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Summary Quantitative Methods for Finance (FEB23006)
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Comprehensive summary of Quantitative Methods for Finance (econometrics EUR)
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Week 1Asset return
"!
Let 𝑟! = ln %" & denote the log return or continuously compounded return, where 𝑃! is the
!"#
asset price at time 𝑡. Asset return exhibit stylized facts:
- Distribtution of returns is not normal: large (and small) returns occur more often than
expected under normality, so excess kurtosis. Large negative returns occur more often
than large positive ones, so negative skewness
- (Almost) no significant autocorrelations in returns
- Small, but very slowly declining autocorrelations in squared and absolute returns
All three stylized facts can be captured by a model with time-varying volatility: 𝑟! = 𝜇 + 𝜎! 𝑧!
where 𝑧! ∼ i. i. d. (0,1)
Volatility clustering
- Volatility is the (annualized) standard deviation of the returns on an asset
- Volatility is of crucial importance in many financial decision problems, such as portfolio
construction and risk management
- Volatility of asset returns is not constant. In particular, periods of large and small
movements in prices alternate: volatility clustering
- Volatility is not directly observable
Risk management
Risk is measured by means of the variance of asset returns: 𝜎 # = 𝐸((𝑟! − 𝜇)# ), 𝜇 = 𝐸(𝑟! ).
The variance is a symmetric measure, in the sense that it weighs positive deviations from the
expected return equally to negative deviations. As an investor, you mostly care about
‘downside risk’, that is, the risk associated with returns below the expected return. Therefore,
we use the Value-at-Risk
Value-at-Risk (VaR)
- VaR can be defined in terms of profits/losses and in terms of returns.
- VaR is the maximum loss that will not be exceeded over a given time period with a
specified probability
- VaR is the minimum return that could occur over a given time period with a specified
probability
- VaR essentially is a quantile of the distribution of profits and losses (P&L) or of the
distribution of returns)
VaR is a quantile
For any 0 < 𝑞 < 1, the VaR at 100 × (1 − 𝑞)% for a period of ℎ days is the return that is
expected to be exceeded with probability 1 − 𝑞. The 𝑉𝑎𝑅! (1 − 𝑞, ℎ) is the 𝑞-th quantile of
the distribution of the ℎ-day return 𝑟!$%,% : 𝑃 %𝑟!$%,% ≤ 𝑉𝑎𝑅! (1 − 𝑞, ℎ)& = 𝑞.
Historical simulation – estimating VaR
Obtain (or reconstruct) returns for the asset (portfolio) under consideration over some
historical time period, and obtain a VaR estimate using the (quantiles of the) empirical density.
Suppose we have historical daily returns 𝑟' , 𝑟# , … , 𝑟( . Let 𝑟(!) denote the corresponding order
statistics. Then then one-day VaR at 100 × (1 − 𝑞)% is equal to 𝑟(+() (assuming that 𝑞𝑇 is
integer). A crucial assumption is that all returns should have the same distribution. This VaR
estimate is constant over the whole period
Normal density – estimating VaR
Assume that one-day returns 𝑟! are i.i.d., following a normal density with mean 𝜇 and variance
𝜎 # . Obtain estimates of 𝜇 and 𝜎 # using historical returns 𝑟! , and obtain a VaR estimate using
the normal density.
, (,!$# -.)
If 𝑟!$' ∼ 𝑁(𝜇, 𝜎 # ), we can write this as 𝑟!$' = 𝜇 + 𝜎𝑧!$' , with 𝑧!$' ∼ 𝑁(0,1). 𝑧!$' ≡ /
(012! ('-+,')-.)
therefore 𝑞 = 𝑃G𝑟!$' ≤ 𝑉𝑎𝑅! (1 − 𝑞, 1)H = 𝑃 %𝑧!$' ≤ /
&. And hence,
𝑉𝑎𝑅! (1 − 𝑞, 1) = 𝜇 + 𝑧+ 𝜎, where 𝑧+ is the 𝑞-th quantile of the standard normal distribution.
A crucial assumption is that all returns should have the same distribution
Square-root-of-time rule – estimating VaR
The assumption that all returns have the same (normal or other) distribution has one big
advantage, which is that we can apply the square-root-of-time rule to obtain VaR-estimates
for a time period of ℎ days. Let 𝑟!$%,% = 𝑟!$' + 𝑟!$# + ⋯ + 𝑟!$% be the ℎ-day return from day
𝑡 + 1 until 𝑡 + ℎ. If the one-day returns 𝑟! ∼ 𝑖. 𝑖. 𝑑. (𝜇, 𝜎 # ), then 𝐸G𝑟!$%,% H = ℎ𝜇 and
𝑉G𝑟!$%,% H = ℎ𝜎 # . So, the ℎ-day VaR at 100 × (1 − 𝑞)% is 𝑉𝑎𝑅! (1 − 𝑞, ℎ) = ℎ𝜇 + 𝑧+ √ℎ𝜎.
