26 
ON THE VALUE OF ANNUITIES. 
39. When interest is convertible m times a year, the amount of £l 
C 
:) 
at the first period of converson is il -] j ; 
C 
) 
1 +—- j , the present value of £l 
l :: l : 
payable at the expiration of the with part of a year, and by reasoning as 
the end of one year, or at the end of m periods of conversion of interest j 
) 
, the present value of £l due at the end of n years, 
or mn periods of conversion of interest. This multiplied by s gives 
, or 
By logarithms, 
This equation is the same as that given in Art. 29, and the expressions 
for s, n, m, and i, derived therefrom, as given in Arts. 28, 30, 31, and 
32, furnish us with the formulae for obtaining those quantities. They 
may also be found by taking the formulae given wdien interest is payable 
yearly, and substituting the number of intervals for the number of years, 
and the interest for one interval instead of the annual interest. 
40. To find the present value. 
The formulae as given above is 
Rule. Find the present value of ¿£l due at the end of the first inter 
val, and raise it to a power equal to the number of times interest is 
convertible before the money becomes due, and multiply by the sum 
due. 
Or when the interest for one interval is equal to any of the annual 
rates for which tables of the present values are given, we have only to 
take from those tables at that rate, the present value of ¿£l due at the 
end of the same number of years as there are intervals of conversion 
during the term, and multiply it by the sum due. 
Example. What is the present value of £350 due 5 years hence f * 
allowing 6 per cent compound interest, payable half-yearly ?