 Some useful concepts in Actuarial Science (including some funny expressions!) - ADITYA MOHAN MATHUR The list below is based on my understanding of the subject. It is also a reflection of what I sometimes heard from my teachers. I am grateful to all of them! Mathematics + Common Sense = Statistics Statistics + Sharper Common Sense = Actuarial Science   Financial Mathematics Security/Instrument: device that ties the issuer/ issuing party to a certain set of cash flows Interest for debt is tax-deductible, dividend for equity/preference share is not. Coupon = interest. Its rate is linked to face value Bonds/debentures and equity/preference shares can be listed on an exchange Index linked security: future cash flows linked to inflation index Loan equated installments: any tenure, equal CFs throughout, interest decreases and capital repayment increases with passage of time Accumulation, A(n), is inverse of Present value, V(n) Actuarial risk is risk of not meeting an obligation when it falls due Interest (i) is paid in arrears, discount(d) is collected in advance Best deal is to borrow on simple interest and lend on compound interest ( at a rate not lower than simple interest rate) Nominal interest= convertible interest = = interest payable p times per year Accumulating factor= effective rate = compound rate =i(p) /p Force of interest = interest payable continuously = ln (1+i) = instantaneous rate of interest i >i(p) > ln(1+i) > d(p) >d (1+effective rate) = (1+inflation rate) (1+real rate) Immediate annuity payable in arrears/advance/continuously = sum of CFs discounted in arrears/advance/continuously Classifications of annuities: 1) immediate or deferred 2) level or increasing or decreasing or customized 3) annual/monthly or, in general, pth ly 4) certain or contingent 5) in arrears or in advance or payable continuously 1 year = 365.25 days = 52.18 weeks Interest rates and bond prices are inversely related APR= annual percentage return is slightly less than two times of Flat rate Flat rate of interest: is simple rate of interest charged on the original amount borrowed for the entire repayment period whereas APR: rate of interest charged on reducing principal amount( reduced by the extent of capital repaid) Payback period: length of time required to recover initial investment without considering interest. If interest is considered it is known as Discounted payback period. Both ignore cash flows after PP/DPP is reached. IRR= internal rate of return from project = rate at which NPV of that project = 0. Generally, if IRR > WACC, project is acceptable Cross over point = rate @ which NPV of 2 projects is equal MWRR: yield earned on the fund over the period by taking into account cash flows and their timings TWRR: also takes into account the growth factors of the CFs in addition to the requirements for MWRR, hence a better indicator of fund manager’s performance Running yield = Coupon rate/ Price of Stock If redemption value > cost price, there is Capital Gain. These can be offset against losses in same year Real yield: yield after adjustment for inflation Equity price inclusive of next dividend is cum dividend price and exclusive of dividend just paid/ immediately payable is Ex dividend Arbitrage opportunity: opportunity to make a risk-free profit Hedging: strategy to minimize future loss Spot rate: rate currently prevailing in the market Forward rate: rate expected to prevail at a future date Factors affecting interest rates: demand, supply, base rates, inflation, money supply, tax rates, rates in other countries Longer dated bonds are more sensitive to interest rate movements comparatively Effective duration/volatility and duration/ DMT: measures for interest rate sensitivity, found out by differentiating PV equations w.r.t. discount rate. They provide measure of life of an investment. Convexity: shows spread between times of payments, double derivative Annual growth factors ~ lognormal distribution     Probability and Mathematical Statistics   Random Variable: variable whose value is subject to chance Actuarial/axiomatic approach to determine probability refer to judging relative chances of occurrence of all outcomes and adjusting them to add up to 1, in order to determine individual probabilities of the outcomes When a series of cards are drawn with replacement or without replacement, from a well shuffled deck of cards, we find that:            P(1st card drawn is a king)          = P(2nd card drawn is a king)          = P(3rd card drawn is a king)             .             .             .          = P(17th card drawn is a king)          =……   If 2 variables are independent, their Covariance has to be 0 but converse is not always true Uniform distribution implies that all values of the variables have equal chances Bernoulli distribution; 1 trial with probability ‘p’ of success.            