1 Preface

These lecture notes provide a comprehensive treatment of Life Insurance Mathematics following the broad approach used in IFoA actuarial education. The notes integrate actuarial theory with practical R implementation.

2 Table of Contents

  1. Introduction to Life Insurance Mathematics
  2. Survival Models
  3. Mortality Tables
  4. Types of Life Insurance
  5. Life Annuities
  6. Premium Calculations
  7. Policy Values
  8. Reserves
  9. Multiple-Life Functions
  10. Multiple Decrement Models
  11. Profit Testing
  12. Unit-Linked Insurance

3 Chapter 1: Introduction to Life Insurance Mathematics

Life insurance mathematics is the actuarial science of quantifying financial risks associated with human life and mortality. It blends compound interest theory and probability to determine fair insurance premiums, evaluate policy reserves, and ensure companies can meet future claims.

Actuarial life contingencies refer to the mathematical and statistical methods used to quantify the financial risks of uncertain human events, like death, survival, or illness. It is the core framework actuaries use to calculate premiums, reserves, and payouts for life insurance policies and pensions.

To understand and explore calculations involving life assurances and contingencies, it is important to study or one has to have a clear understanding of survival model,life Tables and its characteristics, basic theory of interest and basic knowledge in financial mathematics, annuities, probability theory, and calculus.

Life insurance mathematics models complex financial systems where future cash flows depend on human survival or death. Under the syllabus for Subject CM1, these continuous and discrete risks are quantified by combining compound interest theory with specialized probability distributions of life duration.

The structural foundation relies on the , which establishes that at inception: \[\begin{equation} \text{Expected Present Value (EPV) of Income} = \text{Expected Present Value (EPV) of Outgo} \end{equation}\]

i <- 0.05
v <- 1/(1+i)
v
## [1] 0.952381

4 Chapter 2: Survival Models

4.1 Future Lifetime

Let:

\[T_x\]

denote future lifetime of a life aged x. Let \(T_x\) be the continuous random variable representing the remaining future lifetime of a life currently aged \(x\).

The force of mortality \(\mu_{x+t}\) defines the instantaneous mathematical rate of death at an exact age \(x+t\): \[\begin{equation} \mu_{x+t} = -\frac{1}{{}_tp_x} \frac{d}{dt} ({}_tp_x) = -\frac{d}{dt} \ln({}_tp_x) \end{equation}\] Integrating this fundamental differential equation yields: \[\begin{equation} {}_tp_x = \exp\left( -\int_{0}^{t} \mu_{x+s} \, ds \right) \end{equation}\]

When data is provided at integer ages within a traditional life table (\(l_x\)), the following mathematical interpolations are utilized for a fractional period \(0 < s < 1\):

The code block below demonstrates how to construct a standard actuarial table using the S4 class structure inside the library and execute probability calculations.

Life insurance policies pay a designated contract benefit upon the death of the policyholder.

Annuities are contractual agreements that distribute regular periodic streams of capital contingent on human survival.

Under a uniform constant annual interest rate system, standard insurance formulas and annuity-due frameworks are linked via a strict mathematical identity using the effective discount factor \(d = \frac{i}{1+i}\): \[\begin{equation} A_x = 1 - d\ddot{a}_x \quad \Longleftrightarrow \quad \ddot{a}_x = \frac{1 - A_x}{d} \end{equation}\]

Using the fundamental Equation of Value, we can establish formulas for level premiums and prospective reserves.

The Net Level Premium \(P_x\) for a Whole Life policy of 1 is given by: \[\begin{equation} P_x = \frac{A_x}{\ddot{a}_x} \end{equation}\] If accounting for initial expenses \(\alpha\) and renewal expenses \(\beta\) per annum, the Gross Premium \(G\) satisfies: \[\begin{equation} G \cdot \ddot{a}_x = A_x + \alpha + \beta \cdot \ddot{a}_x \implies G = \frac{A_x + \alpha}{\ddot{a}_x} + \beta \end{equation}\]

The mathematical net reserve at policy duration \(t\) is the expected present value of future benefit outgoes minus the expected present value of future incoming net premiums: \[\begin{equation} {}_tV_x = A_{x+t} - P_x \cdot \ddot{a}_{x+t} \end{equation}\]

4.2 Survival Function

\[S_x(t)=P(T_x>t)\]

4.3 Force of Mortality

\[\mu_x(t)=\frac{f_x(t)}{S_x(t)}\]

