Risk Management and Return Prediction
PPortfolio Management and Return Prediction
On Markowitz MPT, Constant Correlation Model, Single Index Model, Multi-factor Model
Qingyin Ge, Yunuo Ma, Rongyu Li, Yuezhi Liao, Tianle ZhuMay 2020
1. Introduction
With the well development in financial industry,market starts to catch people’s eyes, not only bythe diversified investing choices ranging from bondsand stocks, to futures and options, but also by thegeneral ”high-risk, high-reward” mindset prompt-ing people to put money in financial market. Whatwe always show concerns for, is nothing but twoterms, risk, and return. People are interested inreducing risk at a given level of return since thereis no way having both high return and low risk.Many researchers have been studying on this issue,and the most pioneering one is Harry Markowitz’s
Modern Portfolio Theory developed in 1952, whichis the cornerstone of investment portfolio manage-ment. Markowitz’s MPT is one of the most widely-used structure in terms of portfolio construction,which aims at ”maximum the return at the givenrisk”. In contrast to that, fifty years later, E. RobertFernholz’s
Stochastic Portfolio Theory , as opposedto the normative assumption served as the basis ofearlier modern portfolio theory, is consistent withthe observable characteristics of actual portfoliosand markets.In this paper, you will see first some basic theo-ries of Markowitz’s MPT and Fernholz’s SPT. Nextwe step across to application side, trying to figureout under four basic models based on
Markowitz Ef-ficient Frontier , including
Markowitz Model , Con-stant Correlation Model , Single Index Model , and
Multi-Factor Model , what portfolios will be selectedand how do these portfolios perform in real world.Here we also involve
Universal Portfolio Algorithm by Thomas M. Cover to select portfolios as compar-ison. In addition, each portfolio’s
Value at Risk , Ex- pected Shortfall and corresponding
Bootstrap confi-dence interval for risk management will be evalu-ated. Finally, by utilizing factor analysis and timeseries model, we could predict future performanceof our four models.
2. Background Theory
Modern Portfolio Theory assumes that investors arerisk averse, where the risk is measured by the vari-ance of asset price. It basically describes the ”trade-off” between return and risk. Investors who wanthigher return must accept higher risk, but differentpeople may have their own risk toleration, whichlead to different investment strategies, forming theso-called
Efficient Frontier . Under the model, wecan represent the expected portfolio return as E ( R p ) = n (cid:88) i =1 w i E ( R i ) (1)where w i is the weight of asset i and R i is the cor-responding asset return; the portfolio volatility as σ p = n (cid:88) i =1 w i σ i + (cid:88) i (cid:88) j (cid:54) = i w i w j σ i σ j ρ ij (2)where σ i is the individual volatility of asset i and ρ ij is the correlation coefficient between returns onasset i and asset j.If risk-free asset gets involved, we are steppinginto Capital Asset Pricing Model , which is so-called”CAPM”. CAPM provides us a decent way to fairly1 a r X i v : . [ q -f i n . GN ] J un rice portfolios. We will use tangent portfolios alongwith our four models for future investigation. Stochastic Portfolio Theory basically shows that”the growth rate of a portfolio depends not only onthe growth rates of the component stocks, but alsoon the excess growth rate , which is determined bythe stock’s variances and covariances.”(R. Fernholzand I. Karatzas, 2008). The stock capitalisationsare modeled by
Ito Process , dynamically. Roughlyspeaking, n positive stock capitalisation processes X i can be modelled as follows dX i ( t ) = X i ( t ) (cid:32) r i ( t ) dt + d (cid:88) ν =1 σ iν ( t ) dW ν ( t ) (cid:33) (3)for t ≥ i = 1 , ..., n . Here W i are indepen-dent standard Brownian Motions and X i are capi-talisations. Notice that this process is on logarithmscale since SPT uses geometric rate of return in-stead of arithmetic growth rate [1]. It is also worthto mention that r i and σ i are F -progressive andsatisfy sum of integral finite almost surely [3]. Inaddition, Fernholz and Shay (1982) were the firstto observe that portfolio diversification and marketvolatility behave as drivers of a growth in such aframe. The growth of a well-diversified portfolio willdominate strictly the average