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 taxdeductible, 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 riskfree 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(1^{st} card drawn is a king)
= P(2^{nd} card drawn is a king)
= P(3^{rd} card drawn is a king)
.
.
.
= P(17^{th }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 1^{st} 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 n1 to adjust for error in estimation
 Nonparametric tests come handy when sample is small and population also does not obey normal distribution. Nonparametric 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 realworld 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 longterm 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:
E_{x}*qx_{ }>= 5 and E_{x}^{c}*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 overgraduation: 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











