This document provides an overview of the development of bond immunization theory and fuzzy bond immunization strategies over several decades. It discusses early works in 1952 by Redington on bond immunization and Markowitz on portfolio selection. Later developments include using fuzzy numbers to model interest rates and utility functions. The document outlines an approach to designing active bond management strategies in a fuzzy environment using Sengupta's methodology to evaluate portfolio return and risk. It proposes constructing fuzzy return-risk maps to help decision makers choose the best interest rate forecasts and durations.

1. A Short Glimpse Introduction to Multi-Period Fuzzy Bond
Immunization for Construct Active Bond Portfolio
conducted for complete mid-semester evaluation purpose
in Fixed-Income Securities Modelling
by :
Nabih Ibrahim Bawazir 13/351338/PPA/4165
Lecturer :
Dr. Gunardi
PROGRAM S2 MATEMATIKA
JURUSAN MATEMATIKA
FAKULTAS MATEMATIKA DAN ILMU PENGETAHUAN ALAM
UNIVERSITAS GADJAH MADA
YOGYAKARTA
2015

2. A Short Glimpse Introduction to Multi-Period Fuzzy Bond Immunization for Construct
Active Bond Portfolio
A. Chronological Schemes
In chronological schemes below given timeline of contribution in bond imunization
theory, this theoru began 63 years ago. This schemes created for reader convenience
when read the chronologal details part:
B. Chronogical Details
One the most important of early asset management in modern finance theory has been the mean
variance methodology for the portfolio selection problem which was proposed by Markowitz
[A]. This model assumes that given return that is multivariate distributed. In the different paper
1952. Markowitz was
pioneering research in
Portofolio Construction
1991. Quadratic utility
function was developed by
Markowitz.
Konno and Yamazaki was
developing L1 Model
1952. Redington was
pioneering research in
Bond Imunization
1993. Van Der Meer and
Smink revised
Redington’s Work
1995. Several Adjustments
1990. Li
Cayzi says
Probability
Theory
inapropriate
for long term
Interest Rates
1997. Derrig and Ostaszewski uses
fuzzy mathematics
2002. Carlson et All using
possibilities teory
2004. De Wilde create fuzzy
utilities
2007-2014. Nine other remarkable
developments.
2015. A new index for uncertain
bond market using a fuzzy return
risk mapwas created

3. afterwards, Makowitz [B] develop assumtion that the decission maker (DM) has a quadratic
utility function. In contrast, Konno, and Yamazaki [C] develop the first linear model for
portofolio selection, the L1 model, that uses mean-absolute deviation model, which was applied
in Tokyo Stock Market.
Within the asset liability management framework, Imunization is primary method for ascertain
a specific yield in a certain future date that is set off from an initial portofolio. In bond-based
Asset-Liability Management, Immunization can be traced back to Redington [D] who suggests
a parallel treatment to the assets and liablity valuations. Very long time thereafter, Van Der
Meer and Smink [E] and extensive revission of Redingtons’s technique is considered by
dividing analysis in some static methods, such as: cash flow payments, gap analysis,
segmentation and and cas floor machines, passive dynamic ones, such as: immunizations, and
active dynamic ones, such as: contingent immunizations. Afterwards, van Der Meer and
Plantiga [F] was developing adjustment for liability-driven investors, which is applicable in
not only asset-based investments but also asset-liability-based investments.
Even though the intention for passive asset liability management is to construct a portofolio
that will achive a predetermined benchmark performance, active bond portofolio management
is rely on yield rate forecast. Contarily, some authors, like Li Calzy [G], point out that the use
of sthocastics model is not suitable for prediction for the long term interest rates. They
considers that is more realistic to use of discount rates based on fuzzy numbers (FN), since
only data avaliable is the one facilitated by experts.
However, the bibliography on immunization in a fuzzy environtment is very scare. Derrig and
Ostaszewski [H] uses fuzzy mathematics to estimate the effective tax rate and task free discount
rate in an insurance company with a portofolio that combines asset and liabilities. One
reasonable function that is developed by Carlsson, Fuller and Majlender [I] and widely
employed by financial theoris assigns a risky portofolio 𝑃 with the risky rate of return 𝑟𝑝, and
expected rate of return 𝐸(𝑟𝑝) and a variance of return 𝜎(𝑟𝑝), and the utility score is expressed
by equation below:
𝑈(𝑃) = 𝐸(𝑟𝑝) −
𝐴 𝜎(𝑟𝑝)
200
,
with 𝐴 is an index of the DM’s risk aversion.

