# 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
or

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]

and

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 – 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.

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 : MonthsYears 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 : Compound Daily Compound Weekly Compound Fortnightly Compound Monthly Compound Quarterly Compound Yearly Result :

 Compound Interest’s 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

 Years: Interest: Loan Amount: Annual Tax: Annual Insurance: Payment: Tax: 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

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Professor Hossein Arsham