The post Actuarial Science: Life Insurance An Overview appeared first on StepUp Analytics.

]]>*What is Life Insurance?*

A Life insurance contract is an agreement between a policyholder, the insured and an insurer or assurer, the insurance company, where the insurer promises to pay a sum of money (death benefit) in exchange for a premium, upon the death of the insured person i.e. The policyholder. Depending on the contract, other events such as terminal illness or critical illness can also trigger payment. The policyholder typically pays a premium, either regularly or as one lump sum. Other expenses, such as funeral expenses, can also be included in the benefits.

*How Life Insurance Works?*

The Death benefit is the amount of money the insurance company guarantees to the beneficiaries identified in the policy upon the death of the insured. The insured will choose their desired death benefit amount based on estimated future needs of surviving heirs. The insurance company will determine whether there is an insurable interest and if the insured qualifies for the coverage based on the company’s underwriting requirements.

Premium payments are set using actuarially based statistics. The insurer will determine the cost of insurance (COI) which comprises the amount required to cover mortality costs, administrative fees, and other policy maintenance fees. There are eight factors besides medical history, the insured’s age that influences the premium calculation, some of them are occupational hazards, personal risk propensity, geographical location, education. The insurer will remain obligated to pay the death benefit if premiums are submitted as required. With term policies, the premium amount includes the COI. For permanent or universal policies, the premium amount consists of the COI and a cash value amount.

The policyholder pays a premium for a pre-determined fixed period of time (can continue till the death of the policyholder) in return for a sum of money, known as a death benefit, at the time of death of the policyholder. This is a basic life insurance works.

Similarly, there are health assurance schemes in which policyholder receives a sum of money for the treatment of certain diseases or contraction of a terminal illness.

There are certain basic forms of life insurance. The different types of life insurance policies include

Following are the most common life insurance myths:

**Insurance is for saving tax**

Saving tax is just an added advantage of the insurance policy. The main objective of insurance is to provide protection to you and your family and to build an assured corpus for your future needs.**Insurance will benefit only after my death**

Insurance policies provide protection to you and your family. One of the main objectives of taking insurance is to provide a financial cushion to your family in case you are not around but it’s not the only objective. Insurance helps you to build a corpus for yourself; provides you with comfortable retired life and even takes care of your lengthy medical bills.**My group insurance is adequate**

Your group insurance might be adequate but what if you change the job? Once you change the job your group insurance will cancel off and you will not get any insurance benefit. So it is always advisable to take insurance other than the insurance offered by your employer.**Only the Breadwinner of the family needs insurance**

Every family member needs insurance. Your work profile changes the insurance needs but certainly does not eliminate them.**I’m single and don’t have any dependents, therefore I don’t need any coverage**

You might not need a life insurance policy where your nominee is taken care of but you certainly need the policy to take care of your health and retirement worries.**Life insurance is far more important than health coverage**

As the health costs are increasing by the day, taking a health insurance plan has become as important as a life coverage plan. Health plans, disability plans, and critical illness plans provide you with financial cushion and compensate for the financial loss you suffer in case you are not able to work because of illness.

**What does life insurance mean for Actuaries and insurance companies?**

The law underlying life insurance contracts is the law of large numbers. The basic principle in insurance is that the company ensures a diverse group of individuals and the probability of having to pay death benefit early for an insurer is low in that large group.

Policies are designed by actuaries. They estimate the mortality rate of the insured and calculate the appropriate gross premium to be charged. Profit- tests are run to decide an appropriate premium.

Actuaries also calculate reserves to be held by insurance companies to avoid financial problems at the time of payment of death benefits. In life insurance plans, actuaries take note of details like the premium structure, death benefit structure i.e. whether it is paid in the form of an annuity or lump-sum amount after the death of the insured. The premium that is collected is invested in securities to earn income for survival. Since there is no certainty of the time of death, market conditions, certain assumptions are used called experience basis, assumption basis. A way of looking at it is, they put a price on the death and time of death of an individual using basis and probability. There are in a position to do so because they have more information on mortality statistics than an individual.

There are many public and private insurance companies offering various plans with a detailed overview of their policy, which are readily available on their websites. One can also calculate the premium amount for different policies after specifying the requirements like the sum assured, age, term of the policy, premium payment frequency. These help in making smart decisions.

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]]>The post Actuarial Science: Markov Jump Process Models appeared first on StepUp Analytics.

]]>*Do you wonder how a hospital has just the right number of beds in a hospital ward?*

*What if too many workers take sick leave at the worksite and company falls short of employees? How does a company make sure they have the right number of workers at any point in time? *

*Well, they use a Markov Jump model to calculate the probability of a sick patient occupying the bed for a certain number of days and to calculate the probability of a worker becoming sick and thus, keep everything in order.*

*Have you heard of Gambler’s Ruin problem? Markov Jump’s process is at play there.**Do you want to find out how long will you have to wait in a queue before one of the cash counter bills you up at big bazaar? The number of counters can be modeled using the Markov Jump process.*

*Do you have a similar problem to solve?*

*Let me show you how Markov Process can be of help.*

A stochastic process, possessing Markov property, in continuous time with discrete state space (finite or countable) is a **Markov Jump Process.**

A stochastic process is a set of ordered random variables, X_{t }, one for each time t in set J.

The process is denoted {*X*_{t}* *: *t** *∈ *J*} .

A stochastic process can have discrete or continuous state space and move in discrete or continuous time-space. State-space implies the values that the process can take and time-space means the time values at which the process is recorded or observed. A table of various stochastic processes with defining properties is presented below.

*Now, what is Markov’s property?*

Markov property simply means the future evolution of the process can be determined with the current value of the process, independent of past values. Mathematically, This is defined in continuous space,

In discrete space,

So a Markov Jump process is simply a process, taking discrete values, whose future evolution will depend on its current value only and that has continuous state space i.e. it can move at any time. Let’s relate to Markov jump process in a different way,

You might have been told by someone at some point in your life, “Forget the past. Focus on present and Future”, they were asking you to have Markov property.

The role of a die, the number that will appear on your next roll does not depend on how many fours or ones you have rolled before. Every roll is independent of its previous roll. That is Markov.

Similarly, the tossing of the coin is a Markov process. But let me tell you what is not Markov, Marvel’s Infinity saga. To understand the movie, Avengers Endgame, you need to possess the knowledge of all the previous movies released at different times. It depends on past values of the process; hence it does not have Markov property.

