Academic literature on the topic 'Guaranteed Lifetime Withdrawal Benefits'

AbstractWe examine the value of guaranteed lifetime withdrawal benefit (GLWB) options embedded in variable annuities in two different tax regimes. The New Zealand (NZ) system taxes investment income when it is earned, whereas the system in the United States defers taxes on annuity investment income until it is paid out. We examine the effects of these tax differences on the charges collected by the issuer as well as on the value of the contract to the policyholder. We find that the issuer’s charges are typically lower (higher) in the NZ tax regime when the expected fund earnings are low (high) or the fund volatility is high (low). On the other hand, the value to the policyholder is always lower in the NZ tax regime due to the earlier tax payments.We also find that the value of the GLWB in the NZ tax regime is nearly always below the value of an ordinary payout annuity with the same tax rules.

The extant literature on the Guaranteed Lifetime Withdrawal Benefits (GLWB) financial risk is abundant, however, few articles investigate the option offered to the policyholder with respect to the initiation of the contract and examine this impact on the profitability of the product for the insurer. We extend the analysis carried out by Huang et al. (IME, 2014) on the optimal initiation of the product with GLWB. First, we add an additional dimension in the analysis to account for the insurer losses as a function of the age for disbursement chosen by the policyholder. Then, we develop a novel analytical framework to determine by numerical methods the extent to which an insurer, expecting his client to choose when to receive benefits to maximize the value of his variable annuity contract, should change its actuarially fair fee structure. We show that the fair premium is a function of the insured policyholder age when he bought the contract. This result runs counter to the current fee structure and practice in the Canadian insurance industry with insurers charging a uniform level of fees regardless of the policyholder biological age when the contract is issued.

2013/2014
The past twenty years have seen a massive proliferation in insurance-linked derivative products. The public, indeed, has become more aware of investment opportunities outside the insurance sector and is increasingly trying to seize all the benefits of equity investment in conjunction with mortality protection. The competition with alternative investment vehicles offered by the financial industry has generated substantial innovation in the design of life products and in the range of benefits provided. In particular, equity-linked policies have become ever more popular, exposing policyholders to financial markets and providing them with different ways to consolidate investment performance over time as well as protection against mortality-related risks. Interesting examples of such contracts are variable annuities (VAs). This kind of policies, first introduced in 1952 in the United States, experienced remarkable growth in Europe, especially during the last decade, characterized by “bearish” financial markets and relatively low interest rates. The success of these contracts is due to the presence of tax incentives, but mainly to the possibility of underwriting several rider benefits that provide protection of the policyholder’s savings for the period before and after retirement. In this thesis, we focus in particular on the Guaranteed Lifetime Withdrawal Benefit (GLWB) rider. This option meets medium to long-term investment needs, while providing adequate hedging against market volatility and longevity-related risks. Indeed, based on an initial capital investment, it guarantees the policyholder a stream of future payments, regardless of the performance of the underlying policy, for his/her whole life. In this work, we propose a valuation model for the policy using tractable financial and stochastic mortality processes in a continuous time framework. We have analyzed the policy considering two points of view, the policyholder’s and the insurer’s, and assuming a static approach, in which policyholders withdraw each year just the guaranteed amount. In particular, we have based ourselves on the model proposed in the paper “Systematic mortality risk: an analysis of guaranteed lifetime withdrawal benefits in variable annuities” by M. C. Fung, K. Ignatieva and M. Sherris (2014), with the aim of generalizing it later on. The valuation, indeed, has been performed in a Black and Scholes economy: the sub-account value has been assumed to follow a geometric Brownian motion, thus with a constant volatility, and the term structure of interest rates has been assumed to be constant. These hypotheses, however, do not reflect the situation of financial markets. In order to consider a more realistic model, we have sought to weaken these misconceptions. Specifically we have taken into account a CIR stochastic process for the term structure of interest rates and a Heston model for the