The post Project Appraisal: Money-Weighted Rate of Return, Time-Weighted Rate of Return and Linked Rate of Return appeared first on StepUp Analytics.

]]>On a lighter note, appraisal is an act of assessing something or someone. Generally, we come across this word with regards to employees’ performance appraisal at workplace.

**What can be the main motive behind such appraisals?**

The answer is simple- Appraisals demonstrate the need for improvement.

Similarly, **assessing the viability or feasibility of a proposed project by the
lending institutions is called** **project
appraisal**. This is done to know the effect of each project for the company.
This means that **project appraisal is
done to know, how much the company has invested on the project and in** **return how much it is gaining from it.** It
is a tool that company’s use for choosing the best project that would help them
to attain their goal.

This leads us to the introduction of various measures of returns used for project appraisal-

- Money Weighted Rate of Return (MWRR)
- Time Weighted Rate of Return (TWRR)
- Linked Internal Rate of Return (LIRR).

**Money Weighted Rate of Return**

MWRR is a method to calculate the rate of return of a portfolio. It takes into consideration the impact of contributions to (inflow of cashflows), or withdrawals from (outflow of cashflow) the portfolio. It is mainly used to compute individual portfolio returns as timing and amount of contributions and withdrawals can be different for each individual investors’ portfolio.

A money-weighted rate of return is the discount rate at which the

Net present value =0,

or The present value of inflows= present value of outflows.

Money Weighted Rate of Return incorporates the size and timings of cashflows.

Outflows:

- The cost of investment purchased
- Reinvested dividends or interest
- Withdrawals

Inflows:

- The proceeds from any investment sold
- Dividends or interest received
- Contribution

Each inflow or outflow must be discounted back to the present using a rate (r) that will make **PV (inflows) = PV (outflows).**

For example, take a case where we buy one share of a stock for Rs.50 that pays an annual Rs.2 dividend, and sell it after two years for Rs.65. Our money-weighted rate of return will be a rate that satisfies the following equation:

Let’s have a look at the associated cashflows:

PV Inflows = Rs.2/ (1 + r) + Rs.2/ [(1 + r) ^2] + Rs.65/ [(1 + r) ^2].

PV Outflows = Rs.50.**Using PV Inflows = PV Outflows**

Solving for r using

**Time Weighted Rate of Return**

TWRR measures a funds’ compounded rate of growth over a specific time period. While TWRR measures the return of a funds’ investments, it does not consider the effect of investor cash moving in and out of a fund. Thus, TWRR is suitable for measuring the performance of marketable investment managers because they do not control when investor cash enters or exits their funds.

To get a better understanding of TWRR let’s have a look at the associated cashflows:

Investor A invests $1 million into Mutual Fund A on December 31. On August 15 of the following year, his portfolio is valued at $1,162,484. At that point (August 15), he adds $100,000 to Mutual Fund A, bringing the total value to $1,262,484.

By the end of the year, the portfolio has decreased in value to $1,192,328. The holding-period return for the first period, from December 31 to August 15, would be calculated as:

**Return = ($1,162,484 – $1,000,000) / $1,000,000 = 16.25%**

The holding-period return for the second period, from August 15 to December 31, would be calculated as:

**Return = ($1,192,328 – ($1,162,484 + $100,000)) / ($1,162,484 + $100,000) = -5.56%**

The second sub-period is created following the $100,000 deposit so that the rate of return is calculated reflecting that deposit with its new starting balance of $1,262,484 or ($1,162,484 + $100,000).

The time-weighted return for the two time periods is calculated by multiplying each sub period’s rate of return by each other. The first period is the period leading up to the deposit, and the second period is after the $100,000 deposit.

**Time-weighted return = [(1 + 16.25%) x (1 + (-5.56%))] – 1 = 9.79%**

But the point is that both the methods have disadvantages: TWRR requires fund values at all cashflow dates. MWRR may not have a unique solution and fund manager performance cannot be judged. If the fund performance is reasonably stable in the period of assessment, the TWRR and MWRR may give similar results. Then there comes Linked Internal Rate of Return (LIRR).

**Linked Internal Rate of Return **

The main drawback of TWRR is that it requires constant account valuations every time an external cash flow occurs. Money-weighted returns are simpler in that the portfolio only needs to be valued at the beginning and end of the period. The linked internal rate of return (LIRR) was developed to combine TWRR’s immunity to cash flows with MWRR’s ease of calculation.

LIRR attempts to approximate time-weighted returns by chain-linking money-weighted returns over reasonable time intervals. For example, if we calculate the money-weighted return every week in our month evaluation period, we could then just combine them for our month time-weighted return. This turns out to be a very accurate approximation of time-weighted returns, without valuing the portfolio at every cash flow.

Suppose the tenure of a fund is 3 years. We calculate return after every 1 year.

**So the equation is – (1+r1)(1+r2)(1+r3)= [(1+i)^3] **

where r1 represents ^{nd} year, r3 after 3^{rd} year and “i” represents the LIRR. The rate of return over each different sub-period is weighted according to the duration of the sub-period.

Therefore, it can be said that performance measurement is a vital part of overall performance evaluation as it answers the question of how much money we made in a period. I would like to conclude my article by the words spoken by a famous personality named Peter Drucker and I quote-“If you can’t measure it, you can’t manage it!” and that’s what project appraisal is all about.

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]]>The post Actuarial Science: Cyber Insurance a Challenge for Actuaries appeared first on StepUp Analytics.

]]>As part of a risk management plan, organizations routinely must decide which risks to avoid, accept, control or transfer. Transferring risk is where cyber insurance comes into play.

**What is cyber insurance?**

A cyber insurance policy also referred to as cyber risk insurance or cyber liability insurance coverage (CLIC), is designed to help an organization mitigate risk exposure by offsetting costs involved with recovery after a cyber-related security breach or similar event. According to PwC, about one-third of U.S. companies currently purchase some type of cyber insurance.

**The
main issues related to cyber insurance can be summarized as follows: **

• Evolution of information system: The system of an organisation may easily change and new technologies appear, changing the landscape of cyber risks;

• Information asymmetry: There are many obstacles for an insurer to get reliable information about the risk exposure of an insured and it is difficult to know if this exposure will be maintained during the whole period of policy operation;

• Evolution of attacks: It is very hard to determine the rate of occurrences and, as a consequence, the assessment of risk exposure;

• Interdependence of security: Security level of an information system may depend on the security of others;

• Impact determination: Damage for cyber risks is very hard to estimate in advance because of the intangible nature of information assets. Moreover reputation cost, which accounts for a large portion of the whole damage, is very difficult to estimate;

• Lack of statistical data: Data lie at the center of any actuarial project, but data are very limited in this field. Companies often do not want to reveal breaches, since they cause secondary damage, e.g. to reputation.

