Assignment Code: MECE-003/AST/2022-23

Course Code: MECE-003

Assignment Name: Actuarial Economics: Theory and Practice

Year: 2022-2023

Verification Status: Verified by Professor

SECTION A

Answer the following questions in about 700 words each. The word limits do not apply in case of numerical questions. Each question carries 20 marks.

1) Discuss the applications of Chebyshev’s Theorem.

Ans) Chebyshev's Theorem estimates the minimum proportion of observations that fall within a specified number of standard deviations from the mean. This theorem applies to a broad range of probability distributions. Chebyshev's Theorem is also known as Chebyshev's Inequality. Chebyshev’s Theorem helps you determine where most of your data fall within a distribution of values. This theorem provides helpful results when you have only the mean and standard deviation.

Rewrite Chebyshev's Theorem using standard deviation.

Crud, make R a random variable, and make C a positive real number.

That is, the "likely" values of R are clustered in a 0 ()sized region around E[R]. Therefore, this finding supports the notion that the standard deviation measures how evenly distributed the distribution of R is with respect to E[R] its mean.

A One-Sided Bound

A "two-sided bound" is provided by Chebyshev's Theorem. In other words, we determine the boundaries based on the likelihood that a random variable deviates from the mean above or below by a certain amount. Below, we create a one-sided bound for it. Find the likelihood that a random variable deviates above the mean, for instance.

Deviation of Repeated Trials

The demonstration of average of numerous trials approaches the mean in a large number of repeated trials using Chebyshev's Theorem.

Estimation Using Several Trials

Take p, a Bernoulli variable representing the proportion of success of the variable.. G , with G = 1 for success and G = 0 otherwise. In this case, G = G2 so

E[G2}=E{G}=Pr{G=1}p

and

Var[G]=E[G2]-E2[G]=p-p2=p(1-p)

We use a large number of trials, n, and keep track of the success rate to determine p. So, we are using independent Bernoulli variables G, G, G,..., Gn, each of which has the same expectation as G. Adding up their totals

It is important to note that Theorem makes a precise claim regarding how an approaching random variable's average of independent samples. We may generalise it to a wide range of situations in which S is the sum of independent variables, whose means and standard deviations aren't always the same.

Proof of the Weak Law

The probability average deviating from the expectation by more than any specified tolerance approaches zero, according to the Weak Law of Large Numbers.

The Weak Law of Large Numbers Theorem Proving. Let

Probability must be equal to zero at the limit.

It should be noted that neither the Weak Law nor the Pairwise Independence Sampling Theorem Deviation of Repeated Trials give any insight into how one may anticipate that the average value of the observations will fluctuate over the course of repeated tests. Such an aspect of Strong Law of Large Numbers, which deals with oscillations, will be seen. Even if a player's average gains are guaranteed over the long term, these oscillations are crucial in gambling scenarios because large oscillations can bankrupt a person.

2) Explain any two models used in Stochastic Claims Reserving.

Ans) Stochastic models generate predictions about the value of anticipated future payments as well as the variability in those predictions. These models give estimates of both the expected value of the future payments and the variance around that expected value by making assumptions about the random component of the model, which enables the validity of the assumptions to be checked statistically.

A stochastic reserving method's drawback is that it simulates an incredibly complicated chain of events with a small number of parameters, which leaves it vulnerable to accusations that its assumptions are overly optimistic. The availability of measures of the accuracy of reserve estimates is, in general, the main benefit of stochastic reserving models, and in the following, attention is given to the root mean squared error of prediction. The material that follows provides examples of stochastic models that replicate the chain-ladder technique's reserve estimates.

Stochastic Chain-Ladder Models

A stochastic chain-ladder model is defined as a stochastic model that produces the same

estimates of future claims as the chain-ladder method.

In stochastic chain-ladder models we shall assume that the data consist of a triangle of incremental claims when thinking about claims reserving in a stochastic situation.

In light of this, we presumptively possess the following collection of incremental claims data:

Forecasts of claims are the sole claim in the chain-ladder method. The term "Ultimate" refers to the most recent delay year that has been so far recorded and excludes tail influences. In the end, in order to determine whether additional reserves, above and above the 1 anticipated values, should be retained for prudence. Prediction error, which is the standard deviation of the distribution of potential reserve outcomes, is the variable that is measured. I

We develop an underlying statistical model while making assumptions about the data in order to determine the prediction error. To create a stochastic model that is comparable to the chain-ladder technique, the predicted values should be the same as those of the chain-ladder method. We must either specify the first two moments or the distributions of the data for that reason.

Keep in mind that every model's fundamental goal is to generate reserve estimations that are identical to those produced by the chain-ladder method.

Mack's Model

Thomas Mack, In his original paper outlines a method to estimate the standard error of chain ladder estimates. The Method is now generalised with the name of Mack Method. SP7 (formerly ST7) contains an introduction on stochastic reserving where Mack model estimates the standard error.

