simulate
Simulate expected shortfall (ES) test statistics
Description
performs a simulation of ES test statistics. The ebts
= simulate(ebts
)simulate
function simulates portfolio outcomes according to the distribution assumptions
indicated in the esbacktestbysim
object, and
calculates all the supported test statistics under each scenario. The simulated test
statistics are used to estimate the significance of the ES backtests.
adds optional name-value pair arguments.ebts
= simulate(ebts
,Name,Value
)
Examples
Simulate ES Test Statistics
Create an esbacktestbysim
object and run a simulation of 1000 scenarios.
load ESBacktestBySimData rng('default'); % for reproducibility ebts = esbacktestbysim(Returns,VaR,ES,"t",... 'DegreesOfFreedom',10,... 'Location',Mu,... 'Scale',Sigma,... 'PortfolioID',"S&P",... 'VaRID',["t(10) 95%","t(10) 97.5%","t(10) 99%"],... 'VaRLevel',VaRLevel);
The unconditional
and minBiasAbsolute
tests report 1000 scenarios (see the Scenarios
column in the report).
unconditional(ebts)
ans=3×10 table
PortfolioID VaRID VaRLevel Unconditional PValue TestStatistic CriticalValue Observations Scenarios TestLevel
___________ _____________ ________ _____________ ______ _____________ _____________ ____________ _________ _________
"S&P" "t(10) 95%" 0.95 accept 0.093 -0.13342 -0.16252 1966 1000 0.95
"S&P" "t(10) 97.5%" 0.975 reject 0.031 -0.25011 -0.2268 1966 1000 0.95
"S&P" "t(10) 99%" 0.99 reject 0.008 -0.57396 -0.38264 1966 1000 0.95
minBiasAbsolute(ebts)
ans=3×10 table
PortfolioID VaRID VaRLevel MinBiasAbsolute PValue TestStatistic CriticalValue Observations Scenarios TestLevel
___________ _____________ ________ _______________ ______ _____________ _____________ ____________ _________ _________
"S&P" "t(10) 95%" 0.95 accept 0.062 -0.0014247 -0.0015578 1966 1000 0.95
"S&P" "t(10) 97.5%" 0.975 reject 0.029 -0.0026674 -0.0023251 1966 1000 0.95
"S&P" "t(10) 99%" 0.99 reject 0.005 -0.0060982 -0.0039004 1966 1000 0.95
Run a second simulation with 5000
scenarios using the simulate
function. Rerun the unconditional
and minBiasAbsolute
tests using the updated esbacktestbysim
object. Notice that the tests now show 5,000 scenarios along with updated p-values and critical values.
ebts = simulate(ebts,'BlockSize',10000,'NumScenarios',5000,'TestList',["conditional","unconditional","quantile","minBiasAbsolute","minBiasRelative"]); unconditional(ebts)
ans=3×10 table
PortfolioID VaRID VaRLevel Unconditional PValue TestStatistic CriticalValue Observations Scenarios TestLevel
___________ _____________ ________ _____________ ______ _____________ _____________ ____________ _________ _________
"S&P" "t(10) 95%" 0.95 accept 0.0952 -0.13342 -0.17352 1966 5000 0.95
"S&P" "t(10) 97.5%" 0.975 reject 0.0456 -0.25011 -0.24318 1966 5000 0.95
"S&P" "t(10) 99%" 0.99 reject 0.009 -0.57396 -0.38608 1966 5000 0.95
minBiasAbsolute(ebts,"TestLevel",0.99)
ans=3×10 table
PortfolioID VaRID VaRLevel MinBiasAbsolute PValue TestStatistic CriticalValue Observations Scenarios TestLevel
___________ _____________ ________ _______________ ______ _____________ _____________ ____________ _________ _________
"S&P" "t(10) 95%" 0.95 accept 0.0622 -0.0014247 -0.0021797 1966 5000 0.99
"S&P" "t(10) 97.5%" 0.975 accept 0.026 -0.0026674 -0.0032702 1966 5000 0.99
"S&P" "t(10) 99%" 0.99 reject 0.006 -0.0060982 -0.0054814 1966 5000 0.99
Input Arguments
ebts
— esbacktestbysim
object
object
esbacktestbysim
(ebts
) object, which
contains a copy of the given data (the PortfolioData
,
VarData
, ESData
, and
Distribution
properties) and all combinations of
portfolio ID, VaR ID, and VaR levels to be tested. For more information on
creating an esbacktestbysim
object, see esbacktestbysim
.
