1
00:00:00,000 --> 00:00:05,205
So at this point now I wanna walk through
a, a, a fairly lengthy example to

2
00:00:05,407 --> 00:00:10,410
illustrate this very important risk
measure that's known as value at risk.

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00:00:10,410 --> 00:00:18,326
And, valiant risk is, is, It's a risk
measure that is answering the question.

4
00:00:18,326 --> 00:00:22,394
How much money could you lose with a
certain probability?' Okay?

5
00:00:22,394 --> 00:00:25,544
And so, that's a very natural form of
risk, right?

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00:00:25,544 --> 00:00:30,465
You have an investment, you have your
money at risk, you wanna know how much

7
00:00:30,465 --> 00:00:34,008
money could you lose with a certain
probability, right?

8
00:00:34,008 --> 00:00:37,617
And, and so value at risk gives you
exactly that number.

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00:00:37,617 --> 00:00:42,407
So, a value at risk number might be
something like $1,000, and so you might

10
00:00:42,407 --> 00:00:47,000
say with five percent probability, if I
start with a $10,000 investment, I could

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00:00:47,000 --> 00:00:48,050
lose $1,000.
Okay?

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00:00:48,050 --> 00:00:51,334
And, and that's a nice way to talk about
risk because it.

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00:00:51,334 --> 00:00:54,208
You know, it, it hits you right in the
pocket book.

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00:00:54,208 --> 00:00:58,080
It says you know, in, in dollar terms.
You know, what, what's at stake.

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00:00:58,257 --> 00:01:02,528
And this is in contrast to.
From a statistical point of view, we often

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00:01:02,528 --> 00:01:06,562
characterize risk as the standard
deviation over the rate of return.

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00:01:06,562 --> 00:01:11,070
And a lot of people just don't have
intuition about what standard deviation

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00:01:11,070 --> 00:01:13,443
is.
And so even though it turns out that

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00:01:13,443 --> 00:01:18,366
standard deviation and value at risk are
very closely related to one another, value

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00:01:18,366 --> 00:01:23,052
at risk being a dollar loss number is
something that most people, understand

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00:01:23,052 --> 00:01:24,642
much more clearly.
Alright.

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00:01:24,642 --> 00:01:28,954
So where does value of risk come from, and
how do we compute it?

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00:01:28,954 --> 00:01:31,893
So we're gonna walk through a simple
example.

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00:01:31,893 --> 00:01:36,923
We start with a $10,000 investment on
Microsoft and, or Investment Horizons in

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00:01:36,923 --> 00:01:39,275
one month.
So, we buy Microsoft today.

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00:01:39,275 --> 00:01:43,717
We put $10,000 that might buy, you know
100 shares of Microsoft stock.

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00:01:43,717 --> 00:01:48,747
We hold it for a month and we sell it.
And we wanna know, how much can we lose

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00:01:48,747 --> 00:01:53,320
with say, five percent probability, right.
So that would be known as the five percent

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00:01:53,320 --> 00:01:57,740
value of risk or the five percent bar.
Now we're gonna get to that in a sa,

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00:01:57,740 --> 00:02:02,201
slightly systematic way.
Now the random variable of interest is the

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00:02:02,201 --> 00:02:07,048
monthly rate of return, that's the
percentage change in price over the month,

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00:02:07,048 --> 00:02:12,201
and we're gonna assume that the monthly
rate of return has this following normal

33
00:02:12,201 --> 00:02:15,698
distribution, okay?
So the, the mean of the monthly rate of

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00:02:15,698 --> 00:02:20,054
return is gonna be five percent and the
standard deviation is gonna be ten%, all

35
00:02:20,054 --> 00:02:22,140
right?
So mu R is.05, sigma R is.10.

36
00:02:22,140 --> 00:02:25,993
Okay.
And the goal is to calculate how much we

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00:02:25,993 --> 00:02:29,847
can lose with the specified probability
alpha.

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00:02:29,847 --> 00:02:34,960
And alpha is going to be some number.
It's usually five percent or one%.

