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CM1

Actuarial

Mathematics

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ABOUT
Actuarial Mathematics
CM1

CM1 provides a grounding in the principles of actuarial modelling, focusing on deterministic models and their application to financial products. It equips the student with a knowledge of the basic principles of actuarial modelling, theories of interest rates and the mathematical techniques used to model and value cashflows which are either certain or are contingent on mortality, morbidity and/or survival. The subject includes theory and application of the ideas to real data sets using Microsoft Excel.

PATTERN
Actuarial Mathematics

CM1

Duration :-

CM1A

Theoretical Exam

Computer-based

3 hours and 20 minutes
Word

+

CM1B

Computer-based
1 hour and 50 minutes
Excel

Important Note : - Students must sit A + B papers in the same session.

ABOUT ARTHAPOINT
Actuarial Mathematics

CM1

  • Quality video lectures

  • Recorded lectures

  • Quick Doubt Resolution

  • More than 150+ Hours of content

  • Complete syllabus covered

  • 25+ Assignments throughout the course

  • E-Notes

  • Mock Papers

  • Access for 6 months

  • Past year exam discussion

  • Fee - Rs 14,500

DEMO LECTURE
Actuarial Mathematics

CM1

SYLLABUS
Actuarial Mathematics

CM1

Data and basics of modelling

1. Describe the principles of actuarial modelling
Describe why and how models are used, including, in general terms, the use of models for pricing, reserving and capital modelling.
Explain the benefits and limitations of modelling.
Explain the difference between a stochastic and a deterministic model, and identify the advantages/disadvantages of each.
Describe the characteristics, and explain the use, of scenario-based and proxy models.
Describe, in general terms, how to decide whether a model is suitable for any particular application.
Explain the difference between the short-run and long-run properties of a model and how this may be relevant in deciding whether a model is suitable for any particular application.
Describe, in general terms, how to analyse the potential output from a model, and explain why this is relevant to the choice of model.
Describe the process of sensitivity testing of assumptions, and explain why this forms an important part of the modelling process.
Explain the factors that must be considered when communicating the results following the application of a model.


2. Describe how to use a generalised cashflow model to describe financial transactions.
State the inflows and outflows in each future time period, and discuss whether the amount or the timing (or both) is fixed or uncertain for a given cashflow process.
Describe in the form of a cashflow model the operation of financial instruments (like a zero-coupon bond, a fixedinterest security, an index-linked security, a current account, cash on deposit, a credit card, an equity, an interestonly loan, a repayment loan and an annuity certain) and an insurance contract (like endowment, term assurance, contingent annuity, car insurance and health cash plans).




1. Show how interest rates may be expressed in different time periods 
Describe the relationship between the rates of interest and discount over one effective period arithmetically and by general reasoning.
Derive the relationships between the rate of interest payable once per measurement period (effective rate of interest) and the rate of interest payable p (> 1) times per measurement period (nominal rate of interest) and the force of interest.
Calculate the equivalent annual rate of interest implied by the accumulation of a sum of money over a specified period where the force of interest is a function of time.


2. Demonstrate a knowledge and understanding of real and money interest rates


3. Describe how to take into account the time value of money using the concepts of compound interest and discounting.
Accumulate a single investment at a constant rate of interest under the operation of simple and compound interest.
Define the present value of a future payment.
Discount a single investment under the operation of a simple (commercial) discount at a constant rate of discount.


4. Calculate present value and accumulated value for a given stream of cashflows under the following individual or combination of scenarios
Cashflows are equal at each time period.
Cashflows vary with time, which may or may not be a continuous function of time.
Some of the cashflows are deferred for a period of time.
Rate of interest or discount is constant.
Rate of interest or discount varies with time, which may or may not be a continuous function of time.


5. Define and derive the following compound-interest functions (where payments can be in advance or in arrears) in terms i, v, n, d, δ, i(p) and d(p).


6. Show an understanding of the term structure of interest rates 
Describe the main factors influencing the term structure of interest rates.
Explain what is meant by, derive the relationships between and evaluate:
• discrete spot rates and forward rates.
• continuous spot rates and forward rates.
Explain what is meant by the par yield and yield to maturity.


7. Show an understanding of duration, convexity and immunisation of cashflows 
Define the duration and convexity of a cashflow sequence, and illustrate how these may be used to estimate the sensitivity of the value of the cashflow sequence to a shift in interest rates. Evaluate the duration and convexity of a cashflow sequence. Explain how duration and convexity are used in the (Redington) immunisation of a portfolio of liabilities.



1. Define an equation of value
Define an equation of value, where payment or receipt is certain.
Describe how an equation of value can be adjusted to allow for uncertain receipts or payments.
Understand the two conditions required for there to be an exact solution to an equation of value.


2. Use the concept of equation of value to solve various practical problems
Apply the equation of value to loans repaid by regular instalments of interest and capital. Obtain repayments, interest and capital components, the effective interest rate (APR) and construct a schedule of repayments. Calculate the price of, or yield (nominal or real allowing for inflation) from, a bond (fixed-interest or index-linked) where the investor is subject to deduction of income tax on coupon payments and redemption payments are subject to deduction of capital gains tax.
Calculate the running yield and the redemption yield for the financial instrument as described in point 2.
Calculate the upper and lower bounds for the present value of the financial instrument as described in point 2, when the redemption date can be a single date within a given range at the option of the borrower.
Calculate the present value or yield (nominal or real allowing for inflation) from an ordinary share or property, given constant or variable rate of growth of dividends or rents.


3. Show how discounted cashflow and equation of value techniques can be used in project appraisals.
Calculate the net present value and accumulated profit of the receipts and payments from an investment project at given rates of interest.
Calculate the internal rate of return, payback period and discounted payback period and discuss their suitability for assessing the suitability of an investment project.





