In the prior section, we defined profits based on actual experience. It is also helpful to define a profit at the plan design stage, before experience is realized. Instead of using actual experience, we now remove the carats and employ a (textit{profit test basis}) which is the set of assumptions used to examine the incidence and timing of profits.
The profit during the year (at time (k+1)) is
begin{eqnarray*}
Pr_{k+1} &= &left( _k AS + G_k -e_kright) (1+ i_k) -q_{[x]+k}^{(d)} left(b_{k+1} + E_{k+1}right) \
&~~~~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV – p_{[x]+k}^{(tau)} ~_{k+1} AS .
end{eqnarray*}
When summarizing profits, it can be helpful to remind ourselves that profits are only available for those policies in force at the beginning of the year. Thus, we might define a term such as (Pi_{k+1} = ~_k p_{[x]}^{(tau)} Pr_{k+1} ) for profits discounted for survivorship. The term (textit{profit signature}) is used for the vector of discounted profits (boldsymbol{Pi} = left(Pi_0, Pi_1 ldotsright)^{prime}).
Profits depend on all the assumptions, including assumed interest. To summarize profits, one measure used is the (textit{internal rate of return} (textit{IRR})), defined to be the solution of the equation
begin{eqnarray*}
sum_k left(frac{1}{1+j} right)^k Pi_k = sum_k left(frac{1}{1+j} right)^k ~_{k-1} p_{[x]}^{(tau)} Pr_k = 0,
end{eqnarray*}
where profits are calculated using interest rate (i_k equiv j). The textit{IRR} is the solution of a nonlinear equation and so may not exist or may have multiple solutions. For the multiple solution problem, we can use the (textit{hurdle rate}), defined to be the smallest (textit{IRR}) so that the contract is deemed adequately profitable.
We can summarize profits using an assumed discount rate, (r). Define the (textit{net present value}), also known as the textit{expected present value of future profits}, to be
begin{eqnarray*}
NPV = sum_k left(frac{1}{1+r} right)^k Pi_k = sum_k left(frac{1}{1+r} right)^k ~_{k-1} p_{[x]}^{(tau)} Pr_k,
end{eqnarray*}
where profits are calculated using interest rate (i_k equiv r).
Another profit measure is the (textit{discounted payback period}), the first time that the sum of discounted profits is non-negative. Here, the discounting is for (1) survival and (2) for interest, using risk discount rate (r). That is, the discounted payback period is the smallest value of (m) such that
begin{eqnarray*}
sum_{k=0}^m left(frac{1}{1+r} right)^k Pi_k ge 0.
end{eqnarray*}