Estimating Market Risk Measures: A Quick Review (FRM Part 2, Book 1, Market Risk)

Summary

This script delves into market risk estimation, exploring time series analysis of profits/losses and dividends, and adjusting for time value of money. It discusses calculating arithmetic and geometric returns, and introduces historical simulation and parametric methods for Value at Risk (VaR) estimation. The script also covers coherent risk measures, expected shortfall, and general risk measures, emphasizing VaR's limitations and the importance of sub-additivity. It concludes with standard error estimation and QQ plots for distribution fitting and outlier detection.

Takeaways

• 📈 **Time Series Analysis**: The script discusses the importance of time series data for financial instruments, including crisis data for listed instruments and MTM valuations for exotic products.
• 💹 **Profit and Loss (P&L) Calculation**: It explains how P&L is calculated as the difference between the end-of-period receipts and the initial investment at the beginning of the period.
• 💰 **Time Value of Money**: The script covers two methods to adjust P&L for the time value of money: discounting future payments or capitalizing past payments using the risk-free rate.
• 🔢 **Arithmetic vs. Geometric Returns**: It differentiates between arithmetic returns (average return proxy) and geometric returns (considering income reinvestment) and how they relate.
• 📊 **Loss Time Series (LP)**: Introduces the concept of creating a loss time series as the negated version of the P&L time series, important for understanding risk.
• 📚 **Historical Simulation VAR**: Describes the historical simulation technique for VAR estimation, which involves sorting P&L observations and selecting a value at a certain confidence level.
• 📐 **Parametric VAR Methods**: Discusses three parametric approaches to VAR estimation: normal distribution on P&L, normal distribution on returns, and log-normal distribution on returns.
• 🔗 **Coherent Risk Measures**: Defines coherent risk measures and their four conditions, emphasizing the importance of sub-additivity for margin calculations and regulatory capital.
• 📉 **Expected Shortfall (ES)**: Introduces ES as a risk measure that provides insight into tail losses beyond the VAR threshold, calculated as a weighted average of tail losses.
• 🌐 **Generalized Risk Measures**: Explains how VAR and ES can be part of a broader class of risk measures, the generalized risk measures, which are weighted averages of quantiles.
• 📊 **QQ Plots**: Describes the use of QQ plots for comparing empirical distributions with benchmark distributions, identifying outliers, and estimating distribution parameters.

Q & A

What are the two types of time series inputs required for a War estimation approach?

The two types of time series inputs required are the time series of profits/losses and the time series of dividends/coupons, which can be referred to as the income time series.

How is Profit and Loss (P&L) computed in the context of War estimation?

Profit and Loss (P&L) is computed as the receipts at the end of any period t minus the initial investment at the beginning of the period.

What are the two methods to adjust P&L for time value of money?

The two methods to adjust P&L for time value of money are by discounting the payments received at time T or by capitalizing the payments made at time t-1, both using the risk-free rate R.

What is the difference between arithmetic and geometric returns?

Arithmetic returns are calculated by dividing the P&L by the initial investment and are used as a proxy for average or expected return. Geometric returns consider reinvestment of income and a chosen compounding frequency over a designated time period.

How is Value at Risk (VaR) estimated using the historical simulation technique?

In the historical simulation technique, VaR is estimated by sorting the P&L observations in increasing order and picking the VaR at a certain confidence level as the N alpha plus 1/8 observation from the left.

Why is the sub additive property important for risk measures?

The sub additive property is important for risk measures because it is crucial for margin calculations, regulatory capital estimation, and setting a conservative upper bound to the combined risk of multiple positions.

What is the Expected Shortfall (ES) and how is it different from VaR?

Expected Shortfall (ES) is the probability-weighted average of tail losses and provides insight into how bad things can get if losses exceed VaR. Unlike VaR, which only indicates a threshold loss number, ES gives a measure of the average loss in the tail beyond the VaR level.

How can the standard error of an estimate be computed?

The standard error of an estimate can be computed by creating a thin strip around the VaR with a width of H and calculating three probabilities: to the left of the strip, to the right of the strip, and in the middle of the strip. These probabilities are then used in a formula to compute the standard error.