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00:02:37,520 --> 00:02:40,480
Okay?
Now, we're gonna approach this issue in a

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00:02:40,480 --> 00:02:44,467
couple of intermediate steps.
And, and partially this is just to

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00:02:44,467 --> 00:02:49,541
illustrate you know, working with linear
functions and random variables and doing

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00:02:49,541 --> 00:02:54,374
probability calculations and so on.
And then another goal of this is to build

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00:02:54,374 --> 00:02:59,207
more of the intuition about where the
value at risk number that we come up with

44
00:02:59,388 --> 00:03:02,590
comes from.
So, some questions that we're gonna answer

45
00:03:02,590 --> 00:03:07,423
in this example are the following: so,
what's the probability distribution of end

46
00:03:07,423 --> 00:03:12,971
of month wealth?
So end of month wealth is W1, that's our

47
00:03:12,971 --> 00:03:16,832
initial wealth.
Times one plus the rate of return, so

48
00:03:16,832 --> 00:03:19,706
that's the future value of our
investment.'Kay?

49
00:03:19,706 --> 00:03:24,781
And so notice that end of month wealth is
a linear function of R, and R is a normal

50
00:03:24,781 --> 00:03:29,550
random variable, so we know that end of
month wealth is also gonna be a normal

51
00:03:29,550 --> 00:03:34,320
random variable, and we, we just wanna
calculate its mean and its variance.'Kay?

52
00:03:34,320 --> 00:03:39,168
Once we have the distribution for end of
period wealth, then we can answer

53
00:03:39,168 --> 00:03:44,540
questions like, what's the likelihood that
our end of period wealth is less than a

54
00:03:44,540 --> 00:03:45,916
certain amount.
Right.

55
00:03:45,916 --> 00:03:51,222
So this is if we start with $10,000 then
this question is saying what's the

56
00:03:51,222 --> 00:03:54,040
likelihood that we could lose $1000 or
more.

57
00:03:54,040 --> 00:04:00,295
Right.
And then we can ask, another question.

58
00:04:00,295 --> 00:04:06,926
What is the rate of return that produces
our end of period wealth equal to $9,000?

59
00:04:06,926 --> 00:04:10,938
Right?
In percentage terms, you know, what, in

60
00:04:10,938 --> 00:04:17,182
saying, what is the, So, and what rate of
return produces this end of period wealth?

61
00:04:17,182 --> 00:04:22,476
And then finally, you wanna ask, what is
the monthly value at risk on a $10,000

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00:04:22,476 --> 00:04:25,917
investment?
That is, how much could we lose, if the

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00:04:25,917 --> 00:04:29,160
rate of return is, less than its five
percent quantile?

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00:04:30,780 --> 00:04:37,629
Okay, so let's walk through this here, So
our wealth, end of period wealth is our

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00:04:37,629 --> 00:04:41,061
initial investment times one plus the rate
of return.

66
00:04:41,061 --> 00:04:45,012
So that's just the future value of our
initial investment.

67
00:04:45,012 --> 00:04:49,091
It's a linear function of R.
R is normally distributed, so W1 is

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00:04:49,091 --> 00:04:51,876
normally distributed.
What's the mean of W1?

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00:04:51,876 --> 00:04:56,603
Well, it's the expectation of W1.
It's the expectation of 10,000 times one

70
00:04:56,603 --> 00:05:01,460
plus R, and using the rules of the
linearity of expectation, that's equal to

71
00:05:01,460 --> 00:05:06,576
10,000 times one plus the expected return.
The expected return is five%.

72
00:05:06,576 --> 00:05:10,769
So, our expected future wealth.
Is equal to, $10,500.

73
00:05:10,769 --> 00:05:12,052
So, right.
If we.

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00:05:12,052 --> 00:05:16,588
On average if we expect to get a five%.
Rate of return.

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00:05:16,588 --> 00:05:20,439
Then our.
Future wealth is expected to be $500

76
00:05:20,439 --> 00:05:23,520
higher than our initial wealth.
Okay?

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00:05:23,520 --> 00:05:26,263
What about the variance of our future
wealth?

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00:05:26,263 --> 00:05:30,957
Well the variance of our future wealth is
equal to our initial wealth square.