Even when it comes to success, we must be Markov. It is our efforts at present that will determine our future success. The failures in the past cannot overpower the future success. Our future success depends on the hard work at present and not on failures in the past.

I hope Markov Jump process doesn’t seem intimidating anymore.

*Let’s get a little more knowledge of the process before proceeding to apply it in R programming.*

Markov process is named after the Russian mathematician Andrey Markov. Markov processes are the basis for general stochastic simulation methods known as Markov Chain Monte Carlo, which are used for simulating sampling from complex probability distributions, and have found extensive application in Bayesian Statistics. A stochastic process, possessing Markov property, in discrete time with discrete state space (finite or countable) is a Markov Chain. An application of Markov chain is in insurance companies for determining discounts on premiums for motor insurance policies. Depending on the number of accidents in the previous year, a level of discount on premium for the current year is determined for the policyholder. A no-claim-discount (NCD) model is used which is an application of the Markov chain. A model can be used by the shopkeeper to maintain stock of items in inventory.

Markov jump process finds its application in insurance companies for determining premiums for life insurance policies using the Healthy-Sick-Dead model. It is used in pension funds to keep track of employee’s transition from the active state to either ill-health retirement, normal retirement, leaving the company or dead state. The algorithm known as PageRank, which was originally proposed for the internet search engine Google, is based on a Markov process.

A diagram showing states of the process with possible transitions is called a transition diagram. Arrows show the transition from one state to another and the probability of transition is written above or below the arrows.

A Markov jump process can be time-homogeneous and time-inhomogeneous. A time-homogeneous process is one whose probability of moving from one state to another remains the same with time and in time-inhomogeneous process probability changes with time. The probability of changing states in Markov Processes is given by **Chapman-Kolmogorov** Equations. This probability is called transition probability.

Transition probabilities are of major use to the companies. With the help of Chapman-Kolmogorov equations, there are three ways of determining probabilities:

- Differential equations (forward and backward) – they can be solved to get the probability.
- The integrated form of Kolmogorov backward equation
- The integrated form of Kolmogorov forward equation

In continuous time, we consider transition probability over a very short interval of time h. Dividing by *h *expresses this as a probability of transition in unit time. Taking limits as

*h *tends to 0 leads to the concept of a *transition rate*. Transition rate is also called transition intensity. A matrix of transition intensity is called the Generator matrix and a matrix of transition probabilities is called the Transition probability matrix (TPM). The sum of values in a row in a generator matrix must equal zero while the sum of values in a row of TPM must equal one.

Markov models are studied in Actuarial Subject CS2: Risk modeling and Survival Analysis. The Curriculum 2019 requires the topic to be covered in R programming. R has become vital in actuarial work. So we will see the working of the Markov model in R.

We can use R studio which is easier to understand than R. Starting with the basic, creation of a transition intensity matrix and probability matrix

Over a small period of time period h, the transition probability matrix(TPM) is given by **P(h)=1+h*A+ o(h)** where,

I is an identity matrix

A is a generator matrix

o(h) is the order of h

We can approximate P(h) as

P(h)= 1+h*A

Type the following code on your R script

A=matrix(c(m11,m21,-m11,-m21),nrow=2,ncol=2)

I= diag(2)

P=I+h*A

Where m11, m21 are values of transition rates. Now we solve a question in R,

Read more on Actuarial Science: **Click**

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]]>The post Degree In Actuarial Science – IAQS appeared first on StepUp Analytics.

]]>*Through this article, I would like to share some information of potential interest to Actuarial Science, CFA (Certified Financial Analyst) and FRM (Financial Risk Manager) aspirants.*

*Have you been in the dilemma of choosing bachelors or masters course to pursue Actuarial Science? *

*Do you want a taste of Data Science? Have you felt a lack of proper guidance in these fields? Do you feel you lack the job skill set despite completing graduation with a few Actuarial science papers? Do you feel the lack of application in real-world of the concepts taught in these exams?*

IAQS (Institute of Actuarial and Quantitative Studies) is launching two courses in Mumbai with degree recognition from University of Mumbai from the academic year 2019. IAQS is an independent institute formed to form courses for students who want to enter banking, finance, investment, and insurance sector. It is releasing

**B.Sc. in Actuarial Science & Quantitative Finance **

**M.Sc. in Actuarial Science & Quantitative Finance **

*Before getting into details of the course, I’d like to tell you the purpose of the course as told by the designers of the course structure. *

Through the course, they want to connect students to the industry. The goal is to bridge the gap between exam-based knowledge and work-based knowledge. The program is designed to make the students work-ready.

Professional Examinations have failed to equip the successful candidates with skills that make them employable. For this purpose, the Institute and Faculty of Actuaries have introduced programming languages in examinations under the new 2019 Curriculum. The course structure has been formulated under the guidance of eminent people from Finance, Insurance, Actuarial industry. These working professionals have put together their understanding of the field, software skills, and professional skills required to design the course.

*Getting into the details of the course,*

The B.Sc. and M.Sc. program is open to students from any stream. Eligibility for the course is judged by an entrance exam titled QAT(Quantitative Aptitude Test). The test is divided into 70% Maths and Stats and 30% English. An applicant must score 50% to clear the test. After the entrance test, applicants will be screened through a personal interview round where interpersonal skills will be tested. Passing ACET or any Actuarial paper will also make the student eligible for the course.

It will be a three-year full-time program, conducted at KC College, Mumbai. The first-year study will deal with probability, economics, life, and non-life insurance products. The second-year will include knowledge of financial market, derivatives, risk modeling and economics. In the third year, students have to choose between Actuarial Science and Quantitative Finance. The course has a different well-defined syllabus for either elective.

The quantitative finance elective offers a career similar to professional courses like CFA and FRM. Semester five will also focus on soft skills and business communication and semester six will have predictive analysis and machine learning as part of syllabus under both the electives. It has a one-year exchange program with few affiliated international colleges after the second semester. Every student must do an internship after the second year which will be provided by the campus placement cell.

There will be project work on real-world company problems in semester four and compulsory project work for all students in semester six. In the course of three years, students will also learn the application of concepts in excel and software languages like R programming, Python, C++, and Bloomberg. The department also has a placement cell dedicated to counseling, providing internship opportunities and job assistance.

