Multivariate Volatility Modeling
Case study: Developing a Trading Strategy
. Lochstoer
UCLA Anderson School of Management
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Winter 2022
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022
1 Multivariate Volatility Models I MGARCH
F CCC-GARCH F DCC-GARCH
2 Information Ratio
3 Case study
I From two trading signals to full-áedged fund trading strategy
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022
Multivariate Volatility Modeling
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022 3 / 36
The Conditional Covariance Matrix
Consider a vector of asset returns:
rt = μt + εt , where all variables are N 1 vectors
The conditional means (modelled from, e.g., a VAR) are: μt =Et 1[rt]
Denote the conditional N N covariance matrix as: H t = E t 1 ε t ε t0
A mean-variance optimizer would choose N 1 portfolio weights: w = kH 1μ
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022
The Conditional Covariance Matrix (contíd)
DeÖne the mean-zero, unit variance N 1 vector of i.i.d. shocks ηt WN(0N,IN).
This distribution is typically chosen to be Normal or Studentís t
We can then write: as
εt =H1/2η tt
WN0N,H1/20 INH1/2 tt
WN(0N,Ht)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022
Multivariate GARCH-type models take as their starting point rt =μ +H1/2η,
where H1/2 is the Cholesky decomposition of Ht t
These models specify the dynamics of Ht
I Needs to be positive deÖnite for each t
I Note: size of this matrix is of order N2, so need to keep N small for good
performance
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022
Useful decomposition
We can write:
where the diagonal “standard deviations”-matrix is
Ht =DtRtDt,
26qσ21,t q0 0 37
Dt =6 0 σ2,t 0 7
7 40 0qσ2N,t5
6 . . and the symmetric correlation matrix is
2 1 ρ12,t ρ1N,t 3 R =6 ρ21,t 1 ρ2N,t 7 t 4 . . … . 5
ρN1,t ρN2,t 1
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022
Are individual portfolio StDevs and Corrs time-varying?
Consider monthly data from 1970 to 2017 for the Fama-French HML (value) and MOM (momentum) factors
Annualized summary statistics
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022 8 / 36
Are individual portfolio StDevs and Corrs time-varying?
Plot 24-month rolling (annualized) st.devs. and correlation
Yes! These moments look strongly time-varying
MVE portfolio not likely to have constant portfolio weights
0.3 0.2 0.1
0.4 0.3 0.2 0.1
1 0.5 0 -0.5 -1
24-month rolling St.dev. of HML
1980 1985 1990 1995 2000 2005
24-month rolling St.dev. of MOM
2010 2015 2020
1980 1985 1990
1995 2000 2005
2010 2015 2020
24-month rolling corr(HML,MOM)
1980 1985 1990
1995 2000 2005
2010 2015 2020
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StrateWgyinter 2022
MGARCH – SpeciÖcations
The diagonal elements of Dt can be obtained from N univariate GARCH models run on each element in rt
The correlation matrix is tricker, in part as each correlation must be between 1 and 1 and the diagonal has to equal 1
Two standard models: 1 CCC-GARCH
F Constant conditional correlations 2 DCC-GARCH
F Dynamic conditional correlations
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
which implies that
ν =D 1ε. ttt
νt WN (0,Rt).
CCC-GARCH simply assumes that the correlation matrix is constant, thus:
1T0 Rt=R=T ∑νtνt.
where each diagonal element of Dt is obtained from N univariate GARCH
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
DCC-GARCH(1,1)
This model is due to Engle and Sheppard (2001), it is a benchmark model of time-varying conditional covariance matrix estimation
I We still have problem if N is not small
Main idea, put structure (AR(1)-like structure) on how conditional
correlations move over time
I Ensure well-behaved so positive deÖnite conditional correlation matrix and
each element between 1 and 1 Decompose correlation matrix to achieve this:
R = Q 1Q Q 1, tttt
Qt = (1 a b)Q+aνt 1νt0 1 +bQt 1
1T0 Q = T ∑ νtνt
a > 0, b>0, a+b<1
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
DCC-GARCH (contíd)
Denote the ij0th element in Qt as qij,t. Then:
26pq11,t p0 0 37 6 0 q22,t 0 7 Qt=64 . . ... . 75
0 0pqNN,t
The likelihood function depends on the chosen distribution for the shocks (e.g., Normal), but is otherwise found in a way similar to the ARMA likelihood
I Assume initial shocks equal unconditional average (in particular, need Q0 to be positive deÖnite)
I Use multivariate probability density function
I This is all coded up in R/MatLab, so not covered here, though we will
implement this in mini-case to follow
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 13 / 36
Pre-amble to trading strategy example
The Academic "Information Ratio" (The Appraisal Ratio)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 14 / 36
Information Ratio
The information ratio (IR) is a standard performance metric:
IRi =E[Ri Rbenchmark]. σ[Ri Rbenchmark]
Let Ri,t be excess returns to fund i and Rbenchmark = βiMKTt Estimate αi and βi from the usual regression
R i , t = α i + β i0 F t + ε i , t . IRi =E[Ri,t βiMKTt]= αi
σ[Ri,t βiMKTt] σ(εi,t)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Information Ratio and maximal
Assume the fundís information ratio is 0.3 and the market Sharpe ratio is 0.4 What is the maximal Sharpe ratio one could achieve by combining the fund
and the market?