of the individual as-sets growth rate. What really help is its applicationin machine learning framework, especially Function-ally Generated Portfolios . Consider a class of func-tion G ∈ C ( U, R + ) with U an open set. Fernholz’sMaster Equation is a pathwise decomposition of therelative performance of specific portfolios and thatof market, which is free from stochastic integrals: log (cid:18) X π ( T ) X µ ( T ) (cid:19) = log (cid:18) G ( µ ( T )) G ( µ (0)) (cid:19) + (cid:90) T g ( t ) dt (4)where g ( · ) is called the drift process of the portfo-lio π ( · ). G is said to be the generated function ofthe functionally generated portfolio π ( · ). Further-more, one of the most studied FGP is the diversity-weighted portfolios (DWP) with parameter p andsome continuous function f for long only, π fi ( t ) := f ( x i ( t )) (cid:80) nj =1 f ( x j ( t )) , i = 1 , ..., n As Y-L Kom Samo and A. Vervuurt (2016) men-tioned, it is verified by real data that these portfo-lios have potential to outperform the market index,as well as their positive parameter counterparts [2].We try to learn the investment strategy by learn-ing the map f : µ (cid:55)→ µ p , p ∈ [ − , π ∗ (hereis the equally weighted portfolio). We would like tomaximize the objective functions P D ( log ( f )) = SR ( π )= √
252 ˆ E ( r (1) , ..., r ( T ))ˆ S ( r (1) , ..., r ( T ))or P D ( log ( f )) = ER ( π f | EW P )= T (cid:89) t =1 (1 + n (cid:88) i =1 r i ( t ) π fi ( t )) − T (cid:89) t =1 (1 + n (cid:88) i =1 r i ( t ) π ∗ i ( t ))where ˆ E is sample mean and ˆ S is sample sd. As aresult, we will be able to catch argmax p P D ( log ( f ))and thus get the best investment strategy.
3. Application
We have chosen 8 stocks which belong to 6 differ-ent sectors from
Yahoo Finance , including Ama-zon, Apple, Caterpillar, Delta, Google, JP Morgan,Tesla, and Mobil. Our data contain daily prices overthe time period from Jan 1, 2011, to Dec 31, 2019,summing up to total N = 2262 observations.Figure 1: Daily Price of 8 Stocks from 1/1/2011 to12/31/20192rom QQ plots and histograms we find outthat the returns are approximately ”normally” dis-tributed but with heavy tails, meaning that caseswith unexpected high or low values are significantlymore extreme than what would be expected froma normal distribution. Thus we believe the stockreturn follows more or less a t-distribution.Figure 2: Daily Price of 8 Stocks from 1/1/2011 to12/31/2019The correlation plot suggests that each pair ofstocks has around 0.3 correlation between eachother, which lead to careful consideration about theportfolio volatility structure. Let’s begin with
Universal Portfolio Algorithm .”Universal” is in the sense of no statistical assump-tions underlying the market behavior, therefore theconstructed portfolio is robust to real world marketmovements. There are mainly four of them: • Constant Rebalanced Portfolio (CRP), whichuses 1/n as the portfolio weight and rebal-anced it at the beginning of each trading pe-riod • Cover Universal Portfolio (CUP), which cal-culates weights asˆ w k = (cid:82) wS k − ( w ) π ( dw ) (cid:82) S k − ( w ) π ( dw ) (5)where S is the total wealth at current position • Weighted Average of Best CRP, which calcu-lates current portfolio weights as a weightedaverage of the historical best CRP until now • Successively Best CRP, which is a momentum-based strategy using past weight for best CRP for next period. As η → ∞ ˆ w kη = (cid:82) w [ S k − ( w )] η π ( dw ) (cid:82) [ S k − ( w )] η π ( dw ) (6)is the weight for SCRPBased on our data, if we invest the initial wealth$1 on 1/3/2011, the best asset Tesla will bring us$15.57 on 12/31/2019, while the worst asset Cater-pillar will only bring us $1.25.Figure 3: Best Asset and Worst Asset by CRPVia this algorithm, we could build four rebalancedportfolios. The result indicates that CUP, SCRP,CRP provide us better result, which make us endup with $6 or so; while the weighted average CRPonly give us $4.5.Figure 4: Comparison of CRP based on universalportfolio algorithm Markowitz Modern Portfolio Model is a portfolio op-timization model, emphasizing the inherency of risk.It uses historical return and risk as reference, andhelps select the most effective portfolio. Using this3odel, we can construct an efficient frontier of op-timal portfolios offering the maximum possible ex-pected return for a given level of risk.