This will be a two-year program. In the first year, the course will cover probability, statistics, financial mathematics, financial engineering, business economics, insurance products and knowledge of the financial market. The first year will include applications of concepts in Excel and R programming. The second-year will give students the choice between Actuarial Science and Quantitative Finance.

The quantitative finance elective offers a career similar to professional courses like CFA and FRM. In quantitative finance, the course covers financial engineering, enterprise risk management, investment management. In actuarial science, the course covers pricing and reserving of life insurance products, statistical and risk modeling. Placement Cell facility will also be provided to students of this course.

This is the curriculum of the courses at length.

*From the perspective of a student pursuing Actuarial Science,*

Both the courses are recognized by the University of Mumbai. 60%-70% of faculty members of both courses are working professionals. This course is designed by industry professionals so it would encompass the skills and knowledge they expect in us. Since the person teaching me could be someone who would be working in the company apply-in in future, the opportunity of interaction I get in three years will be added advantage. I personally believe learning and working with these professionals in college is great.

Being an Actuarial science student, I had faced the dilemma of choosing a bachelors course and ended up with B.Com. This Bachelor’s course complements Actuarial Science perfectly. You no more have to worry about doing B.Com. , B.B.I. or B.Sc. in Stats or worry about the recognition and acceptance of Degree. You have finally got a course complementing Actuarial Science and Quantitative Finance courses like CFA and FRM.

The program also includes topics that help in competitive exams and exposes students to Data Science. Students registered for Actuarial exams at IFOA usually face exams from the University of Mumbai and IFOA exam at the same time, making it difficult to ace both. They won’t have to face this issue in the Actuarial paper from either IFOA or IAI as of now. In B.Sc. program, in the span of three years, topics of Actuarial papers CS1, CS2, CM1, CM2, CB1, CB2 will be covered.

The program includes project work on real-world problems which prove very useful at the time of employment. Since the internship is compulsory for every student and which will be provided by the placement cell, students get the much-needed work experience before graduating. The B.Sc. program opens up doors to various fields for graduates like Data Science, financial engineering, Actuarial Science, MBA in finance, higher studies in economics, finance.

As far as M.Sc. program is concerned, it will be suitable for individuals who found their interest in Actuarial Science after graduating or maybe while working. It can also be opted by B.Com graduates who want to explore careers in these fields. The wonderful thing about the programs is that these do not certificate courses but a full-time university degree.

For registering for the course, you can check the website https://iaqs.in

Read more on Actuarial Science Click

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]]>The post Actuarial Science: Crop Insurance appeared first on StepUp Analytics.

]]>The variation in the production of agricultural produce is directly affected by many unfavourable conditions such as pest attacks, variations in weather conditions such as rainfall, temperature, humidity, etc. Thus, there is a need to protect farmers from the losses due to those unfavourable events.

Hence, to protect farmers from those unfavourable conditions, “Crop Insurance” came into the picture. Crop Insurance, which is also known as Agriculture Insurance, is purchased by agricultural producers (farmers), and subsidized by the federal government, to protect against either the loss of their crops due to natural disasters, such as hail, drought and floods, or the loss of revenue due to declines in the prices of agricultural commodities.

One of the **most important benefits** of buying agriculture insurance is that farmers get peace of mind, as they know that if any unfavourable event occurs or in other words, if they face any loss, then that loss will be covered up by the insurance, only if they have purchased insurance!

Other benefits of crop insurance include:

**Stability in Income:**It protects farmers against the losses caused by crop failure. It acts like a tool that allows farmers to manage their yield and price risks.**Minimal Debts:**Farmers are able to repay their loans even during the time of crop failure with the support of the right insurance partner.**Technological Advancement:**Insurance companies work along with Agri Platforms, who use the Internet of Things (IoT) (the use of sensors, cameras and other devices to turn every element and action involved in farming into data), to enhance agricultural practices and reduce farmers’ losses. This helps farmers to understand the latest technological advancement and improve their crop production.**Yield Protection:**Crop insurance protects farmers against the loss of production of crops. It also offers preventive planting and replants security.**Provide Awareness:**Insurance companies provide awareness campaigns to help farmers understand the effect of natural calamities and also protect their farms.

One of the major policy issued by the government of India in 2016, to protect farmers from losses due to unfavorable events, was the **Pradhan Mantri Fasal Bima Yojana (PMFBY)**. It has replaced all the prevailing yield insurance schemes in India, with the **objectives** as follows:

- To provide insurance coverage and financial support to the farmers in the event of failure of any of the notified crop as a result of natural calamities, pests, and diseases, i.e. unforeseen events.
- To stabilize the income of farmers to ensure their continuance in farming.
- To encourage farmers to adopt innovative and modern agricultural practices.
- To ensure the flow of credit to the agricultural sector, which will contribute to food security, crop diversification and enhancing growth and competitiveness of the agriculture sector besides protecting farmers from production risks.

Now,
let me give you a brief about the PMFBY scheme by the following **highlights**:

**Eligibility**: All farmers including sharecroppers and tenant farmers growing notified crops in the notified areas are eligible.**Uniform Actuarial premium rate to be paid by farmers:**2% for Kharif crops, 1.5% for Rabi crops and 5% for annual Commercial and Horticulture crops.**No upper limit on government subsidy.**Even if the balance premium is 90%, it needs to be paid by farmers.**Extensive use of technology**(remote sensing, smartphones for uploading pictures of crop cutting, etc.) to have better applicability of the scheme.**Exemption from Service Tax liability**for the implementation of this scheme.**Not only covers the losses suffered by farmers**due to a reduction in crop yield, but it also covers pre-sowing losses, post-harvest losses due to cyclonic rains and losses due to unseasonal rainfall.**Losses arising due to war and nuclear risks**, malicious damage, and other**preventable risks**shall be excluded.