If ‘n’ trials, it advances to Binomial distribution (n, p)            When the event is rare to occur, ie. probability is very small, it becomes Poisson distribution (λ approximately equal to n*p)   Number of trials required to reach to 1st success: Geometric distribution type 2, if the trial in which success is achieved is included: Geometric distribution type1, similarly Negative Binomial Distributions but for ‘k’ th success Whenever we approximate a discrete distribution to continuous, we apply continuity correction Generally, loss amounts in general insurance ~ log normal distribution, insurance claims ~ gamma distribution Pareto distribution is also known as income distribution Point and interval estimations are ways of estimating population values from sample values MLE technique tries to find value of the parameter such that the likelihood of given sample following a distribution is maximum Degree of freedom represents the number of sample values that are free to vary in our sample Only if sample size is large (>=30) and the samples are independent, normal distribution can be used for hypothesis testing. If the sample is small, we generally assume population follows normality Null hypothesis is an assumption while alternative hypothesis is a claim Sample variance has denominator n-1 to adjust for error in estimation Non-parametric tests come handy when sample is small and population also does not obey normal distribution. Non-parametric tests need not follow assumptions of normality Regression is a measure that attempts to determine strength of relationship between one dependent variable and a series of other changing independent variables Run test is used to check whether the sample values exhibit randomness or not   Models  Model: imitation of a real-world process Actuarial model: involves expression(s) describing relationship between response variable and cofactors Models help studying long term events in compressed time, Cost v/s Benefit analysis can be performed to ensure that the cost of building a model is justified by the benefit derived from it Stochastic: involves probability to tackle uncertainty, Process: description of something dynamic in nature Stationary process: A stochastic process whose statistical properties do not change with time Markov Property: To predict future state, currently occupied state is enough Markov chain is a stochastic process with: Discrete state space Discrete time domain Markov property Markov Jump Process(MJP): Discrete state space Continuous time domain Markov property Markov Jump chain = MJP observed only at times of its transitions Poisson process: probability of an event happening in small period of time is approximately proportional to length of time We do P = , while finding the long-term probability of being in a state, as we are multiplying probability of being in a state with probability of transition happening from every other state to the original state Transition rate = number of transitions/ total waiting time Transition rate denotes the number of transitions in a small period of time, it is obtained by differentiating the transition probability w.r.t time Basic concept to go from state A to state B is equal to waiting time in A + time required to go from A to B in all possible scenarios Initial rate of mortality is mortality applicable at beginning of age interval, whereas central rate of mortality is mortality applicable throughout the age interval If complete expectation of life is 57.65, curtate expectation of life is 57 (integer part) Gompertz law: best for middle ages, simple idea was to show that mortality increases with age, Makeham modified the formula with the idea that accidental deaths are not age dependent Right Censoring in an investigation refers to occurrence of an event due to which information about our study stops coming Decrement refers to leaving the investigation due to the reason under study NA model can be regarded as an approximation of KM model as it tried to modify the model by adjusting for continuous time domain of deaths (using approximations) but failed Proportional hazards model is used to find survival probability for a life using the characteristics of a different life without explicitly finding hazard for both the lives. It is divided Into baseline hazard (depends on time) and parameters and covariates Baseline hazard is generally taken for those set of lives for which it is easy to collect sample, corresponds to the zero value of    covariates Initial exposed to risk is number of people at start of an investigation. However, so many people exactly aged same is almost impossible to find in real life, so, central exposed to risk is used, ie. average number of lives that were available during the period of investigation. A more realistic way is to select people of all ages and account their contribution to exposure To make age definition same for population and deaths, we always make adjustments in population data to minimize estimation errors Two state model/ Poisson model are better than binomial model To make the estimated (crude) rates smooth, we graduate the rates. Aim is to make the rate smooth, adherence to crude data be present and the purpose of graduating be fulfilled (avoiding loss by use of crude rates) The mortality rates for people of same age under life tables and pension tables are different because people taking pensions are generally fitter than those taking life insurance The standardized deviation ~ A.N. (0,1), provided:             Ex*qx >= 5 and  Exc*mx  >= 10. These conditions are generally not fulfilled in reality, hence, the test is not a very good evaluator of t                      graduated rates. To check for bias in the graduated rates: signs test/ cumulative deviations test To check for over-graduation: grouping of signs test/ serial correlations test In reality, grouping of signs test and serial correlations test give the most accurate results NCD was initially introduced to discourage small claims as sometimes cost incurred by assigning a surveyor was greater than the benefit derived from that policy

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