**Challenges for cyber risk management **

- Continuous change and digitalization of traditional business models − For example, increased vulnerability of information privacy (e.g., purchase of insurance via online platform)

- Knowledge and data deficits:
- Asset valuation in terms of identifying valuable assets
- Identification and estimation of threats as well as possible losses
- Risk culture is crucial (lack of awareness for cyber risk)

**Actuarial Challenges**

When it comes to determining risk-adequate pricing for cyber insurance contracts, there are many challenges that make it difficult to apply standard actuarial techniques. Actuaries also don’t have experience dealing with digital security incidents, which makes assigning dollar values to any available bits of data even more valuable.

For instance, actuaries aren’t knowledgeable about white hat and black hat hackers, so it would be difficult for them to predict loss propensity or measure cyber risk for corporate networks that oftentimes extend across national borders, grant partner companies some level of access, and consist of technology that’s always changing.

According to a report, it is thought that a good starting point is to determine the costs or expenses the company needs covering and the types of incidents that cyber insurance wants

Instead of general insurance able to cover all cyber attacks, considering the peculiarity and the repercussions behind different attacks, it is thought that each kind of threat can be managed by different insurance policies and furthermore that different companies can exhibit a different risk between these kinds of threats.

Read all the latest articles on Actuarial Science New Curriculum **Click**

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]]>The post Should I Pursue an Actuarial Science Degree or Statistics Degree? appeared first on StepUp Analytics.

]]>The key skills required by both the courses are:

- Strong foundation in Mathematics
- Logical thinking and the ability to comprehend key facts
- Ability to interact with people from various fields to understand problems
- Versatility in problem-solving
- A computer geek

Even though both the courses require the above skills, there are certain lines drawn between them. So, let’s have a look at the differences between these two courses.

**Minimum Eligibility**

Basically, students from Science background having Physics, Chemistry and Mathematics can pursue a Statistics degree, with a minimum of 50% marks (in aggregate) in high school examination. Students from Commerce or Arts background can also pursue a degree in Statistics, but that will depend on the institute providing graduation in Statistics.

Whereas, the minimum eligibility for Actuarial Science degree is qualifying a high school examination with minimum of 60% in aggregate and minimum 70% in Mathematics.

**Study Requirements**

To become an Actuary, one should be good at Mathematics, Statistics and possess good modelling skills. While pursuing a Statistics degree, you’ll learn about programming, modelling and will acquire good analytical skills. Also, you’ll learn to work with large data sets and use statistical tools in analysing data. So, if you pursue Actuarial Science with a Statistics degree, you’ll get a lot of help as you’ll be having a good modelling and analytical skills, which are some of the requirements of Actuarial Science.

Whereas, during Actuarial Science degree, you’ll learn basic concepts of Actuarial Science – economics, financial mathematics, risk management and analysis and many more along with some of the statistical concepts. Thus, in Statistics degree, you’ll learn each topic of Statistics in deep as compared to Actuarial Science degree.

**Time of study**

If you are very clear about seeing yourself as an Actuary for the rest of your life, then you should go for an Actuarial Science degree because it helps you in clearing papers. Since it covers the concepts taught in the initial actuarial examinations and mainly focuses on Actuarial Science throughout your graduation, the time taken to become an actuary reduces. If you are pursuing Actuarial Science from India, you’ll not get any exemption from the Institute of Actuaries of India, until and unless you pursue a master’s degree like M.Stat. or M.Sc. in Statistics from Indian Statistical Institute.

Top institutes offering Statistics degree are:

- Indian Statistical Institute (ISI), Kolkata
- Loyola College, Chennai
- St. Xavier’s College, Mumbai
- Christ University, Bangalore

Top institutes offering Actuarial Science in are:

- Amity School of Insurance and Actuarial Science (ASIAS), Noida
- Bishop Heber College, Tiruchirappalli
- Christ University, Bengaluru
- International School of Actuarial Sciences (ISAS), Hyderabad

**Career Prospects**

Career prospects are ever increasing in both the courses. The prospects of Actuarial Science are majorly in the Insurance sector in India, followed by consultancies. Being a graduate in Statistics, if in future, you think of switching from Actuarial Science, then you can work in industries like agriculture, computer science, health science, automobile, computer software companies and many more. In fact, one can also opt for finance, analytics and software development, to name a few.

Actuarial Science is primarily an applied discipline that combines Math, Statistics and Data Analysis, whereas degree in Statistics tend to focus more on the “pure” theory of the subjects. So to summarise, both Actuarial Science and Statistics degree provides similar skills sets and the use of statistical techniques. They usually differ in the scope of their work. By obtaining graduation in Actuarial Science, you will get to work specifically within insurance industry and handle data related to risk.

Whereas with a Statistics degree, you are not only restricted to the actuarial work, instead you can work as a statistician, business analyst, professor, data analyst, data scientist, consultant, risk analyst and many more. Also, if you pursue a Statistics degree, then you have to study actuarial subjects on your own. Whereas during your graduation in Actuarial Science, you get pertinent training for qualifying actuarial examinations.

Read all the latest articles on Actuarial Science New Curriculum **Click**

Your future depends on your choice, so choose wisely!

Good Luck!

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]]>The post Actuarial Science: Capital Asset Pricing Model appeared first on StepUp Analytics.

]]>**CAPM **is the simplest form of an equilibrium model. The CAPM model is a straight line plot of the excess returns of the stock compared to the excess returns of the benchmark. The capital asset pricing model tells us about the relationship between risk and returns in the security market as a whole, assuming that investors act in accordance with mean-variance portfolio theory and that the market is in equilibrium. Mean-variance portfolio theory, sometimes called modern portfolio theory (MPT), specifies a method for an investor to construct a portfolio that gives the maximum return for a specified risk or the minimum risk for a specified return. It uses just the expected return and variance of the asset.

The standard form of capital asset pricing model was developed by Sharpe, Lintner, and Mossin, hence it is often referred to as the Sharpe-Lintner-Mossin form of CAPM.

**Assumptions of the Model** The real world is sufficiently complex that to understand it and construct models of how it works, one must assume away those complexities that are thought to have an only small effect on its behavior As the physicist builds models of the movement of matter in a frictionless environment, the economist builds models where there are no institutional frictions to the movement of stock prices.

