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: $$\text{EPV}=X{v}^{t_1}{q}_1+X{v}^{t_2}{q}_2+...+X{v}^{t_n}{q}_n.$$

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

$$\begin{array}{rl}& {\pi }_1=p_1-p_2,p_1={\pi }_1+{\pi }_2+...+{\pi }_n\\ & {\pi }_2=p_2-p_3,p_1={\pi }_2+...+{\pi }_n\\ & ......\\ & {\pi }_k=p_k-p_{k+1},p_k={\pi }_k+...+{\pi }_n\\ & ......\\ & {\pi }_n=p_n,p_n={\pi }_n\end{array}$$

Method 1 to calculate EPV #

$$\text{EPV}=v_1{\pi }_1+v_2{\pi }_2+...+v_k{\pi }_k+...+v_n{\pi }_n,$$ where $v_k=x_1{v}^1+x_2{v}^2+...+x_k{v}^k$ 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).