Download CFRM 546 - 0404

CFRM 546 - 0404 - Compressed 2
[00:00:00.00] [MUSIC PLAYING]
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[00:00:09.61] So welcome back, everyone. So as I said last time, we will start doing more
quantitative work in the rest of the class. Now, let's start with the basic concepts in risk
management. We will start doing in these two weeks these risk measures, risk mappings that
evaluates those operators, that evaluate the risk.
[00:00:51.74] First, the basic concepts in risk management-- I will mention risk management for
a financial term-- sorry-- for a financial firm, second, modeling the value and the value change,
and the third is the risk measurement where we will deal with this [INAUDIBLE] risk measures,
OK? Now, risk management for a financial firm, first of all, how does it look like? A balance
sheet, a stylized balance sheet for a bank is what? What are they?
[00:01:27.56] They have some money, and they have some liabilities. They have some surplus, if
any, so SS investments of the firm, and they have some obligations from fundraising, OK? For
example, here we can see that we have some cash, securities, loans, mortgages, and other assets
that corresponds to 200 million pounds, let's say, and the liabilities are deposits, bonds issued-that is the bonds created by the bank-- reserves, short term borrowing. And in total, its debt is a
200 million pounds, let's say.
[00:02:19.46] Now, a balance sheet for an insurer is similar. But we still have these investments,
for example, bonds, stocks, property, investments per unit length, which are contracts, and we
also have other assets and liabilities, similarly resourceful policies, bonds, issues. And in total,
that is 90 million pounds, let's say, and equity is 10 million pounds, OK? So, again, we have a
total of 100 million pounds. Balance sheet equation-- this is the simplest one-- if your assets
equals liabilities, which is equal to depth plus equity, OK?
[00:03:11.39] So what is my debt? 90 million. What is my equity? 10 million. So in total, my
assets are 100 million. My liables are 100 million. Now, this is balance sheet equation. If equity-or, you can check it from here as well. Look here. I have depth of 170 million pounds. I have
equity of 30 million pounds. Total is 200 million pounds, OK? So it's the simplest one, but what
is it? If equity is greater than 0, the company's fine, right? I have money.
[00:03:47.46] If it is called solvent, I am in a good shape. Otherwise, I am not in good shape. I
am insolvent. OK? Valuation of the items on the balance sheet is a non-trivial task. For example,
amortized cost accounting values a position a group value, equities starting at its inception. And
this is carried forward progressively, reduced over time, OK? Now,
[00:04:14.27] Fair value accounting, this fair value you recall from your stochastic calculus
classes, and also from our three-weeks study, this arbitrage, et cetera, this payer value, it's all
where this story revolves around, accounting, values, assets, at prices that they are sold, and
liabilities at prices that would have to be paid in the market. This can be challenging for nontraded or illiquid assets or liabilities.
[00:04:45.45] There's a tendency in the industry to move towards [INAUDIBLE] accounting,
and market consistent valuation insolvency too follows similar principles, OK? Now, I would
suggest you check this solvency one, solvency two, and recall the solvency three will come out
in 2019. But they are still working on how to set up the rules, how to set up the correct models.
Recall, for example-- I will mention it. They are debating whether value at risk should be
continued or whether these expected shortfall or expected value at risk should be replaced at the
value at risk, et cetera, OK?
[00:05:39.77] This is to say not just rules, but also the modeling, the engineering, the financial
engineering they are discussing. Now, risks faced by a financial firm, what are they? Decrease in
the value of the investments on the asset side of the balance sheet, which is very logical, right? I
can happen, for example, losses from securities, trading or credit risk, OK? Maturity mismatch,
what does it mean?
[00:06:11.45] Large parts of the assets are relatively in liquid long term, OK? That is to say I
have some asset, recall. But say I have-- let's exaggerate it-- I have $50 million of asset, OK?
OK, I want to sell in one second. That's not what you can do, OK? Maybe there is a tendency
that people don't want to buy it. So it's not immediately liquid, OK? So if it is not immediately
liquid, what does it mean?
[00:06:43.68] Whenever you make your model, and you say, OK, if I have a depth of this, and I
will sell it, and I will cover it with that, then you cannot do it, because even though you have-think about in the simplest solution, forget about this financial assets. Think about you have a
house, like-- I don't know-- $600,000, worth $600,000. So you say you are in good shape. But if
I need money, then I can sell it, and I can move to a smaller house. I can buy a smaller house.
And the difference, I will cover it with my money that I have sold for the house.
[00:07:22.94] Can you do that? Well, you can, but maybe three, four months, or even one year. It
depends on the money you ask. So it's not liquid, OK-- so the same idea in the bank, in the
assets, in equities as well, OK? So large parts of the liabilities, large parts of the debts are, of
course, shorter, or shorter obligations. So what does this lead to?
[00:07:51.50] This can lead to a default of a solvent bank or even a bankrupt, OK? Now, the
prime risk for an insurer is insolvency. Insolvency means that risk get claims of policyholders
cannot be met. It is to say I insure you, but then I cannot fulfill what I have promised. On the
asset side, risks are similar to those of a bank. But on the liable side, the main risk is that reserves
are insufficient to cover future claim payments, not that the liabilities of life insurer are of a longterm nature.
[00:08:30.02] Why is it so? Because you insure someone, for example, a health policy. Think
about health policy. I don't know. We make the policy, let's say when you are 30 years old, and
its lifelong, OK? So the nature is different than in the bank. And also, for example, think about
working in an insurance company compared to working in a hedge fund. Which one is more
dynamic? Of course, hedge fund is much, much more dynamic.
[00:09:05.78] Insurance, typically the hours are fixed because your scenarios are usually long
term, OK? So there is a big difference between the natures of the two directions are different.
One of them, namely finance side, is, of course, much more faster, but the insurance side is much
more smaller, OK? And on the insurance side, for example, since it is a long term nature, they
are subject to multiple categories of risk, for example, interest rate risk, inflation risk, longevity
risk, et cetera.
[00:09:52.69] So risk is found on both sides of the balance sheet, and thus, risk management
should not focus on the asset side alone. Now, capital, what's a capital? There are different
notions of capital. One distinguishes equity capital, namely the value of assets debt-- sorry-value of assets minus debt. So this is your capital.
