Minimum Risk vs. Capital and Risk Diversification strategies for portfolio construction

Transcription

1 Minimum Risk vs. Capital and Risk Diversification strategies for portfolio construction F. Cesarone 1 S. Colucci 2 1 Università degli Studi Roma Tre francesco.cesarone@uniroma3.it 2 Symphonia Sgr - Torino - Italy stefano.colucci@symphonia.it XVI Workshop on Quantitative Finance Parma, January 29-30, 2015

2 Outline 1 Introduction Motivations Categories of portfolio selection models 2 Models and Methodologies Minimum-Risk strategies Capital Diversification strategy Risk Diversification strategies Description of Performance Measures 3 Computational Results Experimental setup Empirical analysis 4 Conclusions

3 Motivations The risk-gain analysis is a key issue for portfolio construction, but risk-gain optimization can sensitively suffer the changes in the inputs, in particular due to estimation errors of the means of the assets returns (see, e.g., [Best and Grauer(1991)]) However, these approaches can lead to a portfolio poorly diversified in terms of risk [Maillard et al.(2010), Cesarone and Tardella(2014)]. The recent global financial crisis started in 2008 has given rise to a new research stream that aims at diversifying risk. A straightforward method to diversify risk could be the use of the Equally Weighted (EW) portfolio. It is often used in practice and some authors claim that its practical out-of-sample performance is hard to beat on real world data sets [DeMiguel et al.(2009)].

4 Categories of portfolio selection models Classes of models analyzed Minimum-Risk strategies: the use of the expected return is avoided, and only the minimization of risk is performed. Capital Diversification strategy: the invested capital is equally distributed among the assets. Risk Diversification strategies: each asset contributes equally to the total portfolio risk.

5 Minimum-Risk strategies Minimum Variance portfolio min st n n σ ij x i x j i=1 j=1 n x i = 1 i=1 x i 0 i = 1,..., n (1) x the vector of decision variables whose components x i define the assets weights in a portfolio. Σ their covariance matrix, where the generic element σ ij is the covariance of returns of asset i and asset j with i, j = 1,..., n.

6 Minimum-Risk strategies Minimum CVaR portfolio min s.t. CVaR ɛ (x) n x i = 1 i=1 x i 0 i = 1,..., n (2) In the discrete case, where T in scenarios are considered equally likely, we have min CVaR ɛ (x) = min ζ + 1 T in n [ r it x i ζ] + (3) x (x,ζ) ɛt in t=1 i=1

7 Minimum-Risk strategies Minimum CVaR portfolio Using T in auxiliary variables d t, Problem (2) can be reformulated as the following Linear Programming (LP) problem min s.t. ζ + 1 ɛ 1 T in d t T in t=1 n d t r it x i ζ, i=1 d t 0, n x i = 1 i=1 x i 0 ζ R t = 1,..., T in t = 1,..., T in i = 1,..., n (4)

8 Capital Diversification strategy Equally Weighted portfolio The EW portfolio is the one where the capital is equally distributed among the assets: x i = 1/n i = 1,..., n (5) It does not use any in-sample information nor involve any optimization approach. We use this portfolio as a benchmark to compare the performance of the portfolios constructed by the models.

9 Risk Diversification strategies Risk Parity portfolio The RP portfolio is characterized by the requirement of having equal total risk contribution from each asset, where the risk is measured by volatility. Since σ(x) is a homogeneous function of degree 1, σ(x) = n TRCi σ (x), where i=1 TRC σ i (x) = x i σ(x) x i risk contribution of asset i. = x i (Σx) i σ(x) = 1 σ(x) n σ ik x i x k is the total k=1 TRC σ i (x) = TRC σ j (x) x i (Σx) i = x j (Σx) j i, j, (6)

10 Risk Diversification strategies Risk Parity portfolio It can be found by solving the following system of linear and quadratic equations and inequalities: RP portfolio x i (Σx) i = λ n x i = 1 i=1 x i 0 i = 1,..., n i = 1,..., n (7) It has a unique solution (x RP, λ RP ), at least when the covariance matrix Σ is positive definite.