4. In practical aims, the use of utility theory has proved to be problematic, which should be more
serious than having axiomatic problem because of the limitation of probabbility assesment and
utility theory. This condition make us need to use possibility theory. De Wilde [J] said that
fuzzy parameters are assumed to be known membership fuctions in the work of fuzzy utility.
Nonetheless, it is actually not always easy for a DM to specify the membership function or
probability distribution in an inexact environtment [I] . At least in some cases, the use of
interval coefficients may serve the aim better. Therefore, the main goal of presnt work is to
design active management strategies in a fuzzy environtment, using Sengupta’s methodology
to get the return and the risk of a portofolio. Vercher, Bermundez, an Segura [K] uses this
methodology to optimize a fuzzy portofolio under downside risk measures. Interest rate will
have to be forecast by the DM, and as a result, the portofolio duration will have to be modified
in order to escalate the portfolio return, in exchange for a higher risk (false estimation will
decrease return of portfolio). Furthermore, Georgescu [L] developed risk aversion using
possibility theory.
Brotons an Torceno [M], apply immunization strategy in a fuzzy environtment. Nevertheless,
this research is to design active bond managenent in fuzzy environtment, in order to anticipate
changes in interest rates. From a strating immunized portofolio, the DM will have to decide
whether to increase the expected return (modify portofolio duration), increasing the risk as
well, or not. The use of ulitity function and the building of risk return maps will improve
decision making. Conversely, the variation duration incereases portofolio risk, that why the
DM will have choose portfolio, combination of expected return and risk, which has higher
utility.
Georgescu’s theory was developed by some scholar in active asset-liability management
purposses . Shou and Qin [N] developed regret minimazation portolio seltected model. Sadefo,
Mbairadjim, and Terazza [O] developed fuzzy risk adjusted performance measures which was
applied in hege fund Insurance.
Decision-makers are usually provided with information which is characterized by vague
linguistic descriptions such as high risk, low profit, and high interest rate. In these cases, it is
impossible for us to get the precise probability distribution we need. Furthermore, even if we
know all the historical and current data, it is difficult that we predict the future return as a fixed
value. Hence we need to consider that the future return has ambiguousness. Wozabal [P] says
that any several approaches dealing with ambiguous situations. On the one hand, some authors

5. characterize uncertain distributions by defining a confidence region of their first two moments,
so that the portfolio is robust against such uncertainty.
Recent noticable development are developed in Multi-Period Portfolio research. One of those
research was conducted by Zhang, Zhang, and Xiao [Q] for apply possibility measures for
multi-period portfolio optimization. In addition, Liu et all [R] was give another aproach using
Interval Analysis. This reaserch are followed by Zhang [S] by using new fuzzy programming
approach for optimization with return demand and risk control.
In 2015, Brontos, Torseno, and Barbera-Marine [T] published a paper about an index for
uncertain bond market using a fuzzy return risk map. Moreover, the construction of a fuzzy
return risk map will allow the decission maker to know the over risk or over return in regard to
immunization strategy for the DM. The construction of risk return map presents the result
which will help the DM to choose the best duration for interest rate forecast. They were appling
these map for Spanish debt market in 1997 to 2012 data.
C. Some Previous Findings
The effective spot rate at time 𝑡 corresponding to a term [𝑡, 𝑡 + 𝑛] that we will denote 𝑖𝑡 𝑛 is
defined as internat rate of return of zero-coupon bond of maximum credit qulity for thr maturity
𝑡 + 𝑛. If the spot rate is constant value in any term, the expression of the duration at time 0 that
generates the stream of payments, {(𝐶1, 𝑡1), (𝐶2, 𝑡2), … , (𝐶 𝑛, 𝑡 𝑛)} is:
𝐷 =
∑ 𝑡 𝑠 𝐶𝑠(1 + 𝑖)−𝑡𝑠𝑛
𝑠=1
∑ 𝐶𝑠(1 + 𝑖)−𝑡𝑠𝑛
𝑠=1
,which 𝑡 𝑠 is the maturity of cashflow 𝐶𝑠. The maturity, the cashflows and the yield are known
beforehand, and premature payment does not exist. For the case the interest rate are defined by
fuzzy numbers 𝑖̃( 𝑥), the durration expression is transformed into
𝐷 =
∑ 𝑡 𝑠 𝐶𝑠(1 + 𝑖̃( 𝑥))−𝑡 𝑠𝑛
𝑠=1
∑ 𝐶𝑠(1 + 𝑖̃( 𝑥))−𝑡 𝑠𝑛
𝑠=1
and the total duration of a portofolio, denoted by 𝐷 𝐹̃(𝑥), if we assume that it is formed by 𝑁𝑘
bonds of the type 𝑘 , 𝑘 = 1, … , 𝑚 being 𝐶𝑠
𝑘
the cash flow of the bond 𝑘 in the period of 𝑡 𝑠:

