Beta Risklerinin Modellenmesi ve Tahmini: Türkiye'deki Döviz Portföyü Örnegi/Modeling and Forecasting of Beta Risks: The Case of Foreign Currency Portfolio in Turkey

• Published in: Cankiri Karatekin Universitesi Iktisadi ve Idari Bilimler Fakultesi Dergisi Vol. 11; no. 2; p. 467
• Main Authors: NeslIhanoglu, Serdar, Paker, Merve
• Format: Journal Article
• Language: Turkish
• Published: Cankiri Karatekin Universitesi 22.09.2021
• ISSN: 1308-5549
• DOI: 10.18074/ckuiibfd.804693

Arastirmada, Türkiye'deki döviz yatirimcilarinin olusturacaklari döviz portföylerinin modellenmesi ve gelecek tahmininin yapilmasi için; temel model olarak, Sermaye Varliklari Fiyatlandirma Modeli (SVFM) ile tutarli ve duragan beta riskine olanak saglayan, Dogrusal Piyasa Modeli (DPM) kullanilmistir. SVFM'nin performansi ise, Kosullu Sermaye Varliklari Fiyatlandirma Modeliyle (K-SVFM) tutarli ve zamana bagli degisen beta riskine olanak saglayan Zamana bagli degisen Dogrusal Piyasa Modeli (Z-DPM) ile karsilastirilmistir. Z-DPM'nin modellenmesi için, tek degiskenli (GARCH) ve çok degiskenli (DCC-GARCH) GARCH-tipi modeller ve durum uzayi formundaki Kalman filtresi (KFMR) kullanilmistir. Türkiye Cumhuriyet Merkez Bankasi'nda (TCMB) efektif alis-satisa konu olan, 9 ülkenin son 15 yillik haftalik döviz kurlarinin Türk Lirasi (TL) cinsinden fiyatlari, arastirma verisi olarak kullanilmistir. Sonuçta, Z-DPM'nin KFMR ile modellenmesi durumunda, döviz kurlarinin modellenmesi ve gelecek tahmini konusunda digerlerine karsi daha iyi performans gösterdigi; fakat Z-DPM'nin GARCH ve DCC-GARCH ile modellenmesi durumunda ise DPM'ye göre yetersiz kaldigi görülmüstür. Döviz kurlarindaki beta risklerinin duragan olmadigi temel sonucuna ulasilmistir. Anahtar Kelimeler: Beta Riski, DCC-GARCH, GARCH, Kalman Filtresi, Sermaye Varliklari Fiyatlandirma Modeli (SVFM) JEL Siniflandirma Kodlari: C53, G12, G17 In this study, Linear Market Model (LMM) is used which is consistent with the Capital Asset Pricing Model (CAPM) and enables the beta risk as the benchmark model for the purposes of perform the modeling and forecasting of the foreign currency portfolios to be established by the foreign currency investors in Turkey. Performance of Capital Asset Pricing Model is compared with the Time-varying Linear Market Model (Tv-LMM) is used which is consistent with the Conditional Capital Asset Pricing Model (C-CAPM) and enables the time-varying beta risk. For the modeling of Tv-LMM, univariate (GARCH) and multivariate (DCC-GARCH) GARCH-type models and state space form via Kalman filter algorithm (KFMR) are used. The prices of the weekly foreign currency exchange rates in Turkish Liras (TL) of the period of last 15 years for 9 countries subject to effective purchase-sales as an indicator at Central Bank of the Republic of Turkey (CBRT) based on these prices are used as the research data. To sum up, in the case of modeling of Tv-LMM with KFMR, it is shown that it shows much better performance compared to the other models in modeling of foreign currency exchange rates and future estimation; whereas, in case of modeling of Tv-LMM with GARCH and DCC-GARCH, it is shown to be insufficient compared to OLS. It is concluded that the beta risks of exchange rates are not stable. Keywords: Beta Risk, DCC-GARCH, GARCH, Kalman Filter, Capital Asset Pricing Model (CAPM) JEL Classification Codes: C53, G12, G17 1. Introduction For portfolio designs and risk management in accordance with financial markets, the Modern Portfolio Theory (MPT) developed by Markowitz (195) is frequently preferred. Founded on this theory, the Capital Asset Pricing Model (CAPM) was developed by Sharpe (1964), Lintner (1965), and Mossin (1966), and is used frequently in portfolio management due to the feasibility of its application and the flexibility of its parameters. Furthermore, CAPM provides a stable beta risk parameter that is defined as a measurement of systematic risk. CAPM's linearity assumption is modified in research as local linearity, and Conditional CAPM (C-CAPM) is developed providing the modeling of time with varying beta risk parameters (Jagannathan and Wang, 1996). The aim of this research is to model the portfolios of foreign currency investors in Turkey with CAPM and C-CAPM, contributing to the foreign currency portfolio application literature under the subject of future estimations. In this regard, CAPM, being consistent with the Linear Market Model (LMM), is modeled with Ordinary Least Squares (OLS). Additionally, C-CAPM is consistent with the time-varying Linear Market Model (Tv-LMM) and is modeled with univariate (GARCH) and multivariate GARCH-type models (DCC-GARCH) with the state space form via the Kalman filter (KFMR). The weekly foreign currency exchange rates of 9 countries (Australian Dollars (AUD), Canadian Dollars (CAD), Swiss Francs (CHF), Danish Krone (DKK), Euro (EUR), British Pound (GBP), Norwegian Krone (NOK), Saudi Arabian Riyal (SAR), US Dollars (USD)) are contrasted with Turkish Liras (TL), as reported by the Republic of Turkey's Central Bank (TCMB), between 2/01/2005 to 2/01/2020. The currency basket is also created from equally weighted currency values. Both in the modeling and one year forecasting procedure, the performance comparison of LMM and Tv-LMM via the proposed models are performed. The R programing language (R Core Team, 2018) was used in these research analyses. 