The Theory of Interest

Stephen G. Kellison

Chapter 1 The measurement of interest - all with Video Answers

Educators

Chapter Questions

Problem 1

Consider the amount function $A(t)=t^{2}+2 t+3$
ta) Find the corresponding accumulation function $a(t)$
Werify that $a(t)$ satisfies the three properties of an accumulation function. c) Find $I_{n}.$

James Kiss

Numerade Educator

01:04

Problem 2

$a$ ) Prove that $A(n)-A(0)=I_{1}+I_{2}+\cdots+I_{n}$ berbally interpret the result obtained in (a).

Carson Merrill

Numerade Educator

00:15

Problem 3

Find the amount of interest earned between time $t$ and time $n$, where $t$
a) $\quad I_{r}=r$
b) $\quad I_{r}=2^{r}$

Amy Jiang

Numerade Educator

04:04

Problem 4

It is known that $a(t)$ is of the form $a t^{2}+b$. If $\$ 100$ invested at time 0 accumulates to $\$ 172$ at time $3,$ find the accumulated value at time 10 of $\$ 100$ invested at time 5.

Niamat Khuda

Numerade Educator

01:01

Problem 5

Assume that $A(t)=100+5 t.$
a) Find $i_{5}$
b) Find $i_{10}$

Lily An

Numerade Educator

01:01

Problem 6

Assume that $A(t)=10(1.1)^{t}$
a) Find $i_{5}$
b) Find $i_{10}$

Lily An

Numerade Educator

00:36

Problem 7

Show that $A(n)=\left(1+i_{n}\right) A(n-1).$

Amy Jiang

Numerade Educator

00:44

Problem 8

If $A(4)=1000$ and $i_{n}=.01 n,$ find $A(7).$

Tony Ni

Numerade Educator

02:02

Problem 9

$a$ ) At what rate of simple interest will $\$ 500$ accumulate to $\$ 615$ in $21 / 2$ years?
b) In how many years will $\$ 500$ accumulate to $\$ 630$ at $7.8 \%$ simple interest?

Tony Ni

Numerade Educator

02:46

Problem 10

If $i_{k}$ is the rate of simple interest for period $k,$ where $k=1,2, \ldots, n,$ show that $a(n)-a(0)=i_{1}+i_{2}+\cdots+i_{n}.$

Christopher Stanley

Numerade Educator

01:48

At a certain rate of simple interest $\$ 1000$ will accumulate to $\$ 1110$ after a certain period of time. Find the accumulated value of $\$ 500$ at a rate of simple interest three fourths as great over twice as long a period of time.

Gregory Higby

Numerade Educator

01:04

Problem 12

Simple interest of $i=4 \%$ is being credited to a fund. In which period is this equivalent to an effective rate of $21 / 2 \% ?$

Nick Johnson

Numerade Educator

04:08

Problem 13

Assuming that show that:
a) $(1+i)^{2}<1+$ it if $0<t<1$
b) $(1+b)^{\prime}=1+$ if if $t=1$
c) $(1+i)^{2}>1+i t$ if $t>1$
This exercise verifies the relative magnitudes of accumulated values at simple and compound interest over various periods of time.

Joseph Lentino

Numerade Educator

02:50

Problem 14

It is known that $\$ 600$ invested for two years will earn $\$ 264$ in interest. Find the accumulated value of $\$ 2000$ invested at the same rate of compound interest for three years.

