FM for Actuaries

36 CHAPTER 1
(a) loss of 3-month interest;
(b) a reduction in the nominal rate of interest to 6%.
If you wish to withdraw after 9 months, which option would you choose?
Advanced Problems
1.41 (a) For c>0, letf(x) =1+cx and g(x) =(1+c) x ,where0 ≤ x ≤ 1.
(1) Show that f(0) = g(0), andf(1) = g(1).
(2) By using the fact that g(·) is increasing in x, show that f(x) ≥
g(x) for 0 ≤ x ≤ 1.
(3) Letting c = r in (a)(2), what conclusion can you draw?
(b) Let [x] be the integer part of x. For examples, [4.2] = 4, [9.99] = 9
and [1] = 1. By using the result in (a)(2), show that for t ≥ 0,
( ) mt−[mt] ( )
1+ r(m)
mt − [mt]
≤ 1+r (m) .
m
m
(c) Prove that when the loan period is not an integer multiple of the compounding
period, applying the simple-interest method for the remaining
fraction of the conversion part always gives a larger accumulation function
when compared with extending (1.6) to any mt ≥ 0.
1.42 The following graphs plot the force of interest for Fund X and Fund Y for
0 ≤ t ≤ 10.
X (t)
Y (t)
0.1
0.05
10
t
2 10
t
Find the points in time at which the values of the accumulation functions of
the two funds are equal.