Answer: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]
Step-by-step explanation:
