Week 3 Learning Outcomes

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. $${a}_{\angl{n}} = \frac{1-v^n}{i}$$
  2. $$\ddot{a}_{\angl{n}} = (1+i){a}_{\angl n} = \frac{1-v^n}{d}, \quad \text{where } d=1-v .$$
  3. $${a}_{\angl{n}} \approx n v^{\frac{n+1}{2}}$$
  4. $${\ddot{a}}_{\angl{n}} \approx n v^{\frac{n-1}{2}}$$

Formulae with interpretation #

  1. $$1 = i {a}_{\angl{n}} + v^n$$
  2. $$(1+i)^n = i {s}_{\angl{n}} + 1$$

Properties #

  1. $${a}_{\angl{n}} < {\ddot{a}}_{\angl{n}} < n$$
  2. $${a}_{\angl{n}} \downarrow i, {s}_{\angl{n}} \uparrow i$$
  3. $${\ddot{a}}_{\angl{n}} = 1 + {a}_{\angl{n-1}}$$
  4. $${a}_{\angl{n}} \text{ with rate of interest }i\text{ is written as } {a}_{\angl{n} @ i}\text{ or }{a}_{\angl{n} i}$$

\(m\)-year deferred annuities #

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

Other formulae #

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

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

  1. $${a}_{\angl{n}}^{(m)} = v^n {s}_{\angl{n}}^{(m)}$$
  2. $${s}_{\angl{n}}^{(m)} = (1+i)^n {a}_{\angl{n}}^{(m)}$$
  3. $${\ddot{a}}_{\angl{n}}^{(m)} = v^n {\ddot{s}}_{\angl{n}}^{(m)}$$
  4. $${\ddot{s}}_{\angl{n}}^{(m)} = (1+i)^n {\ddot{a}}_{\angl{n}}^{(m)}$$

important formulae #

  1. $${a}_{\angl{n}}^{(m)} = \frac{1-v^n}{i^{(m)}}$$ This is derived from the first principle where \(\left(1+\dfrac{i^{(m)}}{m} \right)^m = 1+i\).
  2. $${\ddot{a}}_{\angl{n}}^{(m)} = \left(1+i\right)^{\frac{1}{m}} {a}_{\angl{n}}^{(m)} \ \ \ \ \ \ \left({\ddot{a}}_{\angl{n}}^{(m)} > {a}_{\angl{n}}^{(m)} \right)$$

Alternative expressions by time unit change #

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

Hence: $${a}_{\angl{n}}^{(m)} = \frac{1-v^n}{i^{(m)}} = \frac{1}{m} \frac{1-(v^{\frac{1}{m}})^{nm}}{\frac{i^{(m)}}{m}} = \frac{1}{m} {a}_{\angl{nm}@ j}.$$

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: $$PV = {a}_{\angl{n}@j} = \frac{1-(v^2)^n}{j}$$ $$PV = {\ddot{a}}_{\angl{n}@j} = (1+j){a}_{\angl{n}@j} = (1+j) \frac{1-v^{2n}}{j}$$

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: $$PV = {a}_{\angl{n_1}@i_1} + (1+i_1)^{-n_1} {a}_{\angl{n_2}@i_2}$$

Accumulated Value: $${s}_{\angl{n_1}@i_1} (1+i_2)^{n_2} + {s}_{\angl{n_2}@i_2}$$

Annuities with varying benefits (payments) #

$$PV = {a}_{\angl{2n}@i} + {a}_{\angl{n}@j} = v {\ddot{a}}_{\angl{n}@j} + 2{a}_{\angl{n}@j}$$

where \(1+j = (1+i)^2\).

Calendar 4 October 2021