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ASX50-excel代写
时间：2023-10-08
Index tracking
For Mean variance investors
Overview
Index tracking
Introduction
Methodology
Application
ASX50 data set
Real time/out of sample analysis
Limitations and extensions
Index tracking: Introduction
Last lecture we saw that the mean-variance portfolio optimisation often
resulted in zero weights for many stocks.
This may be unreasonable for some investors who may still want some exposure to
most (or all stocks) in the index.
To overcome this, we could add additional constraints into our optimiser
eg BHP weight > 0.02.
This week we consider another passive ESG strategy that can overcome this
potential limitation : Index tracking
Index tracking: Introduction
We will consider the same data set as last week.
Top 50 ASX listed stocks based on market capitalisation.
Our objective will be to track the ASX50 index subject to ESG constraints.
The file “Data.xlsx” is the same as last week.
Contains the daily prices and continuously compounded returns on all stocks plus
the ASX200 and ASX50.
Our data set will commence July 4, 2019 and end June 10, 2021.
Only two of the 50 stocks do not have data over the entire sample period: Magellan
Global Fund (MGOC) and TPG Telecom (TPG).
We will use a beta estimate to infer their stock returns based on the ASX200
(highlighted yellow in the spreadsheet).
, ,s t s m tr Beta r= ×
Index tracking: Methodology
We base our methodology on
Andersson, Mats, Patrick Bolton, and Frédéric Samama. "Hedging climate
risk." Financial Analysts Journal 72.3 (2016): 13-32.
They employ a factor approach to estimate the variance-covariance matrix
Factor based approaches (including principle components) may be useful when the
number of assets is large >200.
We will estimate the variance-covariance matrix directly (like last week)
Index tracking: Methodology
The goal is to minimize the tracking error between the benchmark portfolio
and the ESG or “green” portfolio.
Tracking error is the divergence between the price behavior of a position or
portfolio and the price behavior of a benchmark.
Can be viewed as an indicator of how actively a fund is managed and its risk level.
Defined as the standard deviation of the difference in the returns between the
benchmark and the green portfolio.
Index tracking: Methodology
If only seek to minimise
in large negative values
for
We can include >1 ESG constraint
− , this will result
( ) ( )
Min TE sd( )
'
subject to
Portfolio CO2 150T/$1M(USD)in revenue
where
sd is thestandard deviation
is the return on thegreen portfolio
is the return on the benchmark portfolio
is a 1vector of green
g b
g b g b
g
b
g
R R
x x x x
R
R
x N
= −
= − Σ −
≤
× weights, where N is the no of assets
is a 1vector of benchmark weights
is the variance covariance matrix
bx N
N N
×
Σ ×
Index tracking: Methodology
We seek to evaluate the performance of the green portfolio from July 1, 2020
to June 10, 2021 in real time.
We will adopt a rolling window approach
Starting June 30, 2020 estimate the daily covariance matrix using the last 252
trading days of returns (July 4, 2019 to June 30, 2020)
Minimise the TE and set the portfolio weights for the coming day (July 1, 2020).
At the end of the day, re-estimate the covariance matrix using the last 252 trading
days (July 5, 2019 to July 1, 2020).
Minimise the TE and re-balance the portfolio, setting the weight for the next day (July 2,
2020).
This process of daily re-balancing occurs until the last day in the data set (June 10,
2021).
Index tracking: Methodology
The benchmark portfolio weights will be equal to their market cap weights on
June 11, 2020. These weights will be constant over the entire period.
Ideally, we would set these weights equal to the actual weights for each day.
Like last week, we will set to zero the weights on the 10 stocks that do not
report C02.
Acts like a negative screen (removing firms that don’t take climate change
seriously).
This will add tracking error, even before we start the optimization
Index tracking: Spreadsheet
Market weights input into cells
Green weights initially set to arbitrary
values say 1/50
Diff weights = Green weight-Market
weight
Formula for the sum of the green
weights
C02/$1M from last week
Red highlights have weights at zero in
optimiser
Co2 contribution: formula C02*weight
Index tracking: Spreadsheet
CO2 contribution mkt = mkt weight * C02
CO2 mkt portfolio (sum that excludes red
highlights). This sets the upper limit that we
seek to beat.
