# CM1 Actuarial Mathematics

WHY DO WE PAY INTEREST?

Since, lenders provide borrowers with money for a specified period of time, lender will not be able to use that money during that time, so to make up for this inconvenience along with protecting lenders from the risk of default by the borrower, lender demands interest which may either be fixed in monetary terms or in percentage with respect to the principal amount.

WHAT IS FORCE OF INTEREST?

If we consider a nominal interest rate convertible very frequently (e.g. every second), we are no longer thinking of a fund that suddenly acquires an interest payment at the end of each interval, but of a fund that steadily accumulates over the period as interest is earned and added. In the limiting case, the amount of the fund can be considered to be subject to a constant ‘force’ causing it to grow which is known as force of interest.

WHAT DO YOU MEAN BY DISCOUNTING?

When we are interested in finding out the present value of any amount by pulling it back to the present time at a given rate of interest, it is known as discounting of the cash flows.

WHAT IS A TREASURY BILL? WHO ISSUES IT?

Treasury bills are short-term loans made by the government, these are generally issued at discount and redeemed at par value.

STATE THE EQUATIONS OF VALUE?

An equation of value equates the present value of money received to the present value of money paid out: ‘PV income = PV outgo’

HOW DO WE CALCULATE PROSPECTIVE LOAN OUTSTANDING?

Calculating the loan prospectively involves looking forward and calculating the present value at the current point in time of future cash flows, the loan outstanding at time t is the present (or discounted) value at time t of the future repayment instalments.

HOW DO WE CALCULATE RETROSPECTIVE LOAN OUTSTANDING?

Calculating the loan retrospectively involves looking backwards and calculating the accumulated value of past cash flows, the loan outstanding at time t is the accumulated value at time t of the original loan less the accumulated value at time t of the repayments to date.

WHAT IS DISCOUNTED PAYBACK PERIOD?

The DPP tells us how long it takes for the project to move into a position of profit. Other things being equal, a project with a shorter discounted payback period is preferable to a project with a longer discounted payback period because it will start producing profits earlier.

WHAT DO YOU UNDERSTAND BY INTERNAL RATE OF RETURN?

In economics and accountancy terms, the yield per annum is often referred to as the ‘internal rate of return’ (IRR) or the ‘yield to redemption’. The internal rate of return for an investment project is the effective rate of interest that equates the present value of income and outgo, i.e. it makes the net present value of the cash flows equal to zero.

WHAT IS NET PRESENT VALUE OF A PROJECT?

Net present value is defined as the difference between the value of the positive cash flow and the value of the negative cash flow. The net present value is similar to the accumulated profit, the only difference being that we are now looking at the value at the outset (which, by definition, is a fixed date), rather than the value at the end of the project. A higher net present value indicates a more profitable project.

WHAT IS PAYBACK PERIOD?

Payback period refers to the amount of time it takes to recover the cost of an investment. It is calculated by dividing the initial investment by the average cash flows.

WHICH IS BETTER: PAYBACK PERIOD OR DISCOUNTED PAYBACK PERIOD?

Since the discounted payback period factors in the time value of money, it shows how long it will take to recoup an investment based on observing the present value of the project’s projected cash flows. Thus, DPP is better than PP.

IS TREASURY BILL A PART OF MONEY MARKET OR CAPITAL MARKET?

Treasury bills or T-bills, which are money market instruments, are short term debt instruments issued by the Government of India and are presently issued in three tenors, namely, 91 day, 182 day and 364 day. Treasury bills are zero coupon securities and pay no interest.

WHERE EQUATIONS OF VALUE IS USED?

Equations of value are used throughout actuarial work. For example:  the ‘fair price’ to pay for an investment such as a fixed-interest security or an equity (ie PV outgo) equals the present value of the proceeds from the investment, discounted at the rate of interest required by the investor.  Also, the premium for an insurance policy is calculated by equating the present value of the expected amounts received in premiums to the present value of the expected benefits and other outgo.

WHAT ARE THE MAIN COMPONENTS OF LOAN SCHEDULE?

The loan schedule usually consists of original loan amount, the loan balance at each payment, the interest rate, the amortisation period, the total payment amount, and the proportion of each payment that is made up of interest vs. principal.

HOW IRR IS CALCULATED?