11 Risk Diversification strategies Naive RP portfolio If ρ ij = ρ = constant for all i j (in particular when n = 2) then NRP portfolio xi NRP = σ 1 i n k=1 σ 1 k i = 1,..., n (8) i.e. x NRP i is proportional to the inverse of the volatility σ i.

12 Risk Diversification strategies CVaR Equal Risk Contribution portfolio This portfolio allocates (possibly) equal total risk contribution among all assets, where the risk is measured by CVaR. Since CVaR ɛ (x) CVaR ɛ (x) = n i=1 TRCi CVaR (x), where TRCi CVaR CVaR (x) = x ɛ(x) i x i is the total risk contribution of asset i. The CVaR ERC portfolio is obtained by imposing: TRC CVaR i (x) = TRCj CVaR (x) i, j = 1,..., n n i=1 x i = 1 x i 0 i = 1,..., n (9)

13 Risk Diversification strategies CVaR Equal Risk Contribution portfolio Since the existence of a CVaRERC solution is not always guaranteed, we minimize the deviations of relative risk (x) CVaR ɛ(x) contributions TRCCVaR i contribution when TRC CVaR i min n x i=1 s.t. n x i = 1 i=1 x i 0 from 1 n, where 1 n is the relative risk (x) = TRCj CVaR (x) i, j ( TRC CVaR ) 2 i (x) CVaR ɛ(x) 1 n i = 1,..., n (10) The solution of (10) is a feasible portfolio which is as close as possible to a CVaRERC portfolio.

14 Risk Diversification strategies Naive CVaRERC portfolio We present a new naive approach to reach approximatively equal risk contribution of all assets, focusing on the CVaRERC portfolio based on the worst case scenario of CVaR CVaRɛ W. n n CVaR ɛ ( x i r i ) x i CVaR ɛ ( r i ) = CVaRɛ W (11) i=1 i=1 This naive portfolio represents the unique CVaRERC portfolio with x i 0, n i=1 x i = 1, corresponding to CVaR W ɛ x k = CVaR 1 ɛ ( r k ) n j=1 CVaR 1 ɛ ( r j ) (12) Note that the NCVaRERC portfolio is similar to the Naive RP portfolio.

15 Description of Performance Measures Performance measures analyzed Sharpe ratio SR = ˆµout ˆσ out (13) where ˆµ out is the average of out-of-sample portfolio returns, and ˆσ out is their sample standard deviation. We examine the statistical significance of the difference between Sharpe ratios for two given portfolios by using the bootstrapping methodology proposed by [Ledoit and Wolf(2008)]. Turnover Turn = 1 Q Q j=1 i=1 where Q is the number of rebalances realized. n xj,i x j 1,i (14)

16 Description of Performance Measures Performance measures analyzed Maximum Drawdown Drawdowns: dd τ = Wτ max T in +1 s τ (W s) max Tin +1 s τ (W s), where W τ = W τ 1 (1 + Rτ out ) with τ = T in + 1,..., T. Maximum Drawdown Mdd corresponds to the worst drawdown Ulcer Index Mdd = UI = min (dd τ ). (15) T in +1 τ T T τ=t in +1 dd 2 τ T T in (16) UI measures the impact of long and deep drawdowns.

