Complete step-by-step solution:
Now let us first understand interest. When we take a loan the amount of the loan is nothing but the principal amount. Now interest percent is nothing but the percentage of extra money you have to give per annum. And time if for how many years the interest is taken for.
For example, if you take a loan of 100Rs at 10 percent a year for 1 year you have principal amount = 100Rs. Rate of interest as 10 percent and time as 1 year.
Now the total interest is the extra money that you pay and is calculated by
$I=\dfrac{P\times r\times t}{100}$ where P is the principal amount, r is the rate of interest and t is time in years.
Now we have P = 2500, r = 6 percent and t = 4 years hence we get
$I=\dfrac{2500\times 6\times 4}{100}$
Hence I = 25 × 6 × 4 = 600.
Hence the interest after 4 years is 600 Rs
Note: Note that in the formula we have divided by 100 because the value of r that is the rate of interest is in percentage. We have x percent as $\dfrac{x}{100}$ . Hence the formula is nothing but $P\times \dfrac{r}{100}\times t$ .
Also, note that Interest and total amount paid is different. The total amount paid is given by P + I. Where P is the principal amount and ‘I’ is the interest.
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]mtorFV = PV[1 + i]nwhere 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]mtNumerical 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:
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]
andD = 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 ] ] / iwhere 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, whereV = [D/i] + [F - D/i]/[1 + i]NV 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.
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