[00:10:16.95] This is, say, whatever I have minus debt, and it measures the terms value to its
shareholders. It can be split into shareholder capital-- this is to say, insurance capital invested in
the firm-- and retained earnings. This is say, accumulated earnings not paid out to shareholders,
OK? What is the regulatory capital? Regulatory capital is the capital required according to the
regulatory rules.
[00:10:44.01] Well, here is not the banks own decisions, but also, for example, for these
European insurance firms. Recall we had this MCR plus this SCR. Just check from the last week.
And further, regulatory framework also specifies the capital quality. For example, one
distinguished as tier one capital, namely the shareholder capital plus retained earnings can act in
full as buffer. And tier two capital includes other positions on the balance sheet, OK?
[00:11:21.60] Now, the economic capital, the economy capital is the capital required to control
the probability of becoming insolvent, probability of becoming in a risky position, let's say,
typically over one year. Now, the internal assessment of risk capital is also an economy capital
and aims at a holistic view in the sense that assets and liabilities inverse with fair values of
balance sheet items, OK? It is to say this one tries to assess all from all the aspects that it has
toward the holistic view.
[00:11:59.40] Now, almost all of these notions refer to items on the liability side that entail no
obligations to outside creditors. So they can just serve as a buffer against the losses, OK? Now,
OK, how do we model the value and value change? OK, so as I said in the beginning, we need-OK, we have some risk, and we take that risk and change it into a number. That's practically it
OK?
[00:12:38.38] Then I take that thing and change it into a number. Typically, what I mean is if
that risk is larger, the number is larger, OK? If I put some more risk in it, then the number that I
got will be larger, OK? So there will be some axioms that we will put on this risk evaluations,
OK? OK, so for mapping of risk factor, of course, we will need math, OK? To start, we set up a
general mathematical model for changes in value caused by financial risk.
[00:13:19.16] For this, we work on a probability space, omega fp. Let me recall omega is whole
space, OK? F is the sigma algebra. And p is the probability measure, OK? And consider a risk or
loss, as I said, a risk or loss of a random variable x. So x stands for the loss of a position, namely
random risk, OK?
[00:14:22.90] You don't know it in advance. You make an insurance. For example, you are an
insurer. You hope that whatever you make an issue, it will not happen. You It will not have an
earthquake. The guy who you issued an insurance policy will not have a fire in his house. But
there's a risk, OK-- so random risk. And this random risk x comes from abstract whole space to a
real number, OK? This is to say, from abstract, we go to a real number, OK?
[00:15:10.93] Consider a portfolio. Well, you can also denote x as l, OK? L stands for the loss,
OK? Let me close the door. Sorry. OK, consider a portfolio of assets and possibly liabilities,
OK? The value of the portfolio at time t today is denoted by vt. It's a random variable. So OK, vt
is the value of the portfolio a phi t, OK? It is not known the advance at time 0 or starting time,
OK? It is random, OK?
[00:16:20.63] At time t, if I have the information, if I have come to time t, then I observe vt, OK?
I don't know vt at time 0. But this is, again, the idea of information recall filtration, OK? OK,
with the accumulation of information, I will know vt, OK? Now, vt, as I said, is a random
variable assumed to be known at time t. Its distribution function is typically not trivial to
determine.
[00:17:12.23] Now, we consider a given time horizon delta t, OK? Delta t is-- suppose I start at
0. I start at 0. Sorry, I start at time t, OK? Then I go from time t to t plus delta t, OK? And
assume the portfolio composition remains fixed or delta t-- recall I don't change any more of my
assets-- there are no intermediate payments during delta-t. And this is fine for small delta t, but
unlikely to hold for large delta t, right? Because when you have a larger time horizon, that means
probably you will change your portfolio, OK? But for small times, most probably you don't make
any intermediate payments.
[00:18:08.90] Suppose you have an asset, which is like a dividend paying. Suppose you have a
stock. If it is for the next day, probably it will not pay for the next day. But suppose you have one
year of a stock to hold. That probably pays dividends. And also, if you have other assets in your
portfolio, then probably you'll change your portfolio, OK?
[00:18:33.92] So then the change-- so OK, the change in the value of the portfolio is then given
by delta vt plus 1 equals vt plus 1 minus vt. Well, what does this mean? It's my value of the
portfolio is, at time t plus 1, is this. I subtract it, and I'll find a value. Now, what is the loss for it?
It is the negative value. That is to say suppose I gained something.
[00:19:08.03] OK, suppose I gained $2. That means I've lost minus $2, which is good for me,
OK? But since we are dealing in this class risk, so we only consider risk. And that's why, hence
this negative sine, OK? In this class, we will deal with that. OK, now as I said, QRM,
quantitative risk management is mainly concerned with losses.
[00:19:34.06] Now, the distribution of the loss at time t plus 1 is the loss distribution, distribution
function fl, or simply f. Recall fl of x equals probability of all those [INAUDIBLE], omega slash
debt l omega, OK? OK, so this is a set, and this is the measure to measure. This is a probability
measure for the set, OK?
[00:20:35.11] And recall, if I let x goes to infinity, fl of x equals 1, let x goes to minus infinity fl
x equals 0, OK? Fl of x is increasing as x increasing. Why? If I have a larger set, probability of
larger set a1 should be greater than or equal to probability of larger set of a2, which is a2 is
smaller set, OK?
[00:21:35.58] Namely, suppose x1 is less than or equal to x2, OK? If I denote this kind of set as
a x1, then a x1 is included in a x2, right? Everyone sees that? Everyone sees this? OK? That is to
say when I changed this x1 to x2, and x1 is less than or equal to x2, then this set is included
there, OK? But then this means probability of a x1 is less than or equal to a x2, OK?
[00:22:25.84] OK, so practitioners often consider the profit and loss, P&L distribution, which is
the distribution of this thing, minus delta vt plus 1. For a longer time, intervals-- recall from our
previous class this, if you have a risk-free interest rate, are you discounted? And you'll find vt
plus 1 over 1 plus r minus vt. So our value of portfolio at time t plus 1 taken to time t, OK? Here
is the interest rate.
[00:23:45.06] OK, that is to say you cannot substitute from this value this value because you
need an interest rate, and you make that adjustment. OK, this vt is typically modeled as a
function of f of time t and the d dimensional random vector. OK, random vector, namely a vector
of, namely, more than one random variable, OK? So when I say z equals zt1, ztt, this means I
have d random variables, OK? I have d random variables and the revolution at time t, OK?