- All investors have the same one-period horizon.
- All investors can borrow or lend unlimited amounts at the same risk-free rate.
- The markets for risky assets are perfect. Information is freely and instantly available to all investors and no investor believes that they can affect the price of a security by their own actions.
- Investors have the same estimates of the expected returns, standard deviations and covariance of securities over the one-period horizon.
- All investors measure in the same ‘currency’ e.g. pounds or dollars or in ‘real’ or ‘money’ terms.
- All assets are marketable.

There are a few assumptions that underlie many financial models like mean-variance portfolio theory, they are:

- all expected returns, variances and covariance of pairs of assets are known
- investors make their decisions purely on the basis of expected return and variance
- investors are non-satiated
- investors are risk-averse
- there is a fixed single-step time period
- there are no taxes or transaction costs
- assets may be held in any amounts, (with short-selling, infinitely divisible holdings, no maximum investment limits)

The straight line denoting the new efficient frontier is called the capital market line. Its equation is:

Where,

All investors
will end up with portfolios somewhere along the capital market line, and all ** efficient portfolios**would
lie along the capital market line. However, not all securities or portfolios
lie along the capital market line.

**(Expected return) = (Price of time) + (Price of risk) * (Amount of risk)**

**The market price of risk:** It is the extra return that can be gained by increasing the level of risk (standard deviation) on an efficient portfolio by one unit. It is calculated using the formula,

It is also possible to develop an equation relating the expected return on an asset to the return on the market:

Where,

Risk of any stock can be divided into systematic and unsystematic risk. Beta is the index of systematic risk. Security Market Line validates that systematic risk is the only important ingredient in determining expected returns and that non-systematic risk plays no role.

SML states that the expected return on any security is the riskless rate of interest plus the market price of risk times the amount of risk in the security of the portfolio.

Most of the assumptions underlying the CAPM violate conditions in the real world. Although the CAPM may describe equilibrium returns on the macro level, it certainly is not descriptive of micro (individual investor) behavior. For example, most individuals and many institutions hold portfolios of risky assets that do not resemble the market portfolio. the CAPM assumes several real-world influences away, it does not provide us with a mechanism for studying the impact of those influences on capital market equilibrium or on individual decision making.

Only by recognizing the presence of these influences can their impact be investigated. For example, if we assume personal taxes do not exist, there is no way the equilibrium model can be used to study the effects of taxes. All models are wrong but that doesn’t mean they cannot serve any purpose.

Under the assumptions of the CAPM, we say the only portfolio of risky assets that any investor will own is the market portfolio. Market portfolio is a portfolio in which the fraction invested in any asset is equal to the market value of that asset divided by the market value of all risky assets.

The impact of a security on the risk of the market portfolio is given by

SML clearly shows that return is a linearly increasing function of risk. Furthermore, it is the only market risk that affects return. The investor receives no added return for bearing diversifiable risk.

Even if the standard CAPM model explains the behavior of security returns, it obviously does not explain the behavior of individual investors.

The standard form of CAPM can be modified to incorporate more realistic assumptions about each of the following influences:

- Short sales disallowed: the same CAPM relationship is derived here.
- Riskless lending and borrowing: It seems much more realistic to assume that investors can lend unlimited sums of money at the riskless rate but cannot borrow at a riskless rate.
- Personal taxes: The general equilibrium pricing equation for all assets and portfolios, given differential taxes on income and capital gains, is

- Nonmarketable assets:

- Heterogeneous expectations
- Non-price-taking behavior
- Multi-Period Models
- Multi-Beta CAPM

- Inflation model: It is simplest form of multi-Beta model.

- Zero beta CAPM

We have modified one assumption at a time. The derivations and explanations of these models have been detailed in the reference book mentioned below, for curious readers.

Despite having shortfalls, CAPM is widely used to price assets. If the beta of a stock can be estimated from the past data, then the model can be used to find the prospective return that the asset should offer. This return can then be used to discount projected future cash flows and so price the security and determine if it appears to be under-valued or over-valued.

References: Modern portfolio theory and investment analysis by Elton and Gruber

Data Used in the above example can be downloaded from here

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]]>The post Careers in Chartered Accountancy Vs Actuarial Science appeared first on StepUp Analytics.

]]>**What do you want to become when you grow up? **

The most common question asked to almost every child. Career is one of the most important decisions in an individual’s life and it must be taken by accounting various factors like interest, opportunities available, feasibility etc.

In this

**Who are they?**

An **Actuary **is a business professional who deals with the measurement and management of risk and uncertainty. These are true professional statisticians and the ultimate quants that use past data to predict likely future outcomes. Such skills make them exceedingly useful for insurance companies or other businesses that sell insurable products.

An **accountant **is a practitioner of
accounting or accountancy, which is the measurement, disclosure or provision of
assurance about financial information that helps managers, investors, tax
authorities and others make decisions about allocating resource. Accountants
service individuals, businesses and governments to help ensure that everything
is running efficiently and accurately.

As far as the quality of work is concerned, both Actuary and chartered accountant are considered to be

**How can you become**

The Actuarial Science course requires one to pass a series of 13 professional examinations in order to be fully certified. The thirteen papers are divided into 4 groups:

**Core Principles:** 7 compulsory papers**Core Practices:** any 3 papers among the 3**Specialist Principles:** any 2 among the 8 specialization exams**Specialist Practices:** any 1 among the 5 Actuaries can work in the field before they’ve completed all the exams though.

Usually, they’ll do more technical work at the beginning while they’re still passing exams. Actuarial Science is considered a highly professional and difficult course and is usually taken up by individuals who are already working in the industry with different qualifications and after the age of 20. But recently a different trend has been noticed in India where

**A Chartered Accountant:**Similar to actuaries, accountants also need to get a bachelor’s degree. It must be in accounting or business. They also need to have completed a specific number of exams in accounting, auditing, taxation, and business.

It takes at least 4 years to complete the degree.

The CA exam is broken down into 3 stages A score of 40% is required in each subject and they have to attain 50% of the aggregate marks.

Stage 1- CS Foundation

Stage 2-INTER CA

Stage 3-CA Final

An accountant can work in the field before obtaining the CA designation. In fact, they have to do their training and article ship for a period of 30 months before appearing for Final CA. Chartered Accountancy is a course that is studied along with graduation, although there can be exceptions.

