A Short Glimpse Intrododuction to Multi-Period Fuzzy Bond Imunization for Construct Active Bond Portofolio, this paper is made to fullfill Fixed-Income securities mid semester exam
For calculating the crucial value at risk (VaR) numbers, we test several approximation methods against the full revaluation method - delta, delta-gamma and delta-gamma-theta - that save on computational power but lose accuracy, and evaluate when these approximations make sense as substitutes.
Designing and pricing guarantee options in defined contribution pension plansStavros A. Zenios
The shift from defined benefit (DB) to defined contribution (DC) is pervasive among pension funds, due to demographic changes and macroeconomic pressures. In DB all risks are borne by the provider, while in plain vanilla DC all risks are borne by the beneficiary. For DC to provide income security some kind of guarantee is required. A minimum guarantee clause can be modeled as a put option written on some underlying reference portfolio of assets and we develop a discrete model that optimally selects the reference portfolio to minimise the cost of a guarantee. While the relation DB-DC is typically viewed as a binary one, the model can be used to price a wide range of guarantees creating a continuum between DB and DC. Integrating guarantee pricing with asset allocation decision is useful to both pension fund managers and regulators. The former are given a yardstick to assess if a given asset portfolio is fit-for-purpose; the latter can assess differences of specific reference funds with respect to the optimal one, signalling possible cases of moral hazard. We develop the model and report numerical results to illustrate its uses.
For calculating the crucial value at risk (VaR) numbers, we test several approximation methods against the full revaluation method - delta, delta-gamma and delta-gamma-theta - that save on computational power but lose accuracy, and evaluate when these approximations make sense as substitutes.
Designing and pricing guarantee options in defined contribution pension plansStavros A. Zenios
The shift from defined benefit (DB) to defined contribution (DC) is pervasive among pension funds, due to demographic changes and macroeconomic pressures. In DB all risks are borne by the provider, while in plain vanilla DC all risks are borne by the beneficiary. For DC to provide income security some kind of guarantee is required. A minimum guarantee clause can be modeled as a put option written on some underlying reference portfolio of assets and we develop a discrete model that optimally selects the reference portfolio to minimise the cost of a guarantee. While the relation DB-DC is typically viewed as a binary one, the model can be used to price a wide range of guarantees creating a continuum between DB and DC. Integrating guarantee pricing with asset allocation decision is useful to both pension fund managers and regulators. The former are given a yardstick to assess if a given asset portfolio is fit-for-purpose; the latter can assess differences of specific reference funds with respect to the optimal one, signalling possible cases of moral hazard. We develop the model and report numerical results to illustrate its uses.
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
In this paper, the black-litterman model is introduced to quantify investor’s views, then we expanded
the safety-first portfolio model under the case that the distribution of risk assets return is ambiguous. When
short-selling of risk-free assets is allowed, the model is transformed into a second-order cone optimization
problem with investor views. The ambiguity set parameters are calibrated through programming
"Correlated Volatility Shocks" by Dr. Xiao Qiao, Researcher at SummerHaven In...Quantopian
Commonality in idiosyncratic volatility cannot be completely explained by time-varying volatility. After removing the effects of time-varying volatility, idiosyncratic volatility innovations are still positively correlated. This result suggests correlated volatility shocks contribute to the comovement in idiosyncratic volatility.
Motivated by this fact, we propose the Dynamic Factor Correlation (DFC) model, which fits the data well and captures the cross-sectional correlations in idiosyncratic volatility innovations. We decompose the common factor in idiosyncratic volatility (CIV) of Herskovic et al. (2016) into the volatility innovation factor (VIN) and time-varying volatility factor (TVV). Whereas VIN is associated with strong variation in average returns, TVV is only weakly priced in the cross section
A strategy that takes a long position in the portfolio with the lowest VIN and TVV betas, and a short position in the portfolio with the highest VIN and TVV betas earns average returns of 8.0% per year.
Date: Friday, 18-03-2016
Speaker: dr. Drona Kandhai
Title: Challenges in Computational Finance
Abstract: In this talk I will guide you through the world of financial derivatives, their valuation and risk management and, more importantely, how the recent credit crisis has completely changed the landscape.
The impact of this change on the modeling and computational complexity and the related challenges will be discussed. A couple recent research projects in close collaboration with industrial and other academic partners will be highlighted.
These slides were used in an introductory lecture to Computational Finance presented in a third-year class on Machine Learning and Artificial Intelligence. The slides present three examples of machine learning applied to computational / quantitative finance. These include
1) Model calibration (stochastic process) using the stochastic Hill Climbing algorithms.