IRR is calculated as a discount rate where NPV is equal to zero.

STATE THE BENEFITS OF USING NPV FOR PROJECT APPRAISAL?

The most important advantage of NPV is that it takes into account time value of money. Along with this, it makes adjustment for cost of capital the risk inherent in making projections about the future. Also, NPV is easy to calculate with its additive nature where multiple projects’ NPV can be added up to get the actual view of wealth creation.

WHAT DO YOU MEAN BY CONVEXITY OF A BOND?

Convexity gives a measure of the change in duration of a bond when the interest rate changes. Thus, ‘Convexity’ refers to the shape of the graph of the present value as a function of the interest rate.  For a series of cash flows with the same discounted mean term, a series consisting of payments paid close together will have a low convexity, whereas a series that is more spread out over time will have a higher convexity.

HOW VOLATILITY OF BOND IS DETERMINED?

The effects of changes in interest rates on the cash flows generated by an asset or required by a liability can be quantified by calculating the volatility. Both the volatility and the discounted mean term provide a measure of the average ‘life’ of an investment. This is important when considering the effect of changes in interest rates on investment portfolios since an investment with a longer term will in general be more affected by a change in interest rates than an investment with a shorter term.

WHAT IS DISCOUNTED MEAN TERM OF AN INVESTMENT?

One measure of interest rate sensitivity is the duration, also called Macaulay duration or discounted mean term (DMT). This is the mean term of the cash flows weighted by present value.

WHAT IS IMMUNISATION OF A PORTFOLIO?

Immunisation refers to the selection of an asset portfolio that will protect this surplus against small changes in the interest rate.

WHAT ARE THE CONDITIONS IN REDINGTON THEORY OF IMMUNISATION?

In the 1950s, the actuary Frank Redington derived the three conditions that are required to achieve immunisation. The conditions for Redington’s immunisation may be summarised as follows: 1. the value of the assets at the starting rate of interest is equal to the value of the liabilities. 2. The volatilities of the asset and liability cash flow series are equal. 3. The convexity of the asset cash flow series is greater than the convexity of the liability cash flow series.

An index-linked bond is a bond which has payment of interest income on the principal that is related to a specific price index, usually the Consumer Price Index (CPI) or Wholesale Price Index (WPI).

EXPLAIN RUNNING YIELD OF A BOND?

The running yield of a bond is the yield generated by the payment of the coupons alone ignoring the redemption payment.

HOW RUNNING YIELD IS DIFFERENT FROM GROSS REDEMPTION YIELD OF A BOND?

A redemption yield is the yield obtained by an investor who holds the bond until redemption whereas the running yield is the yield generated by the payment of the coupons alone (ignoring the redemption payment).

WHAT IS PAR YIELD? IS IT ALWAYS POSITIVE?

Par yield, also referred to as the par rate, is when the coupon rate and the yield of a bond are equal and the bond price will be the same as its nominal value, also called its par value.A par yield is the coupon rate at which bond prices are zero. A par yield curve represents bonds that are trading at par.

WHAT IS ANNUITY IN PERPETUITY?

‘In perpetuity’ means that the payments continue forever.

WHAT IS SPOT RATE? HOW IS IT DIFFERENT FROM FORWARD RATE?

The n-year spot rate is a measure of the average interest rate over the period from now until n years’ time whereas the discrete-time forward rate is the annual interest rate agreed at time 0 for an investment made at time t for a period of r years.

CAN FORWARD RATE BE NEGATIVE?

Forward interest rates are negative whenever the yield curve is negatively sloped.

WHAT IS TERM STRUCTURE OF INTEREST RATE?

Term structure of interest rates refers to the relationship between interest rates and different terms or maturities of bonds.

WHY I IS GREATER THAN I12?

Due to the effects of accumulation, i is greater than i12.

WHY D IS LESS THAN D12?

Due to the effects of compounding,  we are ‘annualising’ a pthly effective discount rate i.e. d12.  For example, to ‘annualise’ an effective discount rate of 0.5% per month, we would make it twelve times it to get 6% pa. This is not the correct annual effective discount rate. The 6% we have obtained is referred to as a nominal discount rate convertible monthly.

WHAT IS TIME VALUE OF MONEY?

