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无代写-FIN 430

时间：2021-04-02

FIN 430 – Risk Management

Market Risk II

Prof. Ramona Dagostino

Spring 2021

2Agenda

Historical simulation

Monte Carlo simulation

Comparison of advantages and limitations regarding the two approaches

3Historical Simulation

VAR requires assumption on normal distribution of asset

returns

This assumption does not hold in reality – skewness, kurtosis, all

sorts of non-normal distributions

Back simulation avoids this issue

Basic idea: evaluate portfolio based on actual returns (prices) on

the assets that existed yesterday, the day before, etc. (usually

previous 501 days).

Suppose we want to calculate 99% DEAR of a portfolio. We can

simulate it using the following steps.

4Historical Simulation

(1) Identify the market variables affecting the portfolio

exchange rates, interest rates, stock indices, commodity prices and so on

(2) Collect data on movements of those market variables over the most recent 501 days

name them Day 0, Day 1, …, Day 500 (today is 501)

These realizations at each day are the basis for your scenarios/simulations

5Historical Simulation

(3) Create 500 scenarios for returns between today and tomorrow

scenario i (i= 1,…,500) is where the returns (percentage change) of market variables

between today and tomorrow is the same as between Day i-1 and Day i.

the estimated return under ith Scenario is

ିଵ

where and ିଵ are the historical values of day and -1 respectively.

6Historical Simulation

(4) For each scenario, calculate the dollar change of the portfolio between

today and tomorrow, which defines a probability of daily loss

(5) The 99 percentile of the distribution can be estimated as the fifth

worst outcome.

only 1% of larger losses

We are 99% certain that we will not take a loss greater than the VaR

estimate if the changes in market variables in the past 500 days are

representative of what will happen between today and tomorrow.

7Historical Simulation Example

Suppose a US investor owns, on September 25, 2008, a portfolio worth $10

million investments in four stock indices

Dow Jones Industrial Average (DJIA) in US

FTSE 100 in the United Kingdom

CAC 40 in France

Nikkei 225 in Japan

8Historical Simulation Example

Excel spreadsheet containing 501 days of historical data on the closing

prices of the four indices.

First, we convert all asset values to US dollars

for example, the FTSE 100 stood at 5,823.40 on August 10, 2006, when

the exchange rate was 1.8918 USD per GBP. This means that, measured

in U.S. dollars, it was at 5,823.40 × 1.8918 = 11,016.71.

9Historical Simulation Example

Remember today is September 25, 2008

Second, calculate index returns (measured in U.S. dollars) for September

26, 2008, for the 500 scenarios.

for example, scenario 1 estimates the index returns (i.e. percentage

changes in each indices values) on September 26, 2008, assuming that

index returns between September 25 and September 26, 2008, are the

same as between August 7 (Day 0) and August 8 (Day 1), 2006

The DJIA was 11,173.59 on August 8, 2006, down from 11,219.38 on

August 7, 2006. The DJIA return under Scenario 1 is therefore

ଵଵ,ଵଷ.ହଽ

ଵଵ,ଶଵଽ.ଷ଼

10

Historical Simulation Example

Third, calculate estimated portfolio values for the 500 scenarios.

for example, the scenario 1 index returns of the FTSE 100, the CAC 40, and

the Nikkei 225 are 0.9968, 1.0007, and 1.0198, respectively.

the portfolio value under Scenario 1 is

0.9968 1.0007 1.0198

the portfolio therefore has a gain of $14,334 (=10,014.334k - 10,000k) under

Scenario 1

11

Historical Simulation Example

We can plot a histogram of the simulated gains/losses

On the y-axis is the number of times a loss scenario (eg. losses between

-250 and -150, occur)

12

Historical Simulation Example

Fourth, rank the losses in the 500 different scenarios, from the biggest loss

(in absolute value) down to the smallest loss/gain

The 99% DEAR can be estimated as the fifth worst loss, that is, $253,385

(there are 500 scenarios, hence 1%*500=5).

