FM for Actuaries

Interest Accumulation and Time Value of Money 15
provided dt < 1. Hence, we have
A(t) = A(0) , dt < 1 <t,
1 − dt
and
a(t) = 1 , dt < 1 <t, (1.23)
1 − dt
which is the accumulation function for simple discount for dt < 1 <t.
Example 1.11: The discount rate of a 3-month Treasury Bill is 6% per annum.
What is the annual effective rate of interest? What is the accumulated value of 1 in
2 years?
Solution: The rate of interest charged for the 3-month period is
0.06 × 1 4
1 − 0.06 × 1 4
= 1.52%.
Therefore, the equivalent nominal rate of interest compounded quarterly is
and the annual effective rate of interest is
4 × 0.0152 = 6.08%,
(1.0152) 4 − 1 = 6.22%.
The accumulated value of 1 in 2 years is, using (1.20),
a(2) =
Note that this can also be calculated as
[
1 − 0.06 ] −8
=1.13.
4
(1 + i) 2 =(1.0622) 2 =1.13.
Example 1.12: The discount rate of a 6-month Treasury Bill is 8% per annum,
what is the annual effective rate of interest? What is the accumulated value of 1 in
3 years?