Week 3 Detailed Learning Outcomes

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

Actuarial Practice #

Risk Management #

• Understand the classification of risks according to likelihood and impact
• Explain what the top long term risks are according to likelihood and impact
• Explain the main elements of the definition of risk management, and how they translate in the insurance context
• Provide examples of typical actuarial work in relation to risk management in APRA regulated companies
• Understand the basic missions of APRA and ASIC, and how they differ
• Explain what Enterprise Risk Management is
• Discuss evidence of climate change
• List and explain four key relevant areas to actuarial practice, which will likely be impacted by climate change
• Discuss how actuaries can make a difference in addressing climate risk

Actuarial Techniques #

PV and accumulated value of annuity-immediate and annuity-due #

Note that “annuity-immediate” and “annuity in arrears” are equivalent names.

Formulae to memorize #

1. $$a_{\overline{n\hspace{.2em}|}}=\frac{1-v^n}{i}$$
2. $${\ddot{a}}_{\overline{n\hspace{.2em}|}}=(1+i)a_{\overline{n\hspace{.2em}|}}=\frac{1-v^n}{d},\text{where }d=1-v.$$
3. $$a_{\overline{n\hspace{.2em}|}}\approx n{v}^{\frac{n+1}{2}}$$
4. $${\ddot{a}}_{\overline{n\hspace{.2em}|}}\approx n{v}^{\frac{n-1}{2}}$$

Formulae with interpretation #

1. $$1=i{a}_{\overline{n\hspace{.2em}|}}+v^n$$
2. $$(1+i{)}^n=i{s}_{\overline{n\hspace{.2em}|}}+1$$

Properties #

1. $$a_{\overline{n\hspace{.2em}|}}<{\ddot{a}}_{\overline{n\hspace{.2em}|}}<n$$
2. $$a_{\overline{n\hspace{.2em}|}}\downarrow i,s_{\overline{n\hspace{.2em}|}}\uparrow i$$
3. $${\ddot{a}}_{\overline{n\hspace{.2em}|}}=1+a_{\overline{n-1\hspace{.2em}|}}$$
4. $$a_{\overline{n\hspace{.2em}|}}\text{ with rate of interest }i\text{ is written as }{a}_{\overline{n\hspace{.2em}|}@i}\text{ or }{a}_{\overline{n\hspace{.2em}|}i}$$

$m$-year deferred annuities #

1. $${}_{m|}{a}_{\overline{n\hspace{.2em}|}}=v^m{a}_{\overline{n\hspace{.2em}|}}$$
2. $${}_{m|}{\ddot{a}}_{\overline{n\hspace{.2em}|}}=v^m{\ddot{a}}_{\overline{n\hspace{.2em}|}}$$

Other formulae #

1. $${}_{m|}{a}_{\overline{n\hspace{.2em}|}}=a_{\overline{n+m\hspace{.2em}|}}-a_{\overline{m\hspace{.2em}|}}$$
2. $${}_{m|}{\ddot{a}}_{\overline{n\hspace{.2em}|}}={\ddot{a}}_{\overline{n+m\hspace{.2em}|}}-{\ddot{a}}_{\overline{m\hspace{.2em}|}}$$
3. $${}_{m|}{\ddot{a}}_{\overline{n\hspace{.2em}|}}=(1+i{)}_{m|}{a}_{\overline{n\hspace{.2em}|}}$$
4. $${}_{m|}{a}_{\overline{n\hspace{.2em}|}}\approx n{v}^m{v}^{\frac{n+1}{2}}=n{v}^{m+\frac{n+1}{2}}$$ where $$m+\frac{n+1}{2}=\frac{(m+1)+(m+2)+...+(m+n)}{n}.$$
5. $${}_{m|}{\ddot{a}}_{\overline{n\hspace{.2em}|}}\approx n{v}^m{v}^{\frac{n-1}{2}}=n{v}^{m+\frac{n-1}{2}}$$ where $$m+\frac{n-1}{2}=\frac{(m+1)+(m+2)+...+(m+n-1)}{n}.$$