10-day VAR is estimated as

13

Historical Simulation Example

To estimate the 99% one-day expected shortfall,

we average the estimated potential losses in the 1%

tail (the five worst scenarios) of the loss distribution.

In the case of our example, the five worst losses of

the 500 scenarios are scenarios 494, 339, 349, 329,

and 487. The average losses for these scenarios is

$327,181, which is the estimate of expected

shortfall.

14

Historical Simulation Example

We can update the VAR estimation each day by

adding realized new market variables- you roll the

501-day window

For example, on September 26, 2008, we can

collect data from August 8, 2006, to September

26, 2008 as Day 0 to Day 500; then repeat the

procedure to estimate the updated 99% VAR and

expected shortfall.

15

Historical Simulation: Weighting

The basic approach assumes each day in the past is

given equal weight of 1/n

In simple words, each of those scenarios is equally

likely

Researches suggest that more recent observations

should be given more weight

they are more reflective of current volatilities and current

macroeconomic conditions

16

Historical Simulation: Weighting

A natural weighting scheme to use is the one where

weights decline exponentially as time goes back

the weight given to Scenario i is

ି

Intuition: the most current scenario (scenario 500) is given a

weight λ<1, then scenario 499 is given weight λ * λ (so smaller);

scenario 498 is given λ * λ * λ and so on, declining exponentially

all weights add up to one.

n is the number of total scenarios, i.e. 500

17

Historical Simulation: Weighting

Set in the previous example

Scenario 500 weight is 0.995

Scenario 494 weight is= ఒ

ష(ଵିఒ)

ଵିఒ

= .ଽଽହ

ఱబబషరవర(ଵି.ଽଽହ)

ଵି.ଽଽହఱబబ

= 0.00528

18

Historical Simulation: Weighting

To get VAR estimate, we again rank the observations from the worst outcome

to the best in terms of absolute value losses.

Starting at the worst outcome, weights are summed until the required percentile

of the distribution is reached.

For example, a 99% DEAR in the example is $282,204

Reason for this result (different from equal weight case) is that recent

observations are given more weight. Does NOT necessarily imply bigger loss!

If we

weight

equal all

scenarios,

99% dear is

the 5th worst

scenario, ie

# 487

If we weight

recent

scenarios more,

then the VAR is

different

19

Historical Simulation: Weighting

The 0.01 (1%) tail of the loss distribution consists of:

probability 0.00528 of a loss of $477,841

probability 0.00243 of a loss of $345,435

and 0.01 − 0.00528 − 0.00243 = 0.00228 probability of a loss of $282,204

The expected shortfall can therefore be calculated as:

(0.00528 × 477,841 + 0.00243 × 345,435 + 0.00228 × 282,204)/0.01 = $400,914

20

Historical Simulation: Weighting

So we learnt how to give more weight to recent observations

However there is yet another way to weight scenarios, based on volatility

Core idea: even the variance moves Volatility of Volatility:

The variance of changes in prices is typically not constant, but varies over

time

A better approach incorporates varying volatility when estimating return in

historic simulation

21

Historical Simulation: Weighting

Recall we are using past scenarios as plausible representative scenarios of future portfolio

realizations

Take scenario i : it is the change in asset value between Day i − 1 and Day i.

The daily volatility of that particular asset at the end of day (i-1) was σi (estimate it in the data)

Call the current volatility of the asset

Then modify the return estimation by including a volatility weight:

Old way of calculating scenarios returns was : =

௩

௩షభ

New way that adjusts for volatility = 1 +

௩ି௩షభ

௩షభ

× ఙ

ఙ

Intuition:

Suppose today’s volatility σ is twice that of scenario i volatility σi , for a particular market variable.

Scenario i is not representative of today volatility

The changes we expect to see between today and tomorrow are twice as big as changes between

Day i − 1 and Day i (scenario i).

We should use twice of the scenario i‘s net return.

22

Historical Simulation: Weighting

When market variable movements are adjusted for volatility, the rank of the

losses becomes:

The 99% VAR is $602,968 (the fifth worse loss).

