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程序代写案例-ECM10/ECM11

时间：2021-04-02

ECM9/ECM10/ECM11

MPhil in Economics

MPhil in Economic Research

MPhil in Finance and Economics

Tuesday 23 March to Monday 5 April 2021

F540

TOPICS IN APPLIED ASSET MANAGEMENT

Candidates are required to answer two compulsory questions

Write your candidate number (not your name) on the cover of the Project.

1 of 6

Questions 1 is weighted at 25%, Question 2 is weighted at 75%

The answer should be no longer than 3000 words inclusive of footnotes and

appendices but exclusive of bibliography

One A4 Page consisting largely of charts, statistics or symbols will be

regarded as the same as 250 words (pro rata for less than A4 page)

F540 : Topics in Applied Asset

Management

Assessment / Project 2021

Answer both questions; 25% of the final mark will come from the theoretical

question 1 and 75% will come from the empirical exercise- question 2 .

THEORETICAL EXERCISE

1. Factor Risk Budgeting additively decomposes individual asset or portfolio re-

turn risk measures into factor contributions allowing a portfolio manager to

know the sources of factor risk for allocation and hedging purposes and allows

a risk manager to evaluate a portfolio from a factor risk perspective. Assume

an asset or portfolio return Rt can be explained by a factor model (FM), with

k factors, of the form

Rt = α + β

′ft + t

where ft ∼ iid(µf ,Ωf ), t ∼ iid(0, σ2 ), cov(fk,t, s) = 0 ∀ k, t, s . Rewrite the

factor model as

Rt = α + β

′ft + t = α + β′ft + σ × zt

=α + β˜′f˜t

β˜ = (β′, σ)′, f˜t = (ft, zt)′, zt =

t

σ

∼ iid(0, 1)

then

σ2FM = β˜

′Ωf˜ β˜, whereΩf˜ =

(

Ωf 0

0 1

)

denoting the risk measure σFM (the factor model vol) by RM(β˜) show

(a) RM(β˜) is a linearly homogeneous function of β˜

(b) Apply Euler’s theorem to provide an expansion of RM(β˜) into k , β com-

ponents and an asset/portfolio specific risk factor

(c) find simple expressions for an individual factor j’s marginal contribution

to risk, then factor j’s contribution to risk and finally for each factor,

j = 1, ...., k, expressions for their percent contribution to risk and the

asset/portfolio specific factor contribution to risk, ie. j = k + 1

2 of 6

(d) Show the same analysis can be applied for Portfolio Risk Budgeting when

you can additively decompose portfolio risk measures into asset contribu-

tions which allows a risk manager to evaluate a portfolio from asset risk

perspective; ie. with a portfolio return

Rp,t = w

′Rt =

N∑

i=1

wiRit

and let RM(w) denote the portfolio vol, σp. The same analysis can in fact

be applied to any convex risk measure in place of vol, such as Value at

Risk and Expected Shortfall. You may wish to use this analysis in your

empirical exercise.

(e) Consider an investment universe of N assets with R = (R1, ..., RN)

′ ∼

N(µ,Σ) and a portfolio with weights x = (x1, ..., xN)

′ and

∑N

i=1 xi = 1

i. Derive the relationship between the β of the portfolio against a bench-

mark or market portfolio and the βi of the individual assets

ii. Given a quadratic utility function- verify that the optimal portfolio is

a linear function of the risk premium and derive an explicit expression

for the implied risk premium,pi = µ− r where r is the risk free rate.

iii. The investor assumes an ex-ante Sharpe Ratio for their portfolio,

SR(x|r) where r is the risk free rate. Show the risk aversion parameter

φ then satisfies the following relationship

φ =

SR(x|r)√

x′Σx

iv. Deduce then that the implied risk premium of asset i is a linear func-

tion of its marginal volatility.

v. What is the economic interpretation of this previous relationship

vi. Find a new expression of the Sharpe Ratio in terms of marginal volatil-

ities.

EMPIRICAL EXERCISE

The empirical exercise is based on developing strategies that seek to be robust

to different phases of the market or the global macro-economy i.e. “disaster proof”

or defensive strategies and hence factor timing, tilting or style rotation are the

issues to explore. You are free to follow your own path in the project as long as

you demonstrate knowledge of the material and techniques that have been covered

in the course.

3 of 6

The steps outlined below should be covered and developing these beyond that

indicated below would increase the final grade awarded. While the more demand-

ing choices that you may make in your empirical analysis will gain greater credit

you must start simple and only attempt more advanced methods if you have al-

ready completed a basic analysis as far as step(e) for instance below, have the time

and are confident in the potential to deliver further results. Be careful to prevent

look ahead bias in your computations - do not let your strategies use data that

would not have been available at the time of strategy deployment- use suitable

lagged inputs and rolling windows for building and implementing your strategies.

