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: Show
FV = PV(1 + r/m)mtor 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] 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 ) ] / 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 =
Value of a Bond: 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. Compound Interest and DepreciationInterest: It is the additional money besides the original money paid by the borrower to the money lender in lieu of the money used. Principal: The money borrowed (or the money lent) is called principal. Amount: The sum of the principal and the interest is called amount. Thus, amount = principal +interest. Rate: It is the interest paid on Rs 100 for a specified period. Time: It is the time for which the money is borrowed. Simple Interest: It is the interest calculated on the original money (principal) for any given time and rate. Formula: Simple Interest = (Principal x Rate x time)/100 Compound interestCompound interest (abbreviated C.I.) can be easily calculated by the following formula: C.I. = A -P = Remark. Interest may be converted into principal annually, semiannually, quarterly, monthly etc. The number of times interest is converted into principal in a year is called the frequency of conversion, and the period of time between two conversions is
called the conversion period or interest period. Thus "rate of 5%" means a rate of 5% compounded annually; 12% compounded semi-annually means that each interest period of 6 months earns an interest of 6%. Thus the rate of interest per interest period is In solving problems on compound interest, remember the following: 1. A = P where A is the final amount, P is the principal, r is the rate of interest compounded yearly (or every interest period) and n is the number of years (or terms of the interest period). 2. When the interest rates for the successive fixed periods are r1 %, r2
%, r3 %, ..., then the final amount A is given by 3. S.I. (simple interest) and C.I. are equal for the first year (or the first term of the interest period) on the same sum and at the same rate. 4. C.I. of 2nd year (or the second term of the interest period) is more than the C.I. of Ist year (or the first term of the interest period), and C.I. of 2nd year -C.I. of Ist year = S.I. on the interest of the first year. 5. Equivalent, nominal and effective rates of interest 6. Present value or present worth of a sum of Rs P due n years hence at r% compound interest is In particular, present value of sum of a Rs P due one year hence (i.e. n = 1) at r% (compound) interest is P.V.= 7. Equal instalments (with compound interest) P = each equal instalment R = rate of interest per annum (or per interest period) T = time, say 3 years (or 3 interest terms). Note. If T = n years (or interest terms), then there will be n brackets. 8. Formulae for
population If, however, there is annual decrease of r% per annum, the population after n years will be DepreciationAll fixed assets such as machinery, building, furniture etc. gradually diminish in value as they get older and become worn out by constant use in business. Depreciation is the term used to describe this decrease in book value of an asset. The number of years a machine can be effectively used is called its life span. After that it is sold as waste or scrap. Illustrative ExamplesExampleFind the compound amount of Rs 1500 for 6 years 7 months, at 5·2% compounded semi annually. SolutionUsing formula, we could find the value of But in these kinds of problems, generally we use compound interest for full interest period and simple interest for fractional interest period. Here we find compound interest for 13 interest periods and simple interest for 1 month. Required compound amount A = 1500 =1500(1·026)13(1·0043) Taking logs, log A = log 1500 +13 log (1·026) +log (1·0043) = 3·1761 +13(0·0112) +0·0017) = 3·3234. A = antilog (3·3234) = 2144 Thus the required compound amount is Rs 2144. ExampleThe simple interest in 3 years and the compound interest in 2 years on a certain sum at the same rate are Rs 1200 and Rs 832 respectively. Find (i) the rate of interest. (ii) the principal. (iii) the difference between the C.I. and S.I. for 3 years. Solution
ExampleThe population of an industrial town is increasing by 5% every year. If the present population is 1 million, estimate the population five years hence. Also estimate the population three years ago. SolutionPresent population, P = 1 million, rate of increase = 5% per annum = 1000000 = 1276280 Population three years ago = ExampleAvichal Publishers buy a machine for Rs 20000. The rate of depreciation is 10%. Find the depreciated value of the machine after 3 years. Also find the amount of depreciation. What is the average rate of depreciation? SolutionOriginal value of machine = Rs 20000, = 20000 (0·9)³ = 20000 (0·729) = Rs 14580. Amount of depreciation in 3 years = Rs 20000 -Rs 14580 = Rs 5420 Average rate of depreciation in 3 years = (5420/20000) x (100/3) = 9·033% Exercise
Answers1. (i) Rs 784 (ii) Rs 6384 (iii) Rs 894
2. Rs 13240 What would be the compound interest on INR 160000 at 15% per annum for 2 years 4 months compounded annually?194481 - Rs. 160000=Rs. 34481.
What is the compound interest on 160000 at 10% per annum for 2 years compounded half yearly?34481 on the principal amount Rs. 160000 for 2 years at 10% per annum compounded half-yearly is Rs. 1994481.
What is the compound interest on a sum of Rs 16000 at 30% per annum for an year if compounded half yearly?∴ The compound interest is Rs. 5160.
In what time will ₹ 160000 become ₹ 194481 if interest is 10% per annum compounded semi annually?Answer: Time (n) = 2 years.
|