What is a QQ plot and how is it used in risk estimation?

A QQ plot is a quantile versus quantile plot that compares quantiles from an empirical distribution with quantiles from a specified benchmark distribution. It is used to identify if the benchmark distribution is a good fit for the empirical data, to identify outliers, and to estimate parameters such as location and scale.

Why is the normal distribution assumption used in parametric VaR approaches?

The normal distribution assumption is used in parametric VaR approaches to simplify calculations and because it is a commonly used distribution in statistics. However, it may not always accurately represent the true distribution of P&L or returns, especially in the tails.

What is a spectral risk measure and why is it considered coherent?

A spectral risk measure is a special subset of general risk measures where the weighting function satisfies three conditions: non-negative weights, weights sum to one, and weights assigned to more extreme losses are not less than those assigned to less extreme losses. It is considered coherent because it satisfies all four conditions of coherent risk measures, including sub additivity.

Outlines

00:00

📊 Estimation Approaches and Returns

This paragraph introduces the process of estimating market risk, focusing on the inputs required for such estimations. It explains the computation of time series for prophets/losses and dividends/coupons, which are essential for calculating profit and loss (P&L). The P&L is adjusted for time value of money using either discounting or capitalizing methods with the risk-free rate. The paragraph then delves into the calculation of two types of returns: arithmetic returns, which serve as a proxy for expected returns, and geometric returns, which consider reinvestment of income. The relationship between these returns is highlighted, and the importance of understanding the distribution of losses is emphasized. The historical simulation technique for Value at Risk (VaR) estimation is introduced, which involves sorting P&L observations and selecting a specific observation based on the confidence level. The limitations and benefits of this technique are discussed, including its simplicity and its reliance on the assumption that the future will mirror the past.

05:01

📈 Parametric Approaches and Coherent Risk Measures

The second paragraph shifts the discussion to parametric approaches, starting with the normal distribution assumption for P&L to calculate VaR. It contrasts this with the normal world approach, where normality is assumed for arithmetic returns to estimate the Word. The Log-normal VaR approach is also explained, which assumes normality in geometric returns, leading to log-normally distributed prices or valuations. The paragraph then transitions into a discussion on coherent risk measures, which are risk measures that satisfy four conditions: monotonicity, sub-additivity, positive homogeneity, and translational invariance. VaR is noted as not being a coherent risk measure due to its failure to meet the sub-additivity condition. The concept of expected shortfall (ES) is introduced as a risk measure that provides insight into the tail of losses beyond VaR, and it is explained how ES can be estimated using historical simulation or through a formula involving higher confidence levels.

10:02

📉 General Risk Measures and Standard Errors

This paragraph introduces general risk measures, which are weighted averages of various quantiles and can encompass both VaR and ES with appropriate weighting functions. The concept of spectral risk measures is discussed, which are a subset of general risk measures that satisfy three specific conditions to ensure coherence. The paragraph then explains the calculation of standard errors for VaR estimates, which provide a measure of uncertainty. Factors affecting standard error, such as sample size, bin width, and confidence level, are discussed. The chapter concludes with a discussion on QQ plots, which are used to compare an empirical distribution with a benchmark distribution, and their utility in identifying outliers and estimating distribution parameters.

15:04

📋 Summary of Market Risk Estimation

The final paragraph provides a summary of the key takeaways from the chapter on estimating market risk measures. It encapsulates the various methods for calculating VaR, the importance of understanding the distribution of returns and losses, and the use of QQ plots for distribution fitting and outlier detection. The summary underscores the importance of choosing the right distribution for risk estimation and the implications of different risk measures for regulatory capital and margin calculations.

Keywords

💡Time Series

A time series is a sequence of data points collected at successive, equally spaced points in time. In the context of the video, time series are used to represent the historical data of financial instruments, such as crisis or valuations (for exotic products), and dividends or coupons. The script mentions computing the time series of profits/losses using these data points to estimate future financial risks.

💡P&L (Profit and Loss)

P&L refers to the financial performance of a business or investment over a specific period, calculated as the difference between revenue and expenses. The video script discusses computing P&L as the income received at the end of a period minus the initial investment at the beginning of the period, which is crucial for estimating financial risk.

