To get an idea as to how policy values evolve over time, we write the policy value at time (t) in a single equation as
begin{eqnarray*}
~_t V &=& int_0^{infty}
left{ (b_{t+s}+E_{t+s})mu_{[x]+t+s} – (P_{t+s}-e_{t+s})
right}
frac{v(t+s)}{v(t)} ~_s p_{[x]+t+s} ds \
&=& int_t^{infty}
left{ (b_r+E_r)mu_{[x]+r} – (P_r-e_r) right}
frac{v(r)}{v(t)} frac{~_r p_{[x]}}{~_t p_{[x]}} dr
end{eqnarray*}
using (r=t+s) and the relation (~_{r-t} p_{[x]+r}=frac{~_r p_{[x]}}{~_t p_{[x]}}). This yields
begin{eqnarray*}
v(t) ~_t p_{[x]} ~_t V
&=& int_t^{infty} left{ (b_r+E_r)mu_{[x]+r} – (P_r-e_r) right} v(r) ~_r p_{[x]} dr
end{eqnarray*}
Now take a (partial) derivative with respect to (t). On the right-hand side, by the fundamental theorem of calculus, we have
begin{eqnarray*}
frac{partial}{partial t}RHS &=& – left{ (b_t+E_t)mu_{[x]+t} – (P_t-e_t) right} v(t) ~_t p_{[x]}.
end{eqnarray*}
On the left-hand side, by the chain-rule of differentiation (twice), we have
begin{eqnarray*}
frac{partial}{partial t}LHS &=&
v(t) ~_t p_{[x]} left{ frac{partial}{partial t}~_t V – (delta_t + mu_{[x]+t}) ~_t V right}.
end{eqnarray*}
Equating both sides yields Thiele’s differential equation
| (frac{partial}{partial t} ~_t V) | (=P_t-e_t) | (+(delta_t + mu_{[x]+t}) ~_t V) | (- (b_t+E_t)mu_{[x]+t}) |
| change in reserve | = net premium income | + increase in reserve due to interest and mortality | – benefit outgo |