4.3.1 R Example

t <- seq(0,100,1)
S <- exp(-0.02*t)
plot(t,S,type="l")

5 Chapter 3: Mortality Tables

5.1 Life Table Functions

  • lx
  • dx
  • qx
  • px

\[d_x=l_x-l_{x+1}\]

\[q_x=\frac{d_x}{l_x}\]

age <- 20:100
lx <- round(100000*exp(-0.04*(age-20)))
head(lx)
## [1] 100000  96079  92312  88692  85214  81873

6 Chapter 4: Types of Life Insurance

6.1 Whole Life Insurance

\[A_x=\sum_{k=0}^{\infty}v^{k+1}{}_kp_xq_{x+k}\]

6.1.1 Worked Example

Assume:

  • Age 40
  • i=5%
  • qx=0.01
i <- 0.05
v <- 1/(1+i)
qx <- 0.01
px <- 1-qx

Ax <- (v*qx)/(1-v*px)
Ax
## [1] 0.1666667

6.2 Term Insurance

\[A_{x:\overline{n}}=\sum_{k=0}^{n-1}v^{k+1}{}_kp_xq_{x+k}\]

k <- 0:9
sum(v^(k+1)*(0.99)^k*0.01)
## [1] 0.07413131

6.3 Pure Endowment

\[ {}_nE_x=v^n{}_np_x \]

6.4 Endowment Assurance

\[A^{1}_{x:\overline{n}}=A_{x:\overline{n}}+{}_nE_x\]

7 Chapter 5: Life Annuities

7.1 Whole Life Annuity Immediate

\[a_x=\sum_{k=1}^{\infty}v^k{}_kp_x\]

7.1.1 Relationship

\[a_x=\frac{1-A_x}{d}\]

Ax <- 0.25
d <- 0.05/1.05
(1-Ax)/d
## [1] 15.75

7.2 Temporary Life Annuity

7.3 Deferred Life Annuity

7.4 Increasing Annuities

7.5 Decreasing Annuities

8 Chapter 6: Premium Calculations

8.1 Equivalence Principle

\[EPV(Premiums)=EPV(Benefits)\]

8.1.1 Net Premium

\[P=\frac{A_x}{a_x}\]

Ax <- 0.15
ax <- 8.75
100000*Ax/ax
## [1] 1714.286

8.2 Gross Premiums

Include:

  • Initial expenses
  • Renewal expenses
  • Claim expenses

9 Chapter 7: Policy Values

9.1 Prospective Policy Value

\[V_t=EPV(Future Benefits)-EPV(Future Premiums)\]

9.2 Retrospective Policy Value

Detailed derivation and examples.

10 Chapter 8: Reserves

10.1 Prospective Reserve

10.1.1 Example

futureBenefits <- 500000
futurePremiums <- 350000
futureBenefits-futurePremiums
## [1] 150000

10.2 Retrospective Reserve

10.3 Recursive Reserve Formula

\[ (V_t+P)(1+i)=q_{x+t}S+(1-q_{x+t})V_{t+1} \]

10.3.1 Full Numerical Examples

Include: - Whole life policy reserve - Term assurance reserve - Endowment reserve - Gross premium reserve

11 Chapter 9: Multiple-Life Functions

11.1 Joint Life Status

\[T_{xy}=\min(T_x,T_y)\]

11.2 Last Survivor Status

\[T_{\overline{xy}}=\max(T_x,T_y)\]

set.seed(123)
Tx <- rexp(1000,0.03)
Ty <- rexp(1000,0.04)
mean(pmin(Tx,Ty))
## [1] 14.54831

12 Chapter 10: Multiple Decrement Models

Derivations and actuarial calculations.

13 Chapter 11: Profit Testing

13.1 Cashflow Model

\[Profit=Income-Outgo\]

premium <- 1200
claims <- 500
expenses <- 200
premium-claims-expenses
## [1] 500

14 Chapter 12: Unit-Linked Insurance

Fund accumulation models.

premium <- 1000
r <- c(0.05,0.07,0.03)
fund <- numeric(length(r))
fund[1] <- premium*(1+r[1])
for(i in 2:length(r)){
 fund[i] <- fund[i-1]*(1+r[i])+premium
}
fund
## [1] 1050.000 2123.500 3187.205

15 References

  1. Actuarial Mathematics for Life Contingent Risks.
  2. Life Insurance Mathematics.
  3. IFoA Core Reading.
  4. R and actuar package documentation.