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00:05:30,957 --> 00:05:36,129
Times the variance of the rate of return.
And so that's equal to 10,000 square

80
00:05:36,129 --> 00:05:39,628
times, .1 squared.
So that's, that's a $1,000,000.

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00:05:39,628 --> 00:05:44,607
Now notice that the variance of
inter-period wealth is in units of wealth

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00:05:44,607 --> 00:05:49,855
squared, so this is dollars squared, so
this doesn't have a good interpretation.

83
00:05:50,056 --> 00:05:55,439
We take the standard deviation, we take
the square root of this, it turns out the

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00:05:55,439 --> 00:05:59,140
standard deviation of our inter-period
wealth is $1,000.

85
00:05:59,900 --> 00:06:03,326
And that's in the same units as, as our
wealth.

86
00:06:03,326 --> 00:06:09,201
So, wealth distribution is normal and has
mean 10500 and has a standard deviation of

87
00:06:09,201 --> 00:06:12,068
1000.
Okay, again this just using a linear

88
00:06:12,068 --> 00:06:16,404
properties of random variables and the
packet stuff is normal.

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00:06:16,404 --> 00:06:21,929
Alright, so now that we know that this
distribution of'NF' period wealth, we can

90
00:06:21,929 --> 00:06:27,104
answer questions like what's the
probability of'NF' period wealth is less

91
00:06:27,104 --> 00:06:31,231
than 9000 dollars.
Well we just evaluate, this that you want

92
00:06:31,231 --> 00:06:38,007
to know what is This is [inaudible] what's
the area under the normal curve to the

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00:06:38,007 --> 00:06:44,779
left of $9,000 where the normal curve has
this mean and this standard deviation,

94
00:06:44,779 --> 00:06:47,350
okay?
So, we would, draw this.

95
00:06:47,350 --> 00:06:53,951
Here's our normal distribution and our
mean, 10,500, and we wanna know, here is

96
00:06:54,208 --> 00:06:56,865
$9,000.
What is this probability?

97
00:06:56,865 --> 00:07:02,780
This is the probability that end of period
wealth is less than $9,000.

98
00:07:02,780 --> 00:07:06,717
'Kay?
And so we can use say, the Excel NORMDIST

99
00:07:06,717 --> 00:07:13,445
function so we put in $9000, we put in the
mean, the standard deviation, turns out

100
00:07:13,445 --> 00:07:20,160
that this is 6.7%.
So this area right here is 0.067.

101
00:07:21,540 --> 00:07:27,420
So, there's a 6.7 percent chance that our
end of period wealth is less than $9000.

102
00:07:33,040 --> 00:07:37,483
What is the rate of return that produces a
$9,000 loss?

103
00:07:37,483 --> 00:07:41,070
Okay?
The rate of return that produces a $9,000

104
00:07:41,070 --> 00:07:44,422
loss.
Well, if end of period wealth is $9,000,

105
00:07:44,422 --> 00:07:48,710
this is our initial wealth is $10,000
divided by $10,000.

106
00:07:48,710 --> 00:07:51,985
So if we lose ten percent of our
investment, right?

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00:07:51,985 --> 00:07:57,208
We start with $1000, we lose ten%.
That means we've lost $1000, and what

108
00:07:57,208 --> 00:08:02,774
we're left with is $9000.
So, the rate of return now produces this

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00:08:02,774 --> 00:08:06,157
end of period wealth is -ten%.
Kay.

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00:08:06,157 --> 00:08:12,617
What is the likelihood, you know, that
we'll loose more than $9,000?

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00:08:12,617 --> 00:08:16,991
We already calculated that.
It was 6.7%, right?

112
00:08:16,991 --> 00:08:23,352
So another way of saying that is, the 6.7
percent quantile of the return

113
00:08:23,352 --> 00:08:27,747
distribution is -ten%.
So, if we think of the return

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00:08:27,747 --> 00:08:31,020
distribution.
Here.

115
00:08:31,520 --> 00:08:43,700
This has mean 0.05 and if we have, -1.0,
then the area here is -0.067.

116
00:08:43,980 --> 00:08:51,460
So this is the.067 quantile of the return
distribution.