1. Define various assurance and annuity contracts.
Define the following terms:
Whole-life assurance • Term assurance • Pure endowment • Endowment assurance • Whole-life level annuity • Temporary level annuity • Guaranteed level annuity • Premium • Benefit including assurance and annuity contracts where the benefits are deferred.
Describe the operation of conventional with-profits contracts, in which profits are distributed by the use of regular reversionary bonuses and by terminal bonuses. Describe the benefits payable under the above assurance-type contracts.
Describe the operation of conventional unit-linked contracts, in which death benefits are expressed as combination of absolute amount and relative to a unit fund.
Describe the operation of accumulating with-profits contracts, in which benefits take the form of an accumulating fund of premiums, where either: • the fund is defined in monetary terms, has no explicit charges and is increased by the addition of regular guaranteed and bonus interest payments plus a terminal bonus; or • the fund is defined in terms of the value of a unit fund, is subject to explicit charges and is increased by regular bonus additions plus a terminal bonus (unitised with-profits).



2. Develop formulae for the means and variances of the payments under various assurance and annuity contracts, assuming constant deterministic interest rate.
Define the assurance and annuity factors and their select and continuous equivalents. Extend the annuity factors to allow for the possibility that payments are more frequent than annual but less frequent than continuous. Understand and use the relations between annuities payable in advance and in arrear, and between temporary, deferred and whole-life annuities.
Understand and use the relations between assurance and annuity factors using equation of value, and their select and continuous equivalents.
Obtain expressions in the form of sums/integrals for the mean and variance of the present value of benefit payments under each contract, in terms of the (curtate) random future lifetime, assuming: • contingent benefits (constant, increasing or decreasing) are payable at the middle or end of the year of the contingent event or continuously. • annuities are paid in advance, in arrear or continuously, and the amount is constant, increases or decreases by a constant monetary amount or by a fixed or time-dependent variable rate. • premiums are payable in advance, in arrear or continuously and for the full policy term or for a limited period. Where appropriate, simplify the above expressions into a form suitable for evaluation by table look-up or other means.





1. Define and use assurance and annuity functions involving two lives
Extend the techniques of objectives to deal with cashflows dependent upon the death or survival of either or both of two lives.
Extend the technique of point 1 to deal with functions dependent upon a fixed term as well as age.


2. Describe and illustrate methods of valuing cashflows that are contingent upon multiple transition events
Define health insurance, and describe simple health insurance premium and benefit structures.
Explain how a cashflow, contingent upon multiple transition events, may be valued using a multiple-state Markov Model, in terms of the forces and probabilities of transition.
Construct formulae for the expected present values of cashflows that are contingent upon multiple transition events, including simple health insurance premiums and benefits, and calculate these in simple cases. Regular premiums and sickness benefits are payable continuously and assurance benefits are payable immediately on transition.


3. Describe and use methods of projecting and valuing expected cashflows that are contingent upon multiple decrement events.
Describe the construction and use of multiple decrement tables.
Define a multiple decrement model as a special case of multiple-state Markov model.
Derive dependent probabilities for a multiple decrement model in terms of given forces of transition, assuming forces of transition are constant over single years of age.
Derive forces of transition from given dependent probabilities, assuming forces of transition are constant over single years of age.




1. Define the gross random future loss under an insurance contract, and state the principle of equivalence


2. Describe and calculate gross premiums and reserves of assurance and annuity contracts.
Define and calculate gross premiums for the insurance contract benefits as defined in objective under various scenarios using the equivalence principle or otherwise.
State why an insurance company will set up reserves. 
Define and calculate gross prospective and retrospective reserves. 
State the conditions under which, in general, the prospective reserve is equal to the retrospective reserve allowing for expenses.
Prove that under the appropriate conditions, the prospective reserve is equal to the retrospective reserve, with or without allowance for expenses, for all fixed benefit and increasing/decreasing benefit contracts.
Obtain recursive relationships between successive periodic gross premium reserves, and use this relationship to calculate the profit earned from a contract during the period.
Outline the concepts of net premiums and net premium valuation and how they relate to gross premiums and gross premium valuation respectively


3. Define and calculate, for a single policy or a portfolio of policies (as appropriate):
• death strain at risk
• expected death strain
• actual death strain
• mortality profit


4. Project expected future cashflows for whole life, endowment and term assurances, annuities, unit-linked contracts and conventional/unitised with-profits contracts, incorporating multiple decrement models as appropriate
Profit test life insurance contracts of the types listed above and determine the profit vector, the profit signature, the net present value and the profit margin.
Show how a profit test may be used to price a product, and use a profit test to calculate a premium for life insurance contracts of the types listed above. Show how gross premium reserves can be computed using the above cashflow projection model and included as part of profit testing.


5. Show how, for unit-linked contracts, non-unit reserves can be established to eliminate (‘zeroise’) future negative cashflows, using a profit test model.

Single decrement models

Equation of value and its applications

Theory of interest rates

Multiple decrement and multiple life models

Pricing and reserving

FACULTY
Actuarial Mathematics

CM1

Our faculty is a highly qualified former Delhi University Professor who completed Master's in Economics from Delhi School of Economics. She has taught Economics Honours at Delhi University and has been a guest lecturer at many reputed MBA institutes. Further, she has been teaching CB2 for past three years.

For teaching, we have gone through several books and journals both national and international, and gathered all the information so that it can presented and accessible to the student. The entire study material is prepared from very basic level so that every student can understand every topic from the syllabus. Besides the theory being extensively discussed, an enormous collection of problems has been added so that the student can learn to apply the concepts over a diverse spectrum of difficulty. All the problems have been worked out thoroughly so that in case the student gets stuck, we've got you covered! 

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