💡Time Value of Money

The time value of money is the concept that a sum of money is worth more now than the same sum in the future due to its potential earning capacity. The script discusses adjusting the definition of P&L for the time value of money by either discounting future payments or capitalizing past payments using the risk-free rate.

💡Arithmetic Return

Arithmetic return is calculated by taking the P&L and dividing it by the initial investment. It serves as a proxy for the average or expected return on an investment. The script uses this concept to illustrate a simple method of estimating returns from financial data.

💡Geometric Return

Geometric return accounts for the reinvestment of income over a certain period and is calculated using compounding frequencies. If continuous compounding is assumed, the geometric return is defined as the log of the final proceeds divided by the initial investment. The video script contrasts this with arithmetic return to show different ways of measuring investment performance.

💡Historical Simulation

Historical simulation is a technique used to estimate Value at Risk (VaR) by using historical observations of P&L. The script describes sorting these observations and selecting a VaR at a certain confidence level as the N alpha plus 1/8 observation from the left, which is a simple yet effective method of risk estimation.

💡Value at Risk (VaR)

VaR is a measure of the potential loss in the value of a portfolio over a specific time period, given a certain level of confidence. The script explains how VaR can be estimated using historical simulation and also discusses its limitations, such as not being a coherent risk measure.

💡Coherent Risk Measure

A coherent risk measure is one that satisfies four conditions: monotonicity, sub-additivity, positive homogeneity, and translational invariance. The script explains these conditions and why VaR is not a coherent risk measure due to its failure to meet the sub-additivity condition.

💡Expected Shortfall (ES)

Expected Shortfall is a risk measure that provides more information about the tail of the loss distribution beyond VaR. It is defined as the probability-weighted average of tail losses. The script describes how ES can be estimated using historical simulation and is a spectral risk measure, which is a type of coherent risk measure.

💡Standard Errors

Standard errors are estimates that indicate the degree of uncertainty associated with a sample estimate of a population parameter. In the script, standard errors are discussed in the context of estimating VaR, where they help in calculating confidence intervals and understanding the reliability of the VaR estimate.

💡QQ Plot

A QQ plot, or quantile-quantile plot, is a graphical tool used to compare the distribution of two datasets. The script mentions using QQ plots to assess how well a specified distribution fits the empirical data, identify outliers, and estimate parameters such as location and scale.

Highlights

Introduction to inputs required for War estimation: time series of profits/losses and dividends/coupons.

Calculating Profit & Loss (P&L) as the difference between receipts at the end of a period and initial investment.

Adjusting P&L for time value of money by discounting or capitalizing payments using the risk-free rate.

Definition and calculation of arithmetic returns as a proxy for average or expected return.

Explanation of geometric returns considering reinvestment of income and compounding frequency.

Linking arithmetic and geometric returns using a specific expression.

Creation of a time series for losses (LP) as the negated version of the P&L time series.

Historical simulation technique for VaR estimation by sorting P&L observations.

Assumption of future following the past in historical simulation and its benefits.

Transition from nonparametric to parametric world with three approaches for VaR estimation.

Normal distribution assumption on P&L for calculating VaR in the first parametric approach.

Normal distribution assumption on arithmetic returns for the second parametric approach.

Log-normal distribution assumption on geometric returns for the third parametric approach.

Definition of coherent risk measures and their four conditions: monotonicity, subadditivity, positive homogeneity, and translational invariance.

Explanation of why VaR is not a coherent risk measure due to not always satisfying subadditivity.

Introduction to expected shortfall (ES) as a risk measure that provides insight into the tail beyond VaR.

Calculation of ES using historical simulation by averaging losses in the tail.

General risk measure as a weighted average of various quantiles and its relation to VaR and ES.

Spectral risk measure as a special subset of general risk measures that satisfies three conditions to ensure coherence.

Discussion on standard errors in VaR estimation and their relation to sample size, bin width, and confidence level.

QQ plot explanation for comparing empirical distribution with a benchmark distribution and its uses in identifying outliers and determining distribution parameters.