117
00:08:52,560 --> 00:08:59,378
So that means the probability that the
return is ten, is ten%, -ten percent or

118
00:08:59,378 --> 00:09:07,202
less is 6.7%.
Alright, now in a valiant risk context

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00:09:07,202 --> 00:09:12,742
what we do is we specify our probability.
We want to know with five percent

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00:09:12,742 --> 00:09:16,380
probability, you know, how much could we
lose?

121
00:09:16,380 --> 00:09:20,322
Alright, so.
With five percent probability, what is the

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00:09:20,322 --> 00:09:24,294
five percent quantile of the return
distribution, right?

123
00:09:24,294 --> 00:09:29,276
So the five percent quantile return
distribution, we use the quantile

124
00:09:29,276 --> 00:09:34,980
function, we take the inverse CDF of the
normal at the five percent probability

125
00:09:34,980 --> 00:09:41,046
level with the mean and standard deviation
of a return distribution, and we see that

126
00:09:41,046 --> 00:09:44,440
the five percent quantile turns out to be
11.4%.

127
00:09:45,600 --> 00:09:49,320
So if we go to our distribution up over
here.

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00:09:50,580 --> 00:09:58,843
If we want area, this blue area.
To be five percent, then this quantile

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00:09:58,843 --> 00:10:08,663
here is minus zero point one, one four.
So what this means is, with five percent

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00:10:08,663 --> 00:10:16,900
probability or five percent of the returns
are less than 11.4%.

131
00:10:17,420 --> 00:10:22,315
'Kay.
Now, if the return is equal to this

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00:10:22,315 --> 00:10:28,000
quantile, so if the return is -11.4%, how
much money do we lose?

133
00:10:28,000 --> 00:10:32,638
Right.
We lose our initial investment, times the

134
00:10:32,638 --> 00:10:35,894
rate of return.
We lose $1,144, 'kay.

135
00:10:35,894 --> 00:10:40,138
So this is known as the five percent value
at risk.

136
00:10:40,138 --> 00:10:45,960
We calculate the five percent value at
risk by looking at the five percent

137
00:10:45,960 --> 00:10:50,894
quantile of our return distribution, which
is -11.4%.

138
00:10:50,894 --> 00:10:56,815
So we know that the return can be -11.4
percent or smaller, with five percent

139
00:10:56,815 --> 00:11:00,955
probability.
And then we say our dollar loss is our

140
00:11:00,955 --> 00:11:04,441
initial investment times that negative
return.

141
00:11:04,441 --> 00:11:07,876
Okay.
Now, notice when we calculate [inaudible]

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00:11:07,876 --> 00:11:13,915
risk, this is a, a conservative figure.
Because we could lose $1,144 or more with

143
00:11:13,915 --> 00:11:17,120
five percent probability.
Because with five percent, percent

144
00:11:17,120 --> 00:11:22,562
probability, we're here, but we can also
have returns less than that, right?

145
00:11:22,562 --> 00:11:28,154
So this is kind of a conservative number.
When we say five percent probability, it's

146
00:11:28,154 --> 00:11:33,670
not exactly five percent probability.
We're giving ourselves this, this, upper

147
00:11:33,670 --> 00:11:36,130
bound [inaudible].
Or, sorry, sorry.

148
00:11:36,130 --> 00:11:41,509
Lower bound on the loss.
>> Okay.

149
00:11:41,509 --> 00:11:48,918
>> So in general when you calculate value
at risk on an investment the, alpha times

150
00:11:48,918 --> 00:11:55,206
100 percent value at risk for an initial
investment of W dollars and we use this

151
00:11:55,206 --> 00:12:02,185
notation capital V, little a, capital R
and the value at risk is often just called

152
00:12:02,185 --> 00:12:05,803
the VaR.
It's equal to your initial dollar

153
00:12:05,803 --> 00:12:11,920
investment times the alpha quartile of
your simple return distribution.

154
00:12:13,120 --> 00:12:16,580
So this is why quantiles are very
important in finance.

155
00:12:16,580 --> 00:12:21,298
Because when we're calculating risk
measures, for example value at risk, the

156
00:12:21,298 --> 00:12:25,010
value at risk depends on the quantile of
your distribution.