2) Predicting Credit Default rates using a Neural Network
3) Portfolio Optimization using the Particle Swarm Optimization Algorithm.
All of the Python code is available for download on GitHub. Link is available at the end of the slide-show.
Bid and Ask Prices Tailored to Traders' Risk Aversion and Gain Propension: a ...Waqas Tariq
Risky asset bid and ask prices “tailored” to the risk-aversion and the gain-propension of the traders are set up. They are calculated through the principle of the Extended Gini premium, a standard method used in non-life insurance. Explicit formulae for the most common stochastic distributions of risky returns, are calculated. Sufficient and necessary conditions for successful trading are also discussed.
In this paper, the black-litterman model is introduced to quantify investor’s views, then we expanded
the safety-first portfolio model under the case that the distribution of risk assets return is ambiguous. When
short-selling of risk-free assets is allowed, the model is transformed into a second-order cone optimization
problem with investor views. The ambiguity set parameters are calibrated through programming
"Correlated Volatility Shocks" by Dr. Xiao Qiao, Researcher at SummerHaven In...Quantopian
Commonality in idiosyncratic volatility cannot be completely explained by time-varying volatility. After removing the effects of time-varying volatility, idiosyncratic volatility innovations are still positively correlated. This result suggests correlated volatility shocks contribute to the comovement in idiosyncratic volatility.
Motivated by this fact, we propose the Dynamic Factor Correlation (DFC) model, which fits the data well and captures the cross-sectional correlations in idiosyncratic volatility innovations. We decompose the common factor in idiosyncratic volatility (CIV) of Herskovic et al. (2016) into the volatility innovation factor (VIN) and time-varying volatility factor (TVV). Whereas VIN is associated with strong variation in average returns, TVV is only weakly priced in the cross section
A strategy that takes a long position in the portfolio with the lowest VIN and TVV betas, and a short position in the portfolio with the highest VIN and TVV betas earns average returns of 8.0% per year.
Date: Friday, 18-03-2016
Speaker: dr. Drona Kandhai
Title: Challenges in Computational Finance
Abstract: In this talk I will guide you through the world of financial derivatives, their valuation and risk management and, more importantely, how the recent credit crisis has completely changed the landscape.
The impact of this change on the modeling and computational complexity and the related challenges will be discussed. A couple recent research projects in close collaboration with industrial and other academic partners will be highlighted.
These slides were used in an introductory lecture to Computational Finance presented in a third-year class on Machine Learning and Artificial Intelligence. The slides present three examples of machine learning applied to computational / quantitative finance. These include
1) Model calibration (stochastic process) using the stochastic Hill Climbing algorithms.
2) Predicting Credit Default rates using a Neural Network
3) Portfolio Optimization using the Particle Swarm Optimization Algorithm.
All of the Python code is available for download on GitHub. Link is available at the end of the slide-show.
Brief summary and introduction to Rev. Joseph Iannuzzi's Doctoral Dissertation, "The Gift of Living in the Divine Will in the Writings of Luisa Piccarreta."
The dissertation can be bought on his website, in amazon (for kindle), Barnes & Noble (electronic), etc.
http://livinginthedivinewill.com
Rev. Iannuzzi is a world renown theologian that dedicated 12 years of his life to studying the writings of the Servant of God, Luisa Piccarreta.
For more information about S.G. Luisa Piccarreta, visit:
https://www.facebook.com/childrenDivineWill/
Gandhi Glass Introducing New Product Range under Banner Shine9" A]. Smart Switchable Glass B]. Internal Blinds with Insulating Glass C]. Lacquered Glass High Gloss & satin matt Finish D]. Bath Room Mirrors with LED lights and Anti Fogger E]. Decorative vanity Mirror F]. Designer Glass for Doors & window G] Shower Cubical & partition H]. Fabric Laminated Glass I]. Fire Rated Glass for 60 Minutes
Optimal Portfolio Selection for a Defined Contribution Pension Fund with Retu...BRNSS Publication Hub
This paper investigates the optimal investment strategies for a defined contribution pension fund with return clauses of premiums with interest under the mean-variance criterion. Using the actuarial symbol, we formalize the problem as a continuous time mean-variance stochastic optimal control. The pension fund manager considers investments in risk and risk-free assets to increase the remaining accumulated funds to meet the retirement needs of the remaining members. Using the variational inequalities methods, we established an optimized problem from the extended Hamilton–Jacobi–Bellman Equations and solved the optimized problem to obtain the optimal investment strategies for both risk-free and risky assets and also the efficient frontier of the pension member. Furthermore, we evaluated analytically and numerically the effect of various parameters of the optimal investment strategies on it. We observed that the optimal investment strategy for the risky asset decreases with an increase in the risk-averse level, initial wealth, and the predetermined interest rate.
