The record of the sunspot number visible on the sun is regularly collected over the centuries by various observatories for studying the different factors influencing the sunspot cycle and solar activity. Sunspots appear in cycles, and last several years. These cycles follow a certain pattern which is well known. We analyzed monthly and yearly averages of sunspot data observed from year 1818 to 2002 using rescaled range analysis. The Hurst exponent calculated for monthly data sets are 0.8899, 0.8800 and 0.8597 and for yearly data set is 0.7187. Fractal dimensions1 calculated are 1.1100, 1.1200, 1.1403 and 1.2813. From the study of Hurst exponent and fractal dimension, we conclude that time series of sunspots show persistent behavior. The fundamental tool of signal processing is the fast Fourier transform technique (FFT). The sunspot data is also analyzed using FFT. The power spectrum of monthly and yearly averages of sunspot shows distinct peaks at 11 years confirming the well known 11-year cycle. The monthly sunspot data is also analyzed using FFT to filter the noise in the data.

ABSTRAKTujuan dari penelitian ini adalah untuk memilih dan membentuk sebuah portofolio optimal dari aset-aset berisiko (saham) pada Indeks Liquiditas 45 (LQ-45) dengan menggunakan Capital Asset Pricing Model (CAPM). Sebanyak 45 saham di LQ-45 yang diperoleh dari www.finance.yahoo.com kemudian diseleksi untuk didapatkan lima saham terbaik dengan kriteria returnpositif, beta saham agresif, saham undervalued, dan koefisien variasi positif terkecil. Dengan empat kategori sebelumnya kemudian diperoleh lima saham terbaik yaitu saham Unilever Indonesia Tbk. (UNVR), saham Bank Negara Indonesia (Persero) Tbk. (BBNI), saham HM Sampoerna Tbk. (HMSP), saham Adaro Energy Tbk. (ADRO), dan saham Indocement Tunggal Prakasa Tbk. (INTP). Kriteria portofolio optimal pada penelitian ini  adalah portofolio optimal berdasarkan Model Markowitz dengan preferensi return dan risiko dari saham-saham individual, bukan salah satu di antara keduanya. Pembentukan portofolio optimal termasuk menghitung varian, kovarian, dan bobot masing-masing saham, serta  return dan risiko portofolio yang dibantu dengan Microsoft Excel Solver add-ins. Fungsi objektif dari optimalisasi ini adalah meminimumkan varian dari portofolio (atau standar deviasi portofolio) sehingga diperoleh output yaitu bobot kelima saham dalam portofolio optimal. Dari penelitian ini diperoleh sebuah portofolio optimal dengan kombinasi dari kelima saham unggul dengan bobot berturut-turut 65.4 %, 0.0 %, 20.6 %, 14.0 %, dan 0.0 %. Return dan risiko portofolio optimal tersebut berturut-turut 2.4408 % dan 3.7072 %.Kata Kunci: CAPM, LQ-45, Portofolio Optimal, Model MarkowitzABSTRACTThe purpose of this research is to select and to form an optimal portfolio from risky assets (stocks) listed in Indeks Liquiditas 45 (LQ-45) using Capital Asset  Pricing Model (CAPM). There are 45 stocks listed in LQ-45 which obtained from www.finance.yahoo.com to be selected aiming to choose five best stocks categorized by positive return stocks, aggressive beta stocks, undervalued stocks, and stocks with least positive coefficient of variation. Using four categorizes mentioned before, the five best stocks are  stock of  Unilever Indonesia Tbk. (UNVR), Bank Negara Indonesia (Persero) Tbk. (BBNI), HM Sampoerna Tbk. (HMSP),  Adaro Energy Tbk. (ADRO), and Indocement Tunggal Prakasa Tbk. (INTP). The optimal portfolio’s  criteria in this research is based on Markowitz Model using return and risk preferences of the individual stocks. The forming of the optimal portfolio includes calculating variances, covariances, and weights of stocks, and also return and risk of the portfolio utilizing Microsoft Excel Solver add-ins software. The objective function of the optimalization is minimizing the variance of the portofolio (or the deviation standard of the portfolio) such that generated the weights of the five stocks as the outputs of the minimization. Therefore, there is a combination of the stocks forming an optimal portfolio that is including 65.4 % of UNVR, 0.0% of BBNI, 20.6 % of HMSP, 14.0 % of ADRO, dan 0.0 % of INTP. The return and the risk of the portfolio are 2.4408 % and 3.7072 %. Keywords : CAPM, LQ-45, OptimalPortofolio,MarkowitzModel

This study aims at detecting the number, locations and size of deterministic shifts in a financial time series, using Inclan and Tiao (1994)'s algorithm. The algorithm, developed to address the violation of the assumption of constant unconditional variance of GARCH model in order to reduce the persistence of volatility over time, uses the cumulative sums of squares of partitioned series, and is iteratively applied to detect both mean-and variance-changes in the series, hence named Iterated Cumulative Sums of Squares (ICSS) algorithm. A properly normalized version of the maximum of CSS-statistic asymptotically follows normal distribution, the quantiles of which are used in the algorithm. Firm-level data from Karachi Stock Exchange is used to demonstrate the application of the algorithm. An improved form of the algorithm, by Bos and Hoontrakul (2002), is also applied as a sensitivity check to evaluate and rectify the cases where ICSS algorithm might have detected a mean-shift in the series as a variance-shift.