FM for Actuaries

Annuities 67
2.23 Express s(52) n⌉
¨s (12)
n⌉
in terms of the nominal rates of interest and discount.
2.24 A couple want to save $100,000 to pay for their daughter’s education. They
put $2,000 into a fund at the beginning of every month. Interest is compounded
monthly at a nominal rate of 7%. How long does it take for the fund
to reach $100,000?
2.25 Denote the present value and future value (at time n)for1 at time 0; 2 at time
1; ···, n at time n − 1, by(Iä) n| and (I¨s) n| , respectively. Show that
(Iä) n⌉ = än⌉
d
− nvn−1
i
and
(I¨s) n⌉
= ¨s n⌉ − n
d
with the help of formulas (2.36) and (2.37).
2.26 John deposits $500 into the bank at the end of every month. The bank credits
monthly interest of 1.5%. Find the amount of interest earned in his account
during the 13th month.
2.27 Consider lending $1 at the beginning of each year for n years. The borrower
is required to repay interest at the end of every year. After n years, the
principal is returned.
(a) Draw a time diagram and find the present value of all interests.
(b) How much principal is returned after n years?
(c) Formula (2.36) can be written as
ä n⌉
= i(Ia) n⌉
+ nv n .
Explain verbally the meaning of the expression above.
(d) Using (Ia) n⌉ +(Da) n⌉ =(n+1)a n⌉ and (2.36), derive formula (2.40).
2.28 Payments of $500 per quarter are made over a 5-year period commencing
at the end of the first month. Show that the present value of all payments 2
years prior to the first payment is
$2,000(ä (4)
7⌉ − ä(4) 2⌉ ),
where the annuity symbols are based on an effective rate of interest. What
is the accumulated value of all payments 7 years after the first payment?
Express your answers in a form similar to the expression above.