CHAPTER IV
CONTINGENT FUNCTIONS
ENDOWMENTS AND ANNUITIES
29. Pure Endowments.—A pure endowment is an amount of money to be received at the expiration of a term of years, provided the person to receive this amount is then living.
If (x) is to receive $1 at the end of n years, the present value of the mathematical expectation is (Sec. 27)
1 X v" X npz = v" , npz = v lz
This is the present value of a pure endowment to (x) due at the end of n years and is denoted by the symbol nE~.
"E. is the present value of $1 payable in n years to a person who is now x years of age, provided that person is living at the expiration of the term of n years.
We have, then, the fundamental equation
nEz = v" npz = v"llzn    (1)
For example, to compute 10E20 at 3% per cent (American ExpeTable) we have 120 = 92,637, l3° = 85,441 and v10 = (1/1.035)1° = 0.7089188. Hence
0.7089188 X 85,441
10E20 =    92,637    = 0.65384.
A pure endowment of $1000 to a person twenty years of age due in 10 years has a present value 10E20 X $1000 = $653.84, if money is worth 332 per cent. This means that if each of 92,637 persons of age twenty should contribute $653.84, the fund so created would amount to $85,441,000 in 10 years if accumulated at 332 per cent compound interest, or enough to furnish $1000 to each of 85,441 persons.
36
X3O    CONTINGENT FUNCTIONS    37
Without making use of the notion of mathematical expectation, we may reason as follows: In order that each of the persons living at age x + n should receive $1, we must now have on hand a sum equal to v^lx+n dollars. Each of the lx persons living at age x must then contribute v"lx+n/lx dollars. From this point of view, „Ex is the amount of money each of the l= persons living at age x must pay now in order that each of the lx+„ survivors may receive $1. In other words, ~Ex is the net single premium chargeable to a person of age x for an endowment of $1 payable at the end of n years.
„E= may also be regarded as the contingent discount factor for compound interest corresponding to the certain discount factor
v” (cf. Sec. 2).
Exercises
  • With the aid of logarithms, compute the numerical value of IoEso at 3) per cent compound interest. State in words the meaning of the result.
  • Check the result found in the preceding exercise by the method suggested in Sec. 29.
    • 30. Commutation     Symbols.—The amount of computation necessary to determine the numerical values of endowments, annuities, premiums, and so on is often very great. To facilitate this computation, certain symbols have been invented and their values tabulated. These symbols are known as commutation symbols. Four of these are of fundamental importance, namely,M=, and Rx. At present we shall define the symbol D= and illustrate its use. The other symbols will be defined as needed.Resuming Eq. (1) of the preceding section,= v^~~+(I)
and multiplying both numerator and denominator in the right-hand member by vr, we have
v=+^lx+~
nEx = 11=1x    (2)
If we put
v=lx = D.,    (3)

38    MATHEMATICS OF LIFE INSURANCE    [§31 then
v=+„ lz+n = D=+n,    (4)
and Eq. (2) becomes
E = D;    (5)
The value of D. can be computed for each age and for various rates of interest and tabulated once for all (Table III).
Thus, 10E20 = D3o/D2o = 30,440.8/46,556.2 = 0.65384, as in
the preceding section.    i
Exercises
L/f. Check the values in Table III for D20 and D60. / (-2. Compute the numerical value of ,0E30, using cnmmu +ion symy bols. Compare with the result found in Exercise 1 (Sec. 29).
  • Show that the symbol 1/0E. represents an endowment to (x) for n years that can be purchased for $1, or is an endowment whose present value is $1, or may be regarded as a contingent accumulation factor for compound interest.
  • A has $1000 to invest for 10 years. He is now twenty years of ü age and can buy an endowment at rate 0.035 or can loan the amount at 4 per cent compounded annually. Which would yield the greater amount if A is living at age thirty?J
    •
  • If, in Sec. 29, the interest factor is removed from the right " member of Eq. (1), to what does „Er reduce? If the probability factor is removed to what does „E. reduce?
    • 31. Contingent Annuities.—The present value of $1 payable at the end of each year, as long as a person of age x is living, is denoted by the symbol a.. Obviously, a. is the sum of a series of pure endowments 1E., 2E., 3E~, . . . , _sEz where o — x is the complement of life at age x (Sec. 20). Hence we haven =—xa. = 1E. + 2E. + 3E. + . . . + —=Ez = 4 .Ex. (1)n=1In the future, where a summation is to extend over the compleof life at age x, that is, from 1 to elv — x, we shall omit the limits attached to the sign of summation, and shall write Eq. (1) simplyaz = E,,,E..(1)