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A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 1
Blockkurs
Statistical Analysis Statistical Analysis of Financial Dataof Financial Data
Prof. Dr. Andreas RuckstuhlProf. Dr. Andreas RuckstuhlDozent für Statistische DatenanalyseDozent für Statistische Datenanalyse
Institut für Datenanalyse und Prozess Design IDPInstitut für Datenanalyse und Prozess Design IDPZürcher Hochschule für Angewandte Wissenschaften ZHAWZürcher Hochschule für Angewandte Wissenschaften ZHAW
[email protected]@zhaw.ch23. Januar 201723. Januar 2017
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 2
Your Lecturer
Name: Andreas Ruckstuhl
Civil Status: Married, 2 children
Education: Dr. sc. math. ETH
Position: Professor of Statistical Data Analysis, ZHAW WinterthurLecturer, WBL Applied Statistics, ETH Zürich
Connection: More than fifteen years of cooperation with Peter Meier, Centre for Asset Management, ZHAW
One of the central topics is “alternative investments”,but also multi factor models and index construction
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 3
• Lecture 1: Financial Data and Their Properties
• Lecture 2: Model for Conditional Heteroskedasticity and Risk Measures
• Lecture 3: Statistical Issues When Applying Portfolio Theory– Portfolio Theory– Estimating =E(R) and σR
– Effects of Estimation Error on the Optimal Portfolio
• Lecture 4: Financial Factor Models
• Lecture 5: Copulas
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 4
How should we invest our wealth?
Portfolio theory is based upon two principles:
• Maximize the expected return; and
• Minimize the riskdefined as the standard deviation (also called volatility) of the returns in standard modern finance
• These goals are (somewhat) at odds:
riskier assets generally have a higher expected return
• Without risk premiums, few investors would invest in risky assets
risk premium is the difference between expected return of a risky asset and the risk‐free rate
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 5
• A key concept:Reduction of risk by diversifying the portfolio of assets held
➔ Faced with the problem of constructing an investment portfolio that we hold for one time period (=holding period)
➔ Holding period = one day, one month, one quarter, one year, 10 years , …
➔ At the end of the holding period we might want to readjust the portfolio
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 6
A simple case: Two risky assets• With returns R1 and R2
and expected returns
• Mix them in proportion w and (1‐w), respectively
• The return on the portfolio is
• The expected return on the portfolio is
• Let ρ12 be the correlation between R1 and R2
the variance of the returns on the portfolio RP is
where the covariance σ 12 equals ρ12 σ 1 σ 2
• In vector notation:
1 1 2 2andE R E R
1 2(1 )pR wR w R
1 2(1 )pE R w w
2 2 2 2 21 2 12(1 ) 2 (1 )p PVar R w w w w
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 7
Example:
In Fig.: all points are graphed when w[0,1] and is ‐0.4 (green), 0 (black) and +0.4 (blue)→ shape of the por olio fron er depends on the correlation ρ.• If ρ=1 → there is no benefit from diversifica on• If ρ=‐1 → there exists a por olio weight w such that
σp2=0
The left most point on each curve achieves the minimum value of risk and is called minimum variance portfoliominimum variance portfolio
Using differential calculus, one can easily derive the proportion w from the expression for P thereof the minimum variance portfolio (red point on the curve)cf. Exercise 1, or in general Appendix A.
1 2 1 20.14, 0.08, 0.2, 0.15
,P P PE R
The points on each curve that have an expected return at least as large as the minimum variance portfolio are called the efficient frontierefficient frontier
At R1: w=0; at R2: w=1
0.08 0.10 0.12 0.14 0.16 0.18 0.20
0.08
0.09
0.10
0.11
0.12
0.13
0.14
Risk P
Exp
exte
d R
etur
n P
R1
R2
=-0.4 =0 =0.4
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 8
Combining two Risky Assets with a Risk‐Free Asset• Let Rf be the risk‐free asset: E(Rf)=f and Var(Rf) = 0
• Assume f=0.06 In Fig: F represents the risk‐free asset• for a portfolio consisting of a fixed portfolio of two risky
assets and the risk‐free asset is a line connecting F with a point on the efficient frontier
,P P PE R
• The slope of each line is called its Sharpe’s ratioSharpe’s ratio::
• T on the efficient frontier represents the portfolio highest Sharpe’s ratio.