This paper studies an optimal investment and reinsurance problem for a jump-diffusion risk model
with short-selling constraint under the mean-variance criterion. Assume that the insurer
is allowed to purchase proportional reinsurance from the reinsurer and invest in a risk-free asset and a risky
asset whose price follows a geometric Brownian motion. In particular, both the insurance and reinsurance
premium are assumed to be calculated via the variance principle with different p
In this paper, we introduce a universal framework for mean-distortion robust risk measurement and
portfolio optimization. We take accounts for the uncertainty based on Gelbrich distance and another uncertainty
set proposed by Delage & Ye. We also establish the model under the constraints of probabilistic safety
criteria and compare the different frontiers and the investment ratio to each asset. The empirical analysis in the
final part explores the impact of different parameters on the model results.
Project management is an important part of enterprise operation planning. Among numerous indexes
of enterprise investment project evaluation, net present value is particularly accurate and conducive to rapid
decision-making.
lng 1at the 1g out rank-Jigh orma -atten-a li.docxSHIVA101531
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Probability (P) times Outcome = EMV
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$300,000 = +$60,000
-$40,000 = -$32,000
-$50,000 = -$10,000
-$ 20,000 = - $2,000
$60, 000 = $42,000
~
*
Project 2 *
P= .70
Project 1 's EMV = $60,000- $32, 000 = $28, 000
Proj ect 2's EMV = -$10,000- $2,000 + $42, 000 = $30,000
FIGURE 11-7 Expected monetary value (EMV) example
no reimburse ment if it is not awarded the contract. The sum of the probabilities for
outcomes for each project must equal one (for Project 1, 20 percent plus 80 percent) .
Probabilities are normally determined based on expert judgme nt. Cliff or other people
in his firm should have some sense of their likelihood of winning certain projects.
Figure 11-7 also shows probabilities and outcomes for Project 2. Suppose there is a
20 percent probability that Cliffs firm will lose $ 50,000 on Project 2, a 10 percent probabili-
ty that it will lose $20,000, and a 70 percent probability that it will earn $60,000. Again,
experts would ne ed to estimate these dollar amounts and probabilities.
To calculate the expected monetary value (EJvfV) for each proj ect, multiply the proba-
bility by the outcome value (or each potential outcome for each project and sum the results.
To calculate expected monetary value for Project 1, going from left to right, multiply the
probability by the outcome for each branch and sum the results. In this example, the EMV
for Proj ect 1 is $28, 000.
.2($300 ,000) + .8( -$40 ,000) = $60,000- $32,000 = $28 ,000
The EMV for Project 2 is $30,000 .
.2 ( -$50 ,000 ) + .1(-$20 ,000) + .7($60,000) = -$1 0 ,000-$2 ,000 + .$42,000
= $30,000
Beca use the EMV provides an estimate for the total dollar value of a decision , you wa nt
to have a pos itive number; the higher the EMV, the bette r. Since the EJvfV is positive for
both Projects 1 and 2 , Clift"s firm would expec t a positive outcome from eac h and could bid
0n both proj ects. If it had to choose between the two proj ects , perhaps because of limited
resources , Clift"s firm should bid on Project 2 because it has a higher EMV.
443
Project Risk Management
consulting costs might be expanded in the description to say that the organiza-
tion might be able to negotiate lower-than-average costs for a particular consul-
tant because the consultant really e njoys working for that company in that
particular location.
• The categor:y under which the risk event falls: For example, defective server
might fall under the broader category of technology or hardware technology.
• The root cause of the risk: The root cause of the defective server might be a
defective power supply.
• Triggers for each risk: Triggers are indicators or symptoms of actual risk
ev ...
ENTROPY-COST RATIO MAXIMIZATION MODEL FOR EFFICIENT STOCK PORTFOLIO SELECTION...cscpconf
This paper introduces a new stock portfolio selection model in non-stochastic environment.Following the principle of maximum entropy, a new entropy-cost ratio function is introduced as
the objective function. The uncertain returns, risks and ividends of the securities are considered as interval numbers. Along with the objective function, eight different types of constraints are used in the model to convert it into a pragmatic one. Three different models have been proposed by defining the future inancial market optimistically, pessimistically and in hecombined form to model the portfolio selection problem. To illustrate the effectiveness and tractability of the proposed models, these are tested on a set of data from Bombay Stock Exchange (BSE). The solution has been done by genetic algorithm.