The negative binomial model and the normal model both use a recursive methodology, as does the Mack model.

There are only a few distributional assumptions made about the underlying data. Only the first two moments are specified. The mean and variance of Dij are:

Mack produced estimators of the unknown parameters  𝜆 𝑗  and  𝜎 𝑗 2 ;. Making further limited assumptions, Mack also offered formulas for the reserve estimates and expected payment prediction errors.

Given that the complete distribution of the underlying data is unknown, Mack believes the model to be distribution-free. This streamlines the model but only allows for study of the first two moments of the distribution of outstanding reserves. Assumptions must be made if the results are used in a dynamic financial analysis exercise where the distribution of outstanding reserves may be modelled.

Finally, observe that the variance and mean of Dij under the Mack's model is similar to the mean and variance of Dij in the negative binomial model's normal approximation, with the unknown scale parameters ∅j of the normal approximation being replaced by  𝜎 𝑗 2  in Mack's.

SECTION B

Answer the following questions in about 400 words each. Each question carries 12 marks.

3) Write a note on Stopping Times and Martingales with examples.

Ans) Model for discrete time collective risk: We begin with the gambler's ruin dilemma, in which a player wagers z dollars in a casino before participating in a game of chance.. She wins with probability p and her loss is $1 with probability q =l -p. Assume that the gambler will likewise stop gambling if her fortune ever rises to a value greater than z and that she will be compelled to stop by being bankrupt if her fortune reaches 0. Keep in mind that our goal in solving this problem is to determine the likelihood that the gambler would finally lose everything.

Consider a simple case p = q = 1/2. Let X, be the amount won or lost on the Jth play of the game. The gambler's fortune is revealed after k games of the game when there is no prohibition on leaving.

4) What are ‘Yield Curves’? Explain with examples.

Ans) The Yield Curve is a graphical representation of the interest rates on debt for a range of maturities. It shows the yield an investor is expecting to earn if he lends his money for a given period of time. The graph displays a bond’s yield on the vertical axis and the time to maturity across the horizontal axis. The curve may take different shapes at different points in the economic cycle, but it is typically upward sloping.

Bonds come in a variety of classes in the financial markets. We may limit ourselves, for instance, to certain risk categories, such Treasury bonds, municipal bonds, corporate bonds, or junk bonds. Depending on the type of Treasury bond, there are Treasury bills that mature in a year, Treasury notes that mature in one to five years, and Treasury bonds that mature over a 30-year period. The price of a pure discount bond with a maturity of u would be provided by at time t if risk-fee instantaneous interest rates were constant at r.

where B() stands for 100's discounted present value at a rate of r. A pure discount bond has a par value of 100 and makes no coupon payments or contains any implicit options. The function e -r(u-t) at time t plays the role of a discount factor.

At time:

t=u

The exponential function equals 1 as the bond ages.. At all t < u, it is less than 1.

The full range of potential future short rates affects the bond price formula. Therefore, all of the information about future short rates is contained in the yield curve at time t, and bond prices depend either on the entire yield curve or on the term structure of interest rates, as below:

Definition: Assume that there are zero-coupon bonds with a range of maturities at time t.u  [t,T]L. et their price be B(u,t)and their yield be given by . Then the spectrum of yields (, u  [t, T]} is called the term structure of interest rates.

Here the yield  is the quantity that meets the equality

Examples of Yield Curves

There are two ways to proceed in reality. Frequently, a market participant thinks the yield curve has a functional form and then derives the implied forward rates from that. Another approach is to move in the other direction. A yield curve can be generated by assuming that the forward rates will behave dynamically. In the most typical scenario, it is assumed that the yields R() are given by the functional form and depend on a single variable, rt, which stands for the current short rate.

R(rt, u, t)=A(u, t)-C(u, t)rt

Functions A(u,t) and c(u,t) can be constructed in various ways, so that the yield curve can be upward, downward sloping or hump shaped.

5) State the limitations of the Extreme Value Theory (EVT).

Ans) Extreme Value Theory is a branch of statistics dealing with stochastic behavior of extreme events found in the tails of probability distributions. A stochastic model represents a situation where uncertainty is present. In other words, it’s a model for a process that has some kind of randomness. EVT aims to predict probabilities for rare events greater (or smaller) than previous recorded events. For example, EVT might be used in seismology to predict the next mega-earthquake in California, the last of which was in 1857.

EVT originated from astronomy and the need to keep or reject outliers in data. Outliers are stragglers — extremely high or extremely low values — in a data set that can throw off your stats. For example, if you were measuring children’s nose length, your average value might be thrown off if Pinocchio was in the class. EVT has developed into a theory that is applicable to almost every area of science and business. For example, the theory can model and predict a diverse range of phenomena such as the maximum heights of ocean waves or the strength of financial markets. The theory, which uses extreme value distributions, is widely used in economics, finance, materials science, reliability engineering and many other fields.