Constant Correlation Model is a mean-varianceportfolio selection model, where the correlation ofreturns between any pair of different securities isconsidered to be the same. After realizing the pastcorrelation structure hold information about the fu-ture average correlation, we predict future correla-tion with the aggregate technique, by averaging allcorrelation coefficients in the past correlation struc-ture. Below is the formula we use to calculate cor-relation matrix: ρ = (cid:80) Ni =1 (cid:80) Nj =1 ρ ijN ( N − (7) Single Index Model is a regressive model consid-ering the market performance. Our assumption isthat all the securities are related to the market indexas a whole, so here we set S&P500 market return asour index. For each security, we estimate parameter α i and β i to measure their relationship with mar-ket index, which can be represented by the followingformula: R i = α i + β i ∗ R M + (cid:15) i (8)The equation shows that the stock return influencedby the market β and has a specific firm expectedvalue α . Multi Index Model also perform a regression anal-ysis to describe asset returns. The first factor is the excess return of the market portfolio , which is thesole factor in CAPM. The second factor small mi-nus big (SMB), measures the difference in returnson a portfolio of small stocks and a portfolio of bigstocks. The third factor high minus low (HML),measures the difference in returns on a portfolio ofhigh book-to-market value (BE/ME) stocks and aportfolio of low BE/ME stocks.
Fama French threefactors model assumes that excess return on the j th asset for the t th holding period is linearly correlatedwith those three risk factors. The return and riskare estimated below: R it − R ft = α it + β i ( R Mt − R ft )+ β i F + β i F (9)where F ∼ SMB and F ∼ HML σ i = β i σ R M − R f + β i σ SMB + β i σ HML σ ij = β i β j σ R M − R f + β i β j σ SMB + β i β j σ HML (10)We can see from Figure 5, all of four models pro-vide us similar results. During the heyday they canreach around $5 but at last swing down even be-low $1. Generally speaking Constant CorrelationModel performs the best but still shows unsatisfy-ing result. Multi-factor Model ends up with $0.41.Investors certainly can choose to sell it before 2019is coming.Figure 5: Comparison of Model Performance
As we have already generated four basic models,inorder to explore more portfolio risk structure, weutilize two methods, parametric and non-parametricmethods. Here,
Value at Risk (VaR) and
Ex-pected Shortfall (ES) measure the risk, and
Boot-strap method confidence interval can capture itmore precisely.As for parametric method, we calculate VaR andES based on assumption that the our stock re-turns follow t-distribution. By using formulas be-low, where S is the size of the current position and ν is the degree of freedom, (cid:91) V aR t ( α ) = − S × F − ν ( α ) (cid:100) ES t ( α ) = − Sα × (cid:90) F − ν ( α ) −∞ xf ν ( x ) dx we find out that Multi-factor Model has the lowestVaR and ES which are 0.065 and 0.001 respectively.4igure 6: Parametric method resultNon-parametric method is mainly based on thehistorical performance. By ordering the return andfinding the sample quantile, we are able to get theapproximate VaR and ES. The formulas are listedbelow: (cid:91) V aR np ( α ) = − S × ˆ q ( α ) (cid:100) ES np ( α ) = − S × (cid:80) ni =1 R i I { R i ≤ ˆ q ( α ) } (cid:80) ni =1 I { R i ≤ ˆ q ( α ) } It turns out that the Markowitz model has rela-tively the lowest VaR and ES, which are 0.072 and0.140 respectively.Figure 7: Non-parametric method resultNext, we use PerformanceAnalytics package in Rto categorize three Bootstrap methods which areModified, Gaussian and Historical for confidence in-terval.Figure 8: Bootstrap Confidence Interval
