Chapter 6, Problem 69QP

Summary Introduction

To calculate: The faster payment of the loan by Person X if he makes a monthly payment of $225 with his new card and what happens if Person X has 2% of fee charged on any of his transferred balance

Introduction:

The time taken for the repayment of the loan is termed as the number of periods. It is denoted by “t”.

Answer to Problem 69QP

Person X makes the payment faster by 45.10 months and if Person X is charged with a fee of 2% then he takes to pay off the card by 43.28 months.

Explanation of Solution

Given information:

The Christmas ski vacation of person X was good but it ran over the budget. However, everything is not lost. Person X received a mail that states to transfer $12,000 from the current credit card that charges 18.6% of an annual rate and the new credit card of 9.2% charge. The monthly payment made by Person X with his new card is $25. It is assumed that there is a 2% fee charged for the balance transferred.

Note: The number of periods that is essential to pay back the loan with no fees is calculated first. The number of payments is solved using the formulae of the present value of annuity. The without fee and annual rate is 18.60%

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}$

Note: C denotes the payments, r denotes the rate of exchange, and t denotes the period.

Compute the present value annuity for without fee:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$12,000=\$225\left\{\frac{\left[1-\left(\frac{1}{{\left(1+\frac{0.186}{12}\right)}^t}\right)\right]}{\frac{0.186}{12}}\right\}\\ \$12,000=\$225\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.0155\right)}^t}\right)\right]}{0.0155}\right\}\end{array}$

Solving t with this equation:

$\begin{array}{c}\frac{1}{{1.0155}^t}=1-\left(\frac{\$12,000}{\$225}\right)\left(0.0155\right)\\ \frac{1}{{1.0155}^t}=1-\left(53.333333\right)\left(0.0155\right)\\ \frac{1}{{1.0155}^t}=0.1733\\ t=\frac{\ln \left(\frac{1}{0.1733}\right)}{\ln 1.0155}\end{array}$

$\begin{array}{c}t=\frac{\ln \text{5}\text{.769230769}}{\ln 1.0155}\\ t=\frac{1.752538756}{0.015381102}\\ t=113.94\text{ months}\end{array}$

Hence, the number of months without an annual fee at the rate of 18.60% is 113.94 months.

Note: Now the value of t is computed using the formulae of the present value of an annuity without fee and at an annual rate of 9.20%

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}$

Note: C denotes the payments, r denotes the rate of exchange, and t denotes the period.

Compute the present value annuity for without fee:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$12,000=\$225\left\{\frac{\left[1-\left(\frac{1}{{\left(1+\frac{0.092}{12}\right)}^t}\right)\right]}{\frac{0.092}{12}}\right\}\\ \$12,000=\$225\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.0076667\right)}^t}\right)\right]}{0.0076667}\right\}\end{array}$

Solving t with this equation:

$\begin{array}{c}\frac{1}{{1.007667}^t}=1-\left(\frac{\$12,000}{\$225}\right)\left(0.007667\right)\\ \frac{1}{{1.007667}^t}=1-\left(53.333333\right)\left(0.007667\right)\\ \frac{1}{{1.007667}^t}=0.591093333\\ t=\frac{\ln \left(\frac{1}{0.591093333}\right)}{\ln 1.007667}\end{array}$

$\begin{array}{c}t=\frac{\ln 1.691780204}{\ln 1.007667}\\ t=68.84\text{ months}\end{array}$

Hence, the number of months without an annual fee at the rate of 9.20% is 68.84 months.

Calculation of the months to pay off the new card without fee:

$\begin{array}{c}\text{Months quicker to pay off the card}=113.94-68.84\\ =45.10\text{ months}\end{array}$

Note: The quicker months to pay off the card is calculated by subtracting the calculated number of months without an annual fee at 9.20% from the calculated number of months without an annual fee at 18.60%.

Hence, the faster payments made by Person X without a fee on the new card is 45.10 months

Note: It is not necessary to compute the time that is needed to pay back the current credit of Person X with a fee as it incurs no fee. It will take 113.94 months to pay off the current card of Person X.

Calculations of the time taken to pay back the new card with a transfer fee:

The calculations of the time taken to pay back the new card with a transfer fee are made with the help the equations of the present value of the annuity. The annual rate is 9.20%.

Formula to calculate the present value annuity:

$\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{1+r^t}\right)\right]}{r}\right\}$

Note: C denotes the payments, r denotes the rate of exchange, and t denotes the period.

Compute the present value annuity for without fee:

$\begin{array}{c}\text{Present value annuity}=C\left\{\frac{\left[1-\left(\frac{1}{{\left(1+r\right)}^t}\right)\right]}{r}\right\}\\ \$12,240=\$225\left\{\frac{\left[1-\left(\frac{1}{{\left(1+\frac{0.092}{12}\right)}^t}\right)\right]}{\frac{0.092}{12}}\right\}\\ \$12,240=\$225\left\{\frac{\left[1-\left(\frac{1}{{\left(1+0.0076667\right)}^t}\right)\right]}{0.0076667}\right\}\end{array}$

Note: The 2% interest rate is added to the present value of an annuity amount

Solving t with this equation:

$\begin{array}{c}\frac{1}{{1.007667}^t}=1-\left(\frac{\$12,240}{\$225}\right)\left(0.007667\right)\\ \frac{1}{{1.007667}^t}=0.5829152\\ t=\frac{\ln \left(\frac{1}{0.5829152}\right)}{\ln 1.007667}\\ t=\frac{\ln 1.715515396}{\ln 1.007667}\end{array}$

$t=70.66\text{ months}$

Hence, the number of months with an annual fee at the rate of 9.20% is 70.66 months.

Calculation of the months to pay off the new card without a fee:

$\begin{array}{c}\text{Months quicker to pay off the card}=113.94-70.66\\ =43.28\text{ months}\end{array}$

Hence, the faster payments made by Person X with a fee on the new card is 43.28 months.