Week 2 Detailed Learning Outcomes

At the end of your learning for this week, you should be able to master the following contents.

Actuarial Practice #

Data and predictive analytics #

• Describe how actuarial studies interacts with other fields such as computing, mathematics, and statistics
• Define machine learning, and explain how it fits within the two modelling cultures “data modelling culture” and “algorithmic modelling culture”
• Explain the basic thrust, and contrast the two modelling cultures “data modelling culture” and “algorithmic modelling culture”
• Explain what adverse selection is
• Explain how more precise pricing may lead to less mutuality
• Explain how privacy and discrimination issues may arise from the collection and use of data for insurance purposes

Actuarial Techniques #

Under annual effective interest rate $i$: #

$$A=P(1+i{)}^n$$ $$P=A{v}^n,v=\frac{1}{1+i}$$

Under annual nominal interest rate $i^{(m)}$: #

$$A=P{(1+\frac{i^{(m)}}{m})}^{m\times n}$$ $$P=A{(1+\frac{i^{(m)}}{m})}^{-m\times n}$$

The relationship bewteen $i$ and $i^{(m)}$ #

$$1+i={(1+\frac{i^{(m)}}{m})}^m$$ or $$i^{(m)}=m((1+i{)}^{\frac{1}{m}}-1)$$

The force of interest (instantaneous rate of interest) #

$$\delta =\underset{m\to \mathrm{\infty }}{lim}{i}^{(m)}=\log (1+i)$$

Under $\delta$: #

$$A=e^{\delta n}$$ $$P=e^{-\delta n}$$

$$\begin{array}{rl}A=& P{e}^{\delta dt}\\ =& P(1+\delta dt+{\delta }^2(dt{)}^2+...)\\ \approx & P+P\delta dt\end{array}$$

Conclusion: interest earned from $P$ over $(0,dt)$ is approximately $P\delta dt$

Example: For $\delta =5\mathrm{\%}$, interest earned from $100 over one day is approximately: $100\times 0.05\times \frac{1}{365}$

One year discount factor $v$: #

$$v=(1+i{)}^{-1}={(1+\frac{i^{(m)}}{m})}^{-m}=e^{-\delta }$$

Varying interest rates #

$$A=P(1+i{)}^{t_1}{(1+\frac{i^{(m)}}{m})}^{t_2\times m}{e}^{\delta t_3}$$

$$P=A(1+i{)}^{-t_1}{(1+\frac{i^{(m)}}{m})}^{-t_2\times m}{e}^{-\delta t_3}$$

Describe interest rates accurately #

See this.