volatility of the underlying account, analyzing their effect on the fair price of the contract. We have addressed these two hypotheses separately at first, and jointly afterwards. As part of our analysis, we have implemented the theoretical model using a Monte Carlo approach. To this end, we have created ad hoc codes based on the programming language MATLAB, exploiting its fast matrix-computation facilities. Sensitivity analyses have been conducted in order to investigate the relation between the fair price of the contract and important financial and demographic factors. Numerical results in the stochastic approach display greater fair fee rates compared to those obtained in the deterministic one. Therefore, a stochastic framework is necessary in order to avoid an underestimation of the policy. The work is organized as follows. Chapter 1. This chapter has an introductory purpose and aims at presenting the basic structures of annuities in general and of variable annuities in particular. We offer an historical review of the development of the VA contracts and describe the embedded guarantees. We examine the main life insurance markets in order to highlight the international developments of VAs and their growth potential. In the last part we retrace the main academic contributions on the topic. Chapter 2. Among the embedded guarantees, we focus in particular on the Guaranteed Lifetime Withdrawal Benefit (GLWB) rider. We analyze a valuation model for the policy basing ourselves on the one proposed by M. Sherris (2014). We introduce the two components of the model: the financial market, on the one hand, and the mortality intensity on the other. We first describe them separately, and subsequently we combine them into the insurance market model. In the second part of the chapter we describe the valuation formula considering the GLWB from two perspectives, the policyholder’s and the insurer’s. Chapter 3. Here we implement the theoretical model creating ad hoc codes with the programming language MATLAB. Our numerical experiments use a Monte Carlo approach: random variables have been simulated by MATLAB high level random number generator, whereas concerning the approximation of expected values, scenario- based averages have been evaluated by exploiting MATLAB fast matrix-computation facilities. Sensitivity analyses are conducted in order to investigate the relation between the fair fee rate and important financial and demographic factors. Chapter 4. The assumption of deterministic interest rates, which can be acceptable for short-term options, is not realistic for medium or long-term contracts such as life insurance products. GLWB contracts are investment vehicles with a long-term horizon and, as such, they are very sensitive to interest rate movements, which are uncertain by nature. A stochastic modeling of the term structure is thus appropriate. In this chapter, therefore, we propose a generalization of the deterministic model allowing interest rates to vary randomly. A Cox-Ingersoll-Ross model is introduced. Sensitivity analyses have been conducted. Chapter 5. Empirical studies of stock price returns show that volatility exhibits “random” characteristics. Consequently, the hypothesis of a constant volatility is rather “counterfactual”. In order to consider a more realistic model, we introduce the stochastic Heston process for the volatility. Sensitivity analyses have been con- ducted. Chapter 6. In this chapter we price the GLWB option considering a stochastic process for both the interest rate and the volatility. We present a numerical comparison with the deterministic model. Chapter 7. Conclusions are drawn. Appendix. This section presents a quick survey of the most fundamental concepts from stochastic calculus that are needed to proceed with the description of the GLWB’s valuation model.
Negli ultimi venti anni si `e assistito ad una massiccia proliferazione di prodotti de- rivati di tipo finanziario-assicurativo. Gli individui, infatti, sono diventati sempre piu` consapevoli delle opportunita` di investimento esistenti al di fuori del settore as- sicurativo e pertanto richiedono all’impresa di assicurazione non solo la protezione contro il rischio di mortalit`a/longevit`a, ma anche tutti i benefici di un investimento di capitali. Ed `e proprio per soddisfare le esigenze del mercato e per fronteggiare la concorrenza alimentata da altri competitors (banche, ecc.) che il mercato assi- curativo sta cambiando ed ha iniziato a sviluppare nuovi prodotti assicurativi ad elevato contenuto finanziario. Nell’ambito di questi prodotti, particolare interesse rivestono le cosiddette polizze variable annuities. Introdotte per la prima volta negli Stati Uniti nel 1952, esse hanno raggiunto ben presto un notevole sviluppo anche in Europa, soprattutto nell’ultimo decennio caratterizzato da mercati finanziari bearish e da tassi di interesse relativamente bassi. Il successo di questo tipo di contratti `e dovuto al favorevole trattamento fiscale di cui godono, ma soprattutto all’offerta di opzioni implicite che garantiscono