Mathematics of Money:
Compound Interest Analysis With Applications
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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:
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:
Numerical Example: For 4-year investment of $20,000 earning 8.5% per year, with interest re-invested each month, the future value is
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:
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:
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 – P × r)] / log (1 + r)
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
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:
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 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.
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