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Math - College, section 1, Fall 2019
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Federer Vaaler - Mathematical Interest Theory 3/e (Homework)

James Finch

Math - College, section 1, Fall 2019

Instructor: Dr. Friendly

Current Score : 18 / 20

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving... ()

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Question

Points

1	2	3	4	5	6	7	8	9	10
3/3	1/1	1/1	3/4	1/1	3/3	2/2	1/1	2/3	1/1

Total

18/20 (90.0%)

Instructions

Mathematical Interest Theory, Third Edition, by Leslie Jane Federer Vaaler, Shinko Kojima Harper, and James W. Daniel, gives students an introduction of how investments grow over time. This textbook is written for anyone who has a strong high school algebra background and is interested in being an informed borrower or investor. The content is suitable for a mid-level or upper-level undergraduate course or a beginner graduate course, and provides a solid foundation for readers embarking on actuarial careers. This book has been suggested by the Society of Actuaries for people preparing for the Financial Mathematics exam, and updates the previous edition to cover the material in the SOA study notes FM-24-17, FM-25-17, and FM-26-17.

Through partnership with the Mathematical Association of America, WebAssign is pleased to offer online question content with instant feedback from this title. All questions include reading links to the eBook for an integrated student experience.

Question 1 is a multi-part question that has a student find an equation of value, then use that equation to calculate dollar-weighted annual yields.

Questions 2 and 10 utilize the calcPad so students can easily enter any correct form of the complicated functions.

Question 3 gives the student the option to enter more than one interest rate in the case that there are multiple correct answers. Note that this particular question has only one correct answer.

Question 4 is a multi-part question that ultimately has a student find equations for the nominal discount rate.

Questions 5 and 7 require calculus knowledge to answer correctly.

Question 9 steps a student through an alternate proof for the given formula.

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1. 3/3 points | Previous Answers FVIntTheory3 2.6.001. My Notes

Sandra invests $10,831 in the Wise Investment Fund. Three months later her balance has grown to $11,891 and she deposits $2,000. Two months later her fund holdings are $14,296 and she withdraws $7,000. Two years after her initial investment, she learns that her holdings are worth $12,554.

(a)

Write an equation of value involving the exact dollar-weighted annual yield i over the two-year period. (Use the time zero equation of value.)

12554(1 + i)⁻² =

10831+2000(1+i)−(312)−7000(1+i)−(512)

$webMathematica generated answer key$

(b)

Compute the approximate dollar-weighted annual yield over the investment period using the following formula. (Round your answer to two decimal places.)

j ≈

A +

	C_t(1 − t)

t (0, 1)

j ≈

39.82

Compute the approximate dollar-weighted annual yield over the investment period again using the following formula. (Round your answer to two decimal places.)

j ≈

A +

1 −

B −

1 −

A + B − 1

j ≈

34.42

Viewing Saved Work Revert to Last Response

2. 1/1 points | Previous Answers FVIntTheory3 2.7.001. My Notes

On January 1, 1988, Antonio invests $9,800 in an investment fund. On January 1, 1989 his balance is $10,700 and he deposits $2,700. On July 1, 1989 his balance is $14,700 and he withdraws $1000. On January 1, 1992 his balance is $P. Express his annual time-weighted yield

(i_tw)

as a function of P.

i_tw =

((107009800)(1470013400)(P13700))(14)−1

$webMathematica generated answer key$

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3. 1/1 points | Previous Answers FVIntTheory3 1.7.007. My Notes

problem

You have two options to repay a loan. You can repay $6,000 now and $6,480 in one year, or you can repay $12,600 in 6 months. Find the annual effective interest rate(s) i at which both options have the same present value. (Enter your answers as a comma-separated list.)

$webMathematica generated answer key$ %

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4. 3/4 points | Previous Answers FVIntTheory3 1.10.007. My Notes

Let m be a positive real number. Suppose interest is paid once every m years at a nominal interest rate

i^(1/m).

This means that the borrower pays interest at an effective rate of

i^(1/m)

= mi^(1/m)

per m year period.