Ankit Gupta

Numerade Educator

02:16

Problem 15

Show that the ratio of the accumulated value of 1 invested at rate $i$ for $n$ periods, to the accumulated value of 1 invested at rate $j$ for $n$ periods, $i>f$, is equal to the accumulated value of 1 invested for $n$ periods at rate $r$. Find an expression for $r$ as a function of $i$ and $j.$

Priyanka Sadarangani

Numerade Educator

02:09

Problem 16

At a certain rate of compound interest, 1 will increase to 2 in $a$ years, 2 will increase to 3 in $b$ years, and 3 will increase to 15 in $c$ years. If 6 will increase to 10 in $n$ years, express $n$ as a function of $a, b,$ and $c.$

Haley Mortell

Numerade Educator

01:06

Problem 17

An amount of money is invested for one year at a rate of interest of $3 \%$ per quarter. Let $D(k)$ be the difference between the amount of interest earned on a compound interest basis and on a simple interest basis for quarter $k$, where $k=1,2,3,4$ Find the ratio of $D(4)$ to $D(3).$

Nicole Krahulik

Numerade Educator

01:27

Problem 18

Find an expression for the discount factor during the $n$th period from the date of investment, i.e. $\left(1+i_{n}\right)^{-1}$ in terms of the amount function.

Wendi Zhao

Numerade Educator

01:10

Problem 19

The sum of the present value of 1 paid at the end of $n$ periods and 1 paid at the end of $2 n$ periods is $1 .$ Find $(1+i)^{2 n}.$

Khoobchandra Agrawal

Numerade Educator

02:49

Problem 20

Show that the current value of a payment of 1 made $n$ periods ago and a payment of 1 to be made $n$ periods in the future is greater than $2,$ if $i>0.$

Angela Guo

Numerade Educator

02:40

Problem 21

It is known that an investment of $\$ 500$ will increase to $\$ 4000$ at the end of 30 years. Find the sum of the present values of three payments of $\$ 10,000$ each which will occur at the end of $20,40,$ and 60 years.

Narayan Hari

Numerade Educator

00:56

Problem 22

The amount of interest earned on $A$ for one year is $\$ 336,$ while the equivalent amount of discount is $\$ 300 .$ Find $A.$

Emily Himsel

Numerade Educator

01:59

Problem 23

a) Find $d_{5}$ if the rate of simple interest is $10 \%$
b) Find $d_{5}$ if the rate of simple discount is $10 \%$

Manik Pulyani

Numerade Educator

00:34

Problem 24

a) Assuming compound discount, show that $d_{n}$ is constant for all $n$.
b) Assuming simple discount, show that $d_{n}$ is increasing for increasing $n$ if $0<n-1<1 / d.$

Fuzail Shakir

Numerade Educator

02:19

Problem 25

Assuming that $0<d<1,$ show that:
a) $(1-d)^{l}<1-d t$ if $0<t<1$
b) $(1-d)^{f}=1-d t$ if $t=1$
$(1-d)^{\prime}>1-d t$ if $t>1$
This exercise verifies the relative magnitudes of present values at simple and compound discount over various periods of time.

Madi Sousa

Numerade Educator

01:33

Problem 26

If $i$ and $d$ are equivalent rates of simple interest and simple discount over $t$ periods, show that
$$i-d=i d t$$

Carson Merrill

Numerade Educator

01:29

Problem 27

Show that
$$\frac{d^{3}}{(1-d)^{2}}=\frac{(i-d)^{2}}{1-v}$$

Sriram Soundarrajan

Numerade Educator

00:24

Problem 28

a) Express $d^{(4)}$ as a function of $i^{(3)}$
b) Express $i^{(6)}$ as a function of $d^{(2)}$

James Kiss

Numerade Educator

00:29

Problem 29

On occasion, interest is convertible less frequently than once a year. Define $i^{(1 / m)}$ and $d^{(1 / m)}$ to be nominal annual rates of interest and discount convertible once every $m$ years. Find a formula analogous to formula $(1.22 a)$ for this situation.

Alison Rodriguez

Numerade Educator

01:49

Problem 30

Find the accumulated value of $\$ 100$ at the end of two years:
a) If the nominal annual rate of interest is $6 \%$ convertible quarterly.
b) If the nominal annual rate of discount is $6 \%$ convertible once every four years.

Niamat Khuda

Numerade Educator

01:46

Problem 31

Derive formula (1.23).

