Financial Mathematics

Financial mathematics translates a stream of payments into values at one chosen date. It uses exponents, percentages, and geometric series, so familiarity with Exponential and Logarithmic Functions and Sequences and Series is useful. The central rule is simple: state the interest rate per period, count the matching number of periods, and keep money unrounded until the final answer.

Interest, Time, and the Time Value of Money

Interest is the charge for using money. A dollar available today can earn interest, so it is generally worth more than a dollar received later. Let P be principal (the starting amount), r a decimal annual rate, t time in years, A accumulated value, and I interest earned.

Write 6% as 0.06 before calculating. In every formula, the rate and number of periods must use the same unit: a monthly rate goes with a number of months.

Simple interest

Simple interest is calculated only on the original principal. Its calculator-ready formulas are:

I = Prt
A = P(1 + rt)

Simple interest is often used for short-term loans, though most long-term savings and loans compound.

Worked example: a simple-interest investment

Find the interest and final balance on $2,400 invested at 4.5% simple interest for 18 months.

P = 2400, r = 0.045, t = 18/12 = 1.5

I = Prt
  = 2400(0.045)(1.5)
  = 162

A = P + I = 2400 + 162 = $2,562.00

Compound Interest and Effective Rates

With compound interest, each period's interest is added to the balance; later interest is earned on earlier interest. If the nominal annual rate is r, compounded m times each year for t years, then the periodic rate is i = r/m, the number of periods is n = mt, and:

A = P(1 + r/m)^(mt)
  = P(1 + i)^n

“Compounded monthly” means m = 12. Continuous compounding is a different model: A = Pert, where e ≈ 2.71828.

Worked example: quarterly compounding

$5,000 is invested at a nominal annual rate of 6.0%, compounded quarterly, for 3 years.

P = 5000, r = 0.06, m = 4, t = 3

A = 5000(1 + 0.06/4)^(4 · 3)
  = 5000(1.015)^12
  = 5978.09

Interest earned = 5978.09 − 5000 = $978.09

Nominal rate versus effective annual rate

A nominal annual rate quotes r before the effect of repeated compounding. APR (annual percentage rate) is a regulated disclosure term whose definition and included fees depend on the product and jurisdiction; it is often stated as a nominal annual rate, but the terms are not automatically interchangeable. The effective annual rate (EAR) is the actual one-year percentage increase from the stated compounding:

EAR = (1 + r/m)^m − 1

For continuous compounding, EAR = er − 1. Use EAR to compare accounts with different compounding frequencies, provided the rates describe the same type of product and fees are also considered.

Worked example: compare quoted savings rates

Account A pays 5.00% nominal compounded monthly. Account B pays 5.08% nominal compounded annually. Which gives more after one year, before fees?

EAR_A = (1 + 0.0500/12)^12 − 1
      = 0.051162... = 5.1162...%

EAR_B = (1 + 0.0508/1)^1 − 1 = 5.08%

Account A has the larger effective annual rate.

Present Value and Future Value

Future value is the value at a later date; present value is the equivalent value today. Discounting reverses compounding. For one lump sum over n periods at periodic rate i:

FV = PV(1 + i)^n
PV = FV / (1 + i)^n

The factor 1/(1 + i)n is the discount factor. Choose one focal date before adding or comparing cash flows.

Worked example: discount a future payment

How much must be deposited today at 4.8% compounded monthly to have $10,000 in 5 years?

FV = 10000, i = 0.048/12 = 0.004, n = 12(5) = 60

PV = 10000 / (1.004)^60
   = 7870.05

A deposit of $7,870.05 is needed (assuming the stated rate remains available).

Annuities: Equal, Regular Payments

An annuity is a sequence of equal payments. In an ordinary annuity, payments occur at the end of each period. In an annuity due, they occur at the beginning. The timing matters: every annuity-due payment earns interest for one extra period.

For an ordinary annuity with payment PMT, periodic rate i, and n payments:

FV_ordinary = PMT[(1 + i)^n − 1] / i
PV_ordinary = PMT[1 − (1 + i)^(-n)] / i

For an annuity due, multiply the corresponding ordinary-annuity value by (1 + i). These formulas require i ≠ 0; if i = 0, both values are simply PMT × n.

Worked example: saving monthly

At the end of each month, Maya deposits $150 into an account earning 4.8% nominal compounded monthly. Find the balance after 4 years.

PMT = 150, i = 0.048/12 = 0.004, n = 48

FV = 150[(1.004)^48 − 1] / 0.004
   = 7920.25

Total deposits = 48(150) = $7,200.00
Interest earned = 7920.25 − 7200.00 = $720.25

Worked example: value of rent payments

A tenant will pay $1,600 at the end of each month for 12 months. At a monthly discount rate of 0.5%, what is the present value of those payments?

PMT = 1600, i = 0.005, n = 12

PV = 1600[1 − (1.005)^(-12)] / 0.005
   = $18,590.29

This is not the same as $19,200 because later payments are discounted more heavily than earlier ones.