Given that often 𝜎 ≫ |𝜇|, this is approximately equal to √ℎ𝑉𝑎𝑅! (1 − 𝑞, 1)
Shortcomings of normal density
Estimating VaR based on the normal (or some other) density means that we need an estimate
of the expected return and volatility. Using the sample standard deviation based on a set of
historical return observations to estimate 𝜎 is of course very straightforward, but it rests on
the crucial assumption that returns are (independent and) identically distributed, (and, often,
also on the assumption of a normal distribution). Looking at the properties of empirical asset
returns, this assumption might be questionable
VaR with time-varying volatility
A simple model to allow for time-varying volatility is 𝑟!$' = 𝜇 + 𝜎!$' 𝑧!$' , where 𝑧!$' ∼
𝑖. 𝑖. 𝑑. (0,1). This results in a time-varying VaR: 𝑉𝑎𝑅! (1 − 𝑞, 1) = 𝜇 + 𝑧+ 𝜎!$' . Recall that
volatility is not directly observable, hence we need to estimate 𝜎!$'
Historical volatility using rolling windows
A pragmatic approach is to use historical volatility over the past 𝑇 days to estimate volatility
' '
at time 𝑡: 𝜎O!# = ( ∑(34'(𝑟!-3 − 𝑟̅ )# , where 𝑟̅ is the average return, 𝑟̅ = ( ∑(34' 𝑟!-3 . Drawbacks:
- The estimates of 𝜎! exhibit a ‘ghost feature’. That is, 𝜎O! suddenly increases sharply
whenever a large return occurs, and drops sharply whenever this large return leaves the
moving window again. This is due to equal weighting
- Results are sensitive to choice of 𝑇
- Implicitly assumes that returns over the past 𝑇 days have the same variance
Using this method, the VaR estimate changes over time
Exponentially weighted moving average (EWMA)
Another pragmatic approach is to use EWMA to estimate volatility at time 𝑡:
𝜎O!# = (1 − 𝜆) ∑6 34' 𝜆
3-' (𝑟 #
!-3 − 𝑟̅ ) , where 0 < 𝜆 < 1 or 𝜎 O!# = 𝜆𝜎O!-'
#
+ (1 − 𝜆)(𝑟!-' − 𝑟̅ )# .
This is equal to ‘persistence + update/reaction’. For daily data, 𝜆 = 0.94 gives the Riskmetrics
method. Using this method, the VaR estimate changes over time
EWMA and volatility forecasts
#
Forecasts made at time 𝑡 of volatility for future time periods are equal to 𝜎O!$' for any forecast
# # #
horizon: 𝜎O!$' = 𝜆𝜎O! + (1 − 𝜆)(𝑟! − 𝑟̅ ) . A forecast for period 𝑡 + 2 can be obtained as
# # # #
𝜎O!$#|! = 𝐸(𝜎O!$# |ℐ! ) = 𝐸(𝜆𝜎O!$' + (1 − 𝜆)(𝑟!$' − 𝑟̅ )# |ℐ! ) = 𝜎O!$' , as 𝐸((𝑟!$' − 𝑟̅ )# |ℐ! ) =
# # #
𝜎O!$' . In fact, it holds that 𝜎O!$%|! = 𝜎O!$' for any forecast horizon ℎ
Riskmetrics and square-root-of-time rule
The square-root-of-time rule applies when using the Riskmetrics approach to estimate
# #
volatility. The ℎ-period ahead forecast of the variance 𝜎O!$%|! = 𝜎O!$' . Hence, if we want to
%
have a forecast of the cumulative return over the next ℎ days 𝑟!$%,% = ∑(34') 𝑟!$3 , this is equal
# % # #
to 𝜎O!$%|!,% = ∑34' 𝜎O!$3|! = ℎ ∙ 𝜎O!$'
, Simple vs log return
("! -"!"# )
Simple return equals 𝑅! = "!"#
and log return equals 𝑟! = ln(1 + 𝑅! ). A one-period
, 8 , 8
interest rate of 𝑟 compounded 𝑛 times gives a return of %1 + 8& . As 𝑛 → ∞, %1 + 8& → 𝑒 , .
For small and moderate values, log returns and simple returns are almost the same, because
ln(1 + 𝑅! ) ≈ 𝑅! . Large(r) differences occur when returns are more substantial. Advantage of
log returns is that the multi-period return is the sum of the single-period returns. Advantage
of simple returns is that the portfolio return is the weighted sum of individual asset returns
Testing normality
Under the null hypothesis of normality: 𝐽𝐵 ∼ 𝜒##
Autocorrelation tests
The 𝑘-th order autocovariance of returns is defined as 𝛾(𝑘) = 𝐸G(𝑟! − 𝜇)(𝑟!-9 − 𝜇)H, where
:(9)
𝜇 = 𝐸(𝑟! ) for all 𝑡. similarly, the 𝑘-th order autocorrelation 𝜌(𝑘) is defined as 𝜌(𝑘) = :(;).