6. 𝐷 𝐹̃ (𝑥), =
∑ ∑ 𝑡 𝑠
𝑘
𝑁𝑘 𝐶𝑠
𝑘(1 + 𝑖̃( 𝑥))−𝑡 𝑠
𝑘𝑛
𝑠=1
𝑠
𝑘=1
∑ ∑ 𝑁𝑘 𝐶𝑠
𝑘(1 + 𝑖̃( 𝑥))−𝑡 𝑠
𝑘𝑛
𝑠=1
𝑠
𝑘=1
In this way, the interest rate is denoted by 𝑖̃ = (𝑖 𝐶, 𝑙𝑖, 𝑟𝑖) and its membership function
𝜇𝑖(𝑥) =
{
0, 𝑓𝑜𝑟 𝑥 ≤ 𝑖 𝐶 − 𝑙𝑖
1 −
𝑖 𝐶 − 𝑥
𝑙𝑖
, 𝑓𝑜𝑟 𝑖 𝐶 − 𝑙𝑖 ≤ 𝑥 ≤ 𝑖 𝑐
1 −
𝑥 − 𝑖 𝐶
𝑟1
, 𝑓𝑜𝑟 𝑖 𝐶 ≤ 𝑥 ≤ 𝑖 𝑐 + 𝑟𝑖
0. 𝑓𝑜𝑟 𝑖 𝑐 + 𝑟𝑖 ≤ 𝑟𝑖
Therefore,
𝐷̃(𝑥) = (𝐷 𝐶, 𝑙 𝐷, 𝑟𝐷)
where
𝐷 𝐶 = 𝐷(𝑖 𝑐)
𝑙 𝐷 = 𝐷(𝑖 𝑐) − 𝐷(𝑖 𝐶 + 𝑟𝑖)
𝑟𝐷 = 𝐷(𝑖 𝑐 − 𝑙𝑖) − 𝐷(𝑖 𝑐)
𝜇 𝐷(𝑥) =
{
0, 𝑓𝑜𝑟 𝑥 ≤ 𝑖 𝐶 − 𝑙𝑖
1 −
𝐷(𝑥) − 𝐷(𝑖 𝐶)
𝐷(𝑖 𝑐 − 𝑙𝑖)
, 𝑓𝑜𝑟 𝑖 𝐶 − 𝑙𝑖 ≤ 𝑥 ≤ 𝑖 𝑐
1 −
𝐷(𝑖 𝐶) − 𝐷(𝑥)
𝐷(𝑖 𝐶) + 𝐷(𝑖 𝐶 + 𝑟𝑖)
, 𝑓𝑜𝑟 𝑖 𝐶 ≤ 𝑥 ≤ 𝑖 𝑐 + 𝑟𝑖
0. 𝑓𝑜𝑟 𝑖 𝑐 + 𝑟𝑖 ≤ 𝑟𝑖
In our case, if we assume 𝑚 bonds construct the portofolio, the duration of this, 𝐷̃ 𝐹
, would be:
𝐷̃ 𝐹
= 𝑥1 𝐷1
̃ + 𝑥2 𝐷2
̃ + ⋯ + 𝑥 𝑚 𝐷 𝑚
̃
Which is:
𝑥 𝑘: weight of bond 𝑘 in the portofolio
𝐷 𝑘
̃ : duration of the bond 𝑘

