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1

Shifts of the term structure of interest rates against which a given portfolio is preimmunized

Rzadkowski G., Zaremba L. S.

Control and Cybernetics

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2010

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Vol. 39, no 3

857-865

EN

In this paper we formulate an immunization problem, which is rarely stated. Instead of reconstructing an existing bond portfolio B with the aim of securing a desired amount of, say L dollars, q years from now, against uncertain future interest rates shifts (under various, sometimes strong assumptions), we identify the shifts of the current term structure of interest rates against which portfolio B is already preimmunized. We state this problem in two different mathematical settings, and solve it with the help of Proposition 2 from Barber (1999), or, equivalently, Theorem 1 from Rzadkowski and Zaremba (2000). In the first part of this paper shifts are supposed to be polynomials of degree less than a certain number n, while in the second part, where we employ a Hilbert space approach, the shifts are allowed to be continuous functions.

2

On new immunization strategies under random shocks on the term structure of interest rates

Kondratiuk-Janyska A., Kaluszka M.

Badania Operacyjne i Decyzje

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2009

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nr 1

91-101

EN

We introduce new measures of immunization such as exponential duration referring, in particular, to Fong and Vasi.ek [7], Nawalkha and Chambers [14], Balbas and Ibanez [2], and Balbas et al. [3], but under the assumption of multiple shocks in the term structure of interest rates. These shocks are given by a random field. The cases of a single and multiple liabilities are discussed separately.

PL

Przedstawiamy nowe strategie immunizacji portfela przy założeniu wielokrotnych zaburzeń struktury terminowej stóp procentowych, gdzie zaburzenie jest opisane za pomocą sumy pewnego wielomianu i pola losowego [13]. Sformułowano twierdzenia dla ogólnej postaci pola losowego, a w przykładzie analizuje się przypadek płachty Browna. Przy kilku rodzajach zaburzeń struktury terminowej stóp procentowych rozważono zarówno przypadek jednego, jak i wielu zobowiązań [11]. Ponieważ rozważono różne postaci zaburzeń, otrzymano różne dolne oszacowania na wartości strumienia pieniężnego (jako różnica aktywów i pasywów) w chwili H (horyzont inwestycyjny), gdy pojawią się zaburzenia. W konsekwencji strategie uodpornienia zawierają nowe miary ryzyka jak np. wykładniczy czas trwania.

3

Necessary and Sufficient Conditions for the Stability of Interval Matrices

Białas S., Jakubowska M.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2001

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vol. 49, nr 3

467-477

EN

In this paper new necessary and sufficient conditions for the stability of the interval matrix [formula] are presented. These conditions represent a modification of paper [13]. Also, an algorithm for checking the stability of the interval matrix [formula] is proposed. The comparison of efficiency of the new algorithm and the algorithm from paper [13] is demostrated by means of examples.

4

Extension of Khang's Immunization Formula

Zaremba L.S., Rządkowski G.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2001

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vol. 49, nr 3

527-536

EN

The paper presents two theorems on immunization in the continuous context when a liability consists of a single outflow at a known future date. It is assumed in Theorem 1 that interest rates h(O, t) shift to their new values h * (O, t) = h(O, t) + f()..)a(t) where ).. is a random parameter "chosen" by a financial market under consideration, while a(t) is a continuous function and f()..) is twice continuously differentiable with f(O) = O, f' (O) #- O. It is proved that the local immunization is achieved and a formula for the immunizing duration is derived. Theorem 2 provides a formula for the immunizing duration of the portfolio combined with two cash flows. These two theorems extend validity of all similar type results presented by Bierwag (1983).