Loans and Amortization

A fully amortizing loan is repaid by equal periodic payments that reduce both interest and principal, ending at a zero balance. If L is the amount borrowed, solve the present-value annuity formula for the payment:

PMT = L[i(1 + i)^n] / [(1 + i)^n − 1]

For each row of an amortization schedule:

interest for period = previous balance × i
principal repaid    = payment − interest
new balance         = previous balance − principal repaid

Payment amounts are normally rounded to cents, so a final payment may differ by a few cents. Lenders' disclosed payment schedules and compounding conventions control an actual contract.

Worked example: payment and first two loan periods

A $12,000 loan has a 6.0% nominal annual rate compounded monthly and is repaid in 3 years with end-of-month payments.

L = 12000, i = 0.06/12 = 0.005, n = 36

PMT = 12000[0.005(1.005)^36] / [(1.005)^36 − 1]
    = $365.06  (rounded to cents)

Month   Opening balance   Interest   Payment   Principal   Closing balance
1       $12,000.00        $60.00     $365.06   $305.06     $11,694.94
2       $11,694.94        $58.47     $365.06   $306.59     $11,388.35

Although the payment is constant, interest decreases and the principal portion increases. Over 36 payments, the approximate total paid is 36($365.06) = $13,142.16; the approximate finance charge is $1,142.16.

Inflation and Comparing Decisions

Inflation is a rise in the general price level, which lowers the purchasing power of money. If inflation is f per year, a current basket costing C is expected to cost C(1 + f)t after t years.

A nominal return does not directly show the change in purchasing power. The exact real rate q corresponding to nominal return r and inflation f is:

1 + q = (1 + r)/(1 + f)
q = (1 + r)/(1 + f) − 1

Worked example: real return

An investment earns 7.0% over a year while inflation is 3.0% over that year.

q = 1.070/1.030 − 1
  = 0.038835... = 3.88% (approximately)

The common subtraction estimate 7% − 3% = 4% is close but not exact.

Worked example: compare two choices at a common date

Choose between receiving $8,000 now or $9,200 in 2 years. If money can earn 7% annually, compare their values today.

PV of $9,200 = 9200/(1.07)^2 = $8,035.29

$8,035.29 today-value is greater than $8,000 today-value,
so $9,200 in 2 years is financially larger by $35.29.

This conclusion ignores risk, taxes, liquidity, and personal needs; a good decision considers those as well as the calculation.

Common Mistakes to Avoid

  • Using percentages as whole numbers: enter 0.065, not 6.5, for 6.5%.
  • Mismatching periods: monthly compounding requires both i = r/12 and a monthly count such as 12t.
  • Calling a nominal rate an effective rate: 6% nominal compounded monthly is not a 6% annual increase.
  • Putting payments at the wrong time: use ordinary-annuity formulas for end-of-period payments and multiply by (1 + i) for beginning-of-period payments.
  • Rounding every intermediate value: retain calculator precision, then round reported currency to cents.
  • Comparing cash flows on different dates: first move all alternatives to the same present or future date.

Practice Set

  1. Find the simple interest and final amount on $3,600 at 5.25% simple interest for 9 months.
  2. Find the accumulated value of $2,500 at 4.8% nominal annual interest compounded monthly for 6 years.
  3. Find the effective annual rate of 6.4% nominal interest compounded quarterly. Give a percentage to two decimal places.
  4. How much is needed today to have $15,000 in 8 years at 5.5% compounded annually?
  5. $200 is deposited at the end of each month for 5 years into an account paying 3.6% nominal compounded monthly. Find the future value.
  6. Find the present value of receiving $900 at the end of each month for 3 years when the monthly discount rate is 0.4%.
  7. A $18,000 loan is repaid monthly over 4 years at 7.2% nominal annual interest compounded monthly. Find the monthly payment.
  8. For the loan in question 7, use the unrounded payment to find the interest and principal portions of the first payment.
  9. Prices rise by 2.8% while an account earns 5.5% over the same year. Find the exact real rate, approximately.
  10. At a discount rate of 6% per year, which is worth more today: $4,700 now or $5,000 in one year? By how much?
  11. A deposit is made at the beginning of every month rather than the end. In terms of the ordinary-annuity future value, what adjustment is made, and why?

Answer Checks

  1. t = 9/12 = 0.75. I = 3600(0.0525)(0.75) = $141.75, so A = $3,741.75.
  2. A = 2500(1 + 0.048/12)72 = $3,332.48.
  3. EAR = (1 + 0.064/4)4 − 1 = 0.065548... = 6.55%.
  4. PV = 15000/(1.055)8 = $9,773.98.
  5. i = 0.036/12 = 0.003 and n = 60. FV = 200[(1.003)60 − 1]/0.003 = $13,126.32.
  6. PV = 900[1 − (1.004)−36]/0.004 = $30,119.29.
  7. i = 0.072/12 = 0.006 and n = 48. PMT = 18000[0.006(1.006)48]/[(1.006)48 − 1] = $432.70 per month.
  8. Using PMT = 432.705...: first interest = 18000(0.006) = $108.00. First principal = 432.705... − 108 = $324.70, approximately.
  9. q = 1.055/1.028 − 1 = 0.026264... = 2.63%, approximately.
  10. PV of $5,000 = 5000/1.06 = $4,716.98. The $5,000 in one year is worth more today by $16.98.
  11. Multiply the ordinary-annuity future value by (1 + i). Each beginning-of-month deposit earns interest for one additional month.