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,ଵ଻ଷ.ହଽ
ଵଵ,ଶଵଽ.ଷ଼
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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
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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)
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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.
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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
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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
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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
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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
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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
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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
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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೔
௩೔షభ
 New way that adjusts for volatility ෠௜ = 1 +
௩೔ି௩೔షభ
௩೔షభ
× 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.
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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.
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 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.
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 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.
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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.
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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೉

௉ಷ೉
and ஻
௉ಳ

௉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
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 Start by generating 20,000 pairs of independent and standard normal variates (two
random number vectors)
ி௑ ஻

 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

 Recall ఙଶಷ೉௖௢௩ಷ೉,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
ி௑ ஻

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

௜,ଵ
௉೔

௉೔ ி௑
ᇱ = ி௑ ହ
×௭ಷ೉, ஻ᇱ = ஻ ହ
×௭ಳ

Monte Carlo Simulation Example
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 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 € :

ி௑

 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
 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
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 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
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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
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Takeaways
 Historical simulation
 Procedure
 Weights
 Assumptions & Problems
 Monte Carlo simulation
 Procedure
 Assumptions & Problems
 Evolution in Regulation