For our previously developed random variables, expectations and so forth, you can simply replace the flat term structure with a more general expression, e.g., (v^k) becomes (v(k)). Conceptually this is straight-forward although, naturally, it can make a huge difference in practice. Here are a few examples.
Present value random variable of a life annuity due with annual payments issued to a life aged (x) with a yield curve ({y_t}) becomes
begin{eqnarray*}
Y = sum_{k=0}^{K_x} v(k)
end{eqnarray*}
Expected present value for a life annuity due with annual payments issued to a life aged (x) with a yield curve ({y_t}) becomes
begin{eqnarray*}
ddot{a}(x)_y = sum_{k=0}^{infty} v(k) ~_k p_x
end{eqnarray*}
Present value random variable for a whole life insurance with a benefit of 1 payable immediately upon death issued to a life aged (x) with a yield curve ({y_t}) becomes (v(T_x)).
Expected present value for a whole life insurance with a benefit of 1 payable immediately upon death issued to a life aged (x) with a yield curve ({y_t}) becomes
begin{eqnarray*}
bar{A}(x)_y = int_{0}^{infty} v(t) ~_t p_x mu_{x+t} ~ dt .
end{eqnarray*}
The symbols (ddot{a}(x)_y) and (bar{A}(x)_y ) are non-standard but easily understood in this context.
In general, life insurance contracts are long-term. It is common in actuarial practice to use the long-term rate in traditional actuarial calculations.
In traditional non-participating insurance and annuity products, if mortality and survival experience is known, then cash flows from each year are completely predictable. This means that such contracts can be replicated, so that cash inflows (premiums and interest earnings) can be matched exactly with cash outflows (e.g., benefits and expenses).
For example, if my net cash outflow in 10 years is predicted to be 1,000,000, then I can buy today a 10-year bond that pays 1,000,000 in 10 years – and, I know exactly the price of this bond. The future paths of interest rates are not a concern of mine.