maxSR = qSRM2KT+IRF2und = p0.32 + 0.42 = 0.5.
Math will follow in future lecture
Notice that if IR 6= 0 it is possible to increase Sharpe ratio
I Typically, though, we canít short sell a fund...
This is why α is interesting
It means you can improve your Sharpe ratio relative to your benchmark. This is valuable and can therefore justify higher fees
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Mini-Case:
From trading signals to trading strategy
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 17 / 36
Initial idea
You have an idea about how to choose stocks that outperform existing benchmark portfolios
In particular, you use a combination of textual analysis, stock prices, and social media-based data to come up with two trading signals for each stock in your trading universe
1 A valuation signal (of fundemental value): vali,t
2 A sentiment signal (of shorter-term trend): trendi,t
Our task is to:
1 See if the signals have any information
2 If so, implement an e¢ cient portfolio strategy trading based on these signals
Caveat: For this illustrative exercise, we only do in-sample testing
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 18 / 36
Unconditional returns to simple trading strategies
A natural starting point is to sort into portfolios based on the signals
One could sort into decile portfolios for each signal (20 portfolios in total), for instance, and look at average return for each decile portfolio
I Valid approach
I Can see if there is signiÖcant spread in average returns to portfolio 10 vs
portfolio 1 for each signal
I Can also spot nonlinear e§ects (perhaps U-shaped average return pattern
across portfolios)
I Later, we will also use Fama-MacBeth regressions to do this
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Unconditional returns to simple trading strategies
Assume that we have time-series of excess returns to two trading strategies: fHMLtgTt=1 and fMOMtgTt=1
I The value and momentum factors from ́s webpage
I Sample average returns, standard deviation, and Sharpe ratio; all annualized I Also give sample correlation (ρ)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 20 / 36
Constructing a simple joint trading strategy
Recall the conditional mean-variance e¢ cient portfolio: ω =kΣ 1μ,
where ωt is a 21 vector of weights on HML and MOM, Σt is the 22 conditional covariance matrix of HML and MOM, and μt is a 2 1 vector with the conditional expected returns to HML and MOM
I k is a constant we can Önd by matching the unconditional volatility of the resulting portfolio to some desired level of overall risk
In the following we compare performance of portfolio with constant weights (based on unconditional sample covariance matrix and average excess returns) to that of a strategy that uses time-varying weights per the above equation.
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 21 / 36
Constructing a simple joint trading strategy
Use the unconditional moments to form the unconditional mean-variance e¢ cient portfolio based on these two sub-portfolios
w = k 0.12 0.1710.10.156 1 0.039 0.171 0.1 0.156 0.1562 0.064
= k4.74 3.14
Find k be setting portfolio variance to 0.152: 0.152 = k2 4.74 0 0.12
3.14 0.171 0.0156 = k20.385
0.1710.0156 4.74 0.1562 3.14
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 22 / 36
Constructing a simple joint trading strategy (contíd)
Solving for k, we have:
k = 0.385 = 0.24
and so the per-period portfolio weights are:
w =0.24 4.74 = 1.14
So, for a $10M portfolio, put $11.4M in HML, $7.5M in MOM
I If HML and MOM are long-short, zero-investment portfolios, this means long $11.4M in value Önanced by shorting the same amount in growth, and long $7.5M in winners, Önanced by the same amount short in losers. Finally, put $10M in risk-free rate.
I Or, if HML and MOM are long-only versions, you need to borrow $8.9M in the risk-free rate to Önance these positions.
I Maintain these portfolio weights by rebalancing each month
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 23 / 36
Constructing a simple joint trading strategy (contíd)
The expected return and Sharpe ratio of this baseline trading strategy are then:
EhRMVEi = 1.14 0.75 0.039 =9.25%. 0.064
SR RMVE = 9.25% = 0.62 15%
Letís see if we can improve on this baseline by attempting to estimate I The conditional expected returns
I The conditional covariance matrix
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Estimating conditional expected returns
Our tool for this is forecasting regressions.
