CHAPTER V

CONNECTIONS BETWEEN ASSURANCE SINGLE PREMIUMS AND
ANNUITY VALUES

1. From the fundamental bases of interest and mortality, expressions have been obtained independently for the values of a few simple types of assurances and annuities. The values of these single premiums for assurances and annuities may be deduced from one another. The connecting links generally can be written in several forms, and the truth of each such form may be justified by a "verbal explanation".

The student should avail himself of every opportunity to consolidate the ideas that the symbols and the relations between them should convey to him by expressing these relations in his own words and so convincing himself of their truth.

2. If a loan of 1 is made for n years, the lender will receive interest each year and get the 1 back at the end of the nth year, or

1 =tan +vn    (i)

1 =dank +v"    (ii)

Similarly, a loan of 1 made to (x) until the end of the year of his death will produce interest during the period of the loan and a return of the capital at the end of that period, or

1=iax+(1 +i)Ax    (i)

1=dax+Ax    (ii)

1. is on the assumption that interest is paid at the end of each year,

2. is on the assumption that interest is paid at the beginning of each year.

From (i), in the case of the loan for the life of (x), we get

v=viax+Ax

or   1—d =dax+Ax

or   1=dax+Ax.