## 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:

FV = PV(1 + r/m) or^{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)}= $28,065.30Notice 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, theeffective interest rateis:

r _{eff}= (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

nominalrate, and is often denoted as r_{nom}.

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

r _{eff }=(1 + r_{nom}/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, thenand 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:where Log is the logarithm in any base, say 10, or e. 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, thenFV = [ R(1 + r)

^{n}- 1 ] / r

Future Value for an Increasing Annuity:It is anincreasing annuityis 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) where i = r/m is the interest paid each period and n = m × t is the total number of periods.^{n}+ [ R ( (1 + i)^{n}- 1 ) ] / i

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" scenariosby entering different numerical value(s), to make your "good" strategic decision.

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