**Employment Opportunities and Outlook**

**For Actuaries**

Actuaries are most commonly employed by insurance companies. Almost any type of insurance will needs actuaries in the background to assess and manage the risk involved with providing that insurance.

They can also work for pension consulting firms, or the government in areas dealing with public healthcare/retirement systems (like OHIP and CPP in Canada, or Medicare and Medicaid in the U.S.). Any company that has some level of risk that it wants to manage is also a possibility. This isn’t actuarial work in the traditional sense though.

There are fewer jobs for freshers at the entry level. Once you gain some experience in an actuarial position, you likely won’t have a hard time getting other positions in the field.

You may have heard that the unemployment rate for actuaries is
almost 0%. They’re referring to *fully-qualified actuaries*.
There are many actuarial job candidates with some exams passed that are unable to
find entry-level jobs. Technical skills, a good GPA, and related
experience all increase your chances of getting a job.

My personal opinion is that there will be more and more actuarial positions in the future, but since there are also more and more candidates applying for these positions it’s going to continue to be difficult to stand out as “one of the best”. Due to this uncertainty, it’s important to have a back-up plan if you decide to become an Actuary.

**For Accountants**

Almost every company needs an accountant. Many of them have an entire finance department with multiple accountants working together. So, there are many thousands of CA jobs in India and abroad.

The large base of employers that need CAs means that there are jobs in most cities.

With the extensive opportunity paths open to CAs I’m led to believe that intelligent CAs with good communication and technical skills will be able to find a job.

**Actuary Outlook v/s Accountant Outlook** The demand for both actuaries and accountants looks high for the coming years. However, the relatively small number of actuarial positions available compared to CA positions suggests that you’d probably have an easier time finding a job as a CA. If you do go the actuarial route, you should have a back-up plan that’ll allow you to gain experience first and then hopefully switch to an actuarial job.

**Common
thoughts on Actuarial Science**

- Only people who love maths and statistics do Actuarial Science.
- Actuaries earn
very high salary around 1cr. - Actuarial science is too difficult a course to be completed.

To read all the myths and reality on Actuarial Science, click here.

**Common thoughts on Chartered Accountancy**

- CA is an extremely tough course.
- Once you become a CA, you will be highly respected. The truth is, a mere degree doesn’t earn respect, the kind of work you do wherever and whatever that be earns you respect.
- Single strategy to clear CA final works for all.
- Life of a CA student is very difficult with 7am to 10pm studying hours.

There is a common notion that people pursuing these courses don’t have a life because they study all the time , don’t have time for socialising and always think about numbers. This is not true. It is just that these people have a different definition of life, a different thing gives them happiness and they want a different thing from life.

People say, after having the degree in hand, the sky is the limit. Let us see what lies ahead for you.

**For Chartered Accountants**

There are a few courses which you can take up along with a **CWA, CS, CIA, and CISA/DISA.**

**Dip IFRS – ** It is a certification course offered by the ACCA Institute, London. It can help you if you are in audit/ accounting role in a company.**CIMA/ CMA** → CIMA degree is offered by CIMA, London while CMA is offered by IMA, USA. These degrees will help you if you are working in controlling function. Also, these qualifications will be more useful if you are in Foreign MNC and in a business role.

Post qualification courses by ICAI → ICAI offers many post qualification courses which are useful for professionals in their respective service areas. Check more on this at The Institute of Chartered Accountant of India

Following are the areas where CA’s find opportunity: Banks, Public Limited Companies, Auditing firms, finance companies, legal firms, legal houses, patent firms, copyright registers.

**For Actuaries**

Data Science is a field in intersection with the course. People with inclination towards risk specialise doing FRM(Financial Risk Management) , people inclined towards finance also do CFA (Chartered Financial Analyst).

*Finally, I believe*

Accountants and Actuaries are both important in their own ways in the financial world, however, the roles and duties are broader for an accountant, and they are quite specific for an Actuary. One cannot replace the other. While certain job duties of management accountants overlap those of actuaries, the career fields remain largely distinctive.

The broad functions and skills of accountants help to ensure that they have a variety of employment opportunities available to them across all industries. However, actuaries who have skills that are in high demand are mainly employed in the financial services industry exclusively. Subsequently, job function, variety, and technical difficulty primarily sum up the differences between an Actuary and accountant.

Each of these can be a fantastic career for the right individual. Accounting jobs should appeal to those who are interested in finance or business management and who don’t want to deal with constant stress or job uncertainty. Those with a passion for statistics and computer modelling will be hard pressed to find a career better suited than as an Actuary.

Both fields require the ability to work through problems logically and deductively, but accountants normally have simpler and more commonly reoccurring issues. A lot comes down to the desire to perform complex mathematical problem solving. Accountants don’t need to use complicated math concepts like actuaries do, and many might find the intense data harvesting of an Actuary to be a little daunting or too monotonous.

If you are doing a course because that will give you a good life in future but you crib about having to study every night then my friend you don’t want to accept the pain of this journey. There is a different course or field of work for you because you will be able to happily accept the pain that comes along with that, you need to find that destination and start on that path.

If you have thought on working on your passion after you become a CA or an Actuary, then stop and get on with the field that excites you and you love. Don’t let the fear of not excelling stop you from switching careers. What you are made for is calling for you, trust it’s call.

There are many other opportunities available and it really depends on where you want to go and how do you see yourself in the future. If you look at examples like Harsha Bhogle, Suresh Prabhu, Piyush Goyal, Kumar Mangalam Birla you will notice that they identified the area in which they were interested and pursued their path with dedication and commitment.

*Steve Jobs rightly said,**“ Your work is going to fill a large part of your life, and the only way to be truly satisfied is to do what you believe is great work. And the only way to do great work is to love what you do. If you haven’t found it yet, keep looking. Don’t settle. As with all matters of the heart, you’ll know when you find it.” *

Whenever the going gets tough and you feel like you are getting nowhere and you start doubting yourself and you feel like you are incapable of it, then remember

*“He who has a why to live, can bear to live any how ” , says Friedrich Nietzsche. * Because it’s your ‘why’, your ultimate purpose that takes you forward in the journey.You just have to follow your calling.

Read all the latest articles on Actuarial Science New Curriculum **Click**

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]]>The post Correlation Analysis Using R appeared first on StepUp Analytics.

]]>Scatterplots are used to visualize the data and get a rough idea of the existence and degree of correlation between two variables. The plot shows the scatter of the points, hence called scatterplot.

Here, for the birthweight data, we have a look at the scatterplot of Gestation period (in weeks) v/s Birthweight.

