Portfolio Selection using Kernel Regression. J u s s i K l e m e l ä U n i v e r s i t y o f O u l u

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1 Portfolio Selection using Kernel Regression J u s s i K l e m e l ä U n i v e r s i t y o f O u l u

2 abstract We use kernel regression to improve the performance of indexes Utilizing recent price history can improve the returns

3 Portfolio Selection We have assets d S 1 t,..., S d t Return vector R t = ( (S 1 t S 1 t 1)/S 1 t 1,..., (S d t S d t 1)/S d t 1 ) Portfolio vector b t 1 R d Portfolio returns b T t 1R t = b 1 t 1R 1 t + + b d t 1R d t

4 Markowitz and Expected Utility Markowitz: Find portfolio vector b that maximizes ( E b T ) γ ( R t+1 2 Var b T ) R t+1 Expected utility: Find b that maximizes Eu ( b T R t+1 )

5 Utility Functions { CRRA utility functions u(w) = { w 1 γ 1 γ, if γ > 1, log e w, if γ = 1. CARA utility functions u(w) = 1 e αw, α > 0.

6 Conditional Expected utility Define the conditional expected utility f b (x) = E ( u(b T R t+1 ) X t = x ) Find the portfolio vector as b t = argmax b f b (x t ) The explanatory variables can be chosen as X t = (R t,..., R t k+1 ) R kd

7 Data Based Investment We have data for t = 0,..., t 0 1 Y t = u(b T R t+1 ), X t = (R t,..., R t k+1 ) We estimate the regression function by ˆf b (x) E ( u(b T R t+1 ) X t = x ) Define the portfolio vector as ˆbt0 = argmax b ˆfb (x t0 )

8 Literature Ait-Sahalia and Brandt (2001). Variable Selection for Portfolio Choice. Györfi, Lugosi, and Udina (2006). Nonparametric Kernel-based Sequential Investment Strategies. Andriyashin, Härdle, and Timofeev (2008). Recursive Portfolio Selection with Decision Trees.

9 Kernel Regression Kernel estimate is a weighted average ˆf(x) = t 0 1 t=0 p t (x) Y t Weights are defined as p t (x) = K h (x X t ) t0 1 u=0 K h(x X u ) Scaled kernel K : R d R, K h (x) = K(x/h)/h d

10 S&P 500, Nasdaq-100 Data between and S&P 500: annualized mean 8.8%, standard deviation 18.9%, Sharpe ratio 0.47 NASDAQ-100: annualized mean 15.2%, standard deviation 28.8%, Sharpe ratio 0.53 NASDAQ!100 price S&P time

11 S&P 500, Nasdaq-100 Smoothing parameter h = 0.5,1,2 autoregression parameter k = 5,10. Best Sharpe ratio 0.79 and the worst wealth market time

12 Parameter Selection Let us have L parameter combinations that give L sequences of portfolio weights The corresponding wealth processes The portfolio vector obtained by combining strategies is where q (l) t = b (l) 0,..., b(l) t 0, l = 1,..., L W (l) t 0 = W 0 t 0 1 t=0 b t = W (l) t 1 L l=1 W (l) t 1 (R t+1 + 1) T b (l) t L l=1 q (l) t b (l) t, t = 1,..., t 0

13 S&P 500, Nasdaq-100 Long only, without bank account Annualized mean is 18.1%, standard deviation 24.3%, Sharpe ratio wealth portfolio market wealth market portfolio time (a) time (b)

14 Portfolio Weights Simple strategies have mostly 0-1-weights

15 S&P 500, Nasdaq-100 Bank account included Smoothing parameter h = 0.5,1,2 autoregression parameter k = 5,10. Best Sharpe ratio 0.98 and the worst wealth market time

16 S&P 500, Nasdaq-100 Combining strategies Long only, with bank account Annualized mean is 18.1%, standard deviation 20.3%, Sharpe ratio 0.89 wealth portfolio market wealth portfolio market time (a) time (b)

17 Portfolio Weights Simple strategies have mostly 0-1-weights

18 SP 500, Nasdaq-100 Shorting allowed Smoothing parameter h = 0.7,1,1.5,2 autoregression parameter k = 10. Best Sharpe ratio 0.87 and the worst wealth market time

19 SP 500, Nasdaq-100 Combining strategies Shorting allowed Annualized mean is 20.4%, standard deviation 22.3%, Sharpe ratio wealth portfolio market wealth portfolio market time (a) time (b)

20 !0.5! !1.0 S&P 500, Nasdaq Combining strategies, shorting allowed Figure 11: The distribution of the weights of the combined S&P 500 NASDAQ-100 allowing Leftportfolio and right tailshorting. plots The period during 1 October 1985 until 4 May red=s&p 500, blue=nasdaq-100, black=portfolio!0.20!0.15!0.10!

21 Quantiles % -2.0% -1.3% 0.06% 1.4% 2.1% 4.4% Table 6: Quantiles of the SP500 NASDAQ-100 portfolio allowing shorting % -1.4% -1.0% 0.04% 1.0% 1.4% 2.6% Table 15: Empirical quantiles of the S&P 500 returns % -2.8% -1.9% 0.13% 1.9% 2.8% 5.2% Table 18: Empirical quantiles of the NASDAQ-100 returns.

22 Biggest losses -10.9% -9.2% -8.3% -7.7% -7.0% % -6.7% -6.7% -6.5% -6.5% Table 4: The 10 biggest daily losses of the SP500 NASDAQ-100 portfolio allowing shorting and their dates % -9.0% -8.9% -8.8% -8.3% % -6.9% -6.8% -6.8% -6.7% Table 13: The 10 biggest daily losses of S&P 500 and their dates Table 16: The 10 biggest daily losses of NASDAQ-100 and their dates.

23 Biggest Daily Gains 20.5% 18.8% 12.6% 10.6% 9.8% % 9.10% 9.09% 8.8% 8.4% Table 5: The 10 biggest daily gains of the SP500 NASDAQ-100 portfolio allowing shorting and their dates Table 14: The 10 biggest daily gains of S&P 500 and their dates Table 17: The 10 biggest daily gains of NASDAQ-100 and their dates.

24 S&P 500, Nasdaq Combining strategies, shorting allowed Left and right tail plots for different risk aversion parameters gamma black: γ = 1, red: γ = 25, blue: γ = 100 Sharpe ratios: 0.91, 0.87, and 0.61!0.15!0.10!

25 5: e! Expected Utility Curves!0.15!0.10! !5e!04 5e!04!5e!04 Figure 14: The left and right tail plot of the combined S&P 500-NASDAQ100 portfolios that allow shorting for different values of the risk aversion parameter γ. The period during 1 October 1985 until 4 May gamma gamma Figure 15: Expected utility curves for the portfolio with γ = 1 (black), for the portfolio with γ = 10 (red), for the S&P 500 (green), and for the NASDAQ-100 (blue). The portfolio is the S&P 500 NASDAQ-100 portfolio with shorting allowed. The period extends from 1 October 1985 to 4 May Expected utility curves for the portfolio with γ = 1 (

26 Euro, UK Sterling Shorting allowed, combining strategies to Mean ret 4.8%, sd 7.4%, Sharpe ratio 0.65 portfolio wealth portfolio market wealth market time (a) time (b)

27 U.S. 10 yr, 5 yr Bond Shorting allowed, combining strategies Ret 11.4%, sd 9.8%, Sharpe ratio 1.16 wealth portfolio market wealth market portfolio time (a) time (b)

28 Summary We can improve market portfolios using kernel regression The dependence between indexes can be utilized