Name-Value Arguments
Specify optional pairs of arguments as
Name1=Value1,...,NameN=ValueN
, where Name
is
the argument name and Value
is the corresponding value.
Name-value arguments must appear after other arguments, but the order of the
pairs does not matter.
Before R2021a, use commas to separate each name and value, and enclose
Name
in quotes.
Example: ebts =
simulate(ebts,'NumScenarios',1000000,'BlockSize',10000,'TestList','conditional')
NumScenarios
— Number of scenarios to simulate
1000
(default) | positive integer
Number of scenarios to simulate, specified using the comma-separated
pair consisting of 'NumScenarios'
and a positive
integer.
Data Types: double
BlockSize
— Number of scenarios to simulate in single simulation block
1000
(default) | positive integer
Number of scenarios to simulate in a single simulation block,
specified using the comma-separated pair consisting of
'BlockSize'
and a positive integer.
Data Types: double
TestList
— Indicator for which test statistics to simulate
["conditional","unconditional","quantile","minBiasAbsolute","minBiasRelative"]
(default) | character vector with value of 'conditional'
,
'unconditional'
, 'quantile'
,
'minBiasAbsolute'
, or
'minBiasRelative'
| string with value of "conditional"
,
"unconditional"
, "quantile"
,
"minBiasAbsolute"
, or
"minBiasRelative"
Indicator for which test statistics to simulate, specified as the
comma-separated pair consisting of 'TestList'
and a
cell array of character vectors or a string array with the value
conditional
, unconditional
,
quantile
, minBiasAbsolute
or
minBiasRelative
.
Data Types: char
| cell
| string
Output Arguments
ebts
— Updated esbacktestbysim
object
object
esbacktestbysim
(ebts
), returned as
an updated object. After running simulate
, the updated
esbacktestbysim
object stores the simulated test
statistics, which are used to calculate p-values and
generate test results.
For more information on an esbacktestbysim
object, see
esbacktestbysim
.
More About
Simulation of Test Statistics and Significance of the Tests
The VaR and ES models assume that for each period
t, the portfolio outcomes
X
t have a cumulative probability
distribution P
t.
Under the assumption that the distributions
P
t are correct (the null hypothesis),
test statistics are simulated by:
Simulating
M
scenarios ofN
observations each, for example, , with ,t
=1
,…,N
, ands
=1
,…,M
.For each simulated scenario
X
s, compute the test statistic of interestZ
s =Z
(X
s),s
=1
,…,M
.The resulting
M
simulated test statistic valuesZ
1,…,Z
M from a distribution of the test statistic assuming the probability distributionsP
t are correct.
The p-value is defined as the proportion of scenarios for which the simulated test statistic is smaller than the test statistic evaluated at the observed portfolio outcomes: :
where I
(Z
s ≤
Z
obs) is an indicator function
with a value of 1
if
Z
s ≤
Z
obs, and 0
otherwise. If P
test is
1
minus the test confidence level, the test result is to
'reject'
if .
The critical value is defined as the minimum simulated test statistic
Z
critwith a
p-value greater than or equal to
P
test.
References
[1] Acerbi, C., and B. Szekely. Backtesting Expected Shortfall. MSCI Inc. December, 2014.
Version History
Introduced in R2017b
See Also
summary
| runtests
| conditional
| unconditional
| quantile
| minBiasRelative
| minBiasAbsolute
| esbacktestbysim
| esbacktestbyde
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