So PMFBY was one of the policies issued by the government. Before that, there were two major schemes launched by the government of India, which were:

- National Agriculture Insurance Scheme (NAIS), 1999
- Modified National Agriculture Insurance Scheme (MNAIS), 2010

There
are many differences between the above two schemes and the PMFBY, of which the **main differences** are:

**Uniformity of Premiums:**The uniform premium rates are specified in PMFBY, which was missing in the old schemes.**Compensation in case of premium default:**Under PMFBY, the government will pay the premium in case the farmer fails to pay the same. It was not possible under the old schemes.**Premium Rate:**It is lower in PMFBY than the old schemes.**Insurance amount cover:**Full amount of insurance is covered under PMFBY, which was not fully covered under MNAIS.**Localized risk coverage:**Inundation is also covered under PMFBY along with hail storm and landslide, which were covered under MNAIS.**Use of technology:**It is mandatory under PMFBY scheme to use technology.**Post-harvest losses coverage:**These losses are covered in all over India – for cyclonic as well as unseasonal rains under PMFBY, whereas only Coastal areas – for cyclonic rains are covered under MNAIS.**Prevented sowing coverage:**It is available under PMFBY and MNAIS, except NAIS.**On account payment:**It is provided under PMFBY and MNAIS, except NAIS.**One season, one premium:**In one season, only one premium is to be paid under PMFBY and NAIS but this facility was not available under MNAIS.**Awareness:**Under PMFBY, insurance companies provide awareness campaigns to help farmers understand the effect of natural calamities and also protect their farms as compared to the old schemes of crop insurance.

To sum up, there were two old schemes which were replaced by the new one, the Pradhan Mantri Fasal Bima Yojana. The new scheme has many objectives and is better than the old schemes of crop insurance.

I hope that I am able to brief you about the Crop Insurance and the PMFBY scheme.

Read more on Actuarial Science Click

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]]>The post Actuarial Techniques in General Insurance appeared first on StepUp Analytics.

]]>Have you ever thought of how that premium is calculated? How the price of a particular policy is decided and who helps in valuation those policies?

That’s where actuaries play a significant role. The major role of an actuary in general insurance is in pricing and reserving. Other than these two roles, actuaries help in risk and reinsurance, investment and regulatory requirements.

In this article, we are going to discuss mainly the techniques used in pricing and reserving. Some of the techniques which are commonly used in general insurance companies by actuaries, in the valuation of premiums and claims reserve requirement, are:

- Generalized Linear Model
- Chain Ladder Method

Now, let’s see how premiums are calculated.

One of the models which are commonly used by the pricing team of the general insurance company for premium calculation is Generalised Linear Model (GLM). This model can accommodate many factors on the basis of which premium is calculated, resulting in more

accuracy in the premiums calculated.

As depicted by the above pictorial representation, the steps to calculate the actual premium to be charged by general insurance companies are as follows:**Step 1:** The benchmark premium is decided taking profit margin, expected claim cost and claim related expenses such as commission, into consideration.

**Step 2:** Underwriting adjustments are made in the benchmark premium. These adjustments differ for different groups of homogeneous policyholders, depending upon factors of the policy.

**Step 3:** In this step, actuary comes into the picture. The actuary calculates the technical premium and this premium is calculated by considering the past experiences of the claims.

**Step 4:** In this step, the premium is aligned with the sum assured in such a way that they remain parallel at all levels of sum assured.

**Step 5:** Actuary lowers the premium by lowering the expenses due to fierce competition. The premium calculated in this step is known as the target premium.

**Step 6:** A walkaway premium is decided by the company by reducing the target premium due to the motive of sustaining in the market.

**Step 7:** Now, the CEO of the company comes into play. Due to competition, the underwriter wants to reduce the walkaway premium, but the actuary disagrees on this. To solve this dispute, the CEO decides the actual premium being charged.

Thus, this is how the actuary works in deciding the premium of the policies, to be charged by the company.

This is a method used for calculating the claims reserve requirement in a general insurance company. It is used by insurers to forecast the number of reserves that must be established in order to cover future claims.

This actuarial method is one of the popular reserve methods. It calculates Incurred But Not Reported (IBNR) loss estimates, using run-off triangles of paid losses and incurred losses. Run-off triangle (also known as Delay Triangle) is a technique used to calculate the reserve in both Property & Casualty and Health Insurance. The run-off triangles are used to estimate

how many claims have been incurred in a reporting period but are not yet reported and a reserve is held for this.

Insurance companies usually set aside a portion of the premiums received, called reserves, to pay for the claims that may be filed in the future. The difference between the number of claims forecasted and the number of claims that are actually being paid, determine the amount of profit received by an insurer.

It is assumed that patterns in claim activities in the past will continue to be seen in the future. In order for this assumption to hold, data from past loss experiences must be accurate. If the assumptions differ from observed claims, the insurer may have to make adjustments to the model.

The basic steps to apply the chain ladder method to calculate the number of reserves are:**Step 1:** Compile claims data in a loss development triangle. Here, a loss development triangle is a methodology to track how claims, both known and unknown, change over time.

**Step 2:** Calculate age-to-age factors, also called loss development factors or link ratios, which represent the ratio of loss amounts from one valuation date to another, and they are intended to capture growth patterns of losses over time. These factors are used to project where the ultimate amount of losses will settle.

**Step 3:** Calculate averages of the age-to-age factors.

**Step 4:** Select claim development factors.

**Step 5:** Select the tail factor.

**Step 6:** Calculate cumulative claim development factors.

**Step 7:** Project ultimate claims.

Thus, the actuary mostly uses this method to calculate reserves in a general insurance company.

It is to be noted, that there are many more complex methods and techniques used by actuaries in a general insurance company other than the two discussed in this article.

Read more on Actuarial Science Click

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]]>The post Black-Scholes model in valuing options appeared first on StepUp Analytics.

]]>But if a letter arrived from the college stating that you have an 18% chance of catching the sickness, a 12% chance of missing at least a week of college and a 2% risk of fatality. Wouldn’t you consider it odd and would want to know about the college’s doctor and the kind of medicine he practiced that gave such statistics. The Black-Scholes formula did this for Long-Term capital management in investing. It is much harder to calculate the odds in investing. Investing confronts us with risk and uncertainty.

Long -Term Capital Management, a private fund responsible for the 1998 financial crisis, did something like this in a letter to its investors in 1995. In an attachment penned down by his academic stars, Merton, and schools, Long-Term did not merely concede the possibility of loss, it calculated the supposed odds of its occurring. The letter stated, “investors may experience a loss of 5% or more in about one month in five, and a loss of 10% or more in about one month in ten.”

**How could they have the odds?**

They key for Long-Term was the volatility, in bond prices. By plugging in thousands of bond prices into the formula, they found the historic volatility i.e. how much the bonds fluctuated in the past and they used it to assess the future risk.

Black-Scholes formula was behind the huge success of the fund in its early years.

I believe the story very well tells the importance of the formula we are going to study.

