately familiar to me, or what I really mean, is that I was instantaneously weighing different interpretations, based on experience answering questions.

The delicious irony in what you just wrote is that my response to your original question (to itemize some basic definitions) was precisely because I perceived your question to be imprecise . Instinctively, I answered with definitions because I was not settled in my interpretation of your question.

In any case, cross-sectional (https://en.wikipedia.org/wiki/Cross-sectional_data), which is a valid term is statistics and econometrics, refers to point-in-time. So if you have a typical dataframe (aka, matrix or tibble is a cute word for that simply because I happen to prefer R's tidyverse to python. But "tibble" would be a valid but teasing example of how contextual can be language. If a quant hadn't heard of tibble because they'd only used python and never used the tidyverse, is tibble imprecise? No. It's just unfamiliar) of portfolio returns where the columns are (eg) daily returns and the rows are position components (eg., assets), then the can add the components cross-sectionally (ie., the portfolio's return on a given day is the sum of its components or positions) or we can add them over time. In a typical portfolio matrix setup, a cross-sectional operation is column-wise. Log returns are time-additive, simple returns are cross-sectionally additive. But it's all just words, the ultimate language (imo) is in the numbers and/or code.

It seems to me that you misinterpreted the last sentence in my previous post. What I was saying is that I believe that these types of discussions should be done in the mathematical language of formulas, not in spoken word. I wasn't claiming that my previous post were using this approach, instead I was suggesting that we move away from words to formulas in order to avoid further impression and confusion.

I am perfectly aware of what cross-sectional means and of other concepts you are referring to (I have a PhD in Theoretical Physics), but despite your long post, it remains unclear to me what you are trying to say, which is due to us still exchanging in sentences rather than formulas. Is there a support for MathJax in this comments?

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