The debate that follows is credited to Bensalah. The majority of articles in the literature take the same tack. It is challenging to define the extreme observations for n-dimension vectors (n > 1), which is a fundamental issue when using the EVT results in a multivariate scenario. This is because there is no accepted definition of order in a vectorial space with dimensions greater than 1. Estimating the extreme marginal distribution for each asset is suggested as a solution to this issue (use the maxima for the short positions w, and the minima for the long positions). A solution for the extreme p-quantiles VaR, computation of the correlations q between the series of maxima and minima, and determination of the extreme VaR of a portfolio of IV assets are attempted:

Unfortunately, the distribution of the extremes for the total position is not always determined by the joint distribution of the extreme marginal distributions. In other yards, big changes in the log change of the values for the various assets may not always translate into extreme changes in the portfolio as a whole. The mix of the portfolio (position on each instrument) and the connections (dependencies or correlations) between the different assets will determine this.

6) Discuss the special features of the insurance sector which necessitates its regulation?

Ans) The special features of the insurance sector which necessitates its regulation are as follows:

India with about 200 million middle class household shows a huge untapped potential for players in the insurance industry. Saturation of markets in many developed economies has made the Indian market even more attractive for global insurance majors. The insurance sector in India has come to a position of very high potential and competitiveness in the market. Indians, have always seen life insurance as a tax saving device, are now suddenly turning to the private sector that are providing them new products and variety for their choice.

Consumers remain the most important centre of the insurance sector. After the entry of the foreign players the industry is seeing a lot of competition and thus improvement of the customer service in the industry. Computerization of operations and updating of technology has become imperative in the current scenario. Foreign players are bringing in international best practices in service through use of latest technologies. The insurance agents still remain the main source through which insurance products are sold. The concept is very well established in the country like India but still the increasing use of other sources is imperative.

At present the distribution channels that are available in the market are listed below:

1. Direct selling

2. Corporate agents

3. Group selling

4. Brokers and cooperative societies

5. Bancassurance

Internet and technology has helped a lot to insurer. Now policy procuring through online is economical than buying the same plan from agent. The major problem is not getting the support from the agent for that policy if there is a claim or maturity. The person has to keep direct contact with the company.

From 2010, the no of advisors have decreased in the industry. The no of agents declined by 29% from March 2010 to March 2013. Also, it is expected that more agents will leave the industry. Under this situation, Claim management will be tougher for the companies. As people buy insurance because of the face value of agents, assistance of them is highly essential for good business.

From the year 2013, it is very clear that traditional plans have gained more weightage over ULIP. As traditional plans are long term products, insurer need to focus more on this. Customer retention and servicing is the key to remain in business. Even if in new pension plan, the capital protection features demands more policy servicing. Here investment and servicing are important for the companies. Above all, Policy administration is the most difficulty area to provide customer servicing.

Customer satisfaction on service levels of life insurers has improved on several counts, positively impacting insurers’ customer loyalty scores, a study by conducted by market research firm IMRB International has found. As a result, customer loyalty scores have improved in terms of services as well as product-related aspects. “The study shows that close to 60% of the customers are ‘truly loyal’ to their insurance providers, which is significantly better than 2014,” the survey noted.

7) Differentiate between deterministic and stochastic chain reserving techniques

Ans) The differences between deterministic and stochastic chain reserving techniques are as follows:

Deterministic Chain Reserving Techniques

Using the ratios of the payments in subsequent development years rather than the ratios of the cumulative payments is an alternative variant of the chain-ladder approach.

When this method is utilised, some stability is lost, especially in the higher development years where there are fewer open claims and more erratic pay-outs. However, since all of the payment data from earlier origin years may be used, additional information is available. The chain-ladder ratios for later development periods are based on a very small sample size. The additional data points may be useful in achieving stability.

Additionally, the ratios of payments in succeeding years can be transformed into cumulative ratios to maintain the same structure of the valuation base as for the chain-ladder technique. The original or adjusted for inflation payments that we provided in the previous scheme are both subject to the ratios technique. Finally, as with the chain-ladder method, the unpaid claim estimates may be discounted or undiscounted.

Stochastic Chain Reserving Technique

The chain ladder method, which we previously covered, is a deterministic approach to estimating claim amounts.

Stochastic models generate predictions about the value of anticipated future payments as well as the variability in those predictions. These models give estimates of both the expected value of the future payments and the variance around that expected value by making assumptions about the random component of the model, which enables the validity of the assumptions to be checked statistically.

A stochastic reserving method's drawback is that it simulates an incredibly complicated chain of events with a small number of parameters, which leaves it vulnerable to accusations that its assumptions are overly optimistic.

The availability of measures of the accuracy of reserve estimates is, in general, the main benefit of stochastic reserving models, and in this regard, emphasis is placed on the root mean squared error of prediction.