1. A section on the course Moodle site has been set up to help you download

your data from CRSP but you may want to get data from other sources. The

development of your data base is your own responsibility but start by down-

loading stock price for the 100 random stocks (PERMNOs) allocated to you

from CRSP for the longest period you can with a common sample size with

no NAs. You can also if you wish download a subset of associated set of stock

characteristics (fundamentals) which will necessarily be for a shorter period-

see the data note on Moodle, a risk free rate, S&P500 index as well as data

on factors of your choice from a variety of sources that will probably include

Ken French’s and/or AQR’s web site, volatility factors can also be found on

Robeco’s web site - https://www.robeco.com/uk/themes/datasets/ . If you

want to use ETF’s they can be downloaded from CRSP- notice in particular

a number of potential hedge ETFs- Bonds, defensive sector indices etc.- again

feel free to expand on this set of data as you wish- for instance you could

also decide to construct technical indicators along the lines of the Neely et al

paper we considered in the lectures. For constructing technical indicators see

the bookdown book Technical Analysis in R by Chiu Yu Ko and R packages

Quantmod, Quantstrat or TTR. Equally you will most probably want to con-

sider ie. construct or download various “risk alarms” such as VIX or use macro

signals from the FRED web site. Your empirical work will be carried out on a

monthly basis. Locate periods of macroeconomic recession and market- “good”

and “bad” periods- crisis, expansion or downturn in your sample possibly using

the NBER indicators. Build your data set.

(a) Diversification: Examine the diversification of your universe of assets to be

considered for inclusion in your portfolio and from these select an investible

and well diversified set with consideration of the objective of building a

defensive strategy. Consider how this set may have changed through time

by exploring diversification over relevant sub periods using tools covered

in the course in a rolling window manner. Comment on what you find.

(b) Return Prediction:Next consider several different approaches to forming

4 of 6

conditional forecasts of the expected returns for your assets- perhaps de-

velop the code for this step with just a few assets at the outset before

generalizing your analysis to a larger set whose size may be limited by

your own local computer power. Hopefully you will be able to consider

strategies for at least 30 diversified assets. At this point you need to build

the data set of your chosen features- predictor variables- factors, technical

indicators, characteristics..... If you include stock characteristics then your

sample period will be necessarily truncated as discussed on the data down-

load document as different consistent sets of characteristics are available

without NAs on a monthly basis over different periods. You may wish to

consider conflating variables as in the Brandt and Santa Clara paper on

Parametric Portfolio Policies we have studied, where they used the slope

of the yield curve to control for different regimes or alternative approaches

to factor timing. You could examine the use and the relative performance

of relaxation methods such as LASSO or ENET, Random Forests, factors

from PLS- Partial Least Squares, OLS, PCA,.... to compare their relative

forecasting ability along the lines covered in the paper by Kelly Pruitt and

Xiu, Empirical Asset Pricing via Machine Learning that we have consid-

ered in the course- you may wish to use this paper as a template, in some

respects, for your own analysis. Provide in your report an analysis of the

relative forecasting performance of the methods you have selected.

(c) Portfolio Construction: Drawing on your analysis in the sections above

build one or more defensive strategies designed to survive bad times- you

could start by simply building a conditional volatility targeting strategy

as we considered in the course with the paper - Conditional Volatility

Targeting, Bongaerts, Kang,van Dijk, Financial Analysts Journal, (2020 )

and then move on to a conditional strategy based on your findings in section

(b). You could consider min variance, mean variance or PPP and replicate

a similar analysis to Brandt and Santa Clara with Parametric Portfolio

Policies. Carry out an (comparative) analysis of the performance of the

defensive strategy or strategies you have developed including downside

risk measures and statistical comparisons where relevant.

(d) Performance Evaluation: Carry out a comparison with your preferred

defensive strategy from (c) with non defensive alternatives such as a stan-

dard Mean Variance portfolio based on FF3, 1

N

based on a set of your

original assets and the SP500. You might want to compare different ap-

proaches to covariance estimation. Graphical plots of backtests should be

on a common scale and use statistical performance comparisons to draw

conclusions regarding the value of the preferred defensive strategy com-

pared with the alternative non-defensive strategies you have considered.

5 of 6

(e) A report should be drawn up, preferably in Markdown, containing your

results and conclusions specifically addressing the original objective- Can

we build a portfolio strategies that are robust to different cycles in the

markets and macro economy?

(f) All code and data used must be submitted with the report and your em-

pirical work must be able to be reproduced by a third party.

(g) As indicated above you are free to follow your own path at each stage but

this path should, in some form, cover the steps outlined above. Limit

your objectives initially to do just the simplest analysis and then

as time allows develop your ideas further.