The Black-Scholes Formula is a model that **determines the price of European options**. It was developed in 1973 by Fisher Black, Robert Merton and Myron Scholes. When applied to a stock option, the model incorporates the constant price variation of the stock, the time value of money, the option’s strike price and the time to the option’s expiry. The model is most widely used to **find the implied volatility** using market rice of an option.

The Black-Scholes Formula goes as follows,

Without dividends

With dividend yield(q)

The Black -Scholes formula is applied to both call and put options ,yielding and not yielding, dividends.

The terms used in the formula are as follows:

The result can be proved by two methods: PDE (Partial Differential Approach) and the martingale process (using the 5-step Binomial Method). The CM2 material has detailed proof.

**Black Scholes PDE**

Paul Samuelson, the first financial economist to win a Nobel Prize, noted, “ The essence of the Black-Scholes formula is that you know, with certainty, not what the deal of the cards will be but what kind of universe is being sampled, which gives you the assumption of the lognormal process.”

**Assumptions of the model **

**The price of the underlying share follows a geometric Brownian motion.***i.e.*the share price changes*continuously*through time according to the stochastic differential equation:

*dS _{t }= S_{t }(µdt + *σdZt) This is the same as the

- To Black, Scholes, and Merton, price changes in financial markets were random. No one could predict any particular change, but over a long period, they assumed that the distribution of all such prices would mirror the pattern of other random events like coin flips, dice rolls or the heights of high school students.
- Merton assumed that volatility was so constant that prices would trade in continuous time ie. without any jumps. Merton’s markets were as smooth as well-brewed java, in which prices would indeed flow like cream.
- Unlimited short selling (that is, negative holdings) is allowed.
- There are no taxes or transaction costs.
- There are no risk-free arbitrage opportunities.
- The risk-free rate of interest is constant, the same for all maturities and the same for borrowing or lending.
- The underlying asset can be traded continuously and in infinitesimally small numbers of units.

The key general implication of the underlying assumptions is that the market in the

underlying share is complete: that is, all derivative securities have payoffs which can be replicated.

**Criticisms of the model**

Mitchell Kapor, a friend and partner of Merton while publishing Tiny TROLL( a desktop graphics and statistics program) wasn’t the only one who wondered if were also wondering the same.

- The volatility parameter σ may not be constant over time.
- The long-term drift parameter may not be constant over time. In particular, interest rates will impact the drift.
- The distribution of security returns log(
*Su/ St*) has a taller peak in reality than that implied by the normal distribution. This is because there are more days of little or no movement in the share price. - The distribution of security returns log(
*Su/ St*) has fatter tails in reality than that implied by the normal distribution. This is because there are more extreme movements in security prices. - The sample paths of security prices are not continuous, but instead, appear to jump occasionally.

**The Garman-Kohlhagen Model**

Mark Garman and Steven Kohlhagen were the founders of the Garman‐Kohlhagen model. The Garman-Kohlhagen Formula is a variant on the Black-Scholes option pricing formula, applied to find the prices of currency options.

This model alleviates the restrictive assumption used in the Black-Scholes model that borrowing and lending is performed at the same risk free rate. In the foreign exchange market there is no reason that the risk free rate should be identical in each country. This model can cope with the presence of two risk-free rates.

The formula goes as follows:

where,

r_{d} is the domestic risk-free rate

r_{f} is the foreign risk-free rate

The risk-free foreign interest rate, in this case, can be thought of as a continuous dividend yield being paid on the foreign currency. Since an option holder does not receive any cash flows paid from the underlying instrument, this should be reflected in a lower option price in the case of a call or a higher price in the case of a put.

The Garman Kohlhagen model provides a solution by subtracting the present value of the continuous cash flow from the price of the underlying instrument. The model has the same assumptions and limitations as the Black -Scholes model.

Today every financial company, fund managers use Black-Scholes formula on computers for derivative pricing.

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]]>The post Random Variable and Distribution: The Concept appeared first on StepUp Analytics.

]]>In this article, first, we’ll discuss the properties of a random variable, then types of random variables along with their probability functions, in brief.

So, the properties of a random variable are:

- It only takes the real value.
- If X is a random variable and C is a constant, then CX is also a random variable.
- If X
_{1}and X_{2}are two random variables, then X_{1}+ X_{2}and X_{1}X_{2}are also random. - For any constants C
_{1}and C_{2}, C_{1}X_{1}+ C_{2}X_{2}is also random. - |X| is a random variable.

**Note**: We use a capital letter, example X, to stand for the random variable and its equivalent lower case, example x, to stand for a value that it takes.

Now, there are two types of random variables.

- Discrete Random Variable
- Continuous Random Variable

Let’s discuss about them in brief.

**Discrete Random Variable**

A random variable that can only take certain numerical values (i.e. discrete values) is called a discrete random variable. For example, the number of applicants for a job or the number of accident-free days in one month at a factory.

The function f_{x}(x) = P(X=x) for each x in the range of X is the **probability function (PF)** of X – it specifies how the total probability of 1 is divided up amongst the possible values of X and so gives the probability distribution of X. Probability functions are also known as ‘pdf’.

Note the requirements for a function to qualify as the probability function of a discrete random variable, for all x within the range of X:

Other than the probability function, the **cumulative distribution function (CDF)** of X is also very important. It is given by:

for all real values of X, gives the probability that X assumes a value that doesn’t exceed x.

The graph of F_{x}(x) against x starts at a height of 0 then increases by jumps as values of x are reached for which P(X=x) is positive. Once all possible values are included F_{x}(x) takes its maximum value of 1. F_{x}(x) is called a step function.

Let’s understand the above concepts more clearly with the help of an example. Suppose we roll a fair die, then it’s probability distribution would be:

From this table, it can be shown that the sum of all the probabilities is 1. Also, technically we should write the CDF as:

Now, let’s understand what continuous random variable is.

**Continuous Random Variable**

A random variable that can take any numerical value within a given range is called a continuous random variable. For example, the temperature of a cup of coffee served at a restaurant or the weight of refuse on a truck arriving at a landfill.

The probability associated with an interval of values, (a, b) say, is represented as P(a<x<b) or P(a ≤x ≤b) – these have the same values – and is the area under the curve of the **probability density function (pdf)** from a to b. So probabilities can be evaluated by integrating the pdf, f_{x}(x). Thus,

The conditions for a function to serve as pdf are as follows:

for -∞≤x≤∞.

