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 \(i\): #

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

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

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

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

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

The force of interest (instantaneous rate of interest) #

$$ \delta = \lim \limits_{m \rightarrow \infty } i^{(m)} = \log (1+i)$$

Under \(\delta\): #

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

\begin{align*} A =& Pe^{\delta dt} \\ =& P(1+\delta dt + \delta^2 (dt)^2 + ...) \\ \approx & P + P \delta dt \end{align*}

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

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

One year discount factor \(v\): #

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

Varying interest rates #

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

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

Describe interest rates accurately #

See this.