The scatterplot indicates a positive degree of correlation between the Gestation period (in weeks) v/s Birthweight. Now, to quantify this relationship, we use the coefficient of correlation.

The coefficient of linear correlation provides a measure of how well a linear regression model explains the relationship between two variables. To put it simply, it gives a quantification for the relationship between the variables under study.

The degree of association between the *x *and *y *values is summarised by the value of an appropriate correlation coefficient each of which take values from -1 to +1. A negative value indicates that the two variables move together in opposite directions, the eg. speed of the train and time taken to reach the destination exhibits a negative correlation. A positive value indicates that the two variables move together in the same direction, eg. the height and weight of a human being.

In this

Pearson correlation coefficient* (also called Pearson’s product-moment correlation coefficient) *measures the strength of ** the linear **relationship between two variables.

The correlation between two variables is calculated in R using cor( ) function.

For the Birthweight data, we observe a moderate positive correlation between Gestation period (in weeks) and Birthweight.

Spearman’s rank correlation measures correlation based on the ranks of observations. If data are quantitative, then it is less precise than Pearson’s correlation coefficient as we use actual observations for Pearson’s correlation coefficient which gives more information than their ranks.

**Q. Find the Spearman’s rank correlation between Mathematics and Statistics marks scored by 2nd-year college students.**

There is a moderate positive correlation between Mathematics and Statistics marks scored by 2nd-year college students.

Kendall’s rank correlation coefficient τ measures the strength of dependence of rank correlation between two variables. Any pair of observations (Xi, Yi) ; (Xj, Yj) and where i≠j, is said to be concordant if the ranks for both elements agree, i.e. Xi > Xj and Yi> Yj or Xi <Xj and Yi < Yj, otherwise it is said to be discordant.

**Q. Two judges ranked 10 contestants in a fancy dress competition. The ranks are from most favorite to least favorite. Calculate Kendall’s rank correlation**

There is a positive high degree correlation between ranks given by 2 judges.

The sample correlation coefficient (studied so far) measures the extent of the linear relationship between the two variables (X, Y) for the sample data. The population parameter, ρ, measures the extent of the linear relationship between the variables (X, Y) in the population.

We are usually interested in testing whether the population correlation coefficient is significant or not. The hypothesis is stated as,

H_{0}: ρ = 0 and is tested against one of the following alternatives,

H_{1}: ρ ≠ 0

H_{1}: ρ > 0

H_{1}: ρ < 0

For the Birthweight data, we test the hypothesis:

H_{0}: The population correlation coefficient between birthweight and gestation period is equal to 0**v/s**

H_{1}: The population correlation coefficient between birthweight and gestation period is not equal to 0

The test is carried out in R using cor.test() function in R.

As the p-value is
less than 0.05, reject H_{0} and conclude that the population
correlation coefficient between birthweight and gestation period is not equal
to 0.

Since we are using ranks rather than the actual data, no assumption is needed about the distribution of *X*, *Y *or (*X*, *Y*), i.e. it is a non-parametric test. For the data giving Mathematics and Statistics marks scored by 2nd-year college students, test the following hypothesis:

H_{0}: The population correlation coefficient between Mathematics and Statistics marks is equal to 0**v/s**

H_{1}: The population correlation coefficient between Mathematics and Statistics marks is not equal to 0

As p-value < 0.05, reject H_{0} and conclude that the population correlation coefficient between Mathematics and Statistics marks is not equal to 0.

For the data giving ranks of 10 contestants in a fancy dress competition by 2 judges, test the following hypothesis:

**H**_{0}**:** The population correlation coefficient between rank given by Judge 1 and Judge 2 is equal to 0**v/s****H**_{1}**:** The population correlation coefficient between rank given by Judge 1 and Judge 2 is not equal to 0

As the p-value < 0.05, reject H_{0 } and conclude that the population correlation coefficient between rank given by Judge 1 and Judge 2 is not equal to 0.

Data Used in the above example can be downloaded from here

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]]>The post Degree in Actuarial Vs Degree in Economics appeared first on StepUp Analytics.

]]>Today we all stand at the crossroads- the dilemma of choosing a right career.

Though human progress and advancement has opened doors for innumerable opportunities but it would be tedious if I jot down all the career choices that we currently have.

Being an Actuarial Science student let me take this opportunity to share with you all what it means to have a degree in Actuarial Science and is it better than majoring in Economics?

Firstly for a non-professional, Actuarial Science is simply the study of risks. It’s the need to predict, assess and control risks and uncertainties at the heart of the economy. An actuary identifies the possibility of a bad event or a catastrophe and takes all the necessary steps to either reduce it’s the possibility of occurrence or will focus on minimizing the losses if in case such event occurs.

To be a fellow actuary, one has to clear 15 papers. The examination is held twice a year which I’m sure everyone is well versed with. But there is also another route to it. To prepare for an actuarial career, one can even do his under graduation in Actuarial Science. There are some renowned universities that offer such courses namely Columbia University(US), Florida State University, Pennsylvania State University(US), Christ University(Bangalore) etc. After graduation one gets an exemption from a few papers. There are students who do their major in Actuarial Science

While majoring in Actuarial Science or math in college will benefit someone wanting to be an actuary, you are not required to major in these fields to be an actuary. It is the ability to pass the actuarial exams that determine if one is eligible to enter the profession. It is generally recommended to clear a few papers and enter into the working world so that you get a practical exposure of the theoretically learned concepts and simultaneously give your papers. Though knowledge in any form is never harmful it makes more sense to work rather than pursuing a post-graduate degree that consumes 2 years.

Conventionally, actuaries used to work in the Insurance sector specifically life and non-life. But as can be seen from the diagram actuarial scope has increased. Now actuaries have entered into fields like Banking, Cyber Insurance, Rating agencies etc.

On the other hand, Economics is simply the study of production, distribution and consumption of goods and services. An economist conducts research, collect and analyse data, monitor trends and develop forecasts related to inflation, interest rates, employment etc.

The basic qualification a person needs in order to become an economist is a graduate degree in the allied field. There are many colleges in India itself that offer under graduation and post-graduation programmes in economics. If you’re interested in majoring in economics, you’ll be studying (unsurprisingly) economic theory in all its various forms. Microeconomics, macroeconomics, currency systems, international economics, and more will all be slotted into your course of study. Common career paths for economic graduate include:

Both the degrees have their pros and cons and nobody can judge which one is better. In India, both the fields are emerging at a high pace and have an endless scope. At last, I would like to say it’s all upon one’s interest and understanding. If interested in any one of them, then only pursue that as a career else just don’t because it won’t be worth forcing yourself to do a task you ain’t interested in doing.

