# Thiele's differential for reserves in multiple state

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$$\frac{d}{dt} (_tV)=\delta (_tV)+(P_t-e_t)-\mu_{x+t}(S_t+E_t-_{t}V)$$

This formula makes sense to me, except for the last term. The book says we need to increase our policy value by $$_tV$$ if the life dies. I understand that when the life dies, we no longer need to hold that amount in reserve for that policyholder. But $$\frac{d}{dt} (_tV)$$ represents the change in the reserve fund value over a very short time. So it makes sense that the fund increases by the interest earned, the premiums received (less the expenses) and decreases by the expected payout of $$\mu_{x+t}(S_t+E_t)$$.

But shouldn't the fund decrease by the expected amount of $$\mu_{x+t}(_{t}V)$$ since when the individual dies, we can release the money which was held as reserve?

I think perhaps where my knowledge is faulty is in the definition of $$_tV$$. Is this the amount required that the insurer needs to hold for future claims or is it the value of the fund (after premiums, expenses and expected payout of benefits)

$$(S_{t} + E_{t}) – _{t}V$$ can be seen as the extra amount required to increase the policy value to the death benefit (and claims expenses).
$$_{t}V$$ is the value of the fund that has already been built up and should be used to fund any future outgo. Therefore, from the insurer’s perspective, only this “extra” amount will be needed if the policyholder dies; so $$_{t}V$$ is just seen as a part of the death benefit (and claims expenses).