The sooner you start to save, the more you'll earn with compound interest. Show
How compound interest worksCompound interest is the interest you get on:
For example, if you have a savings account, you'll earn interest on your initial savings and on the interest you've already earned. You get interest on your interest. This is different to simple interest. Simple interest is paid only on the principal at the end of the period. A term deposit usually earns simple interest. Save more with compound interestThe power of compounding helps you to save more money. The longer you save, the more interest you earn. So start as soon as you can and save regularly. You'll earn a lot more than if you try to catch up later. For example, if you put $10,000 into a savings account with 3% interest compounded monthly:
Compound interest formulaTo calculate compound interest, use the formula: A = P x (1 + r)n A = ending balance How to calculate compound interestTo calculate how much $2,000 will earn over two years at an interest rate of 5% per year, compounded monthly: 1. Divide the annual interest rate of 5% by 12 (as interest compounds monthly) = 0.0042 2. Calculate the number of time periods (n) in months you'll be earning interest for (2 years x 12 months per year) = 24 3. Use the compound interest formula A = $2,000 x (1+ 0.0042)24  Lorenzo and Sophia compare the compounding effect Lorenzo and Sophia both decide to invest $10,000 at a 5% interest rate for five years. Sophia earns interest monthly, and Lorenzo earns interest at the end of the five-year term. After five years:
Sophia and Lorenzo both started with the same amount. But Sophia gets $334 more interest than Lorenzo because of the compounding effect. Because Sophia is paid interest each month, the following month she earns interest on interest. Simple Interest: A = P(1+rt)P: the principal, the amount invested A: the new balance t: the time r:the rate, (in decimal form)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) b) 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: A = 5000(1 + 0.06/365)(365)(4) = 5000(1.00016)(1460) = $6356.12Continuous Compound Interest:Continuous compounding means compound every instant, consider investment of 1$ 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? Ex7: A bank offers an effective rate of 5.41%, what is the nominal rate? er - 1 = 0.0541 er = 1.0541 r = ln 1.0541 then r = 0.0527 or 5.27%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: 160,000What is the compound interest on USD 10000 in 2 years at 4 per annum?10000; Rate = 2% per half-year; Time = 2 years = 4 half-years. Amount = Rs [10000 * 1+2/1004 ] = Rs10000 * 51/50 * 51/50 * 51/50 * 51/50 = Rs. 10824.32.
What is the compound interest on 10000 in 2 years at 4% per annum if the interest is compounded semi annually?= ₹(10824.32 - 10000) = ₹824.32. Q. Find the compound interest on Rs. 10,000 in 2 years at 4% per annum , the interest being compounded half yearly.
What is the difference between the compound interests on 10000 for 2 years at 5% per annum compounded annually and half yearly?∴ Difference between interest compounded yearly and half yearly is Rs. 57.81.
What is the compound interest on 10000 for 2 years at 10% per annum?Compound Interest would be 12100rs.
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What is the compound interest on USD 10000 in 2 years at 4% per annum if the interest is compounded semi annually?
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