In the example, volatility is much higher at the end of the period. So

estimated VAR is much larger. The estimated VAR would instead be smaller

if current vol is smaller.

23

Historical Simulation Advantages

Simplicity

easy to calculate

can involve many risk factors

Does not require distribution assumption of returns

Does not necessarily need correlations or standard deviations of individual

asset returns.

24

Historical Simulation Disadvantages

500 observation is not large from statistical standpoint.

Increasing number of observations by going back further in time is not

always desirable.

The way of adjusting volatility is artificial.

25

Monte Carlo Simulation

Assume risk factor returns follow certain joint distribution.

Employ historical information to back out parameters of the

distribution.

Use number generator to simulate observations and estimate potential

losses.

26

Monte Carlo Simulation Example

A US based FI has a long position in a one-year zero-coupon

€1 million bond. The current price of the bond is €909,091.

The current $/€ exchange rate is 0.65.

The FI wants to evaluate the VAR for this bond based on

interest rate and FX rates over the next 5 days.

27

Monte Carlo Simulation Example

Assume daily gross returns of FX and bond price are jointly log-normal with

mean zero and ி , , ி,

This means that the logarithm of the returns is normally distributed

ி

ಷ

ᇲ

ಷ

and

ಳ

ᇲ

ಳ

are joint normally distributed with variance-covariance matrix

ఙಷ

௩ಷ,ಳ

௩ಷ,ಳ

ఙಳ

Where ிᇱ and ᇱ are the prices in the future

We want to generate simulated scenarios based on the joint distribution of

returns

28

Start by generating 20,000 pairs of independent and standard normal variates (two

random number vectors)

ி

ᇱ

Now transform these two random vectors into two vectors of shocks that exhibit

the same variance and covariance of the FX and B

Apply Cholesky decomposition to get symmetric matrix such that

ᇱ

Recall ఙଶಷ௩ಷ,ಳ

௩ಷ,ಳ

ఙଶಳ

ఙಷ

௩ಷ,ಳ/ಷ

ఙଶಳି(௩ಷ,ಳ/ಷ)

మ

[You do not need to memorize A]

Obtain values for the joint (correlated) normal variates

ி

ᇱ

Obtain future FX and bond price at n-horizon (recall n=5 days and z is in logs)

,

,ଵ

ᇲ

ி

ᇱ = ி ହ

×௭ಷ, ᇱ = ହ

×௭ಳ

Monte Carlo Simulation Example

29

Then in 5 days the €-denominated bond value, expressed in $ terms, will be estimated

as the future estimated exchange rate $/ € times the future estimated bond value in € :

ᇱ

ி

ᇱ

ᇱ

The simulated losses are: ᇱ

Rank the losses from large to small and obtain 95% VAR as the 1000th largest loss (5%

of 20k) in the 20,000 simulation.

Monte Carlo Simulation Example

30

Monte Carlo Simulation Advantage

Don’t need a large set of historical data.

Incorporates updated estimates of volatility and correlations.

Can take into account

many risk factors

assets whose value is a complicated function of underlying

31

Monte Carlo Simulation Disadvantage

Assumes (log) market returns have multivariate-normal

distributions.

in practice, daily changes in market variables can be very

different from normal

Hoping some form of law of large number applies so that

daily return on a large portfolio is still normally distributed

32

Application to Market Risk: Basel Rules

Basel used to rely on VAR

Crisis showed limitations of VAR – we care of extreme

events & expected losses in those tails

Basel III moved towards Expected Shortfall approach

ES typically relied on a 10 day rule of thumb

Recall you need to know how long you will hold the position

Crisis showed liquidity can dry up entirely

Basel III moved to a more refined approach, to incorporate

liquidity risk

As we noticed from class 1.. risks are intertwined

33

Takeaways

Historical simulation

Procedure

Weights

Assumptions & Problems

Monte Carlo simulation

Procedure

Assumptions & Problems

Evolution in Regulation

学霸联盟

Market Risk II

Prof. Ramona Dagostino

Spring 2021

2Agenda

Historical simulation

Monte Carlo simulation

Comparison of advantages and limitations regarding the two approaches

3Historical Simulation

VAR requires assumption on normal distribution of asset

returns

This assumption does not hold in reality – skewness, kurtosis, all

sorts of non-normal distributions

Back simulation avoids this issue

Basic idea: evaluate portfolio based on actual returns (prices) on

the assets that existed yesterday, the day before, etc. (usually

previous 501 days).

