Mathematics of Money:
Compound Interest Analysis With Applications

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Professor Hossein Arsham   

Compound Interest: The future value (FV) of an investment of present value (PV) dollars earning interest at an annual rate of r compounded m times per year for a period of t years is:

FV = PV(1 + r/m)mt

FV = PV(1 + i)n

where i = r/m is the interest per compounding period and n = mt is the number of compounding periods.

One may solve for the present value PV to obtain:

PV = FV/(1 + r/m)mt

Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is

FV = PV(1 + r/m)mt   = 20,000(1 + 0.085/12)(12)(4)   = $28,065.30

Notice that the interest earned is $28,065.30 - $20,000 = $8,065.30 -- considerably more than the corresponding simple interest.

Effective Interest Rate: If money is invested at an annual rate r, compounded m times per year, the effective interest rate is:

reff = (1 + r/m)m - 1.

This is the interest rate that would give the same yield if compounded only once per year. In this context r is also called the nominal rate, and is often denoted as rnom.

Numerical Example: A CD paying 9.8% compounded monthly has a nominal rate of rnom = 0.098, and an effective rate of:

r eff =(1 + rnom /m)m   =   (1 + 0.098/12)12 - 1   =  0.1025.

Thus, we get an effective interest rate of 10.25%, since the compounding makes the CD paying 9.8% compounded monthly really pay 10.25% interest over the course of the year.

Mortgage Payments Components: Let where P = principal, r = interest rate per period, n = number of periods, k = number of payments, R = monthly payment, and D = debt balance after K payments, then

R = P r / [1 - (1 + r)-n]


D = P (1 + r)k - R [(1 + r)k - 1)/r]

Accelerating Mortgage Payments Components: Suppose one decides to pay more than the monthly payment, the question is how many months will it take until the mortgage is paid off? The answer is, the rounded-up, where:

n = log[x / (x Ė P r)] / log (1 + r)

where Log is the logarithm in any base, say 10, or e.

Future Value (FV) of an Annuity Components: Ler where R = payment, r = rate of interest, and n = number of payments, then

FV = [ R(1 + r)n - 1 ] / r

Future Value for an Increasing Annuity: It is an increasing annuity is an investment that is earning interest, and into which regular payments of a fixed amount are made. Suppose one makes a payment of R at the end of each compounding period into an investment with a present value of PV, paying interest at an annual rate of r compounded m times per year, then the future value after t years will be

FV = PV(1 + i)n + [ R ( (1 + i)n - 1 ) ] / i

where i = r/m is the interest paid each period and n = m t is the total number of periods.

Numerical Example: You deposit $100 per month into an account that now contains $5,000 and earns 5% interest per year compounded monthly. After 10 years, the amount of money in the account is:

FV = PV(1 + i)n + [ R(1 + i)n - 1 ] / i =
5,000(1+0.05/12)120 + [100(1+0.05/12)120 - 1 ] / (0.05/12) = $23,763.28

Value of a Bond: Let N = number of year to maturity, I = the interest rate, D = the dividend, and F = the face-value at the end of N years, then the value of the bond is V, where

V = (D/i) + (F - D/i)/(1 + i)N

V is the sum of the value of the dividends and the final payment.

You may like to perform some sensitivity analysis for the "what-if" scenarios by entering different numerical value(s), to make your "good" strategic decision.


  1. Monthly Payment
  2. Future Value
  3. Compound Annual Rate
  4. Remaining Debt
  5. Monthly Payment with Possible Tax and/or Insurance
  6. Periodic Compound Interest
  7. Compound Interestís Factors
  8. Compound Interest & Effective Rate
  9. Mortgage Payments
  10. Mortgage Payments Schedule
  11. Accelerating Mortgage Payments
  12. Future Value of an Annuity
  13. When Will Your Retirement Capital Run Out?
  14. What Should Be the Present Value of a Bond You Need?
  15. Your Loan's Monthly Payment
  16. Retirement Planner's Calculator
  17. Buying/Selling Stocks with Commissions

Replace the existing numerical example, with your own case-information, and then click one the Calculate.

Monthly Payment

Number of Payments :
Interest Rate :
Principal :
Monthly Payment :

Future Value

Total invested each month :
Number of month/years to aquire sum :
Interest rate on savings (%):
Total :

Compound Annual Rate

Initial Sum :
Number of Years Invested :
Final Sum:
Compound Annual Rate (%) :

Remaining Debt

Principal ($) :
Annual Rate of Interest (%) :
Years :
Number of (monthly) Payments to Date :
Monthly Payment ($) :
Debt Remaining After Monthly Payments ($) :

Monthly Payment with Possible Tax and/or Insurance

Years :
Interest Rate :
Loan Amount :
Annual Tax :
Annual Insurance :
Monthly Principal + Interest :
Monthly Tax :
Monthly Insurance :
Total Payment :

Periodic Compound Interest

Principal :
Interest Rate :
Period :
days weeks years
Calculated :
Result :

Compound Interestís Factors

Provide numerical values for 3 cells to get the 4th one.

Present Value :
Future Value :
Term :
Periodic Rate :
Result for Empty Cell :

Compound Interest & Effective Rate

Principal ($) :
Annual Interest Rate (%) :
No. of periods per year :
Years :
Amount ($) :
Effective Annaul Rate ($) :

Mortgage Payments

Principal ($) :
Annual Interest Rate (%) :
Years :
K = No. of Payments :
Monthly Payment ($) :
Debt after K Payments ($) :

For detailed Payment Schedule use
Compund Interest Schedule

Accelerating Mortgage Payments

Principal ($) :
Annual Interest Rate (%) :
Monthly Payment ($) :
No. of Payments :

Future Value of an Annuity

Amount of Periodic Payment ($) :
Interest Rate per Period (%) :
Number of Payments :
Future Value of Annuity ($) :

For an All-in-one Annuity Calculator, use
Components of an Annuity Calculator

When Will Your Money Run Out?

Capital ($C):
Annual Expense ($E) :
Interest Rate(i) :
Inflation Rate (j) :
Years (n) :

What Should Be the Present Value of a Bond You Need?

Number of Year to Maturity (N):
Interest Rate ($i) :
The Dividend ($D) :
The Face-value in N years ($F) :
Value of the Bond You Need($V) :

Your Loan's Monthly Payment

Loan Amount:
Annual Tax:
Annual Insurance:
Total Payment:

Retirement Planner's Calculator

Pre-Retirement Factors
Accumulated Principal $
Annual Addition $
Years to Build    
Growth Rate Percentage     %
During Retirement Factors
Years for Payouts    
Growth Rate Percentage     %
Annual Retirement Income $

Buying/Selling Stocks with Commissions

Purchase Price:
Sell Price:
Buy commission:
Sell Commission:
Purchased For:
Sold For:

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