The 𝑘-th order sample autocorrelation 𝜌O(𝑘) of returns 𝑟! can be computed as 𝛾O(𝑘) =
' (-9 < (9)
:
∑!4' (𝑟! − 𝑟̅ )(𝑟!$9 − 𝑟̅ ) and 𝜌O(𝑘) = . Under the null that 𝜌(𝑘) = 0, √𝑇𝜌O(𝑘) → 𝑁(0,1).
( < (;)
:
Box-Pierce test for joint significance of first 𝑚 autocorrelations: 𝑄= = 𝑇 ∑= O(𝑘) → 𝜒 # (𝑚)
94' 𝜌
Volatility models
𝑟! = 𝜇! + 𝜀! , where 𝜀! is not white noise, but has the properties:
- 𝐸(𝜀! |ℐ!-' ) = 0 (conditional mean zero)
- 𝐸(𝜀!# |ℐ!-' ) = 𝜎!# (time-varying conditional variance)
Where ℐ!-' = {𝑟!-' , 𝑟!-# , … } is called the information set available at time 𝑡 − 1.
- It holds that 𝐸(𝑟! |ℐ!-' ) = 𝜇! and 𝑉(𝑟! |ℐ!-' ) = 𝐸((𝑟! − 𝜇)# |ℐ!-' ) = 𝜎!#
- For simplicity, we assume that 𝜇! = 𝜇
- We can also write 𝑟! − 𝜇 = 𝜀! = 𝑧! 𝜎! , where 𝑧! ∼ 𝑖. 𝑖. 𝑑. (0,1)
- The unconditional variance equals 𝐸(𝜀!# ) = 𝐸G𝐸(𝜀!# |ℐ!-' )H = 𝐸(𝜎!# ) = 𝜎h # , so 𝜀! has
conditional heteroskedasticity
A different approach treats volatility as stochastic, that is, 𝑟! = 𝜇 + 𝜎! 𝑧! is combined with a
specific like ln(𝜎!# ) = 𝜔 + 𝜙' ln(𝜎!-'
#
) + 𝜂! , where 𝜂! is a white noise process
GARCH(1,1) model
𝜎!# = 𝜔 + 𝛼𝜀!-' # #
+ 𝛽𝜎!-' . To guarantee that 𝜎!# ≥ 0 for all 𝑡: 𝜔 > 0, 𝛼 > 0 and 𝛽 ≥ 0.
The GARCH(1,1) model is a ARMA(1,1) model for 𝜀!# : 𝜀!# = 𝜔 + (𝛼 + 𝛽)𝜀!-' #
+ 𝜈! − 𝛽𝜈!-' ,
where 𝜈! = 𝜀!# − 𝜎!# . 𝐸(𝜈|ℐ!-' ) = 0, such that we can interpret it as an innovation.
The ARMA(1,1) model is stationary if the AR coefficient is smaller than 1 in magnitude. So,
the GARCH(1,1) model is covariance stationary if 𝛼 + 𝛽 < 1. Under the condition 𝛼 + 𝛽 <
>
1, the unconditional variance equals 𝐸(𝜀!# ) = 𝜎h # = '-?-@. Using this expression, we can write
𝜎!# = 𝜎h # + 𝛼(𝜀!-'
# # )
− 𝜎!-' + (𝛼 + 𝛽)(𝜎!-' #
− 𝜎h # ). So, the current conditional volatility:
#
- Is a weighted average of conditional volatility one period ago 𝜎!-' , the squared shock one
# #
period ago 𝜀!-' , and unconditional volatility 𝜎h
- Is higher if the squared shock at 𝑡 − 1 is above its conditional expectation
- Is higher if conditional volatility one period ago was above the unconditional volatility
GARCH(1,1) model and Riskmetrics
Setting 𝜔 = 0 and 1 − 𝛼 = 𝛽 = 𝜆 gives Riskmetrics
GARCH(1,1) model and non-normality
With the GARCH(1,1) model we specify the conditional distribution of 𝑟! : 𝐸(𝑟! |ℐ!-' ) = 𝜇 and
𝑉(𝑟! |ℐ!-' ) = 𝐸((𝑟! − 𝜇)# |ℐ!-' ) = 𝜎!# . From 𝑟! = 𝜇 + 𝜎! 𝑧! with 𝑧! = 𝑖. 𝑖. 𝑑. 𝐷(0,1), we find
that the conditional distribution of 𝑟! is the same as the distribution of the standardized
unexpected returns 𝑧! , that is 𝑓(𝑟! |ℐ!-' ) = 𝐷(𝜇, 𝜎 # ). The non-normality in the stylized facts
refers to the unconditional distribution of 𝑟! .
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