Suppose we want to calculate 99% DEAR of a portfolio. We can

simulate it using the following steps.

4Historical Simulation

(1) Identify the market variables affecting the portfolio

exchange rates, interest rates, stock indices, commodity prices and so on

(2) Collect data on movements of those market variables over the most recent 501 days

name them Day 0, Day 1, …, Day 500 (today is 501)

These realizations at each day are the basis for your scenarios/simulations

5Historical Simulation

(3) Create 500 scenarios for returns between today and tomorrow

scenario i (i= 1,…,500) is where the returns (percentage change) of market variables

between today and tomorrow is the same as between Day i-1 and Day i.

the estimated return under ith Scenario is

ିଵ

where and ିଵ are the historical values of day and -1 respectively.

6Historical Simulation

(4) For each scenario, calculate the dollar change of the portfolio between

today and tomorrow, which defines a probability of daily loss

(5) The 99 percentile of the distribution can be estimated as the fifth

worst outcome.

only 1% of larger losses

We are 99% certain that we will not take a loss greater than the VaR

estimate if the changes in market variables in the past 500 days are

representative of what will happen between today and tomorrow.

7Historical Simulation Example

Suppose a US investor owns, on September 25, 2008, a portfolio worth $10

million investments in four stock indices

Dow Jones Industrial Average (DJIA) in US

FTSE 100 in the United Kingdom

CAC 40 in France

Nikkei 225 in Japan

8Historical Simulation Example

Excel spreadsheet containing 501 days of historical data on the closing

prices of the four indices.

First, we convert all asset values to US dollars

for example, the FTSE 100 stood at 5,823.40 on August 10, 2006, when

the exchange rate was 1.8918 USD per GBP. This means that, measured

in U.S. dollars, it was at 5,823.40 × 1.8918 = 11,016.71.

9Historical Simulation Example

Remember today is September 25, 2008

Second, calculate index returns (measured in U.S. dollars) for September

26, 2008, for the 500 scenarios.

for example, scenario 1 estimates the index returns (i.e. percentage

changes in each indices values) on September 26, 2008, assuming that

index returns between September 25 and September 26, 2008, are the

same as between August 7 (Day 0) and August 8 (Day 1), 2006

The DJIA was 11,173.59 on August 8, 2006, down from 11,219.38 on

August 7, 2006. The DJIA return under Scenario 1 is therefore

ଵଵ,ଵଷ.ହଽ

ଵଵ,ଶଵଽ.ଷ଼

10

Historical Simulation Example

Third, calculate estimated portfolio values for the 500 scenarios.

for example, the scenario 1 index returns of the FTSE 100, the CAC 40, and

the Nikkei 225 are 0.9968, 1.0007, and 1.0198, respectively.

the portfolio value under Scenario 1 is

0.9968 1.0007 1.0198

the portfolio therefore has a gain of $14,334 (=10,014.334k - 10,000k) under

Scenario 1

11

Historical Simulation Example

We can plot a histogram of the simulated gains/losses

On the y-axis is the number of times a loss scenario (eg. losses between

-250 and -150, occur)

12

Historical Simulation Example

Fourth, rank the losses in the 500 different scenarios, from the biggest loss

(in absolute value) down to the smallest loss/gain

The 99% DEAR can be estimated as the fifth worst loss, that is, $253,385

(there are 500 scenarios, hence 1%*500=5).