• It is the optimal portfolio for the purpose of mixing with the risk‐free asset.
• T is called the tangency portfoliotangency portfolio
p f
p
E R
0.00 0.05 0.10 0.15 0.20
0.06
0.08
0.10
0.12
0.14
Risk P
Exp
exte
d R
etur
n P
RF
R1
R2
T
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 9
Optimality ResultThe optimal or efficient portfolios mix the tangency portfolio with the risk‐free asset. Each efficient portfolio has two properties:
• It has a higher expected return than any other portfolio with the same or smaller risk, and
• It has a smaller risk than any other portfolio with the same or higher expected return.
How to find the tangency portfolio?
The proportion wT for the tangency portfolio is
where V1= 1 ‐ f and V2= 2 ‐ f (called excess expected returns)
Can maximize expected returns subject to a lower bound on the risk, or minimize the risk subject to a lower bound on the expected returns by mixing F and T (i.e., find the point on [F,T] which satisfies the constraint)
2
1 2 2 12 1 22 2
1 2 2 1 1 2 12 1 2T
V VwV V V V
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 10
Risk‐free interest rate f:
• While the term "risk‐free interest rate" is an established concept in the theory of financial markets, there are no regulations on how to determine it and, as such, it is not officially established.
• Proxies for the risk‐free rate: Use the returns on domestically held short‐dated government bonds(e.g., US Treasury bill rates, Geldmarktbuchforderungen (GMBF) der Schweizerischen Eidgenossenschaft)However, government bonds may be risky as well ….
• In practice, of course, very few (if any) borrowers have access to finance at the risk free rate.
Note that the issue of “risk‐free interest rate” is not a statistical issue but rather a conceptual issue in finance
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 11
What should we use as the values of =E(R) and σR?
• If returns on the asset are assumed to be stationary, then we can take a time series of past returns and use the
– sample mean and – standard deviation.
Whether the stationary assumption is realistic is always debatable …
• How far back in time should one gather data?A long series, say 10 or 20 years, will give much less variable estimates.However, if the series is not stationary but rather has slowly shifting parameters, then a shorter series (maybe 1 or 2 years) will be more representative of the future…… and almost every time series of returns is nearly stationary over short enough time periods.
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 12
Example: Allocate capital to the equity markets of ten different countries:
MSCI Hong Kong, MSCI Singapore, MSCI Brazil, MSCI Argentina, MSCI UK, MSCI Germany, MSCI Canada, MSCI France, MSCI Japan, S&P 500 (USA).
We use monthly data from January 1988 to January 2002, inclusive. “MSCI” = “Morgan Standley Capital Index”; “S&P” = “Standard and Poors.”
In R:# Prepare Data (be careful with log-returns/returns/%)> dat <- read.table("EquityMarkets.txt",header=T)[,4:13]> n <- nrow(dat)> R <- (dat[2:n,]/dat[1:(n-1),] - 1) # log return, cf tutorial
# Mathematics to optimise a portfolio with more than # two risky assets# (see Appendix and my R function plot.efficientFrontier() # which is built on R package quadprog)
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 13
Example (cont.) source("SAFD_RFn_Lecture3.R")EM.sol <- plot.efficientFrontier(100*R, mufree=1.3/12)# 2002str(EM.sol)List of 5 $ mvp : num [1:2] 3.157 0.902 # (sigma, mu) $ Wmvp : num [1:10] 0.0808 0.00169 ... $ tp : num [1:2] 4.42 1.67 # (sigma, mu) $ Wtp : num [1:10] 0.2472 -0.1928 ... $ tpSharpe: num 0.353
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 14
Selling Short• With the tangency portfolio we obtain the following proportions:
> EM.sol$Wtp 0.2489 ‐0.1950 0.0438 0.0723 0.0716 0.0319 0.0521 0.1781 ‐0.1752 0.6714
• There are negative proportions!!