Sarah Lewites

Numerade Educator

02:27

Problem 32

a) Show that $i^{(m)}=d^{(m)}(1+i)^{1 / m}.$
b) Verbally interpret the result obtained in (a).

Ekaveera Kumar

Numerade Educator

00:32

Problem 33

Given that $i^{(m)}=.1844144$ and $d^{(m)}=.1802608,$ find $m.$

Ankit Gupta

Numerade Educator

04:48

Problem 34

It is known that
$$1+\frac{i^{(n)}}{n}=\frac{1+\frac{i^{(4)}}{4}}{1+\frac{i^{(5)}}{5}}$$
Find $n$

Aman Gupta

Numerade Educator

00:41

Problem 35

If $r=\frac{i^{(4)}}{d^{(4)}},$ express $v$ in terms of $r.$

Brandon Fox

Numerade Educator

01:39

Problem 36

Derive formula (1.37).

Jerrah Biggerstaff

Numerade Educator

01:19

Problem 37

Use formula (1.23) to give a third proof of the result that $\delta^{\prime}=\delta.$

Carson Merrill

Numerade Educator

03:37

Problem 38

Rank $i, i^{(m)}, d, d^{(m)},$ and $\delta$ in increasing order of magnitude, assuming $m>1.$

Caroline Basil

Numerade Educator

01:17

Problem 39

Show that $\frac{d}{d t} \delta_{t}=\frac{A^{\prime \prime}(t)}{A(t)}-\delta_{t}^{2}.$

Bryan Lynn

Numerade Educator

02:00

Problem 40

a) Obtain an expression for $\delta_{t}$ if $A(t)=K a^{t} b^{t^{2}} d^{c^{\prime}}.$
b) Is formula (1.24) or (1.25) more convenient in this case?

Monica Miller

Numerade Educator

06:31

Problem 41

Show that:
a) $\int_{0}^{n} \delta_{t} d t=-\log _{e} v^{n}$
b) $\int_{0}^{n} A(t) \delta_{t} d t=I_{1}+I_{2}+\cdots+I_{n}$

Willis James

Numerade Educator

02:51

Problem 42

Fund A accumulates at a simple interest rate of $10 \% .$ Fund $\mathbf{B}$ accumulates at a simple discount rate of $5 \%$. Find the point in time at which the forces of interest on the two funds are equal.

Sarah Vo

Numerade Educator

05:55

Problem 43

An investment is made for one year in a fund whose accumulation function is a second degree polynomial. The nominal rate of interest earned during the first half of the year is $5 \%$ convertible semiannully. The effective rate of interest earned for the entire year is $7 \%$. Find $\delta_{.5}.$

Ankit Gupta

Numerade Educator

00:15

Problem 44

Find an expression for the fraction of a period at which the excess of accumulated values computed at simple interest over compound interest is a maximum.

Amy Jiang

Numerade Educator

05:21

Problem 45

If $\delta_{t}=.01 t, 0 \leq t \leq 2,$ find the equivalent annual effective rate of interest over the interval $0 \leq t \leq 2.$

Regina Hays

Numerade Educator

02:22

Problem 46

Find the accumulated value of 1 at the end of 19 years if $\delta_{t}=.04(1+t)^{-2}.$

Linh Vu

Numerade Educator

02:18

Problem 47

Find the level effective rate of interest over a three-year period which is equivalent to an effective rate of discount of $8 \%$ the first year, $7 \%$ the second year, and $6 \%$ the third year.

