[[Actuarial Notes Wiki|Wiki]] / [[Actuarial Techniques]] / ==Actuarial Assumptions== > ==Assumptions== about the mathematical structure of risk are foundational to [[actuarial techniques]]. ### Statistics [[Statistics]] is the science of collecting, analyzing, and inferring conclusions from numerical data. | Assumption | Description | | ------------------------------- | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[Independence]] | Individual risks are often model as independent (i.e. each risk has no causal impact on the other). However in reality, risks can be correlated. | | [[Law of Large Numbers (LLN)]] | Given a sample of independent, identically distributed (I.I.D) values, the sample mean converges to the **true mean.** | | [[Normality]] | A random variable is called normally distributed if it is generated by: $f(x)=\frac{1}{\sigma \sqrt{2\pi}}e^{-\frac{1}{2}(\frac{x-\mu}{\sigma})^2}$The normal distribution arises in nature because it is generated by the sum of many small variations. | | [[Central Limit Theorem (CLT)]] | The sample mean of any I.I.D values converges to a **normal distribution.** | | [[Homoscedasticity]] | Constant variance of errors, or residuals helps ensure stable predictions. | ### Life Insurance In [[Life Insurance]], assumptions are made about the frequency and costs of life and death, modelled as survival and mortality. | Assumption | Description | | -------------------------- | ------------------------------------------------------------------------------------------------------------ | | [[Life Table]] | Life expectancy | | [[Force of Mortality]] | A measure of the instantaneous rate of death at a given age **x**: $\mu_{x+t}=-\frac{S'_0(x+t)}{S_0(x+t)}$ | | [[Makeham's Law]] | Makeham's law models the force of mortality as: $\mu_x = A + Bc^x$ | | [[Mortality Distribution]] | Mortality rates often follow Gompertz or Weibull distributions. | ### Property and Casualty (P&C) Insurance In [[Property and Casualty Insurance (P&C)|P&C Insurance]], assumptions are made about the frequency of accidents and the costs of property damage. | Assumption | Description | | -------------------------------------- | ----------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- | | [[The Fundamental Insurance Equation]] | Determines the premium, based on the principal that "a rate provides for all<br>costs associated with the transfer of risk."<br><br>Premium= Losses + LAE + UW Expenses + UW Profit | ### Investment & Capital Modelling When making [[Investment|Investments]], fund managers try to allocate capital in a way that maximizes their expected returns, given the risk. | Assumption | Description | | ---------------------------------------------------------------------------------------------------------------------------------------------------------------- | ------------------------------------------------------------------------------------------------------------------- | | [[Modern Portfolio Theory]] a.k.a Mean-Variance Framework | A model for selecting assets to **build a portfolio** that maximizes return for a chosen level of risk. | | [[Capital Asset Pricing Model (CAPM)]]^[Arbitrage Pricing Model and Multifactor models ease CAPM assumptions about transaction costs, taxes, and distributions.] | The **expected rate of return** for an asset or investment is calculated as: $E(R) = R_f + \beta_i(E(R_m) - R_f)$ |