In this section we will go further into return predic-tion based on factor analysis and time series analy-sis. Primarily, we split our calculated log return intotraining and testing set. Training data start fromthe first trading day of 2012, to the last trading dayof 2018, and testing data contain all trading days of2019. Afterwards, we collect seven factors includingVolume, Market Return, Inflation Rate, Risk-freeRate, GDP, CPI and Unemployment Rate, initially.In order to see which factors are significantly con-tributed to our model, we perform a model selectionThe result only choose Volume, Market Return andRisk-free Rate as significant factors.One interesting thing about stock return is thatthe residuals do not satisfy the general assumptionsof multivariate regression analysis. Generally we as-sume that the residuals follow independent identi-cally distributed N (0 , σ ), thus after fitting regres-sion model we are done with predicting process,since the predicted value is just the fitted value dueto zero-mean residuals. While this is not the casehere. Financial data have the so-called cluster ef-fect that observations do not follow linear patternbut rather tend to cluster due to heteroskedasticity.Therefore, the conclusions and predicted value onecan draw from the model will not be reliable, whichmotivate us to use the GARCH model to capturethe volatility variations. Based on the log-returnresiduals of the four models and their ACF andPACF plots performance, we have selected differentARMA+GARCH model respectively.Comparing prediction result, Multi-Factor Modelhas adjusted R-squared 0.2503, so the best we cando is merely to explain 25.03% of the variation inthe calculated log return.Models Factor Model Residual ModelMM 9 . × − − . × − V + 1 . R m − . × − R f + (cid:15) ARMA(7,7)+GARCH(1,1)CCM 1 . × − − . × − V + 1 . R m − . × − R f + (cid:15) ARMA(3,3)+GARCH(1,1)SIM 1 . × − − . × − V + 1 . R m − . × − R f + (cid:15) ARMA(3,3)+GARCH(1,1)MFM 7 . × − − . × − V + 1 . R m − . × − R f + (cid:15) ARMA(3,3)+GARCH(1,1)Table 1: Factor Models and Residual Models for four series of log-returns5igure 9: Actual return (in red) and fitted return(in black) from four modelsAlthough the outcomes of return prediction arenot extraordinary, in fact, it is par for the course.To conduct further analysis, there are a few poten-tial reason lie behind. • Portfolio log return based on four models cal-culated is not an actual asset but a virtualone. It may not fit perfectly based on realworld data analysis • Using estimated mean of log return as our pre-diction may not be appropriate since possibleoscillation exists • There are also other hidden factors that we didnot captured and what would actually have in-fluences on the stock returns are inscrutable
4. Conclusion
Return and risk trade-off is an eternal theme, thatinvestors always think about. In this article are arefocusing on dealing with the relationship between them, by utilizing most famous Markowitz’s Mod-ern Portfolio Theory. In addition to that, under fourmodels, Markowitz Model, Constant CorrelationModel, Single Index Model and Multi-factor Model,we develop efficient investment strategy, constructportfolios, thereby seize the volatility structure andpredict future performance. All in all, Multi-factorModel seems to be the outstanding one in both riskmanagement and return prediction, however, the fi-nal result is still embarrassing.As you may wonder, why cannot the models cap-ture the reality happen in the real world well? Webelieve that financial market has more variabilitybeyond the scope of the whole bunch of fundamen-tal theory. After further study by analyzing morecomplex model we could do better.