Priyanka Sadarangani

Numerade Educator

04:27

Problem 48

a) Find the accumulated value of 1 at the end of $n$ periods where the effective rate of interest for the $k$ th period, $1 \leq k \leq n,$ is defined by
$$i_{k}=(1+r)^{k}(1+i)-1$$
b) Show that the answer to ( $a$ ) can be written in the form $(1+j)^{n}$. Find $f.$

Anurag Kumar

Numerade Educator

01:14

Problem 49

The force of interest at time $t$ is $t^{3} / 100$. Find $a^{-1}(3).$

Brad Thornton

Numerade Educator

06:05

Problem 50

In Fund $X$ money accumulates at a force of interest
$$\delta_{t}=.01 t+.1 \text { for } 0 \leq t \leq 20$$
In Fund $Y$ money accumulates at an annual effective interest rate $i$. An amount of 1 is invested in each fund for 20 years. The value of Fund $X$ at the end of 20 years is equal to the value of Fund $Y$ at the end of 20 years. Calculate the value of Fund $\mathbf{Y}$ at the end of 1.5 years.

Linh Vu

Numerade Educator

03:42

Problem 51

You are given $\delta_{t}=\frac{2}{t-1}$ for $2 \leq t \leq 10$. For any one year interval between $n$ and $n+1,$ with $2 \leq n \leq 9,$ calculate the equivalent $d^{(2)}.$

Nick Johnson

Numerade Educator

00:42

Problem 52

If the effective rate of discount in year $k$ is equal to $.01 k+.06$ for $k=1,2,3,$ find the equivalent rate of simple interest over the three-year period.

Hossam Mohamed

Numerade Educator

01:04

Problem 53

a) Show that $\delta=\frac{i^{(m)}+d^{(m)}}{2},$ where $_{5}$ denotes approximate equality.
b) Assuming $m=1,$ find an exact expression for the error in (a) expressed as a series expansion in $\boldsymbol{\delta}$

Tyler Moulton

Numerade Educator

01:39

Problem 54

Show that
\[
\sum_{m=1}^{\infty}(-1)^{m-1} i^{m}\left(\frac{1}{d^{(m)}}-\frac{1}{i^{(m)}}\right)=\delta
\]

John Nicolle

Numerade Educator

01:07

Problem 55

Find the following derivatives:
a) $\frac{d}{d i} d$
b) $\frac{d}{d i} \delta$
c) $\frac{d}{d d} d^{(m)}$
d) $\frac{d}{d v} \delta$
e) $\frac{d}{d \delta} d$

Joseph Liao

Numerade Educator

02:24

Problem 56

Find the following in the form of series expansions:
a) $i$ as a function of $d$
b) $d$ as a function of $i$
c) $v$ as a function of $\delta$
d) $i^{(m)}$ as a function of $i$
e) $\delta$ as a function of $d$

Aman Gupta

Numerade Educator

01:07

Problem 57

Show that
\[
\delta=\frac{d+i}{2}+\frac{d^{2}-i^{2}}{4}+\frac{d^{3}+i^{3}}{6}+\cdots
\]

Edward Downes

Numerade Educator

01:33

Problem 58

Show that
\[
\frac{d^{n}}{d v^{n}}\left(v^{n-1} \delta\right)=-(1+i)(n-1) !
\]

Aman Gupta

Numerade Educator

03:25

Problem 59

a) (1) Derive an expression for $a(t)$ assuming $\delta_{r}$ is linear and positive,
i.e. $\delta_{r}=a+b r,$ where $a>0$ and $b>0$
(2) Find the accumulation factor during the $n$ th period from the date of investment, i.e. $1+i_{n}$.
b) (1) Derive an expression for $a(t)$ assuming $\delta_{r}$ is exponential and positive, i.e. $\delta_{r}=a b^{r},$ where $a>0$ and $b>0.$
(2) Find the accumulation factor during the $n$ th period from the date of investment, i.c. $1+i_{n}.$

Adrian Co

Numerade Educator

00:18

Problem 60

Stoodley's formula for the force of interest is
\[
\delta_{t}=p+\frac{s}{1+r e^{s t}}
\]
Show that
\[
a^{-1}(t)=\frac{1}{1+r} v_{1}^{t}+\frac{r}{1+r} v_{2}^{t}
\]
where $v_{1}=e^{-(p+s)}$ and $v_{2}=e^{-p}$

Catt Huth

Numerade Educator

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