# ACTSC231 Study Guide - Quiz Guide: Effective Interest Rate, Problem Set, Royal Institute Of TechnologyExam

by OC2719308

School

University of WaterlooDepartment

Actuarial ScienceCourse Code

ACTSC231Professor

Chengguo WengStudy Guide

QuizThis

**preview**shows half of the first page. to view the full**3 pages of the document.**Actsc 231 - Problem Set 7

This problem set is composed mainly of recommended problems from the course textbook.

For the textbook problems, at the beginning of the question, a reference will be made to the

exercise from which it was taken.

1. Find the present value of a 10-year continuously paying annuity if the payment rate at

time tis t5and the force of interest is 0.002t,0≤t≤10.

2. Surprise! You’re back in the Den and Kevin wants to invest money today in your busi-

ness. In exchange, he wants a royalty on your sales of $1 per unit sold until the $10,000

is repaid (so in this case, you need to sell 10,000 units) and then a royalty of $0.50 per

unit sold thereafter (forever). Assume money is worth i(1) = 5%.

(a) Assume that you will sell 1,000 units each year forever. Assuming that the royalties

will be paid at the end of each year, what is the minimum investment you will ac-

cept from Kevin?

(b) Assume that you will sell 1,000 units each year forever. Assuming that the royalties

will be paid continuously throughout each year, what is the minimum investment

you will accept from Kevin?

(c) Assume that you will sell 1,000 units in the ﬁrst year, 2,000 in the second year, 3,000

in the third year, . . . ,10,000 in the tenth year and 0 units thereafter. Assuming that

the royalties will be paid at the end of the year, what is the minimum investment

you will accept from Kevin?

(d) Assume that you will sell 1,000 units in the ﬁrst year, 2,000 in the second year, 3,000

in the third year, . . . ,10,000 in the tenth year and 0 units thereafter. Assuming that

the royalties will be paid continuously throughout each year, what is the minimum

investment you will accept from Kevin?

3. Find the present value, (¯

D¯a)n i of an annuity which pays continuously at an annual rate

of n−tat time tfor the next nyears. The annual effective interest rate is i.

4. (a) Show that d

di an i =−(1 + i)−1·(Ia)n i.

(b) Let’s treat the effective annual interest rate ias a function of δ; we will write i(δ)to

make this understood (that it is a function of δ). We know that i(δ) := i=eδ−1.

Show that d

dδ an i(δ)=−(Ia)n i.

5. A loan is repaid by yearly payments of $100, $150, $200, . . . , $500. Assume payments are

made at the end of the year, with the ﬁrst payment being made 1 year after the loan is

taken out. Determine the total interest paid using an annual effective rate of 5%.

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