una protezione dei risparmi degli investitori prima e dopo il pensionamento. In questo lavoro di tesi, ci siamo concentrati in particola- re sull’opzione Guaranteed Lifetime Withdrawal Benefit (GLWB). Essa permette di soddisfare esigenze di investimento di medio/lungo periodo e nello stesso tempo offre una discreta copertura al rischio dovuto alla volatilit`a dei mercati e al longevity risk. Infatti, a fronte di un capitale iniziale investito, garantisce all’assicurato un flusso di pagamenti futuri indipendente dalla performance della polizza sottostante per tutta la durata della sua vita. Piu` precisamente, in questo lavoro proponiamo un modello di valutazione per questo tipo di contratto, facendo ricorso a processi stocastici per descrivere la componente finanziaria e quella legata alla mortalità dell’assicurato. Analizziamo la polizza considerando sia il punto di vista del cliente che quello della compagnia di assicurazione. La nostra valutazione si è basata sul modello proposto da M. C. Fung, K. Ignatieva e M. Sherris nell’articolo “Systematic mortality risk: an analysis of guaranteed lifetime withdrawal benefits in variable annuities” (2014). Tuttavia le ipotesi alla base di questa analisi non trovano giustificazione nel mercato; in effetti, considerare un tasso di interesse ed una volatilità costanti sembra poco sensato. Proprio per proporre un modello più fedele al mercato, si è pensato di indebolire questi assunti, prendendo in considerazione un processo stocastico a sé stante per descrivere la dinamica del tasso di interesse e della volatilità. Dapprima abbiamo analizzato separatamente l’impatto dei due processi sul prezzo equo dell’opzione, per poi considerare anche il loro effetto congiunto. Come parte integrante del lavoro, abbiamo implementato il modello teorico proposto impiegando un approccio Monte Carlo. A questo scopo abbiamo creato codici ad hoc utilizzando il linguaggio di programmazione MATLAB, sfruttando al meglio tutte le sue potenzialità di calcolo matriciale. Sono state condotte analisi di sensitività per analizzare l’impatto sul prezzo equo dell’opzione di alcuni importanti parametri finanziari e demografici. I risultati numerici mostrano come effettivamente l’impiego di un approccio stocastico sia più capace di descrivere le fluttuazioni del mercato e quindi permetta di ottenere risultati più realistici. Il valore equo delle commissioni applicate dalla compagnia di assicurazione per l’attivazione della garanzia GLWB aumenta quando si passa da un approccio deterministico ad uno stocastico (soprattutto se quest’ultimo considera congiuntamente tassi di interesse e volatilità stocastici), rivelando come un adeguato modello stocastico sia necessario per evitare una sottovalutazione di tali polizze. Il lavoro è strutturato come segue: Capitolo 1. Questo capitolo ha un ruolo introduttivo e mira a fornire una descrizione delle caratteristiche principali delle polizze variable annuities. Si analizza l'evoluzione storica di tali polizze ed il loro sviluppo nei principali mercati internazionali. Segue una breve rassegna dei principali contributi accademici sulla valutazione di tali contratti e si spiegano le ragioni alla base di questo lavoro. Capitolo 2. Tra le varie garanzie implicite nei contratti variable annuity ci soffermiamo sull'opzione Guaranteed Lifetime Withdrawal Benefit. In questo capitolo analizziamo il modello di valutazione del contratto proposto da M. Sherris (2014); introduciamo le due componenti del modello (il mercato finanziario e l'intensità di mortalità) dapprima descrivendole separatamente, poi combinandole. Nella seconda parte del capitolo studiamo le formule per il calcolo del prezzo equo del contratto considerando due punti di vista, quello dell'assicurato e quello dell'assicuratore. Capitolo 3. In questo capitolo implementiamo il modello teorico creando codici ad hoc con il linguaggio di programmazione MATLAB. Le nostre valutazioni sono state realizzate utilizzando un approccio Monte Carlo. Diverse analisi di sensitività sono state condotte per analizzare l’impatto sul prezzo equo dell’opzione di alcuni importanti parametri finanziari e demografici. Capitolo 4. In questo capitolo si propone una generalizzazione del modello deterministico indebolendo l'ipotesi di struttura a termine dei tassi di interesse costante. Per descrivere la dinamica del tasso di interesse si introduce in particolare un processo Cox- Ingersoll- Ross. Capitolo 5. In questo capitolo si indebolisce l'ipotesi che considera costante la volatilità del fondo d'investimento prevedendo una dinamica descritta dal processo di Heston. Capitolo 6. Si descrive un modello che considera congiuntamente un processo stocastico per i tassi di interesse (CIR) e per la volatilità (Heston). Si conducono analisi di sensitività e si mostrano i risultati ottenuti. Capitolo 7. In questo capitolo traiamo le conclusioni del nostro lavoro. Appendice. Proponiamo una breve rassegna delle principali nozioni di calcolo stocastico necessarie per meglio comprendere la descrizione del modello di valutazione.
