FM for Actuaries

Annuities 69
2.37 A paused rainbow is a cash-flow stream which pays $1 at time 1 and 2n; $2
at time 2 and 2n − 1; ···; $(n − 1) at time n − 1 and n +2;and$n at time
n and n +1. Find the present value of the paused rainbow.
2.38 (a) Find the present value of a perpetuity of $1 at time 0, $2 at time 3, $3
at time 6, $4 at time 9, ··· at an effective rate of interest of 3%. [Hint:
Use the result in Exercise 2.25.]
(b) Hence, find the present value of a perpetuity which pays $1 at the end
of the 3rd, 4th and 5th year, $2 at the end of the 6th, 7th and 8th year,
$3 at the end of the 9th, 10th, 11th year and so on, at an effect rate of
interest of 3%.
2.39 Express (D¨s) n⌉
in terms of i and n under simple interest. [Hint: The symbol
(D¨s) n⌉
is defined similarly as in Exercise 2.25.]
2.40 Eric makes deposits into a retirement fund earning an annual effective rate
of 7%. The first deposit of $1,000 is made on his 38th birthday and the
last deposit is made on his 64th birthday. Every year his deposit increases
by 3%. When he attains age 65, he will withdraw all the money in the retirement
fund to purchase an annuity-immediate which provides him with
monthly payment for 25 years. The nominal rate of interest on the annuity is
convertible monthly at 6%. Find the amount of each monthly payment.
2.41 Albert purchased a house for $1,500,000 fifteen years ago. He put 30% down
and financed the balance by a 20-year real estate mortgage at 6%, convertible
monthly. Albert decides to pay the remaining loan balance in full by a single
payment together with the installment just due. Find the prepayment penalty,
which is one-third of the lender’s interest loss.
2.42 Prove (2.43) and (2.44).
2.43 An investor wishes to accumulate $1,000 at the end of year 5. He makes level
deposits at the end of each year. The deposits earn a 6% annual effective rate
of interest, which is credited at the end of each year. The interests on the
deposits earn 5% effective interest rate annually. How much does he have to
deposit each year?
2.44 A principal of $20,500 generates income of $3,000 at the end of every 2
years at an effective rate of interest of 3% per annum for as long as possible.
Calculate the term of the annuity and discuss the possibilities of settling the
last payment.