The time value of money is the concept of recognising the earning potential of money over time where a sum of money is worth more now than the same sum will be at a future date. The time value of money is also referred to as the present discounted value.

DESCRIBE THE PROCESS OF DATA ANALYSIS?

The key steps in a data analysis process can be described as follows:

1. Develop a well-defined set of objectives which need to be met by the results of the

data analysis.

1. Identify the data items required for the analysis.
2. Collection of the data from appropriate sources.
3. Processing and formatting data for analysis, eg inputting into a spreadsheet,

database or other model.

1. Cleaning data, eg addressing unusual, missing or inconsistent values.
2. Exploratory data analysis, which may include:

(a) Descriptive analysis; producing summary statistics on central tendency and

(b) Inferential analysis; estimating summary parameters of the wider population

of data, testing hypotheses.

(c) Predictive analysis; analysing data to make predictions about future events

or other data sets.

1. Modelling the data.
2. Communicating the results.
3. Monitoring the process; updating the data and repeating the process if required.
4.

DIFFERENCE BETWEEN REPRODUCIBILITY AND REPLICATION.

Reproducibility is linked to the concept of replication which refers to someone repeating an experiment and obtaining the same (or at least consistent) results. Due to the possible difficulties of replication, reproducibility of the statistical analysis is often a reasonable alternative standard.

DIFFERENCE BETWEEN CENSORING AND TRUNCATED DATA.

Censored data occurs when the value of a variable is only partially known. Truncated data occurs when measurements on some variables are not recorded so are completely unknown.

WHY MODELS ARE USED?

A model is an imitation of a real-world system or process. Models of many activities can be developed, for example:  the economy of a country, the workings of the human heart, and  the future cash flows of the broker distribution channel of a life insurance company.

DIFFERENCE BETWEEN DETERMINISTIC AND STOCHASTIC MODELS.

A stochastic model is one that recognises the random nature of the input components. A model that does not contain any random component is deterministic in nature. In a deterministic model, the output is determined once the set of fixed inputs and the relationships between them have been defined. By contrast, in a stochastic model the output is random in nature like the inputs, which are random variables. The output is only a snapshot or an estimate of the characteristics of the model for a given set of inputs. Several independent runs are required for each set of inputs so that statistical theory can be used to help in the study of the implications of the set of inputs. A deterministic model is really just a special (simplified) case of a stochastic model.

WHAT IS SCENARIO BASED TESTING? HOW IS IT DIFFERENT FROM SENSITIVITY TESTING?

A scenario-based model would take into consideration a particular scenario; that is a series of input parameters based on this scenario. Different scenarios would be useful in decision analysis as one can evaluate the expected impact of a course of action. Whereas, sensitivity testing refers to the approach of using a deterministic model with changes in one or more of the assumptions to see the range of possible outcomes that might occur.

WHAT IS ANNUITY CERTAIN?

An annuity-certain provides a series of regular payments in return for a single premium (i.e. a lump sum) paid at the outset.

WHAT IS A PURE ENDOWMENT PLAN?

A pure endowment is an insurance policy which provides a lump sum benefit on survival to the end of a specified term usually in return for a series of regular premiums. For example, a payment of £50,000 is made if a life now aged 30 survives to age 60, but no payment is made if this life dies before age 60.

WHAT ARE VARIOUS ASSURANCE PRODUCTS?

whole life assurance, term assurance, pure endowment & endowment assurance.

WHAT ARE SOME FEATURES OF ANNUITIES?

An annuity is a regular series of payments. An annuity-certain is an annuity payable for a definite period of time: the payments do not depend on some factor, such as whether a person is alive or not. If payments are made at the end of each time period, they are paid in arrears. If they are made at the beginning of each time period, they are paid in advance. An annuity paid in advance is also known as an annuity-due. Where the first payment is made during the first time period, this is an immediate annuity. Where no payments are made during the first time period, this is a deferred annuity. If each payment is for the same amount, this is a level annuity. If payments increase (decrease) each time by the same amount, this is a simple increasing (decreasing) annuity.

RATE ON BASIS OF PREMIUM: TERM ASSURANCE, ENDOWMENT POLICY AND WHOLE LIFE ASSURANCE?

Endowment>whole life>term assurance

HOW NOMINAL RATE OF INTEREST IS DIFFERENT FROM EFFECTIVE RATE OF INTEREST?

