Single Premiums and Annuity Values   49

1 of a year. m

The total value, assuming simple interest, amounts to a year's interest on m2 [(m -1) + (m — 2) + (m — 3) + .... +2+11= 2m    m1.

Therefore the earlier payments during all the completed years

will produce a difference worth m   iax now. 2m

If the life survives the first 1 part of the year of death a payment of 1

in   m

will be made under al"') which would not be made under ax. The probability of such a payment being m -1, its value at the end of

the year of death is m -1   1 C1+m—1 i / .   Similarly at the

nz   m   m

end of the year of death the /value of the second possible additional

payment of 1 is m   -2 1 1 1+m-2i j , and so on. The total
m m m \ m

value at the end of the year of death of all such possible payments is

The value now of these possible additional payments is n2m1 C1+2    3m 1 i ~A

x

plus interest on    1 for m -3 of a year

m   m

etc.   etc.   etc.

plus interest on

1- for

m

1 [m@n—i)+i

2   m

m—1 C1+2m—1i 1

2m   3m

m

z (m—1)+(m -2)+(m—3)+ ....+2+1

+ (m—1 +m—22+...+22+12)

   2   

m

(m—1)m(2m—1

6

4