[00:24:56.32] Now, this, our class f, our plus across rd, recall d stands for the number of random
variables, OK? This plus sign here stands for the time, OK? Time starts at 0 and increases or
starts at 1, or you can take just t0. But the point is it's positive, OK? And I increment one by one.
Time increases one by one. So the risk factor changes. Xt plus 1, I denoted as zt plus 1 minus zt.
[00:25:39.57] Now, when I say zt plus 1 minute zt, when I say this, zt plus 1 minus zt, I mean
component-wise, subtraction, namely zt plus 1, 1 minus zt1 zt plus 1 2 minus zt2. And this is zt
plus 1, d minus zt d, OK? OK, and the loss is, of course, this one here, which corresponds to
since I denote vt as this one, I have this, OK? This is OK for everybody?
[00:26:48.67] Now, we see that the loss distribution function is determined by the loss of
distribution function of xt plus 1. We will thus also write lt plus 1 is l xt plus 1, which
corresponds to exactly lx equals this one, OK? Why am I saying this? Because if I denote this as
xt plus 1, this is the change of it, right? So I say zt plus 1 equals zt plus, of course, xt plus 1.
That's all I'm saying, OK? Is this OK? OK.
[00:27:27.72] This is to say loss is for x, is this situation plus the remaining ones, OK? It is
known as the loss operator. Now, what is it? If f, if this f is differentiable, it's first order Taylor
approximation is-- let me-- f of y is approximately equal to f of y0 plus gradient f of y0 y minus
y0.
[00:28:29.23] Now, dot product here-- and this is called gradient. Well, gradient of f of x means
if f is the dimensional, it means partial derivative. This is called partial derivative with respect to
x1, OK? Gradient f-- oh, sorry-- partial f with respect to x2 up to partial f with respect to xt. And
note that this is a vector, OK?
[00:29:25.59] Now, this is an another vector, OK? This is a number. That product of two vectors
gives me another number, and [INAUDIBLE]. It gives me another number, OK? OK. So then
since for y we have this of this form. By the way, when I say a dot product, gradient f gradient
x1, up to gradient f gradient xt times, let's say this is our y, and y, this is y1 minus y0. Maybe I
should say something like yt plus one yt first coordinate, y1 plus 1 yt, second coordinate, yt plus
1 yt d-th coordinate.
[00:30:35.04] That product corresponds to i equals one sum. Gradient f, gradient y1 times yt plus
1-- so this is i, i, i of this, OK? So this is exactly here. From From this one here is this expression
plus f. OK, derivative t, where does it come from? It comes from the following. We will not use
it in the few coming slides, that partial derivative. But recall gradient-- sorry-- suppose you have
d ft x, OK? Denote this as infinite-- stands for infinite, small increment of function f of f, OK?
[00:31:51.88] This corresponds to dfbt plus dfdx dx, OK? Is this OK? If you have more than one
variable, then the increments is exactly in each derivative. It increases, so if this is your starting
point, fd is the derivative with respect to t. Times 1 is not necessary because ft, it's not a vector,
so this is not necessary. Plus, j equals 1 up to d are the d components of this vector. The
coordinates of this f [INAUDIBLE] stands for the derivative times the xt plus 1j, which is the
differences.
[00:32:48.09] Is this OK? Now, read us-- approximate this loss as the linearized loss. Why is it
linearized? Because it's first order. Now, when I say first order, I mean you just take derivative 1,
OK? First order means this first order approximation, Taylor approximation. And does this one,
if I denote this as ct, and this expression here as btj, so then this loss is a function of xt plus 1. It's
a [INAUDIBLE] function, or linear function. You can see it's a linear function of this one, OK?
[00:33:30.68] When I say fi, well, you can take it as linear. Fi means linear function plus
subconstant, OK? It's a linear function of this, OK? And it is linear. It's an fi function, but in that
it's linear in this linear form. It is this constant [INAUDIBLE] class, the coefficient times the xt
plus 1. And the approximation is best if the risk factor changes are small in absolute value. Why
is it so? Because it's just first order. And first order-- and you only take this approximation. And
for better ones, we need second-order approximation, OK?
[00:34:16.52] Now, think about this example. Consider a portfolio, p of stocks. Suppose at time t
my stock, my first stock at time t, my second stock up to stock at time t, my d stock, it's value of
the stock j at time t-- let's denote them as y lambda j-- is the number of stocks by the number of
stocks j in this portfolio p. Now, in finance and risk management, one typically uses so-called in
finance notation, low price, logarithmic price, or its risk factors. Why is that so? Why do you
think?
[00:35:02.43] It comes back to Black and Scholes. OK, typically, for example, the simplest one-suppose zt is a normal with mean, let's say, sum 0, and variance, so suppose it's standard-normal, OK? Then supposed stock price, of course, you cannot take-- it cannot be negative, OK?
Stock price sp usually take ezt, for example, OK? So it is always greater than 0, OK? And this is
actually real Black and Scholes, just as an example of this kind.
[00:35:57.75] Now, one period ahead loss is then given by lt plus 1 equals-- OK, vt at time t,
your value at time t is given by a time t of this mapping f zt, which corresponds to exactly-- since
zt-- OK. Since if I know zt, then I know st, or if I know st, I know zt, I transform this log. But I
still can't have it as a function of the zt, right? So ft zt equals sum of j equals 1 up to d lambda j st
j, which corresponds to j equals 1 up to d. Lambda j is e to the ztj, OK? Is this OK?
[00:36:51.96] Now, the one period ahead loss is then given by alt plus 1 equals negative gain.
Negative gain means loss, recall, or negative loss means gain. But recall, we are dealing with
loss, risk. We are dealing with bad things. That's why, hence this negative sign. Vt plus 1 minus
vt, which is equal to minus j equals 1 up to d lambda j times e to the ztj plus up to this-- where
does this come from? Well, this is exactly-- since ztj plus xt plus 1 j, j is the coordinate, OK? J is
the coordinate.
[00:37:54.11] T is time. I'm sorry. Let me write here. T is time. T corresponds to zt plus 1j. X
stands for the loss. Z stands for the position, OK? This is OK? When I add them up, I have zt
plus 1j here minus this, right? I take this expression here out, right? This is purely basic, basic
algebra.
[00:38:24.58] Take this out. I have this expression, OK? So far so good? Then just this one
corresponds to exactly-- since this form here denote this as wtj, and so then-- so I have this wtj
times this e to the xt plus 1j minus 1 term, which corresponds to nonlinear in xt plus 1j. Why is it
nonlinear? Because there is an exponential term here, OK?