17 Description of Performance Measures Performance measures analyzed Risk Diversification Index It provides a description of the portfolio risk concentration structure, based on the Herfindahl index HI = n i=1 (RCR i) 2, where RCR i = TRCCVaR i (x) CVaR ɛ(x) represents the relative contribution of asset i to total risk. A normalized version of HI is NHI = 1 HI. 1 1 n We consider the average of NHI on Q rebalances realized ANHI = 1 Q Q NHI j (17) j=1

18 Experimental setup Data sets We select investment universes which consist of equities, bonds and mixed assets. Data set of assets time interval abbreviation 1 Global diversified portfolio 6 03/01/ /10/2014 GDP-Mix1 2 Italian Bond and Global Equity Portfolio 7 03/01/ /10/2014 IBGEP-Mix2 3 Worldwide Asset 28 01/01/ /10/2014 WWA-Mix3 4 Stock Picking on Eurostoxx /01/ /10/2014 Euro-Eq1 5 World Equity Sectors Portfolio 9 03/01/ /10/2014 WES-Eq2 6 Equity Emerging Countries 11 03/01/ /10/2014 EEC-Eq3 7 Euro Government Bond Portfolio 10 03/01/ /10/2014 Euro-Bond Sources of risk that characterize the data sets GDP-Mix1, IBGEP-Mix2, WWA-Mix3 : interest rate, credit, equity, inflation, currency, real estate risk. Euro-Eq1, WES-Eq2, EEC-Eq3 : market and country risk. Euro-Bond : interest rate and credit risk (from 2010).

19 Experimental setup In the empirical analysis we adopt both historical and simulated (Historical Filtered Bootstrap) scenarios (see, e.g., [Colucci and Brandolini(2011)]), namely from a Risk Management viewpoint we use two different methods to evaluate the future portfolio return distribution. Our analysis is based on a rolling time windows approach. We find the optimal portfolios obtained by all models, using an in-sample time window of T in = 500 days. We evaluate the performance of such portfolios on the following 20 days (out-of-sample), during which no rebalances are allowed. For each portfolio strategy the rolling time windows approach generates T T in daily out-of-sample portfolio returns on which we estimate the performance measures described above.

20 Empirical analysis Expected Return, Volatility, Sharpe ratio, Robust test

21 Empirical analysis Max Drawdown, Ulcer Index, Turnover, Risk Diversification Index

22 Introduction Empirical analysis Numerical Models and Methodologies Computational Results Conclusions

23 Conclusions and further research The out-of-sample Sharpe ratio (SR) of many strategies is often higher than that of the EW portfolio (due to high volatility). Minimum-Risk models generally have very good SR in almost all assets universes, while Risk Diversification models tend to have an intermediate position. Performing extensive Robust tests on SRs, we find that on equity markets the SRs obtained with the models are not statistically different from the EW portfolio; conversely in the other investment universes. When one invests in a universe with a single source of risk, there is no a clear dominance between Minimum-Risk and Risk Diversification models, and the EW portfolio. On investment universes with multiple sources of risk, the Minimum-Risk and the Risk Diversification strategies tend to have better performance than the EW one.

24 Conclusions and further research It seems that each tested model responds to different requirements that can be related to diverse investors attitudes. As expected, Minimum-Risk models are advisable for risk adverse investors. The EW portfolio seems to be advisable for risk lover investors, which try to maximize the return without worrying about periods of deep drawdowns. Risk Diversification strategies seem to be appropriate for investors mildly adverse to the total portfolio risk. These investors tend to be available to waive a bit of safety and of return to achieve a more balanced portfolio in terms of risk. Future research: provide a single asset allocation model that is a combination of the three different classes of portfolio strategies analyzed.

25 Thank you for your attention!

26 Bibliography MJ Best and RR Grauer. On the sensitivity of mean-variance-efficient portfolios to changes in asset means: some analytical and computational results. Rev Financ Stud, 4: , F Cesarone and F Tardella. Equal risk bounding is better then risk parity for portfolio selection. Advanced Risk & Portfolio Management Research Paper Series, 4, Available at SSRN: S Colucci and D Brandolini. A risk based approach to tactical asset allocation. Available at SSRN: V DeMiguel et al. Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy? Rev Financ Stud, 22: , ISSN O Ledoit and M Wolf. Robust performance hypothesis testing with the Sharpe ratio. J Empir Financ, 15: , S Maillard et al. The Properties of Equally Weighted Risk Contribution Portfolios. J Portfolio Manage, 36:60 70, 2010.