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, $4,000, 22​%; Card​ 2, $5,700, 26%; and Card​ 3, ​$3,300​,19​%. Interest compounds monthly on all loan balances. A credit card loan consolidation company has captured​ Mary's attention by stating they can save Mary 23​% per month on her credit card payments. This company charges 15.5​% APR. Is the​company's claim​ correct? Assume a 10-year repayment period.

Answer 1

Effective interest rate of Card 1 with monthly compounding i.e n=12 and I=22%

EAR = (1+i/n)^12-1 = (1+22%/12)^12-1 =(1+1.83%)^12-1=1.0183^12-1=1.2436-1=0.2436 or 24.36%

Monthly EAR = 24.36%/12=2.03%

Monthly Repayment for $4000 at 2.03% for 10*12=120 periods will be

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

or, 4000=A*(1-(1+2.03%)^-120)/2.03%

or, 4000=A*(1-(1.0203)^-120)/0.0203

or, 4000=A*(1-0.0897)/0.02203

or, 4000=A*0.9103/0.02203

or, A = 4000*0.0203/0.9103

or, A = $89.20

Effective interest rate of Card 2 with monthly compounding i.e n=12 and I=26%

EAR = (1+i/n)^12-1 = (1+26%/12)^12-1 =(1+2.17%)^12-1=1.0217^12-1=1.2933-1=0.2933 or 29.33%

Monthly EAR =29.33%/12=2.44%

Monthly Repayment for $5700 at 2.44% for 10*12=120 periods will be

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

or, 5700=A*(1-(1+2.44%)^-120)/2.44%

or, 5700=A*(1-(1.0244)^-120)/0.0244

or, 5700=A*(1-0.0554)/0.0244

or, 5700=A*0.9446/0.0244

or, A = 5700*0.0244/0.9406

or, A = $147.24

Effective interest rate of Card 3 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 $3300 at 1.73% for 10*12=120 periods will be

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

or, 3300=A*(1-(1+1.73%)^-120)/1.73%

or, 3300=A*(1-(1.0173)^-120)/0.0173

or, 3300=A*(1-0.1279)/0.0173

or, 3300=A*0.8721/0.0173

or, A = 3300*0.0173/0.8721

or, A = $65.46

Total Monthly Payments = 89.20+147.24+65.46=$301.9

Total Loan Amount = 4000+5700+3300=$13000

Effective interest rate of Consolidation with monthly compounding i.e n=12 and I=15.5%

EAR = (1+i/n)^12-1 = (1+15.5%/12)^12-1 =(1+1.29%)^12-1=1.0129^12-1=1.1665-1=0.1665 or 16.65%

Monthly EAR =16.65%/12=1.39%

Monthly Repayment for $13000 at 1.39% for 10*12=120 periods will be

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

or, 13000=A*(1-(1+1.39%)^-120)/1.39%

or, 13000=A*(1-(1.0139)^-120)/0.0139

or, 13000=A*(1-0.1914)/0.0139

or, 13000=A*0.8086/0.0139

or, A = 13000*0.0139/0.8086

or, A = $223.46

Hence Monthly payment on consolidation is $223.46

Net savings in monthly payment = 301.90-223.46=$78.44

% savings = 78.44/301.9=25.98%

Hence the claim is correct in fact the monthly saving on consolidation is 25.98% which is more than the claim