[00:38:55.00] And my loss function is exactly this expression here, exactly this one. Minus j
equals 1 up to d, wtj e to the xj minus 1, OK? OK. Now how do we linearize it? Well, if you take
this ct term to be 0, namely recalled what was ct, if you go back, just one more here, as you can
see, if you don't take the derivative with respect to t, that is to say, if you take this as constant, if
this function, for example, this one here,-- OK, this function here, I said st equals e to the zt,
OK? I said this, right?
[00:39:52.20] OK, if I take this, this one means f of zt, OK, which this is to say f of x equals e to
the x. I doesn't have any t on it, OK? This is just a function of zt. Well, it contains t, but t is
inside zt. Is this OK? OK. So suppose this function f does not depend on t. Suppose it's 0.
[00:40:25.18] Then if I go back here, see if this is 0, then this expression here corresponds to the
fz jt, which is lambda j e to the ztj. And if I denote this as wtj, which is the linearized loss, which
is just to denote just the notation, which corresponds to minus wt, xt plus 1, note that another
rotation is this one here xt plus 1. Let me get rid to the minus sign. This is a dot product again,
OK? Namely, this corresponds to-- if this sign here, this sign here, st dot product, xt plus 1, OK?
[00:41:25.37] You bounce over, and you check the slides, you will see that notation. So don't get
confused. OK, so then this, we have this approximation for lt plus 1. So, for example, suppose
you have, as an example, mu is-- you note, note that-- I recall note that everything is random
here, OK? You don't know in advance. Everything is random here, OK?
[00:42:03.11] This was for every realization, for every omega, OK? But when I take the
expectation of this, that is to say here there is this omega here, omega here, omega here, and here
omega, right? This is for every realization. This is OK? This is for every realization. When I take
the expectation of this expression, when I take the expectation of this operation, it gives me
what-- e expectation of minus wt. Let's use that notation now, xt plus 1, which corresponds to
expectation minus j equals 1 up to d wtj xt plus 1j.
[00:43:10.57] Is this OK? Now, take minus sign out and summation out. Then this is equal to
minus sum j equals 1 up to d, expectation wtj xt plus 1j. Is this OK? I just took the expectation.
And this is equal to what? Minus j equals 1 d. Take this out, wtj expectation xt plus 1j. And
going back to vector notation, this corresponds to equal to minus wt mu. Is this OK?
[00:44:21.99] Well, just to emphasize maybe mu t plus 1, it's better notation because mu most
probably will change with the time as well. So maybe I should mention here sigma t plus 1. OK,
now, covariance, OK, I should mention who knows this covariance matrix? Who doesn't know
the covariance matrix? Everyone knows the covariance matrix? OK.
[00:44:51.10] So then the covariance matrix, even I take the variance of this expression here,
covariance responds to exactly variance of this expression here since-- which corresponds to wt
prime, wt in this notation times wt in that with the covariance matrix of this, OK? Is this OK for
everybody? Maybe I should just mention a few things about variance and covariance, at least
variance of constant, c is constant. What is variance of constant?
[00:45:37.74] [INAUDIBLE]
[00:45:40.17] OK, everyone knows. Good. So suppose covariance of x plus z-- x plus c, sorry-y. X is random variable. C is constant, OK. Y is a random variable, OK? What does this equal
to? So this corresponds to-- right? Let's check it. This corresponds to x expected plus c y minus
notation of x plus c, expectation of y, which corresponds to expectation of xy plus c times
expectation of y minus expectation of x times expectation of y plus c times expectation of y,
which corresponds to expectation of xy plus c times the expectation of y minus expectation of x
times expectation of y minus c times expectation of y.
[00:47:05.04] Now, this cancels. This cancels, which corresponds to [INAUDIBLE] of xy minus
expectation of x times expectation of y, OK? So this expression disappears if this is a constant,
OK? Is this OK? Covariance, when I multiply this, x plus c times y, this is exactly by definition,
so-- OK, so then this is equal to xy. And I take c out, right? Minus-- then I have here a
multiplication, which is ex ey plus c times ey.
[00:48:03.71] Now, this was here, c times ey. These two cancel, and I have exactly this, which is,
by definition, covariance of xy, OK? Is this OK? Now, what if I have this covariance x1 plus x2
of y, OK? What if I have this? Then this corresponds to what? Expectation of x1 plus x2 times y
minus expectation of x1 plus x2 times expectation of y, which corresponds to expectation of x1 y
plus expectation of x2 y minus expectation of x1 of y minus expectation of x2 expectation of y.
[00:49:14.46] Well, this is equal to covariance of x1 y plus covariance of x2 y. So note the
difference. When a story is random, you cannot make it disappear like that one. When it's a
constant, you can make it disappear, OK? So back to this one here. When variance c is constant,
this equals 0, OK? How is it so?
[00:50:00.63] Covariance of c equals expectation c squared minus expectation of c squared,
which is c squared minus c squared is equal to 0. What's an expectation of a constant? A constant
itself, so it's 0, OK? Let's give a 10 minute break after this very [INAUDIBLE]-- maybe I should
mention-- OK, where were we here? Oh, OK, here. Oh, sorry. OK, technology wants us to give a
break. Let's give a break. [INAUDIBLE]
[00:50:33.37] [MUSIC PLAYING]
[00:00:00.00]
[00:50:43.12] All right. Continuing where we stopped, maybe I should also mention this
covariance matrix because you will not just see in this class. But covariance matrix is like if you
do probability, and if you deal with more than one in the variable, you will see it. Most probably,
you've already seen it. But covariance matrix is-- let's say you have, let's say, two by two. So that
means two random variables I have, two random variables, so which corresponds-- so this is the
covariance of two random variables, let's say, x1 and x2.
[00:51:41.40] This corresponds to covariance x1 x2, OK? The 2, 2, which corresponds to
covariance-- I'm sorry, this, what did I say? Covariance 1, 1, which corresponds to covariance
x1, x1. But what does covariance x1, x1 mean, by the way?
[00:52:06.79] [INAUDIBLE]
[00:52:07.78] Its variance, right? So it's just this exact variance 1. I am sorry. This is sigma 1,
covariance x1, x2. This is, to be more precise, sigma 1, 2, sigma 2, 1. Covariants x1, x2 equals
covariants x2, x1. Is this OK for everybody? Why? Because the expected of this corresponds to
x1, x2, minus expected x1, expected x2, which is exactly equal to-- just interchange these
numbers here, x1, x2 times x1, and interchange this.