6 of 6

END OF PAPER

学霸联盟

MPhil in Economics

MPhil in Economic Research

MPhil in Finance and Economics

Tuesday 23 March to Monday 5 April 2021

F540

TOPICS IN APPLIED ASSET MANAGEMENT

Candidates are required to answer two compulsory questions

Write your candidate number (not your name) on the cover of the Project.

1 of 6

Questions 1 is weighted at 25%, Question 2 is weighted at 75%

The answer should be no longer than 3000 words inclusive of footnotes and

appendices but exclusive of bibliography

One A4 Page consisting largely of charts, statistics or symbols will be

regarded as the same as 250 words (pro rata for less than A4 page)

F540 : Topics in Applied Asset

Management

Assessment / Project 2021

Answer both questions; 25% of the final mark will come from the theoretical

question 1 and 75% will come from the empirical exercise- question 2 .

THEORETICAL EXERCISE

1. Factor Risk Budgeting additively decomposes individual asset or portfolio re-

turn risk measures into factor contributions allowing a portfolio manager to

know the sources of factor risk for allocation and hedging purposes and allows

a risk manager to evaluate a portfolio from a factor risk perspective. Assume

an asset or portfolio return Rt can be explained by a factor model (FM), with

k factors, of the form

Rt = α + β

′ft + t

where ft ∼ iid(µf ,Ωf ), t ∼ iid(0, σ2 ), cov(fk,t, s) = 0 ∀ k, t, s . Rewrite the

factor model as

Rt = α + β

′ft + t = α + β′ft + σ × zt

=α + β˜′f˜t

β˜ = (β′, σ)′, f˜t = (ft, zt)′, zt =

t

σ

∼ iid(0, 1)

then

σ2FM = β˜

′Ωf˜ β˜, whereΩf˜ =

(

Ωf 0

0 1

)

denoting the risk measure σFM (the factor model vol) by RM(β˜) show

(a) RM(β˜) is a linearly homogeneous function of β˜

(b) Apply Euler’s theorem to provide an expansion of RM(β˜) into k , β com-

ponents and an asset/portfolio specific risk factor

(c) find simple expressions for an individual factor j’s marginal contribution

to risk, then factor j’s contribution to risk and finally for each factor,

j = 1, ...., k, expressions for their percent contribution to risk and the

asset/portfolio specific factor contribution to risk, ie. j = k + 1

2 of 6

(d) Show the same analysis can be applied for Portfolio Risk Budgeting when

you can additively decompose portfolio risk measures into asset contribu-

tions which allows a risk manager to evaluate a portfolio from asset risk

perspective; ie. with a portfolio return

Rp,t = w

′Rt =

N∑

i=1

wiRit

and let RM(w) denote the portfolio vol, σp. The same analysis can in fact

be applied to any convex risk measure in place of vol, such as Value at

Risk and Expected Shortfall. You may wish to use this analysis in your

empirical exercise.

(e) Consider an investment universe of N assets with R = (R1, ..., RN)

′ ∼

N(µ,Σ) and a portfolio with weights x = (x1, ..., xN)

′ and

∑N

i=1 xi = 1

i. Derive the relationship between the β of the portfolio against a bench-

mark or market portfolio and the βi of the individual assets

ii. Given a quadratic utility function- verify that the optimal portfolio is

a linear function of the risk premium and derive an explicit expression

for the implied risk premium,pi = µ− r where r is the risk free rate.

iii. The investor assumes an ex-ante Sharpe Ratio for their portfolio,

SR(x|r) where r is the risk free rate. Show the risk aversion parameter

φ then satisfies the following relationship

φ =

SR(x|r)√

x′Σx

iv. Deduce then that the implied risk premium of asset i is a linear func-

tion of its marginal volatility.

v. What is the economic interpretation of this previous relationship

vi. Find a new expression of the Sharpe Ratio in terms of marginal volatil-

ities.

EMPIRICAL EXERCISE

The empirical exercise is based on developing strategies that seek to be robust

to different phases of the market or the global macro-economy i.e. “disaster proof”

or defensive strategies and hence factor timing, tilting or style rotation are the

issues to explore. You are free to follow your own path in the project as long as

you demonstrate knowledge of the material and techniques that have been covered

in the course.

3 of 6

The steps outlined below should be covered and developing these beyond that

indicated below would increase the final grade awarded. While the more demand-

ing choices that you may make in your empirical analysis will gain greater credit

you must start simple and only attempt more advanced methods if you have al-

ready completed a basic analysis as far as step(e) for instance below, have the time

and are confident in the potential to deliver further results. Be careful to prevent

look ahead bias in your computations - do not let your strategies use data that

would not have been available at the time of strategy deployment- use suitable

lagged inputs and rolling windows for building and implementing your strategies.

