Chapter 3, Problem 10P

(a)

To determine

The applicable interest rates for a sum of $160 if it is to become $170 in one year’s time and if it was $150 a year ago.

Answer to Problem 10P

The applicable interest rate for a sum of $160 if it is to become $170 in one year’s time is 6.6%.

The applicable interest rates for a sum of money that was worth $150 one year ago to become $170 in one year’s time is 6.25%

Explanation of Solution

  $FV=PV{(1+r)}^n$

  $160=150{(1+r)}^1$

  $\frac{160}{150}={(1+r)}^1$

  $1.0666={(1+r)}^1$

Next, the first square root of 1.0666 must be calculated in order to find the value of ‘r’.

  $\begin{array}{c}\sqrt[1]{1.0666}=1+r\\ 1.0666=1+r\\ r=1.0666-1\\ r=0.0666\\ r=6.6\%\end{array}$

Next, the same equation would be used to calculate the future value of the second sum of money.

  $FV=PV{(1+r)}^n$

  $170=160{(1+r)}^1$

  $\frac{170}{160}={(1+r)}^1$

  $1.0625={(1+r)}^1$

The next step involves in calculating the first square root of 1.0625 in order to derive the value of ‘r’.

  $\begin{array}{c}\sqrt[1]{1.0625}=1+r\\ 1.0625=1+r\\ r=1.0625-1\\ r=0.0625\\ r=6.25\%\end{array}$

Both the interest rates for the past year and the year ahead must be derived by using the future value equation. In the first equation, the future value would be $160 and the present value, $150. As the interest rate is being calculated for a period of one year, the “n” value becomes one. Solving the equation, the final answer would be 6.6%. In the second equation, the present value is $160 and the future value is $170. Further, the timespan involved is again one year, making the “n” value one. The answer would be 6.25%.

Economics Concept Introduction

Introduction:

As money has a value that changes with the passage of time, calculating the rate of interest is essential in the field of investments. It is said to be the income received for amounts lent. The value of a certain amount of money today will not be the same in one year’s time or one year ago. Hence, the interest rate is helpful in bringing these values to a common platform.

(b)

To determine

The interest rate applicable for a sum of $150 invested one year from today, that would become $170 one year from today.

Answer to Problem 10P

The interest rate applicable for a sum of $150 invested one year from today that would become $170 one year from today is 6.4%.

Explanation of Solution

  $FV=\text{}PV{(1+r)}^n$

  $170=150{(1+r)}^2$

  $\frac{170}{150}\text{}={(1+r)}^2$

  $1.133={(1+r)}^2$

Next, the second square root of 1.133 should be found in order to determine the value of ‘r’.

  $\begin{array}{c}\sqrt[2]{1.133}=1+r\\ 1.0644=1+r\\ r=1.0644-1\\ r=0.0644\\ r=6.4\%\end{array}$

As the amount of $150 has been invested one year ago and shall be grown into $170 one year from now, the total timespan involved would be two years. The sum is done with the standpoint that $150 would be invested today, to be grown into $170 in two years. When solved, the applicable interest rate would be 6.4%

Economics Concept Introduction

Introduction:

Time value of money is the reason why interest is being calculated for moneys that are invested over a span of time. If one invests a particular amount of money today, the future value of that amount would either appreciate or depreciate depending on the economic and other conditions. In order to prevent investors from losing, interest is being calculated.