[00:52:58.29] And this corresponds to the sigma 2, 2, OK? So this corresponds to variants of x2,
OK? And note that this is a symmetric matrix, OK? In fact, most of semi-definite matrix, but
that's not important, just symmetric because these two are equal to each other, OK? Anyway, so
when I say this, so you can take this as constant out, and you have this covariance matrix times
wt, which corresponds to this term. Now, let me continue.
[00:53:51.34] OK. OK, oops. Yes. Let's have an example. Consonant example from the famous
Black and Scholes formula-- of course, I won't go that much into detail [INAUDIBLE] it-- Black
and Scholes formula says-- this stands for Black and Scholes-- the prize-- so OK. We have a
stock st, maturity at time t, and exercise plus k, the Black and Scholes formula says that today's
value is this one where this is the norm distribution function of n 0, 1, OK?
[00:54:36.91] R is the continuously compounded risk-free interest rate. OK, while this one here
assumes r sigma to be constant, this is often not true in real markets. Hence, usually the risk
factors, zt, other than log st-- we also consider rt and sigma t-- that is to say your risk factor is,
again, a vector. you can take this rt to random and the volatility itself to random. That's what I'm
trying to say, OK?
[00:55:13.01] And so xt plus 1, this loss corresponds to the difference of this, two interest rates,
different of the sigma volatility term, but also this log. Recall low of st plus 1 of st equals log of
st plus 1 minus log of st. OK? Recall log price, OK? OK, now, this implies that the mapping f
has a term that rztzz details three fold terms, and the linearized one they had most is then given
by this term from our previous framework.
[00:56:12.16] Ft corresponds to the time derivative times delta t, plus gradient times the vector,
which corresponds to the derivative with respect to st, derivative with respect to rt, derivative
with respect to sigma t of these formulas here, OK? Now, if our risk management horizon is one
day, as opposed to one year-- for example, you need to introduce delta t as 1 over 250. Why is it
1 over 250, by the way?
[00:56:48.86] [INAUDIBLE]
[00:56:49.75] So that's 250 days. You just take 250 days. And note that the Greeks enter. And
this is the usual-- I think it is good for you to know this. Derivative of t is called the theta of the
option. Derivative of this formula with respect to st as the delta, and derivative of the interest rate
is called raw, and derivative of sigma d is called to vega, OK?
[00:57:17.96] Now, for portfolios of derivatives, this expression, this first term expression can be
a rather poor approximation. So we need higher order Taylor approximations, such as the deltagamma approximation. Note that for the gamma, for this term, you need a second-order
approximation, OK? That is to say this is very rough, very crude approximation, OK?
[00:57:49.92] Now, as I said, I won't go into detail of Black and Scholes formula, and equivalent
martingale process. This is a topic of another class. So this was just an example for you. Now,
variation methods, what are they? The fair value accounting-- the fair value of an asset liability is
an estimate of the price, which would be received, paid on an active market. We'll now
[INAUDIBLE] this mark to something.
[00:58:19.74] OK, what are these level one, mark-to-market? This is to say fair value of an
investment is determined from the quoted prices for the same instrument. Let's just say, example
2.2 in this slides. Martin model with object of inputs, the fair value of an instrument is
determined using quoted prices in active markets for similar instruments, or by using valuation
techniques models with inputs based on observable market data, OK? And Martin model, with
subjective inputs, which is the fair value of an instrument, is determined using valuation
techniques, models, for which some inputs are not observable in the market.
[00:59:02.40] For example, determining default risk of portfolios, of loans to companies, for
which no [INAUDIBLE] this. You see the s spreads r available, OK? Now, this risk-neutral
valuation is, of course, widely used for pricing financial products, for example, derivatives, as I
said, option pricing. He said these are like-- I think it's also used. It's also studied extensively in
other classes, like [INAUDIBLE] et cetera.
[00:59:38.73] The value of a financial instrument today equals expected discounted y. Is it what's
expected. Always keep in mind we are always dealing with randomness. Expected discounted
values of future cash flows-- the expectation is taken with respect to the risk-neutral price of
measure q equal to Martingale measure. Have you heard this term before? Equal of Martingale
measure? OK, from your previous classes, in terms of discount-- but as I said, I won't go into that
idea or failed price and arbitrage in this class.
[01:00:15.87] It turns discounted prices into Martingale, so fair bets, OK? And suppose to the
world, physical measure p, this is a change. It practically takes something and changed into a
Martingale. Martingale, who knows Martingale? Who doesn't know Martingale? Who doesn't
know Martingale? OK. OK, I will mention very briefly Martingales too.
[01:00:43.28] So and risk-neutral pricing measure is a probability q such that the expectation of
the discounted payoff with respect to the q equals 3, 0. Now, maybe you will get confused with
respect to et cetera. What I'm trying to say is the following. When I say expectation, I mean I
take expectation using a probability measure p, OK? OK, for example, expectation of x with p
means-- suppose x is [INAUDIBLE] x [INAUDIBLE] values 1 and 2 with probably, 1/2 half
with probability, 1/2, OK?
[01:02:04.40] When I say expectation of x, with this measure p, I mean 1/2 times 1 plus 1/2
times 2, which corresponds to 1/2 plus 1, which corresponds to 1.5, OK? But when I say another
measure q, and I change this 1/3 and this is to 2/3, OK? This is OK? And I change 2/3, then 1/3
times 1 plus 2/3 times 2, which corresponds to 1/3 plus 4/3, which is 5 over 3. And this
corresponds to now expectation with respect to q, OK?
[01:02:54.28] Same random variable in the sense that it's taking same values. But the
probabilities have changed, with respect to, so back to this expression here. A risk-neutral
pricing measure is a probability q such that the expectation of this discounted payoff random
variable in respect to q equals 3,0, OK? I changed the measure, and I take the expectation. Then
the fair value corresponds to that value, to that number. Is this OK? Is this OK for everybody?
[01:03:31.34] OK, now risk-neutral valuation at t of a client h, at time capital t, is done via the
risk-neutral pricing rule, which corresponds to p0h equals expectation qt of e to the minus r times
capital t minus t, OK? This corresponds to the discounting factor. I start at the time t, OK? H is
my payoff, but I don't know it, OK? Varying qt denotes expectation with respect to q given the
information up to and including time t.