157
00:12:25,010 --> 00:12:30,040
The example that we did here, we computed
this quantile by assuming that the rate of

158
00:12:30,040 --> 00:12:32,794
return followed the normal distribution,
right?

159
00:12:32,794 --> 00:12:36,267
Remember, the normal distribution has a
certain tail shape.

160
00:12:36,267 --> 00:12:40,818
If we use a student's t distribution, then
we would get a different quantile.

161
00:12:40,818 --> 00:12:45,310
We would get a quantile that's generally
larger in, in absolute value.

162
00:12:45,310 --> 00:12:46,985
Okay.
So we calculate Bar.

163
00:12:46,985 --> 00:12:51,142
It's our initial investment times
quantaltive distribution.

164
00:12:51,142 --> 00:12:56,638
And whatever distribution we assume then
determines what this, this quantal number

165
00:12:56,638 --> 00:12:59,003
is.
Very often, and, and this is true in

166
00:12:59,003 --> 00:13:03,146
practice in the real world.
People just assume that rates of return

167
00:13:03,146 --> 00:13:08,216
are normally distributed, and that gives
them an easy way to compute that quantile.

168
00:13:08,401 --> 00:13:13,595
And what we'll find when we look at data.
The normal distribution often isn't a very

169
00:13:13,595 --> 00:13:17,737
good assumption for the data.
And so if we use a normal quantile, we

170
00:13:17,737 --> 00:13:20,196
will often underestimate the loss.
Right.

171
00:13:20,196 --> 00:13:25,075
Because if the, if the true distribution
has fatter tails than the normal, then the

172
00:13:25,075 --> 00:13:28,703
five percent quantile of the true
distribution is gonna be a smaller

173
00:13:28,703 --> 00:13:31,500
negative number, than the quantile for a
normal.

174
00:13:31,500 --> 00:13:38,080
And then just to illustrate that point.
So, if this is the normal distribution.

175
00:13:38,640 --> 00:13:42,600
Kay, here's the five percent quantile of
the normal.

176
00:13:43,300 --> 00:13:46,570
Say this is equal to, you know, minus
11.4%.

177
00:13:46,570 --> 00:13:52,592
If you have a distribution with fatter
tails than normal, then you're gonna get a

178
00:13:52,592 --> 00:13:59,726
distribution that's gonna look like this.
The five percent quantile of this

179
00:13:59,726 --> 00:14:06,220
distribution.
You know it might be something like minus

180
00:14:06,220 --> 00:14:09,375
20.6%.
So when you have a fatter tail

181
00:14:09,375 --> 00:14:16,474
distribution from the normal the left tail
quantiles are bigger negative numbers.

182
00:14:16,474 --> 00:14:22,087
So Valiant risk is based upon taking a
quantile from a distribution.

183
00:14:22,087 --> 00:14:28,818
And the previous example, we, assumed that
the simple rate of return was normally

184
00:14:28,818 --> 00:14:33,044
distributed.
And then that gave us the computation for,

185
00:14:33,748 --> 00:14:40,244
the quantile and the valiant risk number.
Now, at the beginning of lecture, I gave

186
00:14:40,244 --> 00:14:45,958
you an example about why the normal
distribution may not be good for the

187
00:14:45,958 --> 00:14:47,445
simple return.
Okay?

188
00:14:47,445 --> 00:14:50,747
So.
And, and I gave an example to show that

189
00:14:50,747 --> 00:14:55,786
the normal distribution may be more
appropriate for the continuously

190
00:14:55,786 --> 00:15:00,021
compounded return.
So let's go back and do the valued risk

191
00:15:00,021 --> 00:15:05,717
computation, where we assume that the
continuously compounded return follows a

192
00:15:05,717 --> 00:15:11,048
normal distribution, and then deduce what
is the value at risk under this

193
00:15:11,048 --> 00:15:14,627
assumption.
Because when we, when the continuously

194
00:15:14,627 --> 00:15:20,056
compounded return is Normally distributed,
one plus the simple return has a log

195
00:15:20,056 --> 00:15:24,620
normal distribution, and so that impacts
the quantile that we use in the

196
00:15:24,620 --> 00:15:27,220
computation.
So let's, see what happens.