Tracking error: formula (previous slide)
Cov matrix: Formula based on excess
returns matrix(same as last week)
Excess returns matrix input: 252 days of
returns (D564:BA815), plus average returns
vector (D817:BA817)
Index tracking: Spreadsheet
For the code to work, we
need to have the following
details in the Name manager
Index tracking: Spreadsheet
Note that the stocks with no
CO2 reporting have their
weights set to zero.
We can now solve for the
green weights given the
covariance matrix.
We now seek to do this with a
rolling window each day. To do
this we will use a VBA program
Index tracking: VBA code
Index tracking: VBA code
Results matrix now contains in
each row:
tracking error
portfolio returns
weight vector
for each day
Index tracking results
Average tracking error 6.07%.
Green portfolio has outperformed the index
i) Higher total return (24.4% v 20.3%) –
difference is significant at 5%
ii) Higher daily Sharpe ratio (0.1095 v 0.0907)
iii) Highly correlated with index (0.9928)
Index tracking
We now seek to remove an additional 5 stocks from the portfolio.
Re-perform the analysis after removing BHP, CSL, NAB, RIO, SYD
What effect does this have on the asset allocation and tracking
error?
Now reduce the total CO2 from 150 to 135 and re-perform the
analysis. Are the results consistent with your expectations?
Index tracking: Limitations & extensions
Same limitations as last lecture
Consider impact of re-balancing at a lower frequency (eg weekly).
Likely to increase tracking error.
Use an industry average (median) for the stocks without CO2 reporting.
Likely to reduce tracking error but may materially mis-represent the CO2 footprint
of the portfolio.
Update benchmark portfolio weights each day (as opposed to the fixed
weights as at June 11, 2021).
More computationally intensive but likely to reduce tracking error
Rather than track a regular index, this approach can also be used to track any
of the growing number of ESG indices available. This would enable you to
track an ESG index but tailor your portfolio to impose some additional ESG
constraints.
Summary
The last two lectures have illustrated two passive ESG strategies
Efficient frontier estimation with and without ESG
Index tracking with ESG
Results have tentatively shown that the incorporation of ESG into
standard portfolio allocation decisions may not necessarily come at a
cost to the investor and may improve outcomes
For Mean variance investors
Overview
Index tracking
Introduction
Methodology
Application
ASX50 data set
Real time/out of sample analysis
Limitations and extensions
Index tracking: Introduction
Last lecture we saw that the mean-variance portfolio optimisation often
resulted in zero weights for many stocks.
This may be unreasonable for some investors who may still want some exposure to
most (or all stocks) in the index.
To overcome this, we could add additional constraints into our optimiser
eg BHP weight > 0.02.
This week we consider another passive ESG strategy that can overcome this
potential limitation : Index tracking
Index tracking: Introduction
We will consider the same data set as last week.
Top 50 ASX listed stocks based on market capitalisation.
Our objective will be to track the ASX50 index subject to ESG constraints.
The file “Data.xlsx” is the same as last week.
Contains the daily prices and continuously compounded returns on all stocks plus
the ASX200 and ASX50.
Our data set will commence July 4, 2019 and end June 10, 2021.
Only two of the 50 stocks do not have data over the entire sample period: Magellan
Global Fund (MGOC) and TPG Telecom (TPG).
We will use a beta estimate to infer their stock returns based on the ASX200
(highlighted yellow in the spreadsheet).
, ,s t s m tr Beta r= ×
Index tracking: Methodology
We base our methodology on
Andersson, Mats, Patrick Bolton, and Frédéric Samama. "Hedging climate
risk." Financial Analysts Journal 72.3 (2016): 13-32.
They employ a factor approach to estimate the variance-covariance matrix
Factor based approaches (including principle components) may be useful when the
number of assets is large >200.
We will estimate the variance-covariance matrix directly (like last week)
Index tracking: Methodology
The goal is to minimize the tracking error between the benchmark portfolio
and the ESG or “green” portfolio.
Tracking error is the divergence between the price behavior of a position or
portfolio and the price behavior of a benchmark.
Can be viewed as an indicator of how actively a fund is managed and its risk level.
Defined as the standard deviation of the difference in the returns between the
benchmark and the green portfolio.