Annuities payable $m$-thly (more frequently) ( $m=2,4,12,52,365,\dots$ ) #

1. $$a_{\overline{n\hspace{.2em}|}}^{(m)}=v^n{s}_{\overline{n\hspace{.2em}|}}^{(m)}$$
2. $$s_{\overline{n\hspace{.2em}|}}^{(m)}=(1+i{)}^n{a}_{\overline{n\hspace{.2em}|}}^{(m)}$$
3. $${\ddot{a}}_{\overline{n\hspace{.2em}|}}^{(m)}=v^n{\ddot{s}}_{\overline{n\hspace{.2em}|}}^{(m)}$$
4. $${\ddot{s}}_{\overline{n\hspace{.2em}|}}^{(m)}=(1+i{)}^n{\ddot{a}}_{\overline{n\hspace{.2em}|}}^{(m)}$$

important formulae #

1. $$a_{\overline{n\hspace{.2em}|}}^{(m)}=\frac{1-v^n}{i^{(m)}}$$ This is derived from the first principle where ${(1+{\displaystyle \frac{i^{(m)}}{m}})}^m=1+i$.
2. $${\ddot{a}}_{\overline{n\hspace{.2em}|}}^{(m)}={(1+i)}^{\frac{1}{m}}{a}_{\overline{n\hspace{.2em}|}}^{(m)}({\ddot{a}}_{\overline{n\hspace{.2em}|}}^{(m)}>a_{\overline{n\hspace{.2em}|}}^{(m)})$$

Alternative expressions by time unit change #

• New time unit: $\frac{1}{m}$th of a year
• $j={\displaystyle \frac{i^{(m)}}{m}}$: effective interest rate per new time unit
• $v^{\frac{1}{m}}={\displaystyle \frac{1}{1+\frac{i^{(m)}}{m}}}$: discount factor per new time unit

Hence: $$a_{\overline{n\hspace{.2em}|}}^{(m)}=\frac{1-v^n}{i^{(m)}}=\frac{1}{m}\frac{1-(v^{\frac{1}{m}}{)}^{nm}}{\frac{i^{(m)}}{m}}=\frac{1}{m}{a}_{\overline{nm\hspace{.2em}|}@j}.$$

Annuities payable less frequently #

• Change time unit: 2-year
• 2-year discount factor: $v^2$
• 2-year effective interest rate: $1+j=(1+i{)}^2$

Hence: $$PV=a_{\overline{n\hspace{.2em}|}@j}=\frac{1-(v^2{)}^n}{j}$$ $$PV={\ddot{a}}_{\overline{n\hspace{.2em}|}@j}=(1+j)a_{\overline{n\hspace{.2em}|}@j}=(1+j)\frac{1-v^{2n}}{j}$$

Remarks:

1. $PV$ can be derived from first principles.

2. How to find the $PV$ of an annuity-immediate payable every 1.5 years for 30 years.

Annuities with varying interest rates #

For the annuity-immediate: $$PV=a_{\overline{n_1\hspace{.2em}|}@i_1}+(1+i_1{)}^{-n_1}{a}_{\overline{n_2\hspace{.2em}|}@i_2}$$

Accumulated Value: $$s_{\overline{n_1\hspace{.2em}|}@i_1}(1+i_2{)}^{n_2}+s_{\overline{n_2\hspace{.2em}|}@i_2}$$

Annuities with varying benefits (payments) #

$$PV=a_{\overline{2n\hspace{.2em}|}@i}+a_{\overline{n\hspace{.2em}|}@j}=v{\ddot{a}}_{\overline{n\hspace{.2em}|}@j}+2a_{\overline{n\hspace{.2em}|}@j}$$

where $1+j=(1+i{)}^2$.