FM for Actuaries

Interest Accumulation and Time Value of Money 19
Solution: From (1.26), we have
(∫ 2
)
a(2) = exp 0.02sds
0
(
=exp
0.01s 2] 2
0
)
= e 0.04 .
We solve the annual effective rate of interest i over the 2-year period from the
equation (compare this with (1.13))
to obtain
(1 + i) 2 = e 0.04
i = e 0.02 − 1=2.02%.
Similarly,
(
a(5) = exp 0.01s 2] )
5
= e 0.25 ,
0
so that the annual effective rate of interest i over the 5-year period satisfies
(1 + i) 5 = e 0.25
and we obtain
i = e 0.05 − 1=5.13%.
1.7 Present and Future Values
At the effective rate of interest i, a 1-unit investment today will accumulate to
(1 + i) units at the end of the year. In this respect, the accumulated amount (1 + i)
is also called the future value of 1 at the end of the year. Similarly, the future value
of 1 at the end of year t is (1 + i) t . This is the amount of money you can get t years
later if you invest 1 unit today at the effective rate i.
Sometimes it may be desirable to find the initial investment that will accumulate
to a targeted amount after a certain period of time. For example, a 1
1+i-unit payment
1
invested today will accumulate to 1 unit at the end of the year. Thus,
1+i
is called
the present value of 1 to be paid at the end of year 1. Extending the time frame to
1
t years, the present value of 1 due at the end of year t is
(1+i)
. t
Example 1.15: Given i =6%, calculate the present value of 1 to be paid at (a)
the end of year 1, (b) the end of year 5 and (c) 6.5 years.
Solution: (a) The present value of 1 to be paid at the end of year 1 is
1
= 0.9434.
1+0.06