[01:04:12.03] And whenever I say information, you understand the filtration, namely-- suppose I
have flow of information, f1, as I said before, f2, et cetera, OK? Suppose x1, random variable,
x2, random variable, et cetera, OK? When I say expectation, x1 plus x2, with the information f1- this is information f1-- I think I will do some rigorous Martingale too-- information, then this
corresponds to plus x1.
[01:05:20.67] Namely, you know it, you observed it, you observed it at time t1, OK? If you
observed it at time t1, you can take it out. That's what I'm trying to say. You know it. It's not
random anymore. At time t1, I know what it is. This expression here means exactly this, OK? Is
this OK?
[01:06:03.27] OK, this is the information or the sigma algebra information. This is the measure,
probability measure, that I am calculating expectations, OK? And here is the expectation. I take
the expectation. I take expectation, OK? Is this OK? Now, back to here, varying qt, they don't
[INAUDIBLE] the expectation with respect to q given the information up to and including time
t.
[01:06:45.95] Now, this probability p is estimated from historical data, whereas q is calibrated to
market prices, OK? Now, suppose that options with strike k or [INAUDIBLE] t are not traded,
but other options on the same stock are traded. Under p, the stock price st is assumed to follow a
geometric Brownian motion. Who doesn't know geometric Brownian motion? Everybody knows
geometric Brownian motion?
[01:07:27.00] OK, so it is assumed that it follows a geometric Brownian motion, the so-called
Black-Scholes model, with these dynamics. Dst equals mu-- stands for the drift term and for the
mean. Sigma stands for [INAUDIBLE] stdt. Dt is the time difference. And here is the stochastic
term, vwt, OK? For constant mu drift, and sigma [INAUDIBLE] tilde and the standard Brownian
motion. Under the equivalent Martingale measure q, e minus rtst is a Martingale. And st follows
the geometric Brownian motion with drift r and volatility sigma, OK?
[01:08:12.11] Now when I say e minus rtst is a Martingale, everyone understand what I say?
Everybody understand? OK, good. Now, the European co-option, for example, payoff is h. So st
minus k, maximum of-- that's to say whatever I take from the positive side-- and the risk valued
nature of [INAUDIBLE]-- as I said, I won't go into details of this-- is the price of this Black and
Scholes formula, which corresponds to exactly this, expectation, conditional expectation t with
respect to the risk-neutral equivalent Martingale measure q.
[01:08:55.90] And 1t, which corresponds, equals vt. And one important thing that you should
keep in mind is the following. To calculate this formula here, for different-- one uses quoted
prices. That is to say, whatever you have observed in the stock, you take this k star and t star,
OK? You know this at time t sdr. You take this k star. You observe this k star, this strike price,
and you observe this t star, and you infer the unknown sigma, OK?
[01:09:39.88] Then you have this formula here. Then to plug in this formula to infer the
unknown volatility sigma, OK? OK, this is to say you have a model in mind, the Black and
Scholes model, Black and Scholes world, OK? And you check the strike price k from the market.
You check it from the market. And you check the capital t.
[01:10:06.32] And from there, since you have already the formula to infer the sigma, OK? is this
OK? OK, this is called the implied volatility, by the way. OK, now having determined the
mapping f, it may involve valuation models like previously seen, like Black and Scholes
[INAUDIBLE] approximation. We can identify the following key tasks of the quantitative risk
management.
[01:10:35.37] Now, find a statistical model xt plus 1, namely typically a model for forecasting xt
plus 1 based on historical data. We will also see in the following weeks. We will do some time
[INAUDIBLE] analysis as well you will see exactly this estimation procedure and forecasting
procedure there. Again, the whole point is you just assume some models onto-- because in when
you are in industry, you don't see any math. You only see numbers, OK? There are those floating
numbers. And you just make some models based on the numbers, period, OK?
[01:11:21.25] And we should compute a driver distribution function for the loss. For that, we
need to require this xt plus 1. And we should also compute a risk measure from this loss capital f
lt 1, lt plus 1, OK? Now, there are three general methods to approach, to approach these
challenges. First idea is choose a distribution function f and small f such that the loss f at time lt
plus 1 can be determined explicitly.
[01:12:02.07] Now, one of the prime examples is the variance, covariance method. Now, we can
check this risk matrix. Just Google it. Now, for example, assumption one is this xt plus 1 is
normal with mean mu and with the covariance matrix sigma. For example, if the t is a Brownian
motion and the st, the stock is a geometry problem in motion, assumption two is that this
approximation, this approximation here, first order approximation, is a good approximation, OK?
That is to say recall our approximation was of this form, which by this assumption implies that I
have this expression here is normal with mean minus ct minus this term plus vt sigma bt.
[01:13:00.55] Now, where does this come from? If I have this, then I say this. Why is it so?
Well, if this is my random variable-- sorry-- if this is my in the vector, and if I say these terms
here is normally distributed, it's a multivariate normal distribution, multivariate Gaussian
distribution, because Gaussian and normal the same, with mean, mu, and the covariance
[INAUDIBLE] sigma, then that means if this is with this distribution, then-- OK, let's use what
we have said in the first class.
[01:13:59.88] OK, by assumption, I know xt plus 1 is normally distributed with mu and sigma,
OK? Lt plus 1 equals minus ct plus bt dot product xt plus 1, OK? Then if I have this, then a
linear transformation of this, then lt plus 1, being a linear transformation of xt plus 1, is normally
distributed as well, OK? OK, so then expectation of lt plus 1 equals expectation of minus ct plus
bt xt plus 1, which is equal to-- since this is constant, this is another constant, OK?
[01:15:36.51] So this is equal to minus ct plus bt prime, expected of xt plus 1. Well, this is equal
to mu t, right? Everybody agrees on this? When I have lt plus 1 and when I take the expectation,
I can take the ct out.
[01:16:02.08] And this is a constant. I take this constant out, OK? But it is exactly minus ct plus
bt dot product mu t. That is to say, minus ct-- I'm sorry, this is a minus sign here-- minus ct,
minus bt prime mu. And when I take the variance, this constant disappears, when I take the
variance. And variance of pt prime xt plus 1 is exactly bt dot product sigma bt, because sigma is
the variance of this expression here, OK?
