Simple Interest: A = P(1+rt) Show
Ex1: If $1000 is invested now with simple interest of 8% per year. Find the new amount after two years. P = $1000, t = 2 years, r = 0.08. A = 1000(1+0.08(2)) = 1000(1.16) = 1160Compound Interest:P:the principal, amount invested A: the new balance t: the time r:the rate, (in decimal form) n: the number of times it is compounded. Ex2:Suppose that $5000 is deposited in a saving account at the rate of 6% per year. Find the total amount on deposit at the end of 4 years if the interest is: P =$5000, r = 6% , t = 4 yearsa) simple : A = P(1+rt) A = 5000(1+(0.06)(4)) = 5000(1.24) = $6200b) compounded annually, n = 1: c) compounded semiannually, n =2: A = 5000(1 + 0.06/2)(2)(4) = 5000(1.03)(8) = $6333.85d) compounded quarterly, n = 4: A = 5000(1 + 0.06/4)(4)(4) = 5000(1.015)(16) = $6344.93e) compounded monthly, n =12: A = 5000(1 + 0.06/12)(12)(4) = 5000(1.005)(48) = $6352.44f) compounded daily, n =365: Continuous Compound Interest:Continuous compounding means compound every instant, consider investment of 1$ for 1 year at 100% interest rate. If the interest rate is compounded n times per year, the compounded amount as we saw before is given by: A = P(1+ r/n)ntthe following table shows the compound interest that results as the number of compounding periods increases: P = $1; r = 100% = 1; t = 1 year
As the table shows, as n increases in size, the limiting value of A is the special number e = 2.71828If the interest is compounded continuously for t years at a rate of r per year, then the compounded amount is given by: A = P. e rtEx3: Suppose that $5000 is deposited in a saving account at the rate of 6% per year. Find the total amount on deposit at the end of 4 years if the interest is compounded continuously. (compare the result with example 2) P =$5000, r = 6% , t = 4 years A = 5000.e(0.06)(4) = 5000.(1.27125) = $6356.24Ex4: If $8000 is invested for 6 years at a rate 8% compounded continuously, find the new amount: P = $8000, r = 0.08, t = 6 years. A = 8000.e(0.08)(6) = 8000.(1.6160740) = $12,928.60Equivalent Value:When a bank offers you an annual interest rate of 6% compounded continuously, they are really paying you more than 6%. Because of compounding, the 6% is in fact a yield of 6.18% for the year. To see this, consider investing $1 at 6% per year compounded continuously for 1 year. The total return is: A = Pert = 1.e(0.06)(1) = $1.0618 If we subtract from $1.618 the $1 we invested, the return is $0.618, which is 6.18% of the amount invested. The 6% annual interest rate of this example is called the nominal rateThe 6.18% is called the effective rate.
Ex6: An amount is invested at 7.5% per year compounded continuously, what is the effective annual rate? the effective rate = er - 1 = e 0.075 - 1 = 1.0079 - 1 = 0.0779 = 7.79%Ex7: A bank offers an effective rate of 5.41%, what is the nominal rate? Present Value:If the interest rate is compounded n times per year at an annual rate r, the present value of a A dollars payable t years from now is: If the interest rate is compounded continuously at an annual rate r, the present value of a A dollars payable t years from now is P = A. e-rtEx8: how much should you invest now at annual rate of 8% so that your balance 20 years from now will be $10,000 if the interest is compounded a) quarterly: P = 10,000.(1+0.08/4)-(4)(20)= $ 2,051.10 b) continuously: P = 10,000.e-(0.08)(20) = $2.018.974.3: The Growth, Decline Model:Same formulas will be applied for population, cost ...:Growth: P(t) = Po . ektDecline: P(t) = Po . e-kt
For compounded continuously, the time T it takes to double the price, population or balance using k as the rate of change, the growth rate or the interest rate is given by: ===>Note: the time it takes to triple it is T = ln3/k and so on..., (only if it is compounded continuously).Ex9: The growth rate in a certain country is 15% per year. Assuming exponential growth : a) find the solution of the equation in term of Po and k. b) If the population is 100,000 now, find the new population in 5 years. c) When will the 100,000 double itself? Answer: a) Po. e 0.15t; b) 211,700; c) 4.62 yearsEx10: If an amount of money was doubled in 10 years, find the interest rate of the bank. Answer: 6.93%Ex11: In 1965 the price of a math book was $16. In 1980 it was $40. Assuming the exponential model : a) Find k (the average rate) and write the equation. b) Find the cost of the book in 1985. c) After when will the cost of the book be $32 ? Answer: a) 6.1%; b) $ 54.19; c) T = 11.36 yearsEx12: How long does it take money to triple in value at 6.36% compounded daily? Answer: 17.27 yearsEx13: A couple want an initial balance to grow to $ 211,700 in 5 years. The interest rate is compounded continuously at 15%. What should be the initial balance? Answer: $100,000Ex14: The population of a city was 250,000 in 1970 and 200,000 in 1980 (Decline). Assuming the population is decreasing according to exponential-decay, find the population in 1990. Answer: 160,000At what rate% per annum will a sum of ₹ 4000 yield compound interest of ₹ 410 in 2 years?∴ Rate =5% p.a.
At what rate percent pa compound interest will Rs 4000 amount to Rs 6000 in 2 years if the interest is compounded half yearly?Hence the rate of interest is 10%.
At what rate of interest will Rs 4000 amount to Rs 6000 in 2 years?In what rate of interest Rs 4000 amounts to Rs 6000 in 2 years? Ans:25%
At what rate per cent will a sum of 4000 yield 1.324 as compound interest in 3 years?Thus, the rate of interest is 10%.
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At what rate per cent per annum compound interest will Rs 4000 amount to Rs 6000 in 2 years if the interest is compounded half yearly?
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