You should have noticed that these conditions are equivalent to those of the probability function for a discrete random variable, where the summation is replaced by integration for the continuous case.

The **cumulative distribution function (CDF)** is defined to be the function:

For a continuous random variable, F_{x}(x) is a continuous, non-decreasing function, defined for all real values of x.

The graph of F(x) can be shown as:

Now we can work out the CDF from pdf. But how can we get back again?

Well, we integrate the pdf, f(x), to get the CDF, F(x), so it makes sense that to go back we differentiate.

We can obtain the pdf, f(x), from the CDF, F(x), as follows:

Like a discrete random variable, a continuous random variable can also be understood more clearly with the help of an example.

Suppose we have the following probability density function:

It can be seen that at x=1, f_{X}(x)=3/7 and at x=2, f_{X}(x)=12/7. Therefore, the given pdf is greater than 0 in the interval [1, 2]. Also, if we integrate the pdf, then we get,

Now, cumulative distribution function can be found as:

Hence, we should write the CDF as:

Now, we should know how to obtain expected values for both, discrete as well as continuous random variables.

**Expected values** are numerical summaries of important characteristics of the distributions of random variables. So, let’s see how mean and standard deviation of a random variable are obtained.

**Mean****E[X]** is a measure of the average/center/ location of the distribution of X. It is called the mean of the distribution of X, or just the mean of X, and is usually denoted by μ.

**E[X]** is calculated by summing (discrete case) or integrating (continuous case) the product:

**value x probability of assuming that value**

over all values which X can assume.

Thus, for the discrete case:

and, for the continuous case:

**Variance and Standard Deviation**

The variance, σ^{2} is a measure of the spread/ dispersion/ variability of the distribution. Specifically, it is a measure of the spread of the distribution about its mean.

Formally,

is the expected value (or mean) of the squared deviation of X from its mean. The standard deviation, σ, is the positive square root of this – hence the term sometimes used “root mean squared deviation”.

Simplifying:

If we take our above example of a discrete random variable, it can be shown how mean and variance are calculated. Let’s calculate these values to understand the calculation of expected values more clearly.

∴ Mean = E[X] = 21/6 = 7/2

and Variance = E[X^{2}] – {E[X]}^{2} = 91/6 – (7/2)^{2} = 35/12

Now, there are some linear functions of X also. Consider changing the origin and the scale of X.

Let Y = aX + b. Let E[X] = μ.

E[Y] = E[aX + b] = aμ + b

So Y – E[Y] = aX + b – [aμ + b] = a[X – μ].

These are important results. The results for the expected value can be thought of simply as “whatever you multiply the random variable by or add to it, you do the same to the mean”. However, the addition of a constant to a random variable does not alter the variance.

This should make sense since the variance is a measure of spread and the spread is not altered when the same constant is added to all values. When you multiply the random variable by a constant you multiply the standard deviation by the same value, so the variance is multiplied by that constant squared.

Now, let’s see what a probability distribution is. The **probability distribution** for a random variable describes how the probabilities are distributed over the values of the random variable. There are two types of probability distributions, discrete and continuous. For a discrete random variable, say X, the probability distribution is defined by a probability mass function.

Whereas for a continuous random variable, say Y, the probability distribution is defined by a probability density function, as discussed above in discrete and continuous random variable heading. There are some special probability distributions.

Two of the most widely used discrete probability distributions are the binomial and Poisson distribution.

First is **Binomial Distribution**. The probability mass function of binomial distribution provides the probability that ‘x’ successes will occur in ‘n’ trials of a binomial experiment. If X ~ Bin(n, p), then

for x = 0, 1, 2, …, n and 0<p<1.

Here, there are two outcomes, success or failure, which are possible on each trial and ‘p’ denotes success on any trial. The trials are independent.

Suppose, X ~ Bin(5, 0.7), then

Second is **Poisson Distribution**. The Poisson distribution is often used as a model of the number of arrivals at a facility within a given period of time. If X ~ Poi(μ), then

for x = 0, 1, 2, … and μ>0.

Here, the parameter ‘μ’ is the mean of a random variable ‘x’.

Suppose, the mean number of calls arriving in a 15 minutes period is 10. Then, to compute the probability that 5 calls arrive within the next 15 minutes period, we have μ = 10 and x = 5, then

The most widely used continuous probability distribution is the normal distribution. The graph of the normal distribution is a bell-shaped curve. The probabilities for the normal distribution can be computed using statistical tables for the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

The normal distribution depends on two parameters, μ, and σ^{2}, which is mean and variance of a random variable. The probability density function of the normal distribution, if X ~ N(μ, σ^{2}), is given by:

for -∞<x<∞.

Suppose X ~ N(μ, σ^{2}), then to convert the random variable X from normal probability distribution to standard normal distribution, we have:

Z is another random variable, which follows a standard normal distribution with mean zero and variance 1.

Suppose X ~ N(50, 3^{2}), then

There are many other discrete and continuous probability distributions, other than the three discussed above. Other discrete probability distributions include uniform distribution, Bernoulli distribution, geometric distribution, hypergeometric distribution, and negative binomial distribution; other commonly used continuous probability distributions include uniform, gamma, exponential, chi-square, beta, log-normal, t and F distribution.

So, to summarise, a random variable is a set of possible values from a random experiment which uses probabilities to decide its value. Then, there are two types of random variables, discrete and continuous; and both have two basic requirements, i.e., the value of probability function should lie between 0 and 1 and sum of all the values of probability should be 1. Next, we have seen how to obtain mean and variance of a random variable followed by some important discrete and continuous probability distributions.

I hope that I am able to tell you about the concept of a random variable in brief.

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]]>The post Actuarial Science: Loan Schedule appeared first on StepUp Analytics.

]]>Have you ever thought of why you pay interest against the loan? How the interest paid on loan is calculated? How the regular payment paid, every year or month, is calculated?

To answer the first question, when people lend money, they can no longer use this money to fund their own purchases. The payment of interest makes up for this inconvenience. This is also known as the time value of money. Also, a borrower may default on the loan.

Before discussing how interest, capital repaid and payments made are calculated, let’s talk about interest.

Interest may be regarded as a reward paid by one person or organization (the borrower) for the use of an asset, referred to as capital, belonging to another person or organization (the lender).