[01:16:44.74] Now, advantage? Of course, it is explicit, easy to implement. But there is a
drawback, which is a big drawback. Assumption one is unlikely to be realistic, for daily,
probably also weekly, monthly data. Let's say, if you just decrease the time interval
[INAUDIBLE] facts about xt plus 1 suggests that it is [INAUDIBLE] namely [INAUDIBLE]
heavier tail. What does heavier tail mean? Recall, we mentioned it last time. For example, we
visually want to see it.
[00:00:00.00]
[01:17:35.74] For example, think about the standard normal distribution, OK? Heavier tail means
something like-- I'm sorry. So if you have this for normal, heavier tail means you have
something more here, maybe something more. So here, in the remaining parts, this is a
distribution 2, distribution 2, distribution 2. Here, the tails stay contained more compared to if
this is normal, for example, heavier tail. It has more volume in the tails, OK? It has more
volume, OK?
[01:18:28.90] So it is like if you have-- the whole tank of water is 1 litre. That's for sure. The
whole probability is 1, but it depends on how much you put the water inside this tank, OK?
Heavier tails means the tails have more volume, have more water, OK? That's it. OK, so it is
thinner body, but heavier tail. It underestimates the tail of flt plus 1. And thus, risk measures
such as value at risk, we will come to that very shortly.
[01:19:03.99] When dynamics models are 4xt plus one are considered, different estimation
methods are possible, depending on whether we focus on conditional distribution or the
equilibrium distribution in a stationary model. Stationary model means it does not depend on t.
As you can see, for example, you can change. You can say x in a station model means you just
don't ignore the time t, OK? So what is a historical simulation?
[01:19:36.89] OK, historical simulation means estimate the distribution function by its empirical
distribution function, OK? Now, to understand this, you can consider what does this-- OK, what
if I take n, go to infinity? This decreases, this increases, right? Right? That is to say, the more I
get data i, the more I get data, I will be more accurate to estimate the distribution function. Is this
OK?
[01:20:22.12] This will give me a more and better accuracy, and an estimation of the real flt plus
1, OK? So then based on that, you can consider this function l as f of this function, where this
expressions here show that what will happen to the current [INAUDIBLE] folder, if the past and
risk factors were changed, were requeued. That is to say, not that I start from t-- I'm sorry. The
most I can go is I equals 1 t minus 1 plus 1. It is t.
[01:21:11.27] But I go to the back, OK? I got from the past, I realize the empirical distributions. I
realize for which of them these losses were less than x. But I only looked to the back. Everybody
sees that? Because from i equals 1 means t. For example, i equals 2 means t minus 1. I equals 3
means t minus 2, et cetera. I go to the back. Everybody sees that?
[01:21:44.90] Now, show that what would happen to the current portfolio if the past and risk
factors change. Well, to recur, advantage is it's very easy to implement because you have all the
data, no estimation of the distribution required, which is perfect. But drawback is sufficient data
is required. Why? Because I only depend on data, so I should increase n, OK? Because to make
it as accurate as possible, I need more and more data, right? Is this OK?
[01:22:15.33] But another thing is only past losses considered. That is to say, I look at the past. I
depend on the past a lot to calculate, model the future. But well, it is a little exaggerated. But
driving a car by looking in the back mirror-- that is to say, whatever happened in the past will
happen in the future. Well, it's too much of a big estimation, too much of an assumption, right?
OK.
[01:22:46.96] Well, [INAUDIBLE] I will just skip this very-- just one slide. We have another
class in that Monte-Carlo methods. Take any model for xt plus 1, simulate xt plus 1, compute the
corresponding losses as in before, and estimate this. Eventually, it's quite general, applicable to
any model, which is easy to sample. It's drawback is unclear how to find the appropriate model
for xt plus 1. Any result is only as good as the chosen fxt plus 1.
[01:23:18.47] And the computational cost is-- every simulation requires [INAUDIBLE] the
mapping f. Expensive-- for example, if the latter contains derivatives, which are priced via
Monte-Carlos themselves, so like nested Monte-Carlo simulations, well, the so-called economic
generators, economic motivated dynamic models for the evolution and interaction of risk factors
used in insurance also fall under the heading of Monte-Carlo methods. But as I said, I will not go
into details of this. I emphasize that we have another class for this specific thing, for this specific
concept.
[01:23:56.18] Now, risk measurement. OK, a risk measure for a financial position with random
loss, not that our loss is random, is a real number, which measures the riskiness of l, OK? In the
Basel or solvency contacts, it is often interpreted as the amount of capital required to make a
position with loss l acceptable to an internal/external regulators, OK? Now, some reasons for
using these measures in practice r, first one is to determine the amount of capital to hold as the
buffer against unexpected future losses on a portfolio, which is in order to satisfy a regulator
manager concerned with the institution's solvency.
[01:24:49.65] Another reason is it is used as a tool for limiting the amount of risk of a business
unit, for example, by requiring that the daily 95% value at risk, namely 95% quantile of a trader's
position should not exceed a given bound, and to determine the riskiness and thus failed
premium of an insurance contract, OK? The existing risk measurement approach are grouped
into three categories. First off them is the notional amount approach, which is the oldest
approach, sometimes the approach of Basel II.
[01:25:48.05] For example, in op risk, you can check it online, the Basel 2 risk of a portfolio is
summed notation, notation of values of the securities times the riskiness factor. It's advantage is
its simplicity. Its drawback is no differentiation between long and short positions and non-acting
in the sense that there is [INAUDIBLE] long position in corporate bonds hatched by an
offsetting position in credit [INAUDIBLE] is counted as twice the risk of the [INAUDIBLE]
bond position, OK?
[01:26:23.13] But this is another drawback of this. No diversification benefits, diversification
namely risk of a portfolio of many loans to many companies equals risk of a portfolio where the
whole amount is lent to a single company, OK? That is to say it just assumes that I have some
amount of money. Then I put all my money into stock A, OK? When I put some of my money
into stock A and some of my money into stock B, and when I evaluate the risk of it, then it
should be less than or equal to the whole amount adjusted with the risk that I put all my money
into stock a and all my money into stock b, OK?
[01:27:20.37] It should be less than that, but it just doesn't take into this account, OK? Problems
for portfolios of derivatives is notional amount of the underlying [INAUDIBLE] widely different
from the economic value of the derivative position. Now, the second one is the risk measures
based on loss distributions, OK? Most model risk measures are characteristics of the underlying
loss distribution or some predetermined time, [INAUDIBLE] delta t.