5. Further Improvement
If time permitted, we are planning to finish modelconstruction based on Stochastic Portfolio Theory,with well-defined machine learning tools mentionedby Vervuurt and Kom Samo.We also made attempt on using copula to fit mul-tivariate joint distribution, based on
Sklar’s Theo-rem , which states that a collection of marginal dis-tributions can be coupled together via a copula toform a multivariate distribution. A copula is a mul-tivariate CDF whose univariate marginal distribu-tions are all Uniform(0,1).[4] Based on the goodnessof fit test we choose to use t-copula and generate thefinal multivariate distribution in dimension 8 by t-copula with degree of freedom 11. The result is un-satisfying, determined by backtesting . In the laterstudy we could investigate what happened in copulautilization.6 . Appendix
Algorithm 1:
Evaluate Basic Model Performance
Result:
Based on CAPM, calculate weight and corresponding wealth for each trading daysi. Initialize total wealth = $1, each stock’s weight = ;ii. Use R f , R i and σ i based on four models, calculate the tangent portfolio which maximize theSharpe’s Ratio as optimal one, derive the weights and final earnings at the end of trading day,and reinvest it at the beginning of next trading day with calculated weight;iii. Iterate (ii) until the end of the trading period, we get a weight matrix W with each rowrepresenting daily weights and a wealth vector S storing the daily earnings ;iv. Return the last entry in S as our final result, and plot evolution of wealth to visualize Algorithm 2:
Bootstrap Confidence Interval Constructioni. Simulate 500 bootstrap samples consisting of log-returns based on our four models;ii. Calculate bootstrap sample (cid:91)
V aR and (cid:100) ES based on simply just quantile (Historical Method);iii. Assume log-return approximately normal and compute (cid:91) V aR and (cid:100) ES (Gaussian Method);iv. Calculate (cid:91) V aR and (cid:100) ES by using Cornish-Fisher Expansion (Modified Method);v. Order the 500 (cid:91) V aR and (cid:100) ES , calculate the sample quantile ˆ q . and ˆ q . , respectively;vi. Lower = 2 (cid:91) V aR ( (cid:100) ES ) − ˆ q . , upper = 2 (cid:91) V aR ( (cid:100) ES ) − ˆ q . Figure 10: Correlation plot for 8 stocksFigure 11: Final performance for four models Figure 12: Sample ACF and PACF plotFigure 13: Time series evaluation7 eferences [1] Ioannis Karatzas, Robert Fernholz. “Stochastic Portfolio Theory: an Overview.” . Handbookof Numerical Analysis, Volume 15, 2009, Pages 89, 91, 93-109, 111-133, 135-147, 149-167, .[2] Yves-Laurent Kom Samo, Alexander Vervuurt. “Stochastic Portfolio Theory: A Machine LearningPerspective.” . eprint arXiv:1605.02654, https://arxiv.org/abs/1605.02654 .[3] Winslow Strong. “Generalizations of Functionally Generated Portfolios with Applications to Sta-tistical Arbitrage.” . SIAM Journal on Financial Mathematics, 2014, Vol.5, No.1 : pp.472492, https://doi.org/10.1137/130907458 .[4] R¨uschendorf L. “Copulas, Sklar’s Theorem, and Distributional Transform.” . Mathematical Risk Anal-ysis. 2013, Chapter 10, pp. 3–34. Springer Series in Operations Research and Financial Engineering, https://link.springer.com/chapter/10.1007/978-3-642-33590-7 [5] Carlos Augusto Zepra Frade “Performance of Return Model: A Port-folio Theoretical Approach.”
Master in Finance, Oct. 2017, pp. 2-24.