I Need predictive variables
I Thus, we need extra instruments/signals to implement conditional strategies I We will keep it simple and estimate a VAR(1) on HML and MOM returns:
HMLt = φ01 + φ11 φ12 HMLt 1 + εHML,t MOMt φ02 φ21 φ22 MOMt 1 εMOM,t
Implementing this in R (see CCLE) , we get:
φˆ01 = 0.30 , φˆ11 φˆ12 = 0.17 0.01 , RH2ML=2.9%
φˆ 02 0.63 φˆ 21 φˆ 22 0.08 0.06 RM2 OM =0.7% . Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Expected Returns from the VAR(1)
The time-series of annualized expected monthly returns is below Lots of variation, economically, even though R2s were small
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 26 / 36
MVE Portfolio Weights
Using the expected returns from the VAR, along with a constant covariance matrix of the residuals, we obtain portfolio weights
I As before, set volatility of portfolio to be 15% annualized
I HML was most predictable and therefore sees biggest changes in portfolio
I Notice economically large changes in weights, despite modest R2s in
forecasting regressions
I Full sample Sharpe ratio for this strategy is 0.78 > 0.61 (from no expected
return timing)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Adding Volatility Timing
Recall that a cursory look at the data indicates substantial time-variation in the conditional covariance matrix
0.3 0.2 0.1
0.4 0.3 0.2 0.1
1 0.5 0 -0.5 -1
24-month rolling St.dev. of HML
1980 1985 1990 1995 2000 2005
24-month rolling St.dev. of MOM
2010 2015 2020
1980 1985 1990
1995 2000 2005
2010 2015 2020
24-month rolling corr(HML,MOM)
1980 1985 1990
1995 2000 2005
2010 2015 2020
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Estimating a DCC-GARCH(1,1)
We will use the ccgarch-package in R #Run GARCH on residuals
a <- c(0.003, 0.005)
A <- diag(c(0.2,0.3))
B <- diag(c(0.75, 0.6)) dcc.para <- c(0.01,0.98)
dcc.GARCH = dcc.estimation(inia = a, iniA = A, iniB = B, ini.dcc = dcc.para,
dvar = data.dt[complete.cases(data.dt), .(eps.log.HML, eps.log.MOM)], model = "extended"
data.dt[!is.na(shift(log.HML, 1)), `:=`(sigma.HML = sqrt(dcc.GARCH$h[, 1]), sigma.MOM = sqrt(dcc.GARCH$h[, 2]),
rho = dcc.GARCH$DCC[, 2],
eta.HML = dcc.GARCH$std.resid[, 1], eta.MOM = dcc.GARCH$std.resid[, 2] )]
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Estimated conditional volatilities
Lots of time-variation, persistent processes
Momentum has most time-variation in vol (least in expected return)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 30 / 36
Estimated conditional correlation
Conditional correlation between HML and MOM
LOTS of time-variation here (-0.7 to 0.7), likely large e§ects on MVE weights
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 31 / 36
MVE weights from vol and return timing
More variation than when only return timing
The vol component is also more persistent (though this is likely a feature of the predictors for returns only being lagged return)
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 32 / 36
Additional strategy: Vol timing only
Also run case where no attempt at forecasting return, only the conditional covariance matrix
I.e., only vol timing. Less frequent trading, no shorting (pretty much).
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 33 / 36
Cumulative Log Returns for four strategies
Strategies are: No timing, Er timing, Vol timing, both Er and Vol timing (dynamic)
Ranking is: Both Er and Vol, only Vol, only Er, no timing Caveat: recall, these are in-sample tests
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
Summary statistics for four strategies
Timing of return and variance-covariance matrix adds substantially to Sharpe ratios
I Marginal SR increase: IR = p0.9432 0.6132 = 0.717 I Caveat: all in-sample and ignoring transaction costs
I Trading is hazardous to your wealth
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022 35 / 36
On timing strategies
In terms of MVE portfolio weights, the ones we calculated move too much I A lot of trading comes with a lot of trading costs (hazardous to your wealth)
In practice, out-of-sample metrics lead to "shrinkage" of weights so they move less
I Likely beneÖt is smaller than what we found in-sample
I Market-timing is not easy
I Neverthess, this should still be a part of your portfolio optimization
. Lochstoer UCLA Anderson School of ManagemLeenctu(r)e 9 Multivariate Volatility Modeling Case study: Developing a Trading StratWegiynter 2022
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