Math of finance helps students understand how money grows, how loans are repaid, and how regular payments build savings over time. This cheat sheet focuses on loans and annuities, which are common in car loans, mortgages, retirement plans, and savings accounts. Students need these formulas to compare financial choices and understand the cost of borrowing or the value of saving.
Key Facts
- Simple interest is calculated by I = Prt, where P is principal, r is annual interest rate as a decimal, and t is time in years.
- Compound amount is calculated by A = P(1 + r/n)^(nt), where n is the number of compounding periods per year.
- The future value of an ordinary annuity is FV = PMT[((1 + i)^n - 1) / i], where payments are made at the end of each period.
- The present value of an ordinary annuity is PV = PMT[1 - (1 + i)^(-n)] / i.
- A loan payment for an amortized loan is PMT = PV[i / (1 - (1 + i)^(-n))].
- The periodic interest rate is i = annual rate / payments per year, and the number of payments is n = years × payments per year.
- Total interest paid on a loan equals total payments minus principal, or Interest = PMT × n - PV.
- For the same annual rate, more frequent compounding usually produces a higher final amount because interest is added more often.
Vocabulary
- Principal
- The original amount of money borrowed, invested, or deposited before interest is added.
- Interest
- The cost of borrowing money or the earnings from investing money.
- Compound Interest
- Interest calculated on both the original principal and previously earned interest.
- Annuity
- A series of equal payments made at regular time intervals.
- Present Value
- The current worth of a future payment or series of payments, discounted by interest.
- Amortized Loan
- A loan repaid with regular equal payments that cover both interest and part of the principal.
Common Mistakes to Avoid
- Using the percent instead of the decimal rate is wrong because formulas require 6% to be written as 0.06.
- Forgetting to match the rate and payment period is wrong because monthly payments need a monthly interest rate, not the annual rate.
- Using the future value annuity formula for a loan is wrong because a loan payment formula is based on present value and repayment of debt.
- Mixing up n and t is wrong because t is time in years while n is the total number of payment or compounding periods.
- Ignoring total interest paid is wrong because a smaller monthly payment can still cost more overall if the loan lasts longer.
Practice Questions
- 1 Find the simple interest on $2,500 invested at 4.8% per year for 3 years.
- 2 A $1,200 deposit earns 5% annual interest compounded monthly for 4 years. What is the final amount?
- 3 A car loan has a present value of $18,000, a monthly interest rate of 0.005, and 60 monthly payments. Use PMT = PV[i / (1 - (1 + i)^(-n))] to find the monthly payment.
- 4 Explain why two loans with the same interest rate and principal can have different total interest costs.
Understanding Math of Finance (Loans & Annuities)
A key idea behind finance math is that money at different times does not have the same value. Money available now can earn interest, so it is usually worth more than the same amount received later. Present value works backward from a future payment or balance to find its value today.
Future value works forward to show what current deposits may become. This time value idea explains why a lender charges interest and why a saver earns it. It is the foundation beneath nearly every loan and savings calculation.
Most long term loans are amortized. This means one regular payment gradually covers both the interest due and part of the original amount borrowed. Early in the loan, the remaining balance is large, so a greater share of each payment goes to interest.
Later, the balance is smaller, so more of the payment reduces the principal. An amortization schedule lists this change payment by payment. It can show why making an extra principal payment early often saves more interest than making the same extra payment near the end.
An annuity is a series of equal payments made at regular times. The timing matters a great deal. In an ordinary annuity, each payment arrives at the end of a period.
A monthly retirement contribution made after each month is one example. In an annuity due, payments arrive at the beginning of each period. Rent is often paid this way.
Each annuity due payment has one extra period to earn interest, so its accumulated value is higher when all other details stay the same. Students should always identify the payment timing before choosing a method.
Rates and time periods must use matching units. A yearly rate cannot be used directly with monthly payments. It must first be changed into a monthly rate.
The total number of periods must match that same monthly pattern. For example, a five year loan with monthly payments has sixty payment periods. This is a common source of wrong answers.
Another important distinction is between a stated annual rate and an effective annual rate. More frequent compounding can make the effective yearly cost or return different from the stated rate.
Real financial agreements contain details that a basic classroom model may leave out. A car loan may include fees, a mortgage may have insurance or property tax payments, and a savings account may set minimum balance rules. Loan advertisements may show a low monthly payment while hiding a longer repayment period and a larger total cost.
When comparing options, students should examine the payment amount, number of payments, total paid, total interest, rate, and any fees. Rounding only at the end of a calculation helps prevent small errors from growing across many periods.