# UnitMath Examples ( Financial ) Last Modified: 2/15/2000

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This page is used to present examples of using UnitMath in Financial situations. I have made a serious effort to ensure that the following examples are correct, but I do not guarantee them.

### Does the % Rate Matter TOP For compound interest does interest rate really matter? The examples used in this example use the compund interest formula. The graph at the right shows the affect that a 1% ( 4% to 5% ) difference in interest rate on \$100 over 30 years.

### Examples

Starting with \$100 invested with several ranges of interest rates find the future value using compound interest.

PV: \$ 100;
interest_period: day;
time: 30 years;

interest_rate: ( 1 to 2 )% per year;
PV ( 1 + interest_rate interest_period ) ^(time / interest_period) ~ ( 134.99 to 182.21 ) \$
(182.21 - 134.99 ) / 134.99 as % ~ ( 34.97 to 34.99 ) %

interest_rate: ( 7 to 8 )% per year;
PV ( 1 + interest_rate interest_period ) ^(time / interest_period) ~ ( 816.45 to 948.55 to 1,102.03 ) \$
(1,102.03 - 816.45 ) / 816.45 as % ~ 34.98 %

interest_rate: ( 27 to 28 )% per year;
PV ( 1 + interest_rate interest_period ) ^(time / interest_period) ~ ( 328,462.43 to 443,277.86 ) \$
(443,277.86 - 328,462.43 ) / 328,462.43 as % ~ 34.96 %

Yes, interest rate matters, no surprise. The surprisingly thing is that it matters as much at the high end ( 27 % to 28 % ) as it does the low end ( 1 % to 2 % ).

### Compound verses Simple Interest TOP Compare the differences in future value between simple and compound interest. The graph at the right shows how the future value varies over time starting with \$100. Below the actual equations ( modified to take advantage of UnitMath's Units ) show specific values.

Simple interest uses the equation: FV = PV ( 1 + n i )
Compound interest uses the equation: FV = PV ( 1 + i )^n

Where
FV: the future value
PV: the present value
n: the interest per year
t: the number of years

To make the units cancel in UnitMath the above equations need to be modified as follows.

Simple interest uses the equation: FV = PV ( 1 + interest_rate time )
Compound interest uses the equation: FV = PV ( 1 + interest_rate interest_period ) ^(time / interest_period)

Where
FV: the future value
PV: the present value
interest_rate: the interest per some time period
time: the time duration
interest_period: the time when the accured interest is considdered part of the principle and begines to draw interest.

### Examples

Starting with \$100 invested at 7% per year, find the amount for both simple and compound interest. For compound interest assume it's compounded daily.

PV: \$ 100;
interest_rate: 7% per year;
interest_period: day;
time: 1 year;

ASI:PV ( 1 + interest_rate time ); ~ 107.00 \$
ACI:PV ( 1 + interest_rate interest_period ) ^(time / interest_period) ~ 107.25 \$
time: 5 years;
ASI; ~ 135.00 \$
ACI; ~ 141.90 \$
time: 10 years;
ASI; ~ 170.00 \$
ACI; ~ 201.36 \$
time: 20 years;
ASI; ~ 240.00 \$
ACI; ~ 405.47 \$

Find the amount of time for the compound interest FV to be double the simple interest FV. The first case is for 7%/year and the second is for 10%/year.

time: ( 1 to 50 ) years as years & months & weeks & days;
solve( 2 *ASI = ACI; time: day) ~ time: ( 23 years + 11 months + 3 weeks + ( 1.1 to 3.6 ) days );

interest_rate: 10% per year;
time: ( 1 to 50 ) years as years & months & weeks & days;
solve( 2 *ASI = ACI; time: day) ~ time: ( 16 years + 9 months + ( 1.8 to 2.0 ) weeks );

### Monthly Loan Payments TOP

Find the monthly payments for a loan.

Monthly payments are calculated from: Payment = Debt / ( ( 1 - ( 1 + i )^n ) / i ) ( 1 + n i )
Where
Debt: Is the original debt
i: the interest per year
n: the number of payments

To make the units cancel the above equation needs to be modified as follows.

Simple interest is given by the equation: payment: Debt /( (1 - ( 1 + IR * IP ) ^(-time / IP))/(IR * IP) )
Where
Debt: Is the original debt
IR: the interest per some time period
IP: the time when the accured interest is considdered part of the principle time: the duration of the loan

### Examples

Find the monthly payment needed to pay off a loan of \$10,000 in 10 years when the interest rate is 12 1/4 %/year.

Debt: \$ 10000;
IR: ( 12 +1/4 ) %/year;
IP: month;
time: 10 years;
payment: P /( (1 - ( 1 + interest_rate interest_period ) ^(-time / interest_period))/(interest_rate interest_period) ) ~ 144.92 \$

Now find the total payed on the loan and the total interest charged.

totalPayments: payment * time / IP ~ 17,390.38 \$
interest: totalPayments - Debt ~ 7,390.38 \$

Find the monthly payment needed to pay off a loan of \$50,000 in 30 years when the interest rate is 13 %/year.

Debt: \$ 50,000;
IR: 13 %/year;
IP: month;
time: 30 years;
payment: Debt /( (1 - ( 1 + IR IP ) ^(-time / IP))/(IR IP) ) ~ 553.10 \$

Now find the total payed on the loan and the total interest charged.

totalPayments: payment * time / interest_period ~ 199,115.91 \$
interest: totalPayments - Debt ~ 149,115.91 \$

### Loan Shark TOP

This example uses the compund interest formula.

Find the amount due to a loan shark on a loan of \$1000 with a rate of 25% per week compounded hourly after 2 weeks, 5 days, and 3 hours. This example shows the power of the generalized compound interest equations, and the danger / foolishness of this kind of debt.

debt: \$ 1000;
interest_rate: 25% per week;
interest_period: hour;
time: 2 weeks + 5 days + 3 hours;

ACI: debt ( 1 + interest_rate interest_period ) ^(time / interest_period) ~ 1,978.87 \$

The following calculations find the time for the debt to double ( about 19 days ), triple ( about 31 days ), ... time: 1 hour to 1 year as days & hours & min;
solve( 2 * P = ACI; time: min) ~ time: ( 19 days + 10 hours + ( 8.0 to 8.5 ) min );
solve( 3 * P = ACI; time: min) ~ time: ( 30 days + 18 hours + 49. min );
solve( 4 * P = ACI; time: min) ~ time: ( 38 days + 20 hours + ( 16. to 17. ) min );
solve( 5 * P = ACI; time: min) ~ time: ( 45 days + 2 hours + ( 20. to 21. ) min );
solve( 6 * P = ACI; time: min) ~ time: ( 50 days + 4 hours + ( 57. to 58. ) min );