Choose wisely.

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]]>The post Actuarial Science: Binomial Model Option Pricing appeared first on StepUp Analytics.

]]>I’ll tell you the reason behind it.

If you buy the share of the company, then you are simply making an investment which can be held for an indefinite time frame. You make your decisions based on the sound fundamentals of that company and you have the right to decide whether to keep holding it or sell it off. But this is not the case with options.

For a commoner, an option is a financial instrument which gives the right to the buyer but not an obligation to buy (call option) or to sell (put option) the underlying asset at an agreed price on a specific date in future (though it can be exercised before maturity in case of American options). Therefore, as you can read, options are only for a stipulated time frame. If you make money during that time period, great. If not, you’ve lost the money you invested to purchase the option and that’s it.

Hence, traders usually spend hours to work out the price of the option that should be charged so as to avoid arbitrage opportunities. In a competitive market, to avoid arbitrage opportunities, assets with identical payoff structures must have the same price. Valuation of options has been a challenging task. Various models are designed to price these complex instruments. Among all models, binomial model is the easiest and the most popular one for pricing options.

Just before discussing this model let me ask you a question.

What do we actually mean by the term ‘pricing of an option’?

As mentioned above, an option gives a privilege to a buyer and hence to get such privilege the buyer has to pay a price. This price is nothing but the premium of the option. Thus, the pricing models designed help us to know the appropriate price of the option.

In this article let’s discuss the simplest of all pricing models- Binomial Option Pricing Model.

**Meaning**:

The name stems from the fact that it calculates two possible values for an option at any given time. The binomial model was developed in 1979 by John Cox (a well respected finance professor), Mark Rubinstein (a financial economist) and Stephen Ross (a finance professor). The aim is to find the value at time t=0 of a derivative that provides a payoff at some future date based on the value of the stock at that future date.

The basic idea is that if we can construct a portfolio that replicates the payoff from the derivative under every possible circumstance, then that portfolio must have the same value as that of the derivative. So by valuing the replicating portfolio we can value the derivative.

**Assumptions:**

Every theory works on certain assumptions. Here we assume that:

In the binomial model, it is assumed that:

- Investors are risk-averse that is when faced with two investments with a similar expected return, they prefer the one with lower risk.
- Investors prefer more to less.
- There are no trading or transaction costs or taxes.
- There are no minimum or maximum units of trading. In other words, unlimited buying and selling are permitted (including short selling).
- There is complete divisibility of assets and investors are allowed to hold the fraction of an asset.
- Stock and bonds can only be bought and sold at discrete times 1, 2, …
- The principle of no arbitrage holds
*.*

As such the model appears to be quite unrealistic. However, it does provide us with good insight into the theory behind more realistic models. Furthermore, it provides us with an effective computational tool for derivatives pricing.

**Notations:****The share price process**

We will use S_{t} to represent the price of a non-dividend-paying stock at discrete time intervals t (t= 0,1,2,.… This means that S_{t} is random.

Here “stock” specifically means a share or equity as opposed to a bond. For the time being, we ignore the possibility of dividends, which would otherwise unnecessarily complicate matters.

Over any discrete time interval from t-1 to t, we assume that S_{t} either goes up or goes down. We also assume that we cannot predict beforehand which it will be and so future values of S_{t }cannot be predicted with certainty. We will, however, be able to attach probabilities to each possibility and we also assume that the sizes of the jumps up or down are known.

**B**_{t- }**Stands for the cash process. It is the risk-free rate at which accumulation happens.**

The above timeline explains the Cash process.
As can be seen, the asset has accumulated at a risk free rate and has grown to
e^{rt} where:

r= Continuously Compounding Risk free rate of interest.

T=Time period.

All the approaches lead to the same price. The only difference is the way price is calculated.

**Replicating Portfolio Approach**

At time 0, suppose that we * ϕ* units (phi units) of stock and ψ units (psi units) of cash. So these phi and psi could be any values and this portfolio could consist of

There is yet another derivative which pays *cu* if the price of the underlying stock goes up and cd if the price of the underlying stock goes down. So, the value of the payment made by the derivative, which we can denote by the random variable *C*, depends on the underlying stock price.

At what price should this derivative trade at time 0?

At
time 0 suppose that we hold *ϕ* units of stock and ψ units
of cash. The value of this portfolio at time 0 is *V*_{0} .

Eg: If the stock price went
up, then the value of one unit of the stock has increased to* S _{0}u
an*d if the price went down then the value of the stock has decreased to S

Therefore, we have the following two equations:

Solving these two equations we can get the value of phi and psi and then by the principle of no arbitrage, we have:

**Risk Neutral Approach**

Under this approach, the investor is risk neutral.

We find the probability of an up move (q) and down move (1-q) in such a way that the payoff is replicated.

Also, it should be noted that the asset grows at a risk-free rate. To better understand this, we will illustrate using excel later.

**Price Deflator Approach**

Under this approach, we find a real-world probability.

This real-world probability is the actual probability by market participants for an up move. This probability will always be greater than the risk-neutral probability.

Here,

Where A_{u }stands for stochastic discounting factor (A_{U}= e^{–r} (q/p) ).

C_{U }stands for

P=

Let us take an example to understand this model better. We will be solving the question using the risk neutral approach which is the most widely used approach of this model.

**Step 1:**

Plot the binomial lattice according to the information provided in the question.

**Step** **2:**

Plot the payoff of the call option. This is nothing but what the option will get at maturity.

So, for _{t}-X, 0)= max( underlying asset- Strike price, 0).

As can be seen in the figure we have plotted the payoff of the call.

**Step** **3:**Find the risk-neutral probability. The formula for the same is given below.

We have found the desired probability using the above-mentioned formulas.

**Step** **4:**

Find the expected payoff of the option by using the below formula.

Expected Payoff= Payoff at maturity * Probability

**Step** **5:**Find the present value factor. This is nothing but exp(-rn). The risk free rate given in the question is continuously compounding therefore we use an exponential function.

Also “n” stands for the time period. In our question, it is 1.

**Step** **6:**

We get the value of the call option at t=0 by discounting the expected payoff at t=0. Here it is nothing but step 4* step 5.

Hence, we derive the value of the option at t=o. This is a small illustration having just one time period. Complexity increases when there are more than two time periods.

