Question

Mary's credit card situation is out of control because she cannot afford to make her monthly payments. She has three credit cards with the following loan balances and​ APRs: Card​ 1, $4,500, 19%; Card 2, $5,500, 23%; and Card 3, $3,200, 16%. interest compounds monthly on all loan balances. A credit card loan consolidation company has captured​ Mary's attention by stating they can save Mary 17% per month on her credit card payments. This company charges 16.5% APR. Is the​ company's claim​ correct? Assume a 10​-year repayment period.

Mary's current minimum monthly payments are ​ $ ??

Answer 1

Effective interest rate of Card 1 with monthly compounding i.e n=12 and I=19%

EAR = (1+i/n)^12-1 = (1+19%/12)^12-1 =(1+1.58%)^12-1=1.0158^12-1=1.2074-1=0.2074 or 20.74%

Monthly EAR =20.74%/12=1.73%

Monthly Repayment for $4500 at 1.73% for 10*12=120 periods will be

PV=A*(1-(1+r)^-n)/r

or, 4500=A*(1-(1+1.73%)^-120)/1.73%

or, 4500=A*(1-(1.0173)^-120)/0.0173

or, 4500=A*(1-0.1279)/0.0173

or, 4500=A*0.8721/0.0173

or, A = 4500*0.0173/0.8721

or, A = $89.27

Effective interest rate of Card 2 with monthly compounding i.e n=12 and I=23%

EAR = (1+i/n)^12-1 = (1+23%/12)^12-1 =(1+1.92%)^12-1=1.0192^12-1=1.2559-1=0.2559 or 25.59%

Monthly EAR =25.59%/12=2.13%

Monthly Repayment for $5500 at 2.13% for 10*12=120 periods will be

PV=A*(1-(1+r)^-n)/r

or, 5500=A*(1-(1+2.13%)^-120)/2.13%

or, 5500=A*(1-(1.0213)^-120)/0.0213

or, 5500=A*(1-0.0795)/0.0213

or, 5500=A*0.9205/0.0213

or, A = 5500*0.0213/0.9205

or, A = $127.27

Effective interest rate of Card 3 with monthly compounding i.e n=12 and I=16%

EAR = (1+i/n)^12-1 = (1+16%/12)^12-1 =(1+1.33%)^12-1=1.0133^12-1=1.1723-1=0.1723 or 17.23%

Monthly EAR =17.23%/12=1.44%

Monthly Repayment for $3200 at 1.44% for 10*12=120 periods will be

PV=A*(1-(1+r)^-n)/r

or, 3200=A*(1-(1+1.44%)^-120)/1.44%

or, 3200=A*(1-(1.0144)^-120)/0.0144

or, 3200=A*(1-0.1808)/0.0144

or, 3200=A*0.8192/0.0144

or, A = 3200*0.0144/0.8192

or, A = $56.25

Total Monthly Payments = 89.27+127.27+56.25=$272.79

Total Loan Amount = 4500+5500+3200=$13200

Effective interest rate of Consolidation with monthly compounding i.e n=12 and I=16.5%

EAR = (1+i/n)^12-1 = (1+16.5%/12)^12-1 =(1+1.375%)^12-1=1.01375^12-1=1.1781-1=0.1781 or 17.81%

Monthly EAR =17.81%/12=1.48%

Monthly Repayment for $13200 at 1.48% for 10*12=120 periods will be

PV=A*(1-(1+r)^-n)/r

or, 13200=A*(1-(1+1.48%)^-120)/1.48%

or, 13200=A*(1-(1.0148)^-120)/0.0148

or, 13200=A*(1-0.1715)/0.0148

or, 13200=A*0.8285/0.0148

or, A = 13200*0.0148/0.8285

or, A = $235.81

Hence Monthly payment on consolidation is $235.81

Net savings in monthly payment = 272.79-235.81=$36.98

% savings = 36.98/272.79=13.56%

Hence the claim is incorrect as the saving in monthly payment is 13.56%