Week 9 Detailed Learning Outcomes

Actuarial Practice #

n/a #

Actuarial Techniques #

EPV of a single contingent payment at (deterministic) time \(n\) #

We have $$\text{EPV} = X v^n p.$$ The expected present value multiplies now \(X\) with two components:

• \(v^n\) is the time value of money;
• \(p\) is the probability of the cash flow \(X\) to occur (hence the “contingency” if \(p<1\)).

EPV of a single payment \(X\) payable at random time \(T\) #

Assume that the time \(T\) of payment \(X\) has the following distribution:

The PV of the payment \(X\) is then

This is a random variable. We still need to take expectations to get the EPV: $$\text{EPV} = Xv^{t_1}q_1 + Xv^{t_2}q_2 + ... + Xv^{t_n}q_n.$$

Note:

1. Payment amount \(X\) is certain.
2. Payment times are random.
3. There is only one payment.

EPV of a series of contingent payments #

Assumptions #

1. \(X_1, X_2, ..., X_k, ..., X_n\) are amounts of contingent payments payable at \(1,2,...,n\).
2. \(p_t\) is the probability \(X_t\) is made at time \(t\), \(t \in [1,n]\).
3. Let \(\pi_k\) be the probability that only \(k\) payments are made, then

\begin{align*} & \pi_1 = p_1-p_2, \quad p_1 = \pi_1 + \pi_2 +...+\pi_n \\ & \pi_2 = p_2-p_3, \quad p_1 = \pi_2 +...+\pi_n \\ & ... \quad\quad ... \\ & \pi_k = p_k - p_{k+1}, \quad p_k = \pi_k +...+\pi_n \\ & ... \quad\quad ... \\ & \pi_n = p_n, \quad p_n = \pi_n \end{align*}

Method 1 to calculate EPV #

$$\text{EPV} = v_1\pi_1 +v_2\pi_2 + ... +v_k\pi_k + ... + v_n\pi_n,$$ where \(v_k = x_1v^1 + x_2v^2 +...+x_kv^k\) is the PV of the payments if there are only \(k\) payments.

Method 2 to calculate EPV #

EPV of a series of contingent payments is the sum of the EPV of each single payment in the series.

$$EPV = \sum_{k=1}^n x_k v^k p_k,$$ where \(x_k v^k p_k\) is the EPV of the \(k\)-th contingent payment.

Comparison of methods #

1. The two methods give the same result.
2. Method 2 is generally preferred (more intuitive as it treats one cash flow at a time, but sometimes Method 1 is more natural).