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

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

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]]>The post Actuarial Science: Mortality Profits and DSAR appeared first on StepUp Analytics.

]]>For a company, in order to survive in the market, it must earn enough revenues to make considerable profits.

Do you know what is the major source of income for these insurance companies?

The premium paid by its policyholders/ customers. Apart from the premiums, it does invest a small portion of the premiums and makes income from the return on these investments. According to the Insurance Bureau of Canada, $0.55 of every dollar collected as a part of the premium goes to pay for the insurance claims.

Another $0.21 goes toward operating costs such as 24 hours’ customer service claims lines, the maintenance of records and making sure websites are up-to-date, while another $0.16 is paid to the government in the form of taxes. Insurance companies only earn $0.08 out of every $1 in profit and this profit margin was consistent over the last seven years, from 2007 to 2013.

Every year the insurance company collects premium from its’ customers which after deducting the aforesaid expenses are invested in the form of reserves which are then used to cover the assurance payouts upon death. For those policyholders who have survived their respective reserve amounts are then accumulated.

The business of these companies become so simple if everything worked in the above way. At times, these companies are burdened with a large number of claims than what they initially expected. In such a scenario, all the estimates of profit go in vain. The other times, it may happen that benefit that the company is liable to pay in the event of the death of its’ policyholders exceeds the amount that it holds in the form of reserves.

Yes, the above two situations can occur while running the business. Then what is the profit that the insurance company will make in such scenarios?

Let me introduce you the second situation in actuarial terms.

**DSAR i.e. Death Strain at Risk.** To put into words, DSAR in a policy year represents the excess sum that needs to be found at the end of the year to fund the death benefit, over the sum being held as a reserve. It represents the excess cost of the policy becoming a death claim.

**DSAR
= Sum Assured in the event of death – Reserves at the end of that year.**

How does DSAR affect the profit?

For understanding this, we must know two important concepts.

- Expected DSAR
- Actual DSAR

**Expected Death Strain at Risk** is the expected amount of death strain that the life insurance company expects to pay in addition to the year-end reserve.

This is calculated as:

**EDSAR= Death Strain at Risk * number of people alive at the start of the year * Probability of people dying during the year.**

**Actual
Death Strain at Risk**, on the other hand, is
calculated in respect of the policies where the death has occurred during the
year. It is the actual amount of benefit paid on death.

Formulating it: **ADSAR= Death Strain at Risk * number of people actually died during the year.**

Now, after understanding these two concepts we can easily calculate the mortality profit.

**Mortality
Profit during the year = EDSAR – ADSAR**

These profits are then used to cover the costs or are reinvested in the form of reserves.

In case, ADSAR > EDSAR, then the insurance company incurs a mortality loss. Let me explain the concept better by taking an example.

Ques: A 20- year special endowment assurance policy is issued to a group of lives aged 45 exact. Each policy provides a sum assured of $10,000 payable at the end of the year of death or $20,000 payable if the life survives until the maturity date. Premiums on the policy are payable annually in advance for 15 years or until earlier death. You are given the following information:

Number of deaths during the 13^{th}
policy year – 4.

Number of policies in force at the end of
the 13^{th }policy year – 195.

Calculate the profit or loss arising from
mortality in the 13^{th} policy year.

Basis: Mortality AM92 Ultimate. Interest 4% per annum. Expenses none.

**STEP 1:** Write down all the given information.

**STEP 2: **Now we have to compute the premium that should be charged.

Since this is an endowment assurance policy, the formula to calculate premium is:

Premium paid at the start of each year = P
* (a due)_{45: 15}

Death benefit = 10,000 * (Term Assurance )_{45:
15}

Survival Benefit = 20,000 * v^{20}
* _{20}p_{45}

Equating,

Premium = Death Benefit in case of death (Term Assurance)

+

Survival Benefit at the end of the term (Pure Endowment)

The “..” after the letter “a” signifies that the premium is paid in due i.e. at the start of the year.

**STEP 3: **Calculating the reserves at the end of the 13^{th} year.

The formula is:

Reserve at the end of 13^{th} year = 10000 * (Term Assurance) _{58:7 }+ 20000* v^7 * _{7}p_{58} – P (a due) _{58:2}

**STEP 4: **Calculating the DSAR and Mortality Profit. The formulas mentioned above in the article are used.

As can be seen, the DSAR is negative signifying that the company already has a sufficient amount of reserves to honor its commitments.

Also, if DSAR is negative and if more people die than expected then the company makes a profit. As from the above information, it can be seen that actual deaths were higher than expected deaths i.e.

4 (Actual Deaths) > 1.12435 (Expected Deaths)

Hence, the company made a profit. This is how this process works and help the insurance companies to find their mortality profits/ losses.

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

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]]>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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