7. An Immunization strategy is intended to tedermine the kind and the number of securities that
a DM should acquire to ensure a capital for a certain term, which we will denominate Investor
Planning Horizon (IPH). If 𝐷 𝑘
̃ and 𝐷 𝑘
̃ 𝐹
can be approximated by triangular fuzzy number, we
can express them as:
𝐷 𝑘
̃ = (𝐷 𝐶
𝑘
, 𝑙 𝐷
𝑘
, 𝑟𝐷
𝑘
)
𝐷̃ 𝐹
= (∑ 𝑥 𝑘 𝐷 𝐶
𝑘
𝑚
𝑘=1
, ∑ 𝑥 𝑘 𝑙 𝐷
𝑘
𝑚
𝑘=1
, ∑ 𝑥 𝑘 𝑟𝐷
𝑘
𝑚
𝑖=1
)
Being the 𝛼-cuts of the duration of a bond 𝑘:
𝐷 𝑘
̃(𝛼) = [𝐷 𝐶
𝑘
− (1 − 𝛼)𝑙 𝐷
𝑘
, 𝐷 𝐶
𝑘
+ (1 + 𝛼)𝑟𝐷
𝑘
]
and the 𝛼-cuts of the portofolio duration is
𝐷̃ 𝐹(𝛼) = [∑ 𝑥 𝑘 𝐷 𝐶
𝑘
𝑚
𝑘=1
− (1 − 𝛼) ∑ 𝑥 𝑘 𝑙 𝐷
𝑘
𝑚
𝑘=1
, ∑ 𝑥 𝑘 𝐷 𝐶
𝑘
𝑚
𝑘=1
+ (1 + 𝛼) ∑ 𝑥 𝑘 𝑟𝐷
𝑘
𝑚
𝑖=1
]
By solving both 𝛼-cuts of the duration of a bond 𝑘 and 𝛼 -cuts of the portofolio duration using
optimization we get the solution of the left side, which is
max
𝑥1,…,𝑋 𝑚
𝛼 =
(∑ 𝑥 𝑘 𝐷 𝐶
𝑘𝑚
𝑘=1 + ∑ 𝑥 𝑘 𝑟𝐷
𝑘𝑚
𝑖=1 − 𝐼𝑃𝐻)
∑ 𝑥 𝑘 𝑟𝐷
𝑘𝑚
𝑖=1
with
𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑚 = 1, and
𝑥 𝑘 ≥ 0, 𝑘 = 1, … , 𝑚
𝐼𝑃𝐻 : Investor Planning Horizon
and the solution of the right side, which is
max
𝑥1,…,𝑋 𝑚
𝛼 =
(𝐼𝑃𝐻 − ∑ 𝑥 𝑘 𝐷 𝐶
𝑘𝑚
𝑘=1 + ∑ 𝑥 𝑘 𝑙 𝐷
𝑘𝑚
𝑖=1 )
∑ 𝑥 𝑘 𝑙 𝐷
𝑘𝑚
𝑖=1

8. with
𝑥1 + 𝑥2 + ⋯ + 𝑥 𝑚 = 1, and
𝑥 𝑘 ≥ 0, 𝑘 = 1, … , 𝑚
𝐼𝑃𝐻 : Investor Planning Horizon
D. Conlussion
Traditional theory of passive bond management immunizes a portofolio in order to assure its
value at IPH. On the other hand, active bond management deals with portofolio aims to increase
its value. The evolution of future interest rates is unknown a priori, so it cannit be said which
kind of management will be preferable. Therefore, this has been aproximated by a fuzzy
number.
Active and passive portfolio management are based on duration. Assuming that in oreder to
calculate the duration we should use future interest rate, which is uncertain, we start by
estimating them through triangular fuzzy number, so the duration would be a fuzzy number
too, but not triangular, however can be approximated by triangular fuzzy number. Based on
some previous research, expression of financial valuations are well approximated by triangular
fuzzy number.
Immunization of a portofolio is obrained by equating duration and IPH. Since the duration of
portofolio in approximated by triangular fuzzy number, our goal is to reach the highest level
of presumption to which the portofolio is immunized. Therefore, we have established an
optimization program which allowa the combination of bonds that maximize the level of
presumption which imunizes the portofolio. Such combination will be calculated of the 𝛼-cut,
being one of them the situation of the IPH respect to the center of the triangular fuzzy number,
which represents the duration of the portofolio.
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