XXVII Ciclo
1986

Thesis (Ph. D.)--Massachusetts Institute of Technology, Sloan School of Management, Operations Research Center, 2008.
This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.
Includes bibliographical references (p. 215-224).
In this thesis we study three problems related to financial modeling. First, we study the problem of pricing Employee Stock Options (ESOs) from the point of view of the issuing company. Since an employee cannot trade or eectively hedge ESOs, she exercises them to maximize a subjective criterion of value. Modeling this exercise behavior is key to pricing ESOs. We argue that ESO exercises should not be modeled on a one by one basis, as is commonly done, but at a portfolio level because exercises related to different ESOs that an employee holds would be coupled. Using utility based models we also show that such coupled exercise behavior leads to lower average ESO costs for the commonly used utility functions such as power and exponential utilities. Unfortunately, utility based models do not lead to tractable solutions for finding costs associated with ESOs. We propose a new risk management based approach to model exercise behavior based on mean-variance portfolio maximization. The resulting exercise behavior is both intuitive and leads to a computationally tractable model for finding ESO exercises and pricing ESOs as a portfolio. We also study a special variant of this risk-management based exercise model, which leads to a decoupling of the ESO exercises and then obtain analytical bounds on the implied cost of an ESO for the employer in this case. Next, we study Guaranteed Withdrawal Benefits (GWB) for life, a recent and popular product that many insurance companies have offered for retirement planning. The GWB feature promises to the investor increasing withdrawals over her lifetime and is an exotic option that bears financial and mortality related risks for the insurance company.
(cont.) The GWB feature promises to the investor increasing withdrawals over her lifetime and is an exotic option that bears financial and mortality related risks for the insurance company. We first analyze a continuous time version of this product in a Black Scholes economy with simplifying assumptions on population mortality and obtain an analytical solution for the product value. This simple analysis reveals the high sensitivity the product bears to several risk factors. We then further investigate the pricing of GWB in a more realistic setting using different asset pricing models, including those that allow the interest rates and the volatility of returns to be stochastic. Our analysis reveals that 1) GWB has insufficient price discrimination and is susceptible to adverse selection and 2) valuations can vary substantially depending on which class of models is used for accounting. We believe that the ambiguity in value and the presence of significant risks, which can be challenging to hedge, should create concerns to the GWB underwriters, their clients as well as the regulators. Finally, many problems in finance are Sequential Decision Problems (SDPs) under uncertainty. We nd that SDP formulations using commonly used financial metrics or acceptability criteria can lead to dynamically inconsistent strategies. We study the link between objective functions used in SDPs, dynamic consistency and dynamic programming. We then propose ways to create dynamically consistent formulations.
by Premal Shah.
Ph.D.

碩士
國立臺灣大學
財務金融學研究所
97
Guaranteed minimum withdrawal benefits (GMWB) is an innovative rider of variable annuity (VA) policies. In recent years GMWB has gained popularity due to it being an investment-linked insurance while guaranteeing minimum return. The pricing method of GMWB can be generally classified in two ways: Monte Carlo simulation and numerical PDE techniques. In this research, the tree model is used to price GMWB rider in a more realistic and intuitive fashion than existing methods. We extend the static model in Milevsky and Salisbury (2006), showing that the product can be decomposed into a discrete down-and-out single barrier option plus a generic term-certain annuity. We follow the idea of stair tree in Dai and Lyuu (2004) and Dai (2009), using bino-trinomial tree (BTT) in Dai and Lyuu (2008) to price the GMWB’s embedded exotic option. Numerical experiments show this method to be more accurate and efficient.