Transcripts

having done this reading in a lot of

detail now let's take a look at its key

takeaways we started off with the

various inputs that are required for any

War estimation approach these inputs are

number one the time series of prophets

slash losses this this time series we

computed it using twos time series the

first of them was the time series of

crisis if it's a listed instrument or

the time series of M TMS or valuations

if it's let's say an exotic product the

second time series was that of dividends

/ coupons which you can refer to as the

income time series so we compute the P&L

as let's say the receipts at the end of

any period t minus the initial

investment let's say as the beginning of

the period we then took a look at two

ways in which we can let's say adjust

this definition of PL for time value of

money either by discounting the payments

received at time T or by capitalizing

the payments made at time t minus 1 both

using let's say the risk-free rate R we

then moved on to defining two kinds of

returns the first return was what we

referred to as arithmetic returns we

calculated these by just taking in the

pl time series take the pl and divided

by the initial investment that's an

arithmetic return remember this

arithmetic return is something which you

use as a proxy for the average or the

expected return the other kind of return

that we took a look at was a geometric

return this return is for a chosen or

designated time period and it takes into

account reinvestment of this income and

a certain chosen compounding frequency

so if you assume that your compounding

happens you with the continuous

compounding then I can define my

geometric return as the log of the final

proceeds divided by the initial

investment the two kinds of returns

arithmetic and geometric can be linked

to one another using this expression

generally speaking the two returns are

close to one another if your period over

which these returns are earned is a very

small period let's say a daily period

then we also took a look at another time

series which is that of losses so I can

create another time series which lets me

call it as LP I define this time series

to be lets say the negated version of

the pl time series and if we are talking

about distributions then we said that

the mean of the LP distribution is

simply the negated value of the mean of

the PL distribution and the standard

deviation of the LP distribution we said

is the same as the standard deviation of

the PL distribution when it comes to the

most sort of simplest technique for var

estimation we said it's the historical

simulation technique all that we do in

this technique is we take the

observations that are given to us assume

that these are let's say the PL

observations we sort them in an

increasing order then let's say from the

smallest to the highest smallest would

therefore refer to negative entries or

losses and we pick the var at a certain

confidence level let's say at a C or a

certain significance level let's call it

alpha as the N alpha plus 1/8

observation from the left here I am

assuming that I am dealing with a PL

time series number one and number two I

am dealing with n observations so if I

pick n alpha plus one observations and

if a plus one at the observation I

should say from the left then that

observation would clearly leave a

probability mass of alpha in the left

tail and I've been working with the LP

series we had said that we will pick our

war again to be the N alpha plus 1 at

observation but this time we'll pick it

from the right okay so this was about

historical simulation keep this thing in

mind that it's a very simple technique

but at its core it makes this assumption

that the future will follow the past a

very important benefit of this technique

is that it accommodates real-life

empirical observed distributions it does

not it does not impose any

distributional Essam

on the PL or the returns now let's move

from this nonparametric world which

historical simulation was in to a

parametric world in this parametric

world we defined three approaches the

first approach was one in which we

impose as we do in any parametric

approach a normal distribution

assumption on this time the PL the

profit and loss so if you do that then I

can calculate the var which in the pl

world remember is in the left tail by

starting with the mean moving to the

left and that's why you have a minus

these many multiples of your standard

deviation of the pl and since we are

talking about a PL and since war is a

lost number we had negated it to create

a war had this been an LP distribution

you would have computed the var by

starting with the mean which is mu LP

and moving to the right this time by

these many multiples of the standard

deviation Sigma PL is a same as Sigma LP

now moving on to the second approach in

this category we called it the normal

word in this approach we impose the

normal distribution assumption on the

arithmetic returns and let's assume that

these returns are distributed normally

with this as the mean and this as the

standard deviation to calculate the word

which in this case would assume that

valuations or prices are normally

distributed I'll start with the mean

return move to the left and that's why

there's a minus these many multiples of

the standard deviation what this gives

me along with the minus is what we call

a VAR percent because it's like a word