The price so calculated is the theoretical price of the option. This price gives useful insights to the traders. It helps them in taking, decisions on whether to buy/sell an option.

Also, it should be noted that the real price of an option may not always be same as the theoretical price derived from the model. The variations can be attributed to the changing demand/supply situations, the incorrect assumptions of up move or down move etc.

Though this result in a different price of the option but the significance of the theoretical price cannot be denied. It plays a crucial role in creating various positions.

Thus, this is how the option’s price is calculated using the Binomial Option Pricing Method.

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]]>The post Actuarial Science: NPV vs IRR, A Comparison appeared first on StepUp Analytics.

]]>**Positive NPV or NPV>0**implies that the value of revenues (cash inflows)is greater than the costs(cash outflows)**Negative NPV or NPV<0**, implies vice versa.

**ADVANTAGES OF NPV**

- It accepts conventional cash flow pattern.
- It takes into consideration both before & after cash flow over the life span of a project.
- It considers all discount rates that may exist at different points of time while discounting back our cash flows.
- Discount rates are used in calculating NPV; the risk of undertaking the project (Business Risk, Financial Risk, and Operating Risk) gets factored into this method.

**DISADVANTAGES
OF NPV**

- It might not give you accurate decision when the two or more projects are of unequal life.
- It does not give
the clarity on how long a project or an investment plan will generate positive NPV due to its simple calculation. - NPV method suggests accepting an investment plan which provides positive NPV but it doesn’t provide an accurate answer to at what period of time you will achieve positive NPV.
- Calculating the appropriate discount rate for cash flows is difficult.

**IRR**: also referred as “yield to redemption” or “yield per annum”. The internal rate of return for
an investment project is the effective rate of interest that equates the
present value of inflows and outflows. Higher IRR represents a more profitable project.
It is a commonly used concept in project and investment analysis, including
capital budgeting. The IRR of a projector or investment is the discount rate
that results in an NPV of Zero.

- However, IRR need not be positive. Zero return implies investor receives no return on investment. If the project has only cash inflows then the IRR is infinity.
- When IRR > cost of capital, NPV will be positive
- When IRR < cost of capital, NPV will be negative.

**ADVANTAGES OF IRR**

- This approach is mostly used by financial managers because the ranking of project proposals is very easy under the internal rate of return since it indicates percentage return.
- IRR method gives you the advantage of knowing the actual returns of the money which you invested today.
- It is very simple to interpret an investment decision or a project after the IRR is calculated. If IRR exceeds the cost of capital, then accept the project otherwise not.

- IRR tells you to accept the project or investment plan where it is greater than the weighted average cost of capital but in case, if the discount rate changes every year then it is difficult to make such comparison.
- If there are two or more mutually exclusive projects (where acceptance of one project rejects the other from concern) then in that case too, IRR is not effective.
- It ignores the actual monetary value of benefits. One should always prefer a project value of $1000000 with 18% rate of return over a project value of $10000 with a 50% rate of return. But in the IRR method, the latter project with less monetary benefit will be given preference, simply because the IRR of 50% is higher than 18%.

**EXAMPLE:**

XYZ Company is planning to invest in a plant; it generates the following cash flows.

From the above information calculate NPV & IRR. The discounting rate is 10%. Suggest whether XYZ ltd should invest in this plant or not.

**NPV formula:**

NPV = CF/ (1+r) ^t-Cash outflow.

Where, CF= Cash inflow,
r=discount rate, t= time, Cash outflow= Total Project Cost.

**IRR formula:**

IRR: CF/ (1+IRR) ^t=Cash Outflow.

Now on comparing the IRR with the discount rate, From the above calculation you can see that the NPV generated by the plant is positive and IRR is 14% which is more than the required rate of return.

This implies, when the discounting rate is 14% NPV will become zero. Here 14% is the investor’s required rate of return. Had the discount rate been higher than the IRR, say the discount rate offered to the company is 16% then the NPV will be a negative amount. A positive NPV makes the investment proposal a viable option.

So, XYZ Company should invest in the plant.

So, if you are evaluating two or more mutually exclusive projects it’s advisable to go for NPV approach instead of IRR approach. For selecting the best investment plan, NPV method is preferred due to its realistic assumptions and better measure of profitability. IRR method is a great compliment to NPV and will provide you accurate analysis for investment decisions.

So why is IRR still commonly used in Capital budgeting? It’s because of its reporting simplicity. The NPV method is considered complex and require assumptions at each stage.

Regardless of its disadvantages, NPV is widely used as it is a better measure of profitability than the IRR.

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]]>The post Theory of Estimation Or What is Estimation appeared first on StepUp Analytics.

]]>**Population**: A group of individuals
under study is called population. The population may be finite or infinite. Eg.
All the registered voters in India.

**Sample**: A finite subset of statistical
individuals in a population. Eg. Selecting some voters from all registered
voters.

**Parameter**: The statistical constants of the
population such as mean (μ), variance (σ^{2}) etc. Eg. Mean of income of all the
registered voters.

**Statistic**: The statistical constants of the sample such as mean (X̄), variance (s^{2}) etc. In other words, any function of the random sample x_{1}, x_{2},…, _{n} that are being observed say, T_{n} is called a statistic. Eg. Mean

**Estimator**: If a statistic is used to estimate an
unknown parameter θ of the distribution,
then it is called an estimator. Eg. Sample mean is an estimator of population
mean.

**Estimate:** A particular value of the estimator is
called an estimate of an unknown parameter. Eg. Mean income of selected voters
is ₹25000 which represents mean income of all
the registered voters.

**Sampling Distribution**: When the total probability is distributed according to the value of statistic then the distribution is said to be sampling distribution. Eg. If we want the average height of a voter, we can randomly select some of them and use the sample mean to estimate the population mean.

**Standard Error**: The standard deviation of the sampling distribution of a statistic is known as its standard error and is denoted by ‘s.e.’ Eg. If we want to know the variability of the height of voters, then standard error is used.

