Terminal Reserve for Net Single Premium

[jonah]

Lots of contingent annuity questions floating around lately. I thought perhaps I’d post an elementary problem (i.e. net premiums) I encountered years ago. Here it goes:

A man aged 30 pays the net single premium for a whole life insurance for $10,000. Twenty years later, what is the terminal reserve?

Familiar as I was only with the solution formats of a few problems in terminal reserves concerning the net annual premiums of whole life, term and endowment insurance, I was stumped. Several hours, simulations, and analyses later, I came up with an assessment. My assessment back then (using the Commutation columns of the COMMISSIONERS' 1958 STANDARD ORDINARY MORTALITY TABLE FOR MALES at 3%, or at any rate of interest for that matter) was

$\displaystyle 10,000\frac{{M_{30} }}{{D_{50} }} - \sum\limits_{t = 0}^{19} {10,000\frac{{C_{30 + t} }}{{D_{50} }}}$

At 3%, this amounted to $5,148.17386350266…
I’m sure Sir TKHunny has a more compact assessment.

 

[tkhunny]

Simplify your life. (10000/D50)*(stuff) is much simpler and wonderfully less prone to round-off error during calcultion.

Definition of Mx is sum af all present and later Cx. This makes all your "stuff" M30 - (C30 + C31 + ... + C49) = M50

Terminal Reserves, most fundamentally, are PV(Future Expected Benefits) less PV(Future Expected Premiums)

If you've paid the one and only premium, that leaves PV(Future Expected Benefits) = Single Premium for that age. In this case, 10,000*(M50/D50)