Mathematics of Finance

MATH104Dr. Quinn

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The best projects will be displayed (with your permission) on the new 60+” big screen TV in the lobby of Ross Hall next semester.

Example: I take a $10,000 cash advance on my credit card.

The card charges 24% interestI make equal payments for 20 yearsHow much interest will I pay? Monthly payment = PMT= = $201.74Total repaid =($201.74)(12)(20) = $48,417.80Total interest paid=$48,417.80 - $10,000 = $38,417.80

rate 24% time 20 total interest $38,417.80debt 10,000 PMT $201.74

month pay to I to P new balance total I$10,000.00

1 $201.74 200.00 $1.74 $9,998.26 $200.002 $201.74 199.97 $1.78 $9,996.48 $399.973 $201.74 199.93 $1.81 $9,994.67 $599.894 $201.74 199.89 $1.85 $9,992.83 $799.795 $201.74 199.86 $1.88 $9,990.94 $999.646 $201.74 199.82 $1.92 $9,989.02 $1,199.46

You can see in the amortization table that only $1.74 of my $201.74 monthly payment goes to pay off the principal the first month. After 240 months of payments, I will have spent $38,417.80 in interest on this $10,000 loan!

To pay less interest, we could:

• Reduce the time on the loan• Reduce the interest rate• Pay more down• Make extra payments towards principal

If we reduce the time from 20 years to 10 years

Monthly payment = PMT= = $220.48Total repaid =($220.48)(12)(10) = $26,457.72Total interest paid

=$26,457.72 - $10,000 = $16,457.72 We will pay $18.74 a month, but we will save $38,417.80

– $16,457.72= $21,960.08 in interest!

rate 24%

time 10 total interest $16,457.72

debt 10,000

PMT $220.48

month pay to I to P new balance total I

$10,000.00

1 $220.48 200.00 $20.48 $9,979.52 $200.00

2 $220.48 199.59 $20.89 $9,958.63 $399.59

3 $220.48 199.17 $21.31 $9,937.32 $598.76

4 $220.48 198.75 $21.73 $9,915.59 $797.51

5 $220.48 198.31 $22.17 $9,893.42 $995.82

6 $220.48 197.87 $22.61 $9,870.80 $1,193.69

In the original example, $1.74 went towards principal. Now, by paying it off faster, $20.48 goes towards principal. We will pay $18.74 more a month, but we will save $38,417.80 – $16,457.72= $21,960.08 in interest!

If we reduce the rate from r=24% to r=12%

Monthly payment = PMT= = $110.11Total repaid =($110.11)(12)(20) = $26,426.07Total interest paid =$26,426.07 - $10,000 =

$16,426.07 Our monthly payment goes down $91.63 and we

save $38,417.80 – $16,426.07= $21,991.73 in interest!

rate 12%

time 20 total interest $16,426.07

debt 10,000

PMT $110.11

month pay to I to P new balance total I

$10,000.00

1 $110.11 100.00 $10.11 $9,989.89 $100.00

2 $110.11 99.90 $10.21 $9,979.68 $199.90

3 $110.11 99.80 $10.31 $9,969.37 $299.70

4 $110.11 99.69 $10.41 $9,958.95 $399.39

5 $110.11 99.59 $10.52 $9,948.44 $498.98

6 $110.11 99.48 $10.62 $9,937.81 $598.46

Our monthly payment goes down $91.63 and we save $38,417.80 – $16,426.07= $21,991.73 in interest!

To save the most on interest, we could pay it off in less time (10 years) and at a lower rate (r=12%)

Monthly payment = PMT= = $143.47Total repaid =($143.47)(12)(10) = $17,216.51Total interest paid =$17,216.51 - $10,000 = $7,216.51 Adjusting both our time and rate makes our monthly

payment go down $58.27, and we save $38,417.80 – $7,216.51= $31,200.57 in interest!

rate 12%

time 10 total interest $7,216.51

debt 10,000

PMT $143.47

month pay to I to P new balance total I

$10,000.00

1 $143.47 100.00 $43.47 $9,956.53 $100.00

2 $143.47 99.57 $43.91 $9,912.62 $199.57

3 $143.47 99.13 $44.34 $9,868.28 $298.69

4 $143.47 98.68 $44.79 $9,823.49 $397.37

5 $143.47 98.23 $45.24 $9,778.25 $495.61

6 $143.47 97.78 $45.69 $9,732.57 $593.39

Adjusting both our time and rate makes our monthly payment go down $58.27, and we save $38,417.80 – $7,216.51= $31,200.57 in interest!

Conclusion

To save interest on a loan:• Reduce the rate• Reduce the time• Pay more down• Make extra payments