1. A section on the course Moodle site has been set up to help you download

your data from CRSP but you may want to get data from other sources. The

development of your data base is your own responsibility but start by down-

loading stock price for the 100 random stocks (PERMNOs) allocated to you

from CRSP for the longest period you can with a common sample size with

no NAs. You can also if you wish download a subset of associated set of stock

characteristics (fundamentals) which will necessarily be for a shorter period-

see the data note on Moodle, a risk free rate, S&P500 index as well as data

on factors of your choice from a variety of sources that will probably include

Ken French’s and/or AQR’s web site, volatility factors can also be found on

Robeco’s web site - https://www.robeco.com/uk/themes/datasets/ . If you

want to use ETF’s they can be downloaded from CRSP- notice in particular

a number of potential hedge ETFs- Bonds, defensive sector indices etc.- again

feel free to expand on this set of data as you wish- for instance you could

also decide to construct technical indicators along the lines of the Neely et al

paper we considered in the lectures. For constructing technical indicators see

the bookdown book Technical Analysis in R by Chiu Yu Ko and R packages

Quantmod, Quantstrat or TTR. Equally you will most probably want to con-

sider ie. construct or download various “risk alarms” such as VIX or use macro

signals from the FRED web site. Your empirical work will be carried out on a

monthly basis. Locate periods of macroeconomic recession and market- “good”

and “bad” periods- crisis, expansion or downturn in your sample possibly using

the NBER indicators. Build your data set.

(a) Diversification: Examine the diversification of your universe of assets to be

considered for inclusion in your portfolio and from these select an investible

and well diversified set with consideration of the objective of building a

defensive strategy. Consider how this set may have changed through time

by exploring diversification over relevant sub periods using tools covered

in the course in a rolling window manner. Comment on what you find.

(b) Return Prediction:Next consider several different approaches to forming

4 of 6

conditional forecasts of the expected returns for your assets- perhaps de-

velop the code for this step with just a few assets at the outset before

generalizing your analysis to a larger set whose size may be limited by

your own local computer power. Hopefully you will be able to consider

strategies for at least 30 diversified assets. At this point you need to build

the data set of your chosen features- predictor variables- factors, technical

indicators, characteristics..... If you include stock characteristics then your

sample period will be necessarily truncated as discussed on the data down-

load document as different consistent sets of characteristics are available

without NAs on a monthly basis over different periods. You may wish to

consider conflating variables as in the Brandt and Santa Clara paper on

Parametric Portfolio Policies we have studied, where they used the slope

of the yield curve to control for different regimes or alternative approaches

to factor timing. You could examine the use and the relative performance

of relaxation methods such as LASSO or ENET, Random Forests, factors

from PLS- Partial Least Squares, OLS, PCA,.... to compare their relative

forecasting ability along the lines covered in the paper by Kelly Pruitt and

Xiu, Empirical Asset Pricing via Machine Learning that we have consid-

ered in the course- you may wish to use this paper as a template, in some

respects, for your own analysis. Provide in your report an analysis of the

relative forecasting performance of the methods you have selected.

(c) Portfolio Construction: Drawing on your analysis in the sections above

build one or more defensive strategies designed to survive bad times- you

could start by simply building a conditional volatility targeting strategy

as we considered in the course with the paper - Conditional Volatility

Targeting, Bongaerts, Kang,van Dijk, Financial Analysts Journal, (2020 )

and then move on to a conditional strategy based on your findings in section

(b). You could consider min variance, mean variance or PPP and replicate

a similar analysis to Brandt and Santa Clara with Parametric Portfolio

Policies. Carry out an (comparative) analysis of the performance of the

defensive strategy or strategies you have developed including downside

risk measures and statistical comparisons where relevant.

(d) Performance Evaluation: Carry out a comparison with your preferred

defensive strategy from (c) with non defensive alternatives such as a stan-

dard Mean Variance portfolio based on FF3, 1

N

based on a set of your

original assets and the SP500. You might want to compare different ap-

proaches to covariance estimation. Graphical plots of backtests should be

on a common scale and use statistical performance comparisons to draw

conclusions regarding the value of the preferred defensive strategy com-

pared with the alternative non-defensive strategies you have considered.

5 of 6

(e) A report should be drawn up, preferably in Markdown, containing your

results and conclusions specifically addressing the original objective- Can

we build a portfolio strategies that are robust to different cycles in the

markets and macro economy?

(f) All code and data used must be submitted with the report and your em-

pirical work must be able to be reproduced by a third party.

(g) As indicated above you are free to follow your own path at each stage but

this path should, in some form, cover the steps outlined above. Limit

your objectives initially to do just the simplest analysis and then

as time allows develop your ideas further.

6 of 6

END OF PAPER

学霸联盟