Interest is of two types:

- Simple Interest: The essential feature of simple interest is that interest, once credited to an account, does not itself earn further interest. Suppose an amount C is deposited in an account that pays simple interest at the rate of (i
*X*100%) per annum. Then after n years, the deposit will have accumulated to:

- Compound Interest: The essential feature of compound interest is that interest itself earns interest. Suppose an amount C is deposited in an account that pays compound interest at the rate of (i
*X*100%) per annum. Then after n years, the deposit will have accumulated to:

A very common transaction involving compound interest is a loan that is repaid by regular installments, at a fixed rate of interest, for a predetermined term. Loans are mostly used by companies or individuals to raise funds, usually to buy buildings or equipment.

Now, let’s see from an example of how interest and regular payments are calculated. Assume a bank lends an individual ₹200000 for 3 years, in return for regular payments at the end of each month. The bank will charge an effective rate of interest of 7% per annum.

Let X be the monthly payment, then the equation of value for the transaction is given by:

So, by using annuities in arrear, we get a monthly payment of ₹6155.78. By using the PMT function in excel, we can find monthly payments. For that, we have to input some values like loan amount, annual interest rate, loan period in years and number of payments per year. For our example, this can be done as follows:

Here, monthly payment is named as “Instalment”. So, by using the PMT function, we get:

You can see that there is some difference in the installment by using annuity function and PMT function in excel.

One important point to note is that each repayment must pay first for interest due on the outstanding capital. The balance is then used to repay some of the capital outstanding. Each payment, therefore, comprises of both interest and capital repayment. It may be necessary to identify the separate elements of the payment.

**∴ Installment = Interest + Capital re-paid**

This can be shown in the form of a table, which is often known as “loan schedules”, having columns:

- Payment Number or Instalment Number
- The loan outstanding in the beginning
- Scheduled payment (which is our monthly installment)
- Interest due
- Capital repaid
- The loan outstanding in the end

Interest due is calculated by using the nominal interest rate. In our example, interest due in the first year would be calculated as:

and capital repaid is calculated as:

**Capital repaid=Installment-Interest due**

Also, loan outstanding is calculated as:

**Loan outstanding in end=Loan outstanding in beginning-Capital repaid**

So, in excel, the first row of payment will look like this:

Here, scheduled payment is our fixed monthly installment, which will be paid at the end of the first month and this will remain fixed till our last payment. Interest due, capital repair, and loan outstanding in the end are calculated by using the above formulae.

Now, while calculating the next row of loan schedule, our loan outstanding in the end will become loan outstanding in the beginning and the rest of the columns are calculated as before. So, our full loan schedule will look like this:

Here you can see that at the end of the third year, i.e., after paying all the 36 installments, our loan outstanding in the end will become zero, since we have paid all the installments due and the full amount of the loan taken, under the fixed period of 3 years.

Notice the pattern of interest due and capital repayment. Interest due is decreasing and capital repaid is increasing during the term of the loan. This happens because as we pay our capital, the loan outstanding decreases, decreasing the value of interest due and increasing the value of capital repaid. You can also see this from a chart for interest due and capital repaid.

So, till now, we have discussed the schedule where we have fixed installments. What happens when we pay extra money in between the period as extra capital against our loan?

Now, we’ll have one extra column of “Extra payment” in our loan schedule. Formulae will remain the same as before, except for capital repair and loan outstanding in the end. New formula for capital repair and loan outstanding in the end will be:

**Capital Repaid=Installment-Interest due+Extra payment**

For calculating the **loan outstanding in the end**, we’ll use IF function in excel as follows:

Now, if we pay ₹2000 extra capital at the end of year 1 and year 3, and ₹1500 at the end of year 2 and apply the above formulae. Then, the loan schedule will look like this:

If we don’t use IF function in calculating the loan outstanding in the end, then we’ll get negative values.

So, we have seen how to make loan schedule in excel and how to consider the extra payment made during the term of the loan.

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]]>The post Actuarial Science: CT8 vs CM2 – A Comparison appeared first on StepUp Analytics.

]]>With the same impression in mind, the IFoA has made a drastic change in its curriculum- Making the students’ industry ready at the time when they are preparing for their exams. The institute has introduced the practical application of the theoretically learned concepts.

While we have already welcomed the 2019 curriculum there is still a lot of skepticism regarding the changes in the course structure, time to be devoted for the preparation and the most important of all- the online test.

Let me take this opportunity to clear your doubts on the subject CT8. CT8 which was named as Financial Economics has now been replaced with CM2 – Loss Reserving and Financial Engineering.

**Aim of the subject**

The aim remains the same i.e. to provide a grounding in the principles of modeling as applied to actuarial work – focusing particularly on stochastic asset liability models and the valuation of financial derivatives.

**Exam Structure**

The examination pattern has changed. Along with the conventional pen-paper test of 3 hours and 15 minutes (CM2A) one also has to appear for an online test of 1 hour and 45 minutes (CM2B). For CM2B one has to learn Advance Excel.

**Fee Structure**

The standard examination fee for CT8 was £225 and for CM2 is £280. However, the reduced fee for CM2 is £165, while it was £120 for CT papers.

**Course**

Under the revised curriculum, topics from CT6 such as Ruin Theory and Run Off Triangles are added. The weight of each unit is given below:

1. Theories of financial market behavior (15%).

2. Measures of investment risk (15%).

3. Stochastic investment return models (10%).

4. Asset valuations (20%).

5. Liability valuations (20%).

6. Option theory (20%).

**Study Hours**

The recommended study hours for CT8 were 125-150 hours. The number of study hours for CM2 has increased to 250 hours, due to the computer-based exam and inclusion of a few new topics.

For the computer-based exam, it is advisable to learn the shortcuts and practice a lot so that one is handy in using the software during the exam.

**Evaluation Pattern**

There is a complete change in the method of evaluation. Earlier CT8 was a 100 marks theoretical exam. Now CM2 will be evaluated at 2 levels: a theory paper (CM2A) of 100 marks and a computer-based exam (CM2B) of 100 marks to be done using Excel. CM2A will be weighted at 70% and CM2B at 30%.

If your combined score is above the passing mark for that diet, you’ll clear the exam, otherwise, you will have to reappear for both the papers.

As rightly said by Charles Darwin,” It is not the strongest or the most intelligent who will survive but those who can best manage change”.

Hence, get ready, embrace the change, prepare heartedly, put all your efforts and ace your exams.

Happy Studying.