本研究主要探討附保證最低提 (Guaranteed Minimum Withdrawal Benefits, GMWB)的變額(Variable Annuity, VA) 在隨機模型下之定價。保證最低提是變額的一種附加約 (rider) 並在市場下跌的情況下為變額持有人提供保障。它保證持有人在合約期內的總提少於一個預先訂的額，而變額的投資表現。一般，這個保證額相等於變額的初始投資額。本研究的融模型假設投資標的基價格符合對維過程 (exponential Lévy process)，而隨機則符合由維過程驅動的瓦西克模型 (Vasiček model)。融模型中的個維過程的相依結構 (dependence structure) 會由維關結構 (Lévy Copula) 描述。這個方法的好處是可描述同型的相依結構。用一個配合維關結構而有效的蒙地卡模擬方法，我們研究在同相依結構及模型下保證最低提的價值變化。在固定的特別情況下，保證最低提的價值能夠透過卷積方法 (convolution method) 而得到半解析解 (semi-analytical solution) 。最後，我們將本研究中的學模型擴展以研究近期出現由保證最低提演化而成的一種保證產品。這個產品名稱為保證終身提 (Guaranteed Lifelong Withdrawal Benefit, GLWB)，而此產品的到期日則與持有人的壽命相關。
In this thesis, we study the problem of pricing the variable annuity(VA) with the Guaranteed Minimum Withdrawal Benefits (GMWB) under the stochastic interest rate framework. The GMWB is a rider that can be elected to supplement a VA. It provides downside protection to policyholders by guaranteeing the total withdrawals throughout the life of the contract to be not less than a pre-specied amount, usually the initial lump sum investment, regardless of the investment performance of the VA. In our nancial model, we employ an exponential L´evy model for the underlying fund process and a Vasiček type model driven by a L´evy process for the interest rate dynamic. The dependence structure between the two driving L´evy processes is modeledby the L´evy copula approach whichis exible to model a wide range of dependence structure. An effcient simulation algorithm on L´evy copula is then used to study the behavior of the value of the GMWB when the dependence structure of the two L´evy processes and model parameters Vry. When the interest rate is deterministic, the value of the GMWB can be solved semi-analytically by the convolution method. Finally, we extend our model to study a recent variation of GMWB called Guaranteed Life long Withdrawal Benefits (GLWB) in which the maturity of the GLWB depends on the life of the policyhodler.
Detailed summary in vernacular field only.
Chan, Wang Ngai.
Thesis (M.Phil.)--Chinese University of Hong Kong, 2012.
Includes bibliographical references (leaves 115-121).
Abstracts also in Chinese.
Abstract --- p.i
Acknowledgement --- p.iv
Chapter 1 --- Introduction --- p.1
Chapter 1.1 --- Variable Annuity & Guaranteed Minimum Withdrawal Benefit --- p.1
Chapter 1.2 --- Literature Review --- p.4
Chapter 1.3 --- Financial Model for GMWB --- p.7
Chapter 2 --- L´evy Copulas and the Simulation Algorithm --- p.12
Chapter 2.1 --- Definitions and Theorem --- p.15
Chapter 2.2 --- Examples of L´evy Copulas --- p.19
Chapter 2.2.1 --- Independence case --- p.19
Chapter 2.2.2 --- Complete Dependence --- p.20
Chapter 2.2.3 --- The Clayton L´evy Copula --- p.21
Chapter 2.3 --- Simulation algorithm for two-dimensional dependent L´evy process --- p.22
Chapter 3 --- Model Formulation for GMWB --- p.26
Chapter 3.1 --- Financial Model for GMWB --- p.27
Chapter 3.2 --- Underlying Fund of VA and the Interest Rate --- p.30
Chapter 3.3 --- A Special Case of Deterministic Interest Rate --- p.34
Chapter 4 --- Numerical Implementation --- p.38
Chapter 4.1 --- The Clayton L´evy Copula --- p.39
Chapter 4.2 --- The Underlying Fund and the Interest Rate Processes --- p.42
Chapter 4.3 --- Kendall’s Tau Coefficient --- p.47
Chapter 4.4 --- The GMWB Option Value --- p.49
Chapter 4.4.1 --- Control Variate for Simulation --- p.49
Chapter 4.4.2 --- Simulation Results --- p.51
Chapter 4.5 --- Deterministic Interest Rate --- p.52
Chapter 5 --- GMWB Pricing Behavior --- p.56
Chapter 5.1 --- L´evy model for the underlying fund --- p.57
Chapter 5.1.1 --- The Skewness --- p.57
Chapter 5.1.2 --- The Kurtosis --- p.65
Chapter 5.2 --- The Vasiček model driven by L´evy process --- p.73
Chapter 5.2.1 --- The Volatility Parameter ôV --- p.73
Chapter 5.2.2 --- The Mean Reverting Parameter aV --- p.77
Chapter 5.3 --- Dependence between the underlying fund and rate processes --- p.81
Chapter 5.3.1 --- The jump direction dependence parameter n{U+1D9C} --- p.83
Chapter 5.3.2 --- The jump magnitude dependence parameter θ{U+1D9C} --- p.90
Chapter 6 --- GMWB for Life --- p.96
Chapter 6.1 --- Model Formulation --- p.98
Chapter 6.1.1 --- Mortality model --- p.99