number for $1 of notional invested this

I have to scale it with the current

valuation of my position which is PT

minus 1 I am assuming I'm standing at

time t minus 1 and this gives me a word

at this confidence level see the next

approach in this parametric category was

a log normal word approach in this

approach I impose the normality

assumption

this time on the geometric return which

is the continuously compounded return if

you do that then we know that the prices

or valuations are log normally

distributed and that's why we refer to

this var as a log normal var in terms of

its var calculation I should say how do

we calculate it start with the mean

return move to the left by these many

multiples of the standard deviation this

gives you some kind of a worst-case

valuation so current valuation minus the

worst-case scaled by the total size of

your position this gives you the bar

keep this thing in mind that normal and

log normal var are close to one another

if the period over which they are being

computed I mean the horizon is small

let's say daily we then moved on to

defining coherent risk measures that

means let's say we have portfolios x and

y and our risk measure is denoted by Rho

which is written we've written it as a

function it's said to be coherent if it

satisfies all of the four conditions

listed here just to name them at this

moment these conditions were

monotonicity sub additive 'ti positive

homogeneity and translational invariance

we described each one of them in detail

in our videos out of these four we

reasoned out that sub additive 'ti is

really important it's important from the

standpoint of margin calculations if

let's say the exchange uses this

particular risk measure for calculating

initial margins it's important from the

standpoint of regulatory capital

estimation if let's say the supervisor

is using this risk measure for

calculating Bank capital and it's also

important from the standpoint of setting

a conservative upper bound to the

combined risk of let's say many

positions put together now in terms of

how var fares on these conditions then

remember that var is not a coherent risk

measure the reason being that it does

not always satisfy this condition of sub

additive 'ti by taking a suitable

example we had reasoned out why this is

not the key

is why is it I mean why is it not

coherent now in terms of making work

coherent then and you know making it sub

additive I should say you should pick a

PL distribution which let's say comes

from this family of distributions which

we call the elliptical distribution

family and this normal distribution

remember is one member of this

elliptical distribution family we then

moved on to this disk measure of

expected shortfall

now this risk measure is something which

helps you peek into the tail the VAR

only tells you a threshold loss number

it doesn't tell you how bad things can

get if the losses were to exceed the war

in terms of how we defined es we defined

it as the probability weighted average

of tail losses in terms of the formula

it's the expected value of the loss

conditional upon this fact that the loss

exceeds the war so this is like a

conditional expectation when you come to

the world of historical simulation which

is very simple approach we know and in

this world the es can be estimated very

simply by a simple average of all losses

that lie in the tail remember we had

picked the N alpha plus 1 at observation

from the right if it's a loss

distribution as my var so if I can take

these n alpha observations which lie in

the tail and take their simple average I

arrive at the expected shortfall now the

expected shortfall be reasoned out you

know with lot of detailed analysis can

also be computed using another formula

and that formula does not deal directly

with losses in the tail but rather it

deals with quantiles or we should say it

deals with VARs which are computed at a

confidence level that is higher than the

confidence level for which we are

computing this es these confidence

levels which are higher than C let's

call them C I I pick them which are you

know these are comfortably in the tail

and I pick them as equally spaced

confidence levels it's like I am slicing

the tail in

- endless slices and then I sort of take

an average of all these words to compute

my es now more are the number of slices

of the stale more accurate with this

approximation be for the expected

shortfall at this point also note that

if your losses are continuously

distributed I can write down two

formulas for the expected shortfall the

first formula remember is like an

integral it's like a conditional

expectation which looks something like

this it's 1 by alpha run this integral

from the var up until infinity and then

compute let's say X FX DX where X refers

to the loss the other formula for this

which gives generally the same result as

this one is 1 by alpha this time the

integral runs over a probability that

I'll run it from C which is the level of

confidence right up till 1 and I'll do

this integration as Q P DP ok so it's

like I am taking in a quantile and I'm

basically integrating it over the

probability values which span from C to

1 both these integrals are basically

computing an integral over the right

tail which is the case for the loss

distribution now let's move to more

genetic risk measures and we defined a

measure which is like some kind of a

weighted average of various quantiles so