Now, before discussing about different methods of finding estimates of unknown population parameter, it is important to know the characteristics of a good estimator. Here, “a good estimator” is one which is close to the true value of the parameter as much as possible. The following are some of the criterion that should be satisfied by a good estimator:

- Unbiasedness
- Consistency
- Efficiency
- Sufficiency

**Unbiasedness**

This is a desirable property of a good estimator. An estimator T_{n }is said to be an unbiased estimator of γ (θ), where γ (θ) is a function of unknown parameter θ, if the expectation of the estimator is equal to the population parameter, i.e.,

**E [T**_{n}**] = γ (θ)**

Example: If X ~ N (μ,σ^{2}),

**Consistency**

An estimator is said to be consistent if increasing the sample size produces an estimate with smaller standard error (standard deviation of sampling distribution of a statistic). In other words, if the sample size increases, it becomes almost certain that the value of a statistic will be very close to the true value of the parameter. Example: Sample mean is a consistent estimator of the population mean, since as sample size n→∞, the sample means converges to the population mean in probability and variability of the sample mean tends to 0.

**Efficiency**

There is a necessity of some further criterion which will enable us to choose between the estimators, with the common property of consistency. Such a criterion which is based on the variances of the sampling distribution of estimators is usually known as efficiency.

It refers to the size of the standard error of the statistic. If two statistic are compared from a sample of same size and we try to decide which one a good estimator is, the statistic that has a smaller standard error or standard deviation of the sampling distribution will be selected.

If T_{1} is the most efficient estimator with variance V_{1} and T_{2}, any other estimator with variance V_{2}, then the efficiency E of T_{2} is given by:

[∵ Efficiency and Variances are inversely proportional]

**Sufficiency**

An estimator is said to be sufficient for a parameter, if it contains all the information in the sample regarding the parameter.

If T_{n} is an estimator of parameter θ, based on a sample x_{1}, x_{2},…, x_{n} of size n from the population with density f(x,θ), such that the conditional distribution of x_{1}, x_{2},…, x_{n} given T_{n}, is independent of θ, then T_{n} is sufficient estimator for θ.

**Methods of Point Estimation**

So far we have been discussing the requisites of a good estimator. Now we shall briefly outline some of the important methods of obtaining such estimators. Commonly used methods are:

- Method of Moments
- Method of Maximum Likelihood Estimation
- Method of Minimum Variance
- Method of Least Squares

**Method of Moments (MoM)**

The basic principle is to equate population moments (i.e. the means, variances, etc. of the theoretical model) to the corresponding sample moments (i.e. the means, variances, etc. of the sample data observed) and solve for the parameter(s).

Let x_{1}, x_{2}, …, x_{n} be a random sample from any distribution f(x,θ) which has m unknown parameters θ_{1}, θ_{2}, …, θ_{m}, where m ≤ n. Then the moment estimators θ ̂ _{1}, θ ̂ _{2}, …, θ ̂ _{m }are obtained by equating the first m sample moments to the corresponding m population moments and then solving for θ_{1}, θ_{2}, …, θ_{m}.

**Method of Maximum Likelihood Estimation (MLE) **

MLE is widely regarded as the best general method of finding estimators. In particular, MLE’s usually have easily determined asymptotic properties and are especially good in the large sample situations. “Asymptotic’’ here just means when the samples are very large.

Let x_{1}, x_{2}, …, x_{n} be a random sample from a population with density f(x,θ). The likelihood function of the observed sample at the function of θ is given by:

Notice that the likelihood function is a function of the unknown parameter θ. So different values of θ would give different values for the likelihood. The maximum likelihood approach is to find the value of θ that would have been most likely to give us the particular sample we got. In other words, we need to find the value of θ that maximizes the likelihood function. In most cases, taking logs greatly simplifies the determination of the MLE θ ̂. Differentiating the likelihood or log likelihood with respect to the parameter and setting the derivative to 0 gives the MLE for the parameter.

It is necessary to check, either formally or through simple logic, that the turning point is a maximum. The formal approach would be to check that the second derivative is negative.

**Method of Minimum Variance**

It is also known as Minimum Variance Unbiased Estimator (MVUE). As the name itself depicts, estimator which is unbiased as well as having minimum variance.

If a statistic T_{n} based on a sample of size n is such that:

- T
_{n}is unbiased - It has the smallest variance among the class of all unbiased estimators
- then T
_{n}is called MVUE of θ.

**Method of Least Squares**

The principle of least squares is used to fit a curve of the form:

where θ_{i}’s are unknown parameters, to a set of n sample observations (x_{i}, y_{i}); i=1,2,…,n from a bivariate population. It consists of minimizing the sum of squares of residuals,

subject to variations in θ_{1}, θ_{2}, …, θ_{n}. The normal equations for estimating θ_{1}, θ_{2}, …, θ_{n} are given by:

**Confidence Intervals and Confidence Limits**

Confidence interval provides an ‘interval estimate’ for an unknown population parameter. It is designed to contain the parameter’s value with some stated probability. The width of the interval provides a measure of the precision accuracy of the estimator involved.

Let
x_{i}, i = 1, 2, … n be a random sample of size n from f(x,θ). If T_{1}(x) and T_{2}(x)
be any two statistics such that T_{1}(x) ≤
T_{2}(x) then,

**P(T**_{1}**(x) < θ < T**_{2}**(x)) = 1 – α**

where α is level of
significance, then the random interval (T_{1}(x), T_{2}(x)) is
called 100(1-α)% confidence interval for θ.

Here, T_{1} is called lower confidence limit and T_{2}
is called upper confidence limit. (1-α) is called the confidence coefficient.

Usually, the value of α is taken as 5% in the testing of hypothesis. Thus, if α = 5%, then there is a 95% chance of the estimate to be in the confidence interval.

*Interval estimate =
Point estimate ± Margin of Error*

The margin of error is the amount of random sampling error. In other words, the range of values above and below the sample statistic.

*Margin of Error =
Critical Value * Standard Error of the statistic*

Here, a critical value is the point (or points) on the scale of the test statistic beyond which we reject the null hypothesis, and is derived from the level of significance α of a particular test into consideration.

Confidence intervals are not unique. In general, they should be obtained via the sampling distribution of a good estimator, in particular, the MLE. Even then there is a choice between one-sided and two-sided intervals and between equal-tailed and shortest length intervals although these are often the same.

So, we have learned what the estimation is, i.e., the process of providing numerical value to unknown population parameter. To test whether an estimate is a good estimator of the population parameter, an estimate should have the following characteristics:

- Unbiasedness
- Consistency
- Efficiency
- Sufficiency

There are different methods of finding estimates such as method of moments, MLE, minimum variance and least squares. Of these methods, MLE is considered as the best general method of finding estimates.

Also, there are two types of estimations, point and interval estimation. Point estimation provides a single value to the estimate, whereas, interval estimation provides confidence interval which is likely to include the unknown population parameter.

Hence, now you have the basic understanding about the theory of estimation.

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