2. Method The benchmark model, LMM, being consistent with CAPM, allows for stable beta risk parameters and are defined as follows: [R.sub.it] - [R.sub.ft] = [[alpha].sub.i] + [[beta].sub.im]([R.sub.mt] - [R.sub.ft]) + [[epsilon].sub.it] [[epsilon].sub.it] ~ N(0, [[sigma].sub.i.sup.2]) (1) Here, [R.sub.mt] - [R.sub.ft] is the excess return of a portfolio at time t and [R.sub.it] - [R.sub.ft] is the excess return of currency i (i = 1,...,9) at time t (t = 1, ..., T). [[beta].sub.im] accounts for the beta risk of currency i.. [[epsilon].sub.it] are residuals, with [[epsilon].sub.it] ~ N(0, [[sigma].sub.i.sup.2]) and E([[epsilon].sub.it] [[epsilon].sub.tk]) = 0 for i [not equal to] k and E([[epsilon].sub.it] [[epsilon].sub.i t+j]) = 0 for j > 0. The constant term of LMM, [[alpha].sub.i], is assumed to be 0. LMM is modeled with the OLS method throughout this research. Tv-LMM, being consistent with C-CAPM, allows for time-varying beta risk parameters, which are defined as follows: [R.sub.it] - [R.sub.ft] = [[alpha].sub.i] + [[beta].sub.imt]([R.sub.mt] - [R.sub.ft]) + [[epsilon].sub.it] [[epsilon].sub.it] ~ N(0, [[sigma].sub.i.sup.2]) (2) Here, [[beta].sub.imt] is defined as the beta risk parameter of currency i at time t. The constant term of Tv -LMM, [[alpha].sub.i], is assumed to be 0 (Choudhry and Wu, 2009). Time-varying beta risk parameters in Tv-LMM (Equation 2) are modeled with univariate and multivariate GARCH-type and state space forms via the Kalman filter (KFMR) throughout this research. The time-varying beta risk parameter in Tv-LMM obtained with GARCH is defined as follows: [Please download the PDF to view the mathematical expression] (3) Here, [[rho].sub.im] is the constant correlation coefficient between currency i and the portfolio. [h.sub.iit] stands for the conditional variances of currency i at time t and [h.sub.mmt] stands for the conditional variances of the portfolio at time t. The time varying beta risk parameter in Tv-LMM, obtained with DCC-GARCH, is defined as follows: [[beta].sub.imt] = [[Cov(R.sub.it], [R.sub.mt])]/[Var(R.sub.mt])]] = [[[h.sub.imt]]/[[h.sub.mmt]]] (4) Here, [h.sub.imt] and [h.sub.mmt] are the conditional variance of the portfolio and the conditional covariance of the portfolio and currency i at time t, respectively. The state space form of Tv-LMM via Kalman filter (KFMR) is defined as follows: [R.sub.it] - [R.sub.ft] = [[beta].sub.imt] (R.sub.mt] - [R.sub.ft]) + [[epsilon].sub.it] [[epsilon].sub.it] ~N (0, [H.sub.i]) (5) [[beta].sub.imt] = [bar.[[beta].sub.im]] + [[phi].sub.i]([[beta].sub.imt-1] - [bar.[[beta].sub.im]]) + [w.sub.it] [W.sub.it] ~ N(0, [Q.sub.i]) (6) Here, the observation ([[epsilon].sub.it]) and state ([w.sub.it]) errors are assumed to be independent of each other and independent at time t. Furthermore, they are normally distributed with a mean of 0 with respect to observation variance ([H.sub.i]) and state variance ([Q.sub.i]), respectively. [[phi].sub.i] also quantifies over the temporal autocorrelation in [[beta].sub.imt] in currency i. 3. Results and Discussion According to the findings of this research in the case of modeling and making future estimations with the C-CAPM-consistent KFMR of the Tv-LMM, it has been demonstrated that the aforementioned model produces a superior performance when compared with the CAPM-consistent LMM. Therefore, it is inferred that the beta parameter, defined as a measurement of systematic risk, is unstable. When time-varying beta risk estimates and risk measurements are all taken into consideration, the investments made on the Australian Dollars (AUD), the Canadian Dollar (CAD), and the Norwegian Krone (NOK), which all had beta coefficient estimates of less than 1, can be said to have lower risk compared to the investments made on the currency basket (SPT); moreover, the investments made on the Swiss Franc (CHF), the Danish Krone (DKK), the Euro (EUR), the British Pound (GBP), the Saudi Arabian Riyal (SAR) and the US Dollar (USD), which all had beta coefficient estimates of greater than 1, can be described as being higher risk investments when compared to the investments made on the currency basket (SPT). 4. Conclusion To conclude, it has been observed in this research that the performance in foreign currency exchange rate data modeling and forecasting when using the state space form of the Tv-LMM via KFMR is apparently superior when compared to the LMM. Thus, it was derived that the beta risk is unstable. Moreover, regarding the modeling of the time-varying beta risk parameters in the Tv-LMM with GARCH and DCC-GARCH, it is observed to be generally insufficient compared to the OLS method, which provides for stable beta risk parameters in the LMM. Finally, in the case of the Tv-LMM's modeling and forecasting with GARCH and DCC-GARCH, it was not possible to make any accurate distinctions. The results acquired in this research have therefore provided significant contributions to the literature of foreign currency portfolio applications.