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.<\/p>\n
The profit during the year (at time \\(k+1\\)) is
\n\\begin{eqnarray*}
\nPr_{k+1} &= &\\left( _k AS + G_k -e_k\\right) (1+ i_k) -q_{[x]+k}^{(d)} \\left(b_{k+1} + E_{k+1}\\right) \\\\
\n&~~~~~~~-& q_{[x]+k}^{(w)} ~_{k+1} CV – p_{[x]+k}^{(\\tau)} ~_{k+1} AS .
\n\\end{eqnarray*}
\nWhen 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 l\\dots\\right)^{\\prime}\\).<\/p>\n
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
\n\\begin{eqnarray*}
\n\\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,
\n\\end{eqnarray*}
\nwhere 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.<\/p>\n
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
\n\\begin{eqnarray*}
\nNPV = \\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,
\n\\end{eqnarray*}
\nwhere profits are calculated using interest rate \\(i_k \\equiv r\\).<\/p>\n
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
\n\\begin{eqnarray*}
\n\\sum_{k=0}^m \\left(\\frac{1}{1+r} \\right)^k \\Pi_k \\ge 0.
\n\\end{eqnarray*} <\/p>\n