10-day VAR is estimated as

13

Historical Simulation Example

To estimate the 99% one-day expected shortfall,

we average the estimated potential losses in the 1%

tail (the five worst scenarios) of the loss distribution.

In the case of our example, the five worst losses of

the 500 scenarios are scenarios 494, 339, 349, 329,

and 487. The average losses for these scenarios is

$327,181, which is the estimate of expected

shortfall.

14

Historical Simulation Example

We can update the VAR estimation each day by

adding realized new market variables- you roll the

501-day window

For example, on September 26, 2008, we can

collect data from August 8, 2006, to September

26, 2008 as Day 0 to Day 500; then repeat the

procedure to estimate the updated 99% VAR and

expected shortfall.

15

Historical Simulation: Weighting

The basic approach assumes each day in the past is

given equal weight of 1/n

In simple words, each of those scenarios is equally

likely

Researches suggest that more recent observations

should be given more weight

they are more reflective of current volatilities and current

macroeconomic conditions

16

Historical Simulation: Weighting

A natural weighting scheme to use is the one where

weights decline exponentially as time goes back

the weight given to Scenario i is

ି

Intuition: the most current scenario (scenario 500) is given a

weight λ<1, then scenario 499 is given weight λ * λ (so smaller);

scenario 498 is given λ * λ * λ and so on, declining exponentially

all weights add up to one.

n is the number of total scenarios, i.e. 500

17

Historical Simulation: Weighting

Set in the previous example

Scenario 500 weight is 0.995

Scenario 494 weight is= ఒ

ష(ଵିఒ)

ଵିఒ

= .ଽଽହ

ఱబబషరవర(ଵି.ଽଽହ)

ଵି.ଽଽହఱబబ

= 0.00528

18

Historical Simulation: Weighting

To get VAR estimate, we again rank the observations from the worst outcome

to the best in terms of absolute value losses.

Starting at the worst outcome, weights are summed until the required percentile

of the distribution is reached.

For example, a 99% DEAR in the example is $282,204

Reason for this result (different from equal weight case) is that recent

observations are given more weight. Does NOT necessarily imply bigger loss!

If we

weight

equal all

scenarios,

99% dear is

the 5th worst

scenario, ie

# 487

If we weight

recent

scenarios more,

then the VAR is

different

19

Historical Simulation: Weighting

The 0.01 (1%) tail of the loss distribution consists of:

probability 0.00528 of a loss of $477,841

probability 0.00243 of a loss of $345,435

and 0.01 − 0.00528 − 0.00243 = 0.00228 probability of a loss of $282,204

The expected shortfall can therefore be calculated as:

(0.00528 × 477,841 + 0.00243 × 345,435 + 0.00228 × 282,204)/0.01 = $400,914

20

Historical Simulation: Weighting

So we learnt how to give more weight to recent observations

However there is yet another way to weight scenarios, based on volatility

Core idea: even the variance moves Volatility of Volatility:

The variance of changes in prices is typically not constant, but varies over

time

A better approach incorporates varying volatility when estimating return in

historic simulation

21

Historical Simulation: Weighting

Recall we are using past scenarios as plausible representative scenarios of future portfolio

realizations

Take scenario i : it is the change in asset value between Day i − 1 and Day i.

The daily volatility of that particular asset at the end of day (i-1) was σi (estimate it in the data)

Call the current volatility of the asset

Then modify the return estimation by including a volatility weight:

Old way of calculating scenarios returns was : =

௩

௩షభ

New way that adjusts for volatility = 1 +

௩ି௩షభ

௩షభ

× ఙ

ఙ

Intuition:

Suppose today’s volatility σ is twice that of scenario i volatility σi , for a particular market variable.

Scenario i is not representative of today volatility

The changes we expect to see between today and tomorrow are twice as big as changes between

Day i − 1 and Day i (scenario i).

We should use twice of the scenario i‘s net return.

22

Historical Simulation: Weighting

When market variable movements are adjusted for volatility, the rank of the

losses becomes:

The 99% VAR is $602,968 (the fifth worse loss).

