Week 10 Learning Outcomes

Week 10 Detailed Learning Outcomes

Actuarial Practice #

Retirement saving and pensions #

  1. Explain what is meant by “Bismarck’s Pension Trap.”
  2. Explain what longevity risk is.
  3. With respect to Pay-As-You-Go and Funded systems, describe and explain
  • Main features and differences;
  • Advantages and disadvantages;
  • Their place within a three pillar retirement system.
  1. Explain at a higher level what the first, second and third pillars of a retirement systems are.
  2. Describe the main features of the first and second pillars in Australia.
  3. Briefly describe the potential role of actuaries in the retirement saving and benefits area.

Superannuation systems #

  1. Describe what the accumulation and decumulation phases
  2. With respect to Defined Contribution and Defined Benefit systems, describe and explain
  • Main features and differences;
  • Advantages and disadvantages.
  1. Describe the main role of actuaries in defined contribution and defined benefit superannuation plans.

Readings #

  1. Discuss some considerations for investment strategy in the decumulation phase.
  2. Explain what we mean by “accounts based pension.”
  3. Explain what an investment-linked lifetime annuity is, how it compares to an accounts based pension, and discuss advantages and disadvantages.

Actuarial Techniques #

Calculating Insurance Premiums #

  • You should understand the following notation used in life insurance and be able to explain what they represent in words.

The principle of equivalence: EPV Premiums = EPV Benefit.

EPV of premiums:

  • \(P \times \ddot{a}_{x:\angl{n}}\) in case of \(n\) year payment period
  • \(P \times \ddot{a}_x\) potentially, but only in case of whole of life payments

EPV of Benefits:

  • \(S \times A_x\) in the case of whole life insurance
  • \(S \times A_{x:\angl{n}}^{1}\) in the case of term life insurance
  • \(S \times A_{x:\angl{n}}^{\,\,\,\,1}\) in the case of pure endowment
  • \(S \times A_{x:\angl{n}}\) in the case of endowment

EPV of Life Insurance contracts #

Notation

Notation Used For:
\(A_{x}\) Whole of Life Insurance
\(A_{x:\angl{n}}^{1}\) Term Life Insurance

Calculating EPV

Recall that the probability of death in \(t^{th}\) year from now is: $$\Pr[K(x) = t] = \Pr[t \leq T(x) < t+1] = _tp_x - _{t+1}p_x = \frac{d_{x+t}}{l_x}$$ Therefore for whole of life insurance, the EPV of the death beenfit becomes: $$A_{x} = \sum_{t=0}^{\infty} v^{t+1} \frac{d_{x+t}}{l_x}$$

For term life insurance, we stop the summation of the benefit payments after \(n\) years instead of calculating the present value to \(\infty\): $$A_{x:\angl{n}}^{1} = A_x - \frac{l_{x+n}}{l_n}v^n A_{x+n} = \sum_{t=0}^{n-1} v^{t+1} \ \frac{d_{x+t}}{l_x}$$

EPV of endowments #

Notation

Notation Used For:
\(A_{x:\angl{n}}^{\,\,\,\,1}\) Pure Endowment
\(A_{x:\angl{n}}\) Endowment

Calculating EPV

For pure endowments, the EPV is simply the discounted probability of survival: $$A_{x:\angl{n}}^{\,\,\,\,1} = v^n _tp_x$$

We also know that \(A_{x:\angl{n}} = A_{x:\angl{n}}^{1} + A_{x:\angl{n}}^{\,\,\,\,1}\), therefore the EPV of one unit of endowments is: $$A_{x:\angl{n}} = \sum_{t=0}^{n-1} v^{t+1} \ \frac{d_{x+t}}{l_x} + v^n _tp_x$$

EPV of life annuities #

Notation

Notation Used For:
\(a_x\) Life annuity
\(\ddot{a}_x\) Life annuity
\(a_{x:\angl{n}}\) Term Life annuity
\(\ddot{a}_{x:\angl{n}}\) Term Life annuity

Calculating EPV

By matching up the timing of the payments (discounting factor) and the expected value of payment (probability of survival), we end up with:

\begin{align*} a_x =& \sum_{t=1}^\infty v^t \ _tp_x \end{align*}

\begin{align*} \ddot{a}_x =& \sum_{t=0}^\infty v^t \ _tp_x \end{align*}

\begin{align*} \ddot{a}_{x:\angl n} =& 1+v p_x+...+v^{n-1} \ _{n-1} p_x \\ =& \sum_{t=0}^{n-1} v^t \ _tp_x \end{align*}

\begin{align*} a_{x:\angl n} =& v p_x+ v^2 \ _2p_x +...+v^{n} \ _n p_x \\ =& \sum_{t=1}^{n} v^t \ _tp_x \end{align*}

Relationship between actuarial functions #

You should be able to derive and explain the following relationships in words.

  • \(\ddot{a}_x = 1 + vp_x \ \ddot{a}_{x+1}\)
  • \(A_{x:\angl{n}} + (1-v)\ddot{a}_{x:\angl{n}} = 1\)
  • \(A_x + (1-v) \ddot{a}_x = 1\)

Other formulae that may help with derivation

Payment Term Payments in Advance Payments in Arrears
Forever \(\ddot{a}_{\angl{\infty}} = \frac{1}{d}\) \(a_{\angl{\infty}} = \frac{1}{i}\)
Deterministic term \(\ddot{a}_{\angl{n}} = \frac{1-v^n}{d}\) \(a_{\angl{n}} = \frac{1-v^n}{i}\)
Forever with
contingencies		 \(\ddot{a}_{x} = \frac{1-A_x}{d}\) \(a_{x} = \frac{1}{i}-\frac{A_x}{d}\)
Deterministic term
with contingencies		 \(\ddot{a}_{x: \angl{n}} = \frac{1-A_{x:\angl{n}}}{d}\) \(a_{x: \angl{n}} = \frac{1}{i}-\frac{A_{x:\angl{n}}}{d}\)

Recognise the underlying pattern between each of the formulae, and understand which formulas to use in which contexts.

Calculation of \(A_x\) under a variable interest rate #

We know that under a constant interest rate:

\begin{align*} A_x =& \sum_{t=0}^\infty v^{t+1} \frac{d_{x+t}}{l_x} \\ =& A_{x:\angl{m}}^1 + v^m \ _mp_x \ A_{x+m} \end{align*}

Therefore we can rewrite \(A_x\) in terms of the components before time \(x+m\) and after time \(x+m\):

\begin{align*} A_x =& \sum_{t=0}^\infty v(t+1) \frac{d_{x+t}}{l_x} \\ =& A_{x:\angl{m} \ @i_1}^1 + (1+i_1)^{-m} \ _mp_x \ A_{x+m \ @ i_2}, \end{align*} where \(v(t) = (1+i_1)^{-t}\) for \(t \leq m\) and \(v(t) = (1+i_1)^{-m}(1+i_2)^{-(t-m)}\) for \(t > m\).

Calendar 12 November 2021