(a)

Find an expression for

i^(1/m)

in terms of i.

i^(1/m)

(i+1)m−1m

$webMathematica generated answer key$

(b)

i^(2/7) = 0.07,

find i. (Round your answer to two decimal places.)

i =

6.46

(c)

Define

d^(1/m)

to be the nominal discount rate payable once every m years. This means that the borrower pays discount at an effective rate of

d^(1/m)

= md^(1/m)

per m year period.

Find a formula that gives

d^(1/m)

in terms of

i^(1/m).

d^(1/m) =

1−mi 1m+1m

$webMathematica generated answer key$

Find a formula that gives

d^(1/m)

in terms of d.

d^(1/m) =

1−(1−d)mm

$webMathematica generated answer key$

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5. 1/1 points | Previous Answers FVIntTheory3 1.R.009. My Notes

Calculus is needed.

Suppose that

δ_t =

t − 1

for

2 ≤ t ≤ 8.

For

2 ≤ n ≤ 7,

let

f(n) = i_{n + 1} + 1.

Write a simple formula for

f(n).

f(n) =

n2(n−1)2

$webMathematica generated answer key$

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6. 3/3 points | Previous Answers FVIntTheory3 1.12.008. My Notes

(a)

Fund I grows according to simple interest at rate r. Find the force of interest

δ_t^(I)

acting on fund I at time t.

δ_t^(I) =

rrt+1

$webMathematica generated answer key$

(b)

Fund D grows according to simple discount at rate s. Find the force of interest

δ_t^(D)

acting on fund D at time t.

δ_t^(D) =

s1−st

$webMathematica generated answer key$

(c)

Suppose

r > s.

Find all t such that

δ_t^(I) = δ_t^(D).

t =

r−s2rs

$webMathematica generated answer key$

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7. 2/2 points | Previous Answers FVIntTheory3 1.R.004. My Notes

Calculus is needed.

(a)

Express

dδ

as a function of d. (Assume compound interest.)

d−1

$webMathematica generated answer key$

(b)

Express

as a function of d. (Assume compound interest.)

d−1

$webMathematica generated answer key$

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8. 1/1 points | Previous Answers FVIntTheory3 1.3.005. My Notes

problem

It is known that for each positive integer k, the amount of interest earned by an investor in the k-th period is k. Find the amount of interest earned by the investor from time 0 to time n, n a fixed positive integer.

12n(n+1)

$webMathematica generated answer key$

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9. 2/3 points | Previous Answers FVIntTheory3 4.3.007. My Notes

Use the following fact to obtain an alternate proof of the formula

a_{n i}^(m) =

1 − vⁿ

m[(1 + i)^1/m − 1]

The expression

a_{nm J}

gives the present value of an annuity that pays

at the end of each m-th of an interest period, for a total of 1 per interest period. Here

J = (1 + i)^1/m − 1 =

i^(m)

is the interest rate for the annuity-payment period.

According to the given fact and the equation

a_{n i} =

1 − vⁿ

a_{n i}^(m) =

a_{nm J} =

1 −

J+1=0

$webMathematica generated answer key$

	−nm

But

J = (1 + i)^1/m − 1

and

(1 + J)^−nm =

i+1

$webMathematica generated answer key$

	−n

v

$webMathematica generated answer key$

	n

Therefore,

a_{n i}^(m) =

1 − (1 + J)^−nm

1 − vⁿ

m[(1 + i)^1/m − 1]

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10. 1/1 points | Previous Answers FVIntTheory3 4.2.007. My Notes

A perpetuity paying $5,000 at the beginning of each two years has the same present value as another perpetuity with level payments, this one having payments at the end of each three years. Express the level payment amount of the second perpetuity, P, as a function of the annual effective interest rate i.

P =

5000((i+1)3−1)1−1(i+1)2

$webMathematica generated answer key$

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Federer Vaaler - Mathematical Interest Theory 3/e (Homework)

Current Score : 18 / 20

Due : Sunday, January 27, 2030 00:00 EST

Last Saved : n/a Saving... ()

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