[01:27:54.06] Now, examples, variance, value [INAUDIBLE] can expect a shortfall, we will see
in this class, and in this actually next class in the next week. Advantages makes sense on all
levels from single portfolios to the overall position of a financial institution. Loss distributions
reflect neglecting, and they also obey diversification, OK? Estimates of loss distributions, as we
saw before, are typically based on past data. It is difficult to estimate loss distribution accurately,
especially for large portfolios, OK?
[01:28:35.67] Risk measures should be complemented by information from scenarios, from
forward looking, OK? Now, what are scenario-basis measures typically considered in-- we also
can have it typically considered in risk measures, these scenario basis measures. One considers
possible future risk factors, OK? For example, we make up a scenario, OK, where scenarios, for
example, risk testing means something really bad happens, something extreme, for example, a
20% drop in a market index, OK? Risk of a portfolio is maximum [INAUDIBLE] loss under all
scenarios, OK?
[01:29:26.77] For example, I make x to be denoted by x1 through xl. I denote the risk factor
changes scenarios with the corresponding y's, OK? The risk is then maximum of this. That is to
say I only consider among the first one, OK? I put some weight to the extreme scenarios, and I
check the risk from it, OK?
[01:29:55.88] Recall that this gives me a number. Maximum of this is the risk of my position, of
my future, OK? OK, it might happen that-- OK, whether I have insured that guy, that guy's house
might burn, or the market fall by 30%, et cetera. So I make these scenarios, and I calculate the
worst of it, OK? [INAUDIBLE] l of loss of x denotes the loss par the loss. The portfolio will
suffer if the hypothetical scenario x were to occur. Numerous measures are of this form, and you
can check online, this CME span, which is the standard portfolio analysis of risk, 2010.
[01:30:39.22] Well, what is the mathematical interpretation of this? Assume as 0 loss at 0 is 0,
OK? It is OK if delta t is small, that is to say, when I start at time 0. And in wi, the weight is
between 0 and 1. Namely, i, for each weight, I make it mathematically means convex
combination of it, OK? I put some weights to wi, some weights to w2, which are all numbers
between 0 and 1. When I add them up, it gives me 1, number 1, OK?
[01:31:18.14] And this wi lxi is exactly-- if the l0 equals 0, then what it says here is actually the
following. wi of l xi equals wi lxi plus 1 minus wi times l0, since we assume l0 equals 0, OK? So
then this corresponds exactly expected of pi of l of xi, which corresponds where we define this
measure pi as wi delta xi plus 1 minus wi delta 0. This is called Dirac measure at xi.
[01:32:44.91] It means del xi of x means I put all my weight. It means I put all my weight to a
single point xi. That's what it means, OK? That is to say when you see xi, then I put all my
weight there, OK? For example, mathematically, this means the following.
[00:00:00.00]
[01:33:40.03] Suppose I have a function. Take the real valued function, value to function f of x.
So I integrated in front of minus infinity to the infinity, OK? F of x, f is from r to r, OK? So then
f of x delta x0, x dx equals f of x 0. [INAUDIBLE], it just takes that point and period, OK? This
means that, OK? That's all.
[01:34:26.28] Of course, you put some assumptions. F is integrable, namely when you take the
integration, it takes [INAUDIBLE] et cetera. But let's just say that we have this minus infinity to
infinity f of x, meaning I take the absolute value. It is finite plus-- let's just even say f of x is
continuous, and bounded, just to make this simple, continuous bounded. Then I take the integral.
It is finite, OK?
[01:34:57.36] So then when I apply this direct delta operator-- it's called direct delta-- at x0, it
gives me practically just choosing that fx 0, OK? So when I defined this, pi of this sort, then it
means I only take xi, OK? Therefore, this one, since it is maximum of this, is maximum of the
expectation of that. This is to say since such a risk measure is known as a generalized scenario,
and it is [INAUDIBLE] to play an important role in the theory of [INAUDIBLE] measures-- we
will see that shortly-- which just sees the worst that can happen, OK?
[01:35:42.27] Practically, this is all it is, OK? I calculate everything, and I just check first off
what can happen, OK? Now, advantages are useful for portfolios with few risk factors, useful
complementary information to risk measures based on loss distributions past data. But drawback
is how they determine the scenario and how they determine the weight, OK? For that, you need,
again, some future imagination.
[01:36:15.22] OK, now where you are at risk that we are also mentioning in the previous slides
value at risk-- for a loss, for a loss a random variable l, distributed as fl, value at risk at
confidence in level alpha is defined by value at risk alpha l just for the definition infimum of x in
r such that fl x is greater than or equal to alpha, OK?
[01:37:21.83] Since I checked my costs, this is like if my-- what does it mean? For example,
recall-- suppose you have this distribution. So let's say your l distribution is positive, not
necessarily negative, but positive, OK? Suppose this is alpha. Say alpha equals 0.95. Alpha is
between 0 and 1, OK?
[01:37:52.10] Then the whole volume of the tank is 1 litre, right? So this is like 0.95 litre. This
number here is x0. OK? Well, this is also called value at risk alpha for l. This is the density
function, density of l, l standing for loss, OK? This number here x0 is called value at risk alpha,
which is a quantile 0.95 quantile, OK? That is to say just check it when this loss exceeds this x
with 0.95%, loss l exceeding that number x with 0.95 with alpha here, like an alpha percent, OK?
[01:39:05.57] Whenever my loss exceeds 0.95%, then I am in trouble, OK? So that number x is
my signal for something bad, OK? This is to say loss l is more probable. I should do something,
OK? In truth, it means that. I repeat this flx recall probability of l less than or equal to x. This
means that is greater than or equal to alpha, OK? Recall this probability, this cumulative
distribution function is between 0 and 1, right?
[01:39:59.64] So when I increase x, I increase x from the first lecture, this number will be
increasing, right? And I increase x, flx is increasing. In the exagerated case, it is 1, right? When I
increase x to the n-th, it is 1. So when I pass this border, alpha, that means you are in trouble,
OK? This is like a red light, OK? You are in trouble.
[01:40:25.87] That means that, OK? So as I have mentioned, this 415 report from last week,
recall this is exactly what JP Morgan and risk metrics are using. It is known since 1994. It is
called Weatherstone 415 report, and; it is the most widely used risk measure by Basel II or
Solvency II. And we will see the properties of these and the remaining risk measures in the next
lecture Thursday as well, and also the week after that, OK? Thanks for your time.
[01:41:11.26] [MUSIC PLAYING]
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