Chapter 6.1.2 --- Financial Model for GLWB --- p.101
Chapter 6.2 --- GLWB product from John Hancock --- p.103
Chapter 6.3 --- GLWB Pricing Behavior --- p.104
Chapter 6.3.1 --- The correlation effect --- p.106
Chapter 7 --- Conclusion --- p.108
A Proofs --- p.113
Chapter A.1 --- Proof of Equation 3.1 --- p.113
Chapter A.2 --- Proof of Equation 3.3 --- p.114
Bibliography --- p.115

碩士
東吳大學
商用數學系
96
In the research, we deal with financial valuation of guaranteed minimum withdrawal benefits（GMWB）. We assume the equity process underlying the GMWB follows a Geometric Brownian Motion（GBM）. GMWB is a kind of Asian option. Under the GBM process, this paper extends the study of Tsao, Chang and Lin（2003）and integrals the Taylor series expansion to derive analytic solutions for GMWB. Numerical experiments show that under some conditions analytic solutions sometimes perform very well and are efficient as benchmarked with large sample Monte Carlo simulation.

The Guaranteed Minimum Withdrawal Benefits (GMWBs) are optional riders provided by insurance companies in variable annuities. They guarantee the policyholders' ability to get the initial investment back by making periodic withdrawals regardless of the impact of poor market performance. With GMWBs attached, variable annuities become more attractive. This type of guarantee can be challenging to price and hedge. We employ two approaches to price GMWBs. Under the constant static withdrawal assumption, the first approach is to decompose the GMWB and the variable annuity into an arithmetic average strike Asian call option and an annuity certain. The second approach is to treat the GMWB alone as a put option whose maturity and payoff are random. Hedging helps insurers specify and manage the risks of writing GMWBs, as well as find their fair prices. We propose semi-static hedging strategies that offer several advantages over dynamic hedging. The idea is to construct a portfolio of European options that replicate the conditional expected GMWB liability in a short time period, and update the portfolio after the options expire. This strategy requires fewer portfolio adjustments, and outperforms the dynamic strategy when there are random jumps in the underlying price. We also extend the semi-static hedging strategies to the Heston stochastic volatility model.

碩士
東吳大學
財務工程與精算數學系
102
In this thesis, we deal with the problem of financial valuation of Variable Annuities (VAs)with guaranteed lifelong withdrawal benefits （GLWB）. We assume the invested fund of the GLWB follows a Geometric Brownian Motion （GBM）. We consider products without accumulation period and discrete withdrawal scheme. This study proposes three variations of Monte Carlo simulation methods. These methods utilize the concept of control variates. The results show that variance can be reduced to two hundred times as much.

碩士
東吳大學
商用數學系
96
Guaranteed minimum withdrawal benefit (GMWB) is a very popular rider on variable annuities (VA). It promises annuitants to earn the entire initial investment, regardless of investment performance, yet spread over an established time period. In this paper, we extend the “static model” in Milevsky and Salisbury (2006) by considering a more realistic situation that VA could lapse during the guaranteed period if the annuitant dies earlier. And if this happens, remaining annuity payments discontinue, whereas a death benefit will be offered to maintain the minimum withdrawal guarantee. Two different death benefits are considered including the market value of the separate account at the annuitant’s death; and the greater of the withdrawal account and the market value of the separate account mentioned above. We decompose the GMWB with death benefits into a certain-year temporary life annuity plus a Quanto Asian put with random maturity date; and then explore their values. Finally, in order to sustain the insurer’s capacity to absorb the downside financial risk and the systematic risk, appropriate a proportional fee under the fair price and risk-neutral framework is evaluated numerically via Monte Carlo simulation, moreover, financial risk measure is evaluated by the same way.