Your Christmas ski vacation was great, but it unfortunately ran a bit over budge

Question

Your Christmas ski vacation was great, but it unfortunately ran a bit over budget. All is not lost: You just received an offer in the mail to transfer your 513,700 balance from your current credit card, which charges an annual rate of 21.5 percent, to a new credit card charging a rate of 12.1 percent. How much faster could you pay the loan off by making your planned monthly payments of $310 with the new card? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.) What if there was a 1 percent fee charged on any balances transferred? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Explanation / Answer

Solution:

Let To denote the number of months it would take to reimburse the $13,700 with the old credit card. Then To is such that

13,700 = 310 * {1 – [1/ (1+0.215/12)]To / (0.215/12)}

Then To = Log (1/ {1 - (13,700 * 0.215/12)/ 310}/ Log (1 + 0.215/12)

= Log (1/ {1 - (13,700 * 0.01791667)/ 310}/ Log (1 + 0.01791667)

= Log (1/ {1 - (245.458333)/ 310}/ Log (1.01791667)

= Log (1/ {1 - 0.791801075}/ Log (1.01791667)

= Log (1/ {0.208198925}/ Log (1.01791667)

= Log (4.803098773)/ Log (1.01791667)

= 0.68152/ 0.0071

= 88.4 Months

Let Tn denote the number of months it would take to repay the loan with the new credit card. Assuming that there is no fee on balances transfer, Tn is found by replacing 0.215 with 0.121 in the last equation, which gives us

Tn = Log (1/ {1 – (13,700 * 0.121/12)/ 310}/ Log (1 + 0.121/12)

= Log (1/ {1 – 0.44561828}/ Log (1 + 0.010083)

= Log (1/ 0.55438172)/ Log (1 + 0.010083)

= Log (1.8038112)/ Log (1.010083)

= 0.25619/ 0.00436

= 58.8Months

If there is a 1% fee on balances transfer, the amount of the loan becomes 13,700 * 1.01 = 13,837, and then the number of months it takes to repay is

Tn,fee = Log (1/ {1 – (13,837 * 0.121/12)/ 310}/ Log (1 + 0.121/12)

= Log (1/ {1 – 0.450074462}/ Log (1 + 0.010083)

= Log (1/ 0.549925538)/ Log (1 + 0.010083)

= Log (1.818428)/ Log (1.010083)

= 0.2597/ 0.00436

= 59.6 Months

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