Chapter 3, Problem 20P

(a)

To determine

The interest rate for an amount of $2000 borrowed, which has become $2600 at the point of re-payment at the end of two years.

Answer to Problem 20P

The interest rate for an amount of $2000 borrowed, which has become $2600 at the point of re-payment at the end of two years is 14.01%.

Explanation of Solution

  $FV=PV{(1+r)}^n$

  $2600=2000{(1+r)}^2$

  $\frac{2600}{2000}={(1+r)}^2$

  $1.3={(1+r)}^2$

The next step is to calculate the second square root of 1.3 in order to determine the value of ‘r’.

  $\begin{array}{c}\sqrt[2]{1.3}=1+r\\ 1.140175=1+r\\ r=1.140175-1\\ r=0.1401\\ r=14.01\%\end{array}$

The relevant data has to be substituted to the future value equation in finding the applicable interest rate. The future value of the borrowed amount has to be divided by the present value as the first step. Next, the second square root of the aforesaid figure has to be derived, in simplifying the steps to find “r”. Finally, “r” could be found by subtracting one from the solved square root figure.

Economics Concept Introduction

Introduction:

Calculating the interest rate for lending and borrowings is essential as the value of money changes with the passage of time. The longer the time period taken for settling a fixed amount of money, the lower the interest rate. This is due to the same amount being spread over a higher number of years.

(b)

To determine

The interest rate for an amount of $ 2000 borrowed, which has become $ 2600 at the point of re-payment at the end of three years.

Answer to Problem 20P

The interest rate for an amount of $ 2000 borrowed, which has become $ 2600 at the point of re-payment at the end of three years is 9.13%.

Explanation of Solution

  $FV=PV{(1+r)}^n$

  $2600=2000{(1+r)}^3$

  $\frac{2600}{2000}={(1+r)}^3$

  $1.3={(1+r)}^3$

Next, the third square root of 1.3 is being calculated to determine the ‘r’ value.

  $\begin{array}{c}\sqrt[3]{1.3}=1+r\\ 1.0913=1+r\\ r=1.0913-1\\ r=0.0913\\ r=9.13\%\end{array}$

The relevant data has to be substituted to the future value equation in finding the applicable interest rate. The future value of the borrowed amount has to be divided by the present value as the first step. Next, the second square root of the aforesaid figure has to be derived, in simplifying the steps to find “r”. Finally, “r” could be found by subtracting one from the solved square root figure.

Economics Concept Introduction

Introduction:

Calculating the interest rate for lending and borrowings is essential as the value of money changes with the passage of time. The longer the time period taken for settling a fixed amount of money, the lower the interest rate. This is due to the same amount being spread over a higher number of years.

(c)

To determine

The interest rate for an amount of $ 2000 borrowed, which has become $ 2600 at the point of re-payment at the end of six years.

Answer to Problem 20P

The interest rate for an amount of $ 2000 borrowed, which has become $ 2600 at the point of re-payment at the end of six years is 4.46%.

Explanation of Solution

  $FV=PV{(1+r)}^n$

  $2600=2000{(1+r)}^6$

  $\frac{2600}{2000}={(1+r)}^6$

  $1.3={(1+r)}^6$

The sixth square root of 1.3 is being calculated next, to determine the value of ‘r’.

  $\begin{array}{l}\sqrt[6]{1.3}=1+r\\ 1.04469=1+r\\ r=1.04469-1\\ r=0.04469\\ r=4.46\%\end{array}$

The relevant data has to be substituted to the future value equation in finding the applicable interest rate. The future value of the borrowed amount has to be divided by the present value as the first step. Next, the second square root of the aforesaid figure has to be derived, in simplifying the steps to find “r”. Finally, “r” could be found by subtracting one from the solved square root figure.

Economics Concept Introduction

Introduction:

Calculating the interest rate for lending and borrowings is essential as the value of money changes with the passage of time. The longer the time period taken for settling a fixed amount of money, the lower the interest rate. This is due to the same amount being spread over a higher number of years.

(d)

To determine

The interest rate for an amount of $ 2000 borrowed, which has become $ 2600 at the point of re-payment at the end of ten years.

Answer to Problem 20P

The interest rate for an amount of $ 2000 borrowed, which has become $ 2600 at the point of re-payment at the end of ten years would be 2.65%.

Explanation of Solution

  $FV=PV(1+r)$

  $2600=2000{(1+r)}^{10}$

  $\frac{2600}{2000}={(1+r)}^{10}$

  $1.3={(1+r)}^{10}$

The tenth square root of 1.3 is being calculated next, in determining the value of ‘r’.

  $\begin{array}{c}\sqrt[10]{1.3}=1+r\\ 1.02658=1+r\\ r=1.02658-1\\ r=0.02658\\ r=2.65\%\end{array}$

The relevant data has to be substituted to the future value equation in finding the applicable interest rate. The future value of the borrowed amount has to be divided by the present value as the first step. Next, the second square root of the aforesaid figure has to be derived, in simplifying the steps to find “r”. Finally, “r” could be found by subtracting one from the solved square root figure.

Economics Concept Introduction

Introduction:

Calculating the interest rate for lending and borrowings is essential as the value of money changes with the passage of time. The longer the time period taken for settling a fixed amount of money, the lower the interest rate. This is due to the same amount being spread over a higher number of years.