In the example, volatility is much higher at the end of the period. So

estimated VAR is much larger. The estimated VAR would instead be smaller

if current vol is smaller.

23

Historical Simulation Advantages

Simplicity

easy to calculate

can involve many risk factors

Does not require distribution assumption of returns

Does not necessarily need correlations or standard deviations of individual

asset returns.

24

Historical Simulation Disadvantages

500 observation is not large from statistical standpoint.

Increasing number of observations by going back further in time is not

always desirable.

The way of adjusting volatility is artificial.

25

Monte Carlo Simulation

Assume risk factor returns follow certain joint distribution.

Employ historical information to back out parameters of the

distribution.

Use number generator to simulate observations and estimate potential

losses.

26

Monte Carlo Simulation Example

A US based FI has a long position in a one-year zero-coupon

€1 million bond. The current price of the bond is €909,091.

The current $/€ exchange rate is 0.65.

The FI wants to evaluate the VAR for this bond based on

interest rate and FX rates over the next 5 days.

27

Monte Carlo Simulation Example

Assume daily gross returns of FX and bond price are jointly log-normal with

mean zero and ி , , ி,

This means that the logarithm of the returns is normally distributed

ி

ಷ

ᇲ

ಷ

and

ಳ

ᇲ

ಳ

are joint normally distributed with variance-covariance matrix

ఙಷ

௩ಷ,ಳ

௩ಷ,ಳ

ఙಳ

Where ிᇱ and ᇱ are the prices in the future

We want to generate simulated scenarios based on the joint distribution of

returns

28

Start by generating 20,000 pairs of independent and standard normal variates (two

random number vectors)

ி

ᇱ

Now transform these two random vectors into two vectors of shocks that exhibit

the same variance and covariance of the FX and B

Apply Cholesky decomposition to get symmetric matrix such that

ᇱ

Recall ఙଶಷ௩ಷ,ಳ

௩ಷ,ಳ

ఙଶಳ

ఙಷ

௩ಷ,ಳ/ಷ

ఙଶಳି(௩ಷ,ಳ/ಷ)

మ

[You do not need to memorize A]

Obtain values for the joint (correlated) normal variates

ி

ᇱ

Obtain future FX and bond price at n-horizon (recall n=5 days and z is in logs)

,

,ଵ

ᇲ

ி

ᇱ = ி ହ

×௭ಷ, ᇱ = ହ

×௭ಳ

Monte Carlo Simulation Example

29

Then in 5 days the €-denominated bond value, expressed in $ terms, will be estimated

as the future estimated exchange rate $/ € times the future estimated bond value in € :

ᇱ

ி

ᇱ

ᇱ

The simulated losses are: ᇱ

Rank the losses from large to small and obtain 95% VAR as the 1000th largest loss (5%

of 20k) in the 20,000 simulation.

Monte Carlo Simulation Example

30

Monte Carlo Simulation Advantage

Don’t need a large set of historical data.

Incorporates updated estimates of volatility and correlations.

Can take into account

many risk factors

assets whose value is a complicated function of underlying

31

Monte Carlo Simulation Disadvantage

Assumes (log) market returns have multivariate-normal

distributions.

in practice, daily changes in market variables can be very

different from normal

Hoping some form of law of large number applies so that

daily return on a large portfolio is still normally distributed

32

Application to Market Risk: Basel Rules

Basel used to rely on VAR

Crisis showed limitations of VAR – we care of extreme

events & expected losses in those tails

Basel III moved towards Expected Shortfall approach

ES typically relied on a 10 day rule of thumb

Recall you need to know how long you will hold the position

Crisis showed liquidity can dry up entirely

Basel III moved to a more refined approach, to incorporate

liquidity risk

As we noticed from class 1.. risks are intertwined

33

Takeaways

Historical simulation

Procedure

Weights

Assumptions & Problems

Monte Carlo simulation

Procedure

Assumptions & Problems

Evolution in Regulation

学霸联盟