197
00:15:27,960 --> 00:15:34,920
So we have so we start with the assumption
that the continuously confounded return is

198
00:15:34,920 --> 00:15:38,440
normal with some mean and standard
deviation.

199
00:15:39,200 --> 00:15:45,150
Then the process that we go through for
computing the value at risk is the

200
00:15:45,150 --> 00:15:50,783
following: we compute the quantile based
on the continuously compounded

201
00:15:51,021 --> 00:15:52,450
distribution.
Okay?

202
00:15:52,450 --> 00:15:58,876
So the we know that, well, I, I stated as
a result previously, that if a variable

203
00:15:58,876 --> 00:16:05,382
has a normal distribution, then the alpha
percent quantile of that distribution is

204
00:16:05,382 --> 00:16:10,540
equal to the mean plus the standard
deviation, times the quantile.

205
00:16:10,540 --> 00:16:16,991
You actually have a homework ex, example,
where you, you, you show this result.

206
00:16:16,991 --> 00:16:22,981
But you can deduce this from just a,
standardization process of, of a random

207
00:16:22,981 --> 00:16:26,514
variable.
So anyway, so the, quantile, we compute

208
00:16:26,514 --> 00:16:29,970
for the continuously compounded
distribution.

209
00:16:29,970 --> 00:16:31,660
Computed this way.
Now.

210
00:16:31,660 --> 00:16:35,938
This is a quantile for the continuously
compounded distribution.

211
00:16:35,938 --> 00:16:40,751
When we compute value at risk, we actually
need to use the simple return.

212
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So we need to convert the continuously
compounded quantile into the quantile for

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the simple return.
And it turns out that the quantile for the

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simple return is E to the quantile for the
continuously compounded return -one.

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Okay and this is because you know this the
function that we use to, to convert the

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continuously compounded return to the
simple return.

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Is monetonically increasing and so the
quantile is preserved under this, this

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00:17:15,737 --> 00:17:19,942
transformation.
So the con-, the simple quantile is E to

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00:17:19,942 --> 00:17:25,688
the continuously compounded quantile -one.
So now that we have this simple return,

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quantile.
Then we compute our value at risk, in the

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usual way.
We take our initial investment, and you

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multiply it by the simple quantile that
we've computed.

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Okay, so let's rawk, walk through an
example to, to see this in, in action.

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So take our previous example here, the
continuously compounded return has the

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mean of five percent and a standard
deviation of ten%.

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Our initial investment is $10,000.'Kay.
So we're gonna compute five percent value

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at risk.
That is, we want to know with five percent

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probability, how much could we lose or
more.

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So we compute the continuously compounded
return quantile.

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00:18:04,814 --> 00:18:09,280
We take the mean plus the standard
deviation times the five percent quantile

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of a standard normal.
So that's equal to.05 plus.1 times the

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five percent quantile of a standard normal
is minus 1.645.

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Now we compute the simple return quantile.
We take e to the continuously compounded

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00:18:25,326 --> 00:18:31,218
return quantile minus one, so we compute e
to the minus.114 minus one.

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That's minus.108.
So the simple quantile is actually less

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than the continuously compounded quantile,
right?'Coz we always know, remember, 'coz

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the continuously compounded return is
always smaller than the simple return.

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So then we calculate our five percent bar.
We calc, , take our initial investment,

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00:18:49,782 --> 00:18:53,234
and you multiply it by the simple return
quantile.

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And that's, -1077.
So if we start with the assumption that

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contingency compatible returns are
normally distributed, and then we do our

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bar calculation.
You know, we get a different quantile.

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00:19:06,145 --> 00:19:09,735
And our bar number tends to be a little
bit smaller.

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That is, it, it looks like we don't lose
as much in that context.

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And one of the reasons why, is, is
because, if the continuously compared to

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00:19:20,657 --> 00:19:26,530
return is normal we know that the simple
return is log normal, and the simple

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return has a right skewness to it, right?
And so the left tail quantile is not as

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big for the log normal return as it is for
the continuously compounded return.

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00:19:38,583 --> 00:19:45,829
So, that's another sort of illustration of
the point about how, asymmetry impacts the

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quantiles of the distribution.