Due to the limited size of the insurance market, insurance companies usually purchase insurance
from a few reinsurance companies with large differences. At this time, using the Vasicek model to describe the
counterparty credit risk will be inaccurate; besides, the insurance company’s understanding of the counterparty
default threshold distribution is incomplete, which makes it difficult to effectively determine the counterparty
default probability.
Multi-dimensional time series based approach for Banking Regulatory Stress Te...Genpact Ltd
Under regulatory paradigm of banking risk management, banks are required to perform stress testing of internally computed risk parameters to ensure holding of adequate amount of capital to offset the effects of downturn events. For this purpose, most of the contemporary stress-testing practices are limited to one dimensionality of the calculation, where endogenous risk parameters are predicted by modeling and scenario based values of exogenous parameters (macroeconomic variables).
Investment portfolio optimization with garch modelsEvans Tee
Since the introduction of the Markowitz mean-variance optimization model, several extensions have been made to improve optimality. This study examines the application of two models - the ARMA-GARCH model and the ARMA- DCC GARCH model - for the Mean-VaR optimization of funds managed by HFC Investment Limited. Weekly prices of the above mentioned funds from 2009 to 2012 were examined. The funds analyzed were the Equity Trust Fund, the Future Plan Fund and the Unit Trust Fund. The returns of the funds are modelled with the Autoregressive Moving Average (ARMA) whiles volatility was modelled with the univariate Generalized Autoregressive Conditional Heteroskedasti city (GARCH) as well as the multivariate Dynamic Conditional Correlation GARCH (DCC GARCH). This was based on the assumption of non-constant mean and volatility of fund returns. In this study the risk of a portfolio is measured using the value-at-risk. A single constrained Mean-VaR optimization problem was obtained based on the assumption that investors’ preference is solely based on risk and return. The optimization process was performed using the Lagrange Multiplier approach and the solution was obtained by the Kuhn-Tucker theorems. Conclusions which were drawn based on the results pointed to the fact that a more efficient portfolio is obtained when the value-at-risk (VaR) is modelled with a multivariate GARCH.
Similar to A Short Glimpse Intrododuction to Multi-Period Fuzzy Bond Imunization for Construct Active Bond Portofolio (20)
Techniques to optimize the pagerank algorithm usually fall in two categories. One is to try reducing the work per iteration, and the other is to try reducing the number of iterations. These goals are often at odds with one another. Skipping computation on vertices which have already converged has the potential to save iteration time. Skipping in-identical vertices, with the same in-links, helps reduce duplicate computations and thus could help reduce iteration time. Road networks often have chains which can be short-circuited before pagerank computation to improve performance. Final ranks of chain nodes can be easily calculated. This could reduce both the iteration time, and the number of iterations. If a graph has no dangling nodes, pagerank of each strongly connected component can be computed in topological order. This could help reduce the iteration time, no. of iterations, and also enable multi-iteration concurrency in pagerank computation. The combination of all of the above methods is the STICD algorithm. [sticd] For dynamic graphs, unchanged components whose ranks are unaffected can be skipped altogether.
Levelwise PageRank with Loop-Based Dead End Handling Strategy : SHORT REPORT ...Subhajit Sahu
Abstract — Levelwise PageRank is an alternative method of PageRank computation which decomposes the input graph into a directed acyclic block-graph of strongly connected components, and processes them in topological order, one level at a time. This enables calculation for ranks in a distributed fashion without per-iteration communication, unlike the standard method where all vertices are processed in each iteration. It however comes with a precondition of the absence of dead ends in the input graph. Here, the native non-distributed performance of Levelwise PageRank was compared against Monolithic PageRank on a CPU as well as a GPU. To ensure a fair comparison, Monolithic PageRank was also performed on a graph where vertices were split by components. Results indicate that Levelwise PageRank is about as fast as Monolithic PageRank on the CPU, but quite a bit slower on the GPU. Slowdown on the GPU is likely caused by a large submission of small workloads, and expected to be non-issue when the computation is performed on massive graphs.
A Short Glimpse Intrododuction to Multi-Period Fuzzy Bond Imunization for Construct Active Bond Portofolio
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:
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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