This means that this asset is to be sold short
• To sell a stock shortsell a stock short, one sells the stock without owning it. The stock must be borrowed from, e.g., a broker. At a later point in time, one buys the stock and gives it back to the lender
• Selling short is a way to profit if a stock prices goes down.
• If short selling is unwanted, you must include the corresponding constraint into the optimisation.
(Results on slide 13 requires short selling)
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 15
Efficient Frontier with and without Selling Short• ## Selling short allowed
EM.sol1 <- plot.efficientFrontier(100*R, mufree=1.3/12)round(EM.sol1$Wtp,4)0.2489 -0.1950 0.0438 0.0723 0.0716 0.0319 0.0521 0.1781 -0.1752 0.6714
• ## Selling short not allowedEM.sol2 <- plot.efficientFrontier(100*R, mufree=1.3/12, noSS=TRUE)round(EM.sol2$Wtp,4)0.1099 0.0000 0.0336 0.0636 0.0613 0.0671 0.0000 0.1013 0.0000 0.5631
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 16
Basic Considerations • Measurable uncertainty/risk (MU) vs unmeasurable uncertainty(UU) (*)
• MU = situations where the probabilities are known (e.g., in gambling (dice), probabilities can be found by simple reasoning and an assumption of symmetry)
• UU = situations where the probabilities are unknownUU is much like the Greek concept of fate; there isn’t much one can say about the future except that it is uncertain
• MU is rather rare outside gambling and random sampling• BUT, statistical inference is the science of estimating probabilities using
data and assuming that the data are representative of what is to be estimated; i.e., statistical inference is a way out of UU
• Hence, if we are willing to assume that future returns will be similar to past returns (i.e., stationarity), we can act as if uncertainty is measurable using statistical inference.BUT: Never drive a car by looking solely in the rear view mirror – unless you want to crash
(*) According to University of Chicago economist Frank Knight
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 17
When N (number of risky assets) is small, the theory of portfolio optimisation can be applied using sample means and sample covariance matrix.
However, the effect of estimation error, especially with larger values of N, can result in portfolio that only appear efficient.
What is the effect of the estimation error on
• Proportions w?
• Tangency portfolio T?
• Sharpe’s Ratio?
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 18
A Bootstrap Simulation Experiment to analyse the effect of estimation on the tangency portfolio
Idea: • The sample of size n is the “true population” of the N=10 assets
→ sample mean and covariance matrix are the “true parameters”
• Bootstrap resampling mimics the sampling process. – Each resample has the same sample size n as the original sample– The resamples are drawn with replacement from the original sample– Calculate all the statistics we are interested in on each resample;e.g., , Sharpe’s ratio*,k, …, for k=1, …, B
• Compare the estimated with the actual Sharpe’s ratio:ActualActual Sharpe’s ratio: Sharpe’s ratio: In each resample we calculate Sharpe’s ratio based on the proportions but use the “true” mean and the “true” covariance matrix
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 19
Example: Allocate capital to the equity markets of ten different countries
Bootstrapping the estimated Sharpe’s ratio of the tangency portfolio (B=250)
In R:> EM.TPboot <- BootTP(Dat=100*R, mufree=1.3/12, maxMu=20, nBoot=250, setSeed=123)> boxplot(EM.TPboot[,1:2], las=1, ylab="Sharpe's ratio")> abline(h=EM.sol$tpSharpe, lty=2, lwd=2, col="gray")
Dashed horizontal line = Sharpe’s ratio of the «true» tangency portfolio (=0.353)
• All resampled tangency portfolio have actual Sharpe’s ratio below 0.352
• The estimated Sharpe’s ratios overestimate the true Sharpe’s ratio quite seriously
• The estimated Sharpe’s ratio is always larger than the actual Sharpe’s ratio
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 20
Bootstrap simulation (cont.)• Effect on the tangency portfolio compared to the efficient frontier and to
the «true» tangency portfolio–Huge variability in both directions (each time skewed to the right)–Overestimation of expected returns
– max(round(sapply(R*100, FUN=mean),3))3.419 ## = Argentina
Why is max(muP) larger than 3.419?(Answer: Because of short selling)→ without short selling see next slide
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 21
Bootstrap simulation (cont.)• What if we know either
– the expected return of the assets ( volatility must be estimated)–Volatility of the assets ( expected return must be estimated)
known / estimated known / estimated
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 22
Conclusion• The efficient frontier is not estimated very accurately.