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]]>The post Actuarial Science: Unit Linked Contracts appeared first on StepUp Analytics.

]]>*What if I tell you I have a
product that invests your premium in stock market?*

*You will probably think the stock market involves risk, you might end with very less and insufficient sum during the claim and so the product is not useful. But, what if I tell you I have a product that gives you the benefit of investment in securities(chosen by you), protection of your Sum Assured and insurance cover? Would it intrigue you?*

*Here is the deal, I put a unit-linked contract on the table.**Confused? **Well, *

*Let me break-down unit-linked contracts ,for you , in the article below,*

Firstly, **insurance contracts** are simply contracts between two parties wherein one party transfers its risk to another party for a price. For general people, it is a way of removing their exposure to risk. People are risk-averse and are ready to pay some amount to transfer their risk, because of which insurance companies exist.

Insurance companies take a bunch of uncorrelated risks together which reduces the probability of risk for them due to which they make a profit and continue the business. Insurance can be Life and Non-Life (health, motor, etc.).

Life Insurance companies offer various types of policies like whole life cover, endowment benefit, and term assurance cover. In these policies, the policyholder (insured) pays a premium and receives a death benefit on death during the term of the policy or maturity benefit on surviving the term of the policy, whichever applicable.

**Unit Linked Insurance Plan** is a life insurance product, which provides risk cover for investments such as stocks, bonds or mutual funds. In this policy, the insured does not receive a fixed amount when the benefit is payable .The premium that the insured pays are invested in an investment fund chosen by the policyholder. The investment fund is divided into units which are priced continuously.

The value at the date of death or survival (i.e. At the time of claim) of the cumulative number of units purchased is the sum assured under the contract. A minimum guaranteed sum assured is specified in the terms of the contract to ensure that the policyholder avoids any difficulties arising from particularly poor investment performance. The insurance company holds reserves to meet the claims timely and ensure its survival.

**UNIT LINKED CONTRACTS= LIFE INSURANCE COVER + INVESTMENT BENEFIT + SAVINGS POLICY**

**Investment Options available for policyholders:**

ULIP offers investors the option to invest in equity and debt. Investors can choose their investment fund as complete equity, complete debt or balanced (50% each), according to their risk appetite. An aggressive investor can pick equity oriented fund option whereas a conservative one can go with debt option.

**Benefits of having a ULIP **

You get market-linked returns as the sum assured. While traditional insurance plans offer 4% to 6% returns, ULIP can offer you double-digit returns if you are invested in equity funds, says Deepak Yohannan, CEO of myinsuranceclub.com

Unit-linked plans offer the twin benefits of life insurance and savings at market-linked returns. Thus, you have the opportunity to invest your money to earn higher returns, while taking care of your protection needs.

Unit Linked Plans offer you a wide range of flexible options such as the option to switch between investment funds to match your changing needs, the facility to partially withdraw from your fund, subject to charges and conditions.

**Structure Of The Contract**

**Allocation percentage:** A certain proportion of premium, taken from the policyholder, is deducted for the company’s expenses. The remaining premium is invested to purchase units.

**Bid-Offer spread: ** The policyholder buys units with his premium at the offer price and at maturity the company buys those units back from the policyholder at the bid price. The bid price is lower than the offer price. This bid-offer spread goes to the company’s fund.

**Charges:** The company charges certain expenses to cover its administration cost like policy fee, fund management charges and other charges to allow for the withdrawal of policy or switching investment funds during the term of the contract.

The unallocated premium amount, bid-offer spread and other charges are recorded in the company’s non-unit fund which is maintained for tracking its profit on policies. A unit fund is maintained for the policyholder. The heads in the respective funds can be understood very well through the diagram.

**Unit
linked policies are relatively complicated compared to traditional insurance
products. **It is difficult to use the traditional
actuarial approach for evaluating the premiums and the reserves of these
contracts. Instead, profit testing is a plausible and popular approach for
evaluating premiums and the reserves for it.

**Let’s understand profit-testing with an example,**

In the question, mortality rates are taken from a tabulated AM92 ULTIMATE table, tabulated for UK and interest rate for discounting is assumed to be 6%. Indian mortality rates are tabulated in CMI Tables (Continous Mortality Investigation). Every insurance company has a record of age-wise mortality rates and interest rate is chosen by them according to their policy.

We will perform the operation on excel, it can be done on other software also like R programming. The formula to be used in the calculation is as follows:

**For unit fund**

Premium allocation= premium received * allocation percentage

Fund at start= fund value at end

Interest = (fund at start + premium allocated – B/O spread)*0.05

Management charge = 0.01*(fund at start+ premium allocated-B/O spread +interest)

Fund value at end = fund at start+ premium allocated- b/o spread + interest – management charge

**For Non-unit fund **

Interest = (unallocated premium + B/O spread – expenses)*0.03

Extra death cost = max (12000- fund value at end) * age mortality

Extra maturity cost = fund value at end of 3^{rd} year *(1-mortality at age 22)

End of year cf= unallocated premium + B/O spread – expenses+interest – extra death cost – extra maturity cost + management charge

The sum assured is maximum of 12000 and the fund value at time of claim.

In the non-unit fund, the end of year cash flow is the profit per policy for the company. The expenses of the company are administration cost and death benefit and maturity benefit also pose some cost to the company. The cost of maturity to the insurance company is just 10% of the fund value at the end of year 3.

We create a profit vector with profit values.

**Profit vector** = (341.79, 771.76, -799.95)

**Profit signature** = profit vector multiplied with the probability of surviving the year

**Profit margin** = (present value of profit signature)/ present value of premiums received

Now, companies decide a profit margin for a set of policies. The profit margin is the profit % the company expects to earn from a policy to continue surviving. Once a profit margin is chosen, companies vary the premiums. Certain assumptions are made during the test. The company runs tests varying their assumptions and profit margin and reaches an appropriate premium level.

This sums up the premium calculation process and working of unit-linked contracts. ULIP is an innovative product that strikes a balance between investment and insurance and balance is determined by personal goals and risk factors. There can be innovations in ULIP, few instances are:

- a regular premium product offering a guaranteed sum assured on death, to give an endowment assurance;
- a single premium product offering a return of premium on death to give a pure savings bond;
- a regular premium contract offering an annuity payment during periods of disability or unemployment; or,
- a single premium product offering an annuity payment until death.

Download the Excel Used above

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