if I run through various quantiles

I have X you know expressed my quantile

as QP so if this is my loss distribution

then a quantile QP would be that number

on the horizontal axis to the left of

which you will have a total accumulated

area of P so if I can you know take

various QP s like this and combine them

in some kind of a weighted average way

then I arrive at a risk measure which I

refer to as a general risk measure the

weighting function is this file you keep

changing the file and the risk measure

keeps changing we had said that both the

var and the es can actually be subsumed

into this definition of the general risk

measure

an appropriate choice of the weighting

functions so these are what are you know

what is given here now we then raised

this question our general risk measures

also coherent and we said not

necessarily so to make them coherent we

moved to a special subset of these risk

measures which we refer to as a spectral

risk measure this subset is one in which

the weighting function satisfies three

conditions the first one is that these

weights are non-negative they can be 0

but they can never be negative these

weights you know in very laypersons

terms sum to one that's like a

requirement which we always impose on

weights when we do a weighted average in

this world since these weights are

continuously defined so it's like saying

the area under the weighting function is

equal to 1 the last one is a very

important condition we had said if you

are risk-averse

then the weight that you assign to a

certain quantile which is further into

the tail that means for this quantile P

2 is greater than a quantile to its left

which is let's say P 1 then the weight

which you assign to a verse or an

extreme loss cannot be less than the

weight that you assign to a less extreme

loss that means they're not drawing it

visually if this is P 2 quite an extreme

loss and this is P 1 then the weight

which you assign to this guy cannot be

lower than the weight you assign to this

guy so es it satisfies this condition

but what does not so es is actually a

spectral risk measure and it's also

coherent we then moved on to defining

standard errors so when we calculate war

or estimated based on a sample of

observed let's say PNL or returns then

it's like a statistical estimate every

statistical estimate we said is is to be

accompanied by some kind of a standard

error estimation because if we have the

standard error let's say it's 4q which

is a point estimate of war and I am

saying standard error is a C of Q if I

have this standard error available with

me I can easily use it to Cal

or estimate a confidence interval for my

bar now let's take a look at how we we

computed the standard error we had said

let's pick the quartile we had used the

case of a PL distribution so my bar was

in the left tail

this was my bar we had created a thin

strip around it

this strip had a width of H and we had

computed three probabilities the

probability to the left of this strip to

let's say the right of the strip and in

the middle of the strip based on these

three we had computed the standard error

using this formula we had then done no

quick rules of thumb in terms of how

this standard error varies with respect

to the size of the sample so as the

sample size increases intuitively you

could agree that thus the standard error

decreases then the other determinant we

had you know reasoned out was the width

of this bin we had said as the bin width

increases the standard error decreases

and lastly the determinant was the

confidence level at which this VAR was

estimated we had said as you estimate

what which are let's say for higher and

higher confidence levels their

uncertainty goes on increasing that

means the standard error for high

confidence VARs is pretty high we had

actually finished then this chapter with

a quick discussion of this plot which we

refer to as a QQ plot expands as the

quantile verses quantile plot this plot

is basically one which plots quantiles

which are read from one distribution

let's refer to it as an empirical

distribution plotted against the

quantiles read from a specified

benchmark distribution if you what you

achieve is a pretty linear graph

something like this that means that the

specified benchmark distribution is a

very good choice to fit the empirical or

observed data if it's not the case then

you switch to another benchmark

distribution and keep trying out

different choices the second thing which

we noted about QQ plots was that if you

plot this QQ plot and the extremities of

the

plot you see them bending towards a

certain axis then the distribution

plotted on that axis has fatter tails so

that was my second observation my third

observation was that QQ plots they can

be used to somehow get a handle on the

parameters such as location which is

another name for mean and scale another

name for Sigma of any distribution if

you were to do a linear transformation

of one of the distributions in the QQ

plot then this transformation goes and

changes both the slope and the intercept

of this QQ plot these changes in slope

and intercept are the ones which can

help you sort of determine the mean and

the Sigma of this empirical distribution

lastly we had the fourth point we had

said that QQ plot is a very good way of

identifying outliers

in your data so this was a very quick

run-through of various takeaways in our

first chapter estimating market risk

measures