• Estimation of and yields a lot of variability in the estimated optimal portfolio
• The optimal portfolio is very sensitive to estimation errors in , i.e., much more than to estimation errors in
Consequence: Avoid considering future returns in the optimisation and use other portfolio construction approaches; e.g.,
• Using minimum risk/variance portfolio (i.e., red points on slide 7)
• Using optimisation approaches like «risk parity» or «risk budgeting»
But in any case,we need a good estimation of the variance‐covariance matrix
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 23
What if we had more data?More data do help, though perhaps not as much as we would like.
And be aware: Suppose that we were considering selecting a portfolio from all 3000 stocks on the Russell index.• there would be (3000*2999/2)4.5million covariances to estimate!• A tremendous amount of data would be need to be collected and, after
that effort, the effects of estimation error in 3000 means and 4.5million covariances would be overwhelming – though the computer computes the estimates easily!
• Of course, using a longer time series carries the danger that the data may not be stationary anymore
• Another problem is that some of the times series may not be available for earlier time periods
Hence, although implementing the theory is computationally feasible, this theory may be used if the number of assets is small (clearly below ten).
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 24
Other ways of out of the problem: improving estimation• Bagging:
Idea: Suppose we are estimating some quantity, e.g., the optimal portfolio weight to achieve an expected return of xxx. We have one estimate from the original sample, and B additional estimate, one from each of the bootstrap samples. The bagging estimates is the average of all of these bootstrap estimates. This may yield improved results since bagging can dramatically reduce the variance of unstable procedures. Richard Michaud has patented methods of portfolio rebalancing that essentially are an application of bagging (around 1998).
• Principal Components and Factor Models (cf. next week)
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 25
Minimum Risk/Variance Portfolio• Bootstrap Simulation with Data from Slide 19:
EM.MVPboot <- BootMVP(Dat=100*R, nBoot=250, setSeed=998876)plot.efficientFrontier(Dat=100*R, mufree=NULL, xlim=c(0, 6), ylim=c(0,1.7))points(EM.MVPboot$out$riskMVP, EM.MVPboot$out$muMVP)
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 26
• Portfolio Theory is a nice piece of mathematicsImplementing the theory is computationally feasible / simple
• The results of the theory are disappointing when the two inputs “returns” and “covariances” are estimated from the data by its sample versionsyou may used it if the number of assets is small (clearly below ten).
• The main troublemaker is the estimated returns. The optimal tangency portfolio is very sensitive to the input values of the returns.(Be aware that the weights result from a nonlinear function of of the estimated parameter)
• Use bootstrap simulation experiments to learn more about the effect of estimation errors
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 27
• Lecture 3 in the book:
• Chapter 16: Portfolio Selection
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 28
Finding the Gobal Minimum Variance PortfolioLet be Σ the covariance matrix, w the vector of weightsthen the problem can be expressed as
The Lagrangian for this problem is
And the first order conditions for a minimum are
A. Ruckstuhl -- WBL 2017, Lecture 3 of SAoFD -- Page 29
Finally, substitute the value for λ back into (1) to solve for :
Remark: To invert Σ is not trivial numerically, particularly if the dimension of Σ is big