Effective rates have interest paid once per measurement period whereas nominal rates are paid more frequently than once per measurement period.

WHAT IS ACCUMULATED PROFIT?

Accumulated profit is the accumulated value of the net cash flows as at the time of the last payment.

WHAT IS SELECT MORTALITY?

Policyholders with a duration of one year, say, will have recently satisfied the company about their state of health. We would therefore expect their mortality to be better than that of policyholders of the same age with duration of, say, 3 years who passed the medical underwriting hurdle several years ago. Thus, the mortality of the recently joined policyholders is called select mortality, and we expect it to be better than that of longer duration policyholders, whose mortality we call ultimate mortality.

HOW UDD IS DIFFERENT FROM CFM?

Under uniform distribution of deaths(UDD) assumption, for an individual aged exactly x , the probability of dying on one particular day over the next year is the same as that of dying on any other day over the next year whereas under constant force of mortality(CFM), for integer ages, we suppose that the force of mortality is constant.

The gross premium is the premium required to meet all the costs under an insurance contract, and is the premium that the policyholder pays.  It is also sometimes referred to as the office premium which contains impact for loaded profits, cost of capital, adjustments for competition and other loadings.

WHEN PROSPECTIVE AND RETROSPECTIVE RESERVES ARE EQUAL?

If the retrospective and prospective reserves are calculated on the same basis, and this basis is the same as the basis used to calculate the premiums used in the reserve calculation, then the retrospective reserve will be equal to the prospective reserve.

WHAT IS JOINT LIFE ANNUITY?

A joint life annuity covers two lives, where the regular payments are contingent on the survival of one or both of those lives.

WHEN LAST SURVIVOR POLICY BEGINS PAYMENT?

Two common types of policy are: an annuity payable to a couple while at least one of them is alive, and  an assurance payable on the second death of a couple. These are both examples of last survivor policies, where the payment is contingent on what happens to the second life to die, rather than the first.

WHAT IS MORTALITY PROFIT?

The mortality profit is defined as the difference between Expected Death Strain & Actual Death Strain where EDS is the amount the company expects to pay out, in addition to the year-end reserve for a policy and ADS is the amount it actually pays out, in addition to the year-end reserve.

HOW MULTIPLE STATE MODELS IS DIFFERENT MULTIPLE DECREMENTS MODEL?

Multiple state models are well suited to valuing cash flows that are dependent on multiple transitions, such as of health insurance contracts. A multiple decrement model is a multiple state model which has:  one active state, and  one or more absorbing exit states.

WHAT IS UNIT FUND?

Unit fund is the total value of the units in respect of the policy at any time.

HOW ULIP WORKS?

With unit-linked contracts the policyholders are entitled units, which represent a portion of a fund or funds of investments managed by the life insurer referred to as the unit fund of the policy. The unit fund value moves in line with the performance of the backing investments. The non-unit fund represents the accumulation of all cash flows paid in that are not used to buy units, less all cash flows paid out that have not arisen from the cancellation of units. As such it represents the accumulation of the company’s profits from the policy at any time.

WHAT ARE THE VARIOUS CONSTITUTES OF NON-UNIT FUND?

Non-unit fund includes unallocated premiums, bid/offer spread and unit fund charges. Non-unit benefits include, for instance, any sum insured payable on death in excess of the value of the units, or any guaranteed maturity value in excess of the value of the units.

The bid price is the cash-in value of each unit and the offer price is the price that has to be paid to purchase a unit in the fund. The difference between the two (with the offer price being greater than the bid price) is called the bid-offer spread.

WHAT ARE THE STEPS OF PROFIT TESTING?

The steps in profit testing starts with deciding on the structure of the product, building a model to project cash flows for the product then deciding on a risk discount rate and profit criterion then decide on some ‘first draft’ premiums (conventional product) / charges (unit-linked product), vary the premiums / charges until our profit criterion is met, keep varying premiums until we have a product that meets the profitability criterion, is marketable, and is resilient to adverse future experience.

HOW PROFIT MARGIN IS CALCULATED?

Profit margin is the NPV expressed as a percentage of the EPV of the premium income where the risk discount rate is used to do the discounting in both the numerator and the denominator.