Index tracking: Methodology
If only seek to minimise
in large negative values
for
We can include >1 ESG constraint
− , this will result
( ) ( )
Min TE sd( )
'
subject to
Portfolio CO2 150T/$1M(USD)in revenue
where
sd is thestandard deviation
is the return on thegreen portfolio
is the return on the benchmark portfolio
is a 1vector of green
g b
g b g b
g
b
g
R R
x x x x
R
R
x N
= −
= − Σ −
≤
× weights, where N is the no of assets
is a 1vector of benchmark weights
is the variance covariance matrix
bx N
N N
×
Σ ×
Index tracking: Methodology
We seek to evaluate the performance of the green portfolio from July 1, 2020
to June 10, 2021 in real time.
We will adopt a rolling window approach
Starting June 30, 2020 estimate the daily covariance matrix using the last 252
trading days of returns (July 4, 2019 to June 30, 2020)
Minimise the TE and set the portfolio weights for the coming day (July 1, 2020).
At the end of the day, re-estimate the covariance matrix using the last 252 trading
days (July 5, 2019 to July 1, 2020).
Minimise the TE and re-balance the portfolio, setting the weight for the next day (July 2,
2020).
This process of daily re-balancing occurs until the last day in the data set (June 10,
2021).
Index tracking: Methodology
The benchmark portfolio weights will be equal to their market cap weights on
June 11, 2020. These weights will be constant over the entire period.
Ideally, we would set these weights equal to the actual weights for each day.
Like last week, we will set to zero the weights on the 10 stocks that do not
report C02.
Acts like a negative screen (removing firms that don’t take climate change
seriously).
This will add tracking error, even before we start the optimization
Index tracking: Spreadsheet
Market weights input into cells
Green weights initially set to arbitrary
values say 1/50
Diff weights = Green weight-Market
weight
Formula for the sum of the green
weights
C02/$1M from last week
Red highlights have weights at zero in
optimiser
Co2 contribution: formula C02*weight
Index tracking: Spreadsheet
CO2 contribution mkt = mkt weight * C02
CO2 mkt portfolio (sum that excludes red
highlights). This sets the upper limit that we
seek to beat.
Tracking error: formula (previous slide)
Cov matrix: Formula based on excess
returns matrix(same as last week)
Excess returns matrix input: 252 days of
returns (D564:BA815), plus average returns
vector (D817:BA817)
Index tracking: Spreadsheet
For the code to work, we
need to have the following
details in the Name manager
Index tracking: Spreadsheet
Note that the stocks with no
CO2 reporting have their
weights set to zero.
We can now solve for the
green weights given the
covariance matrix.
We now seek to do this with a
rolling window each day. To do
this we will use a VBA program
Index tracking: VBA code
Index tracking: VBA code
Results matrix now contains in
each row:
tracking error
portfolio returns
weight vector
for each day
Index tracking results
Average tracking error 6.07%.
Green portfolio has outperformed the index
i) Higher total return (24.4% v 20.3%) –
difference is significant at 5%
ii) Higher daily Sharpe ratio (0.1095 v 0.0907)
iii) Highly correlated with index (0.9928)
Index tracking
We now seek to remove an additional 5 stocks from the portfolio.
Re-perform the analysis after removing BHP, CSL, NAB, RIO, SYD
What effect does this have on the asset allocation and tracking
error?
Now reduce the total CO2 from 150 to 135 and re-perform the
analysis. Are the results consistent with your expectations?
Index tracking: Limitations & extensions
Same limitations as last lecture
Consider impact of re-balancing at a lower frequency (eg weekly).
Likely to increase tracking error.
Use an industry average (median) for the stocks without CO2 reporting.
Likely to reduce tracking error but may materially mis-represent the CO2 footprint
of the portfolio.
Update benchmark portfolio weights each day (as opposed to the fixed
weights as at June 11, 2021).
More computationally intensive but likely to reduce tracking error
Rather than track a regular index, this approach can also be used to track any
of the growing number of ESG indices available. This would enable you to
track an ESG index but tailor your portfolio to impose some additional ESG
constraints.
Summary
The last two lectures have illustrated two passive ESG strategies
Efficient frontier estimation with and without ESG
Index tracking with ESG
Results have tentatively shown that the incorporation of ESG into
standard portfolio allocation decisions may not necessarily come at a
cost to the investor and may improve outcomes
