Compound Interest
Compound interest refers to the interest calculated on the initial principal and also on the accumulated interest from previous periods. It is a powerful concept in finance and investment, as it allows for exponential growth of wealth over time. The formula for compound interest takes into account the compounding periods, interest rate, and time, making it a fundamental concept in financial mathematics.
6 Key excerpts on "Compound Interest"
eBook - ePub- Andrew Baum, David Mackmin, Nick Nunnington(Authors)
- 2017(Publication Date)
- Routledge(Publisher)
...This concept leads naturally to the possibility of continuous compounding. Continuous compounding When interest is added more frequently than annually, the Compound Interest formula (1 + i) n is adjusted to: where m is the number of times per year that interest is added, i is the nominal rate of interest per year and n is the number of years. Here, the more frequently interest is added, the greater the annual effective rate of interest will be. If, for example, the nominal rate of interest is 12% per year and interest is added monthly then the effective rate is 12.68%: The greater the number of times in the year that interest is added, the greater the total sum at the end of the year will be, but there must be a limit, because whilst the number of periods becomes infinitely large, the rate of interest per period becomes infinitely small. Potentially, m might tend towards infinity...
eBook - ePubAn Introduction to Banking
Liquidity Risk and Asset-Liability Management
- Moorad Choudhry(Author)
- 2011(Publication Date)
- Wiley(Publisher)
...Such a deposit would accrue interest of 6 in the normal way, but 1.50 would be credited to the account every quarter, and this would then benefit from compounding. Again assuming that we can reinvest at the same rate of 6%, the total return at the end of the year will be: which gives us, a terminal value of 106.136. This is some 13 pence more than the terminal value using annual compounded interest. In general, if compounding takes place times per year, then at the end of years interest payments will have been made and the future value of the principal is given by: (B.4) As we showed in our example, the effect of more frequent compounding is to increase the value of total return when compared with annual compounding. The effect of more frequent compounding is shown below, where we consider annualized interest rate factors, for an annualized rate of 5%. This shows us that the more frequent the compounding the higher the interest rate factor. The last case also illustrates how a limit occurs when interest is compounded continuously. Equation (B.4) can be rewritten as: (B.5) where. As compounding becomes continuous and and hence approach infinity, equation (B.5) approaches a value known as, which is shown by: (B.6) If we substitute this into (B.5) we get: where we have continuous compounding. In equation (B.6) is known as the exponential function of ; it tells us the continuously compounded interest rate factor. If % and year then: This is the limit reached with continuous compounding. To illustrate continuous compounding from our initial example, the future value of 100 at the end of 3 years – when the interest rate is 6% – can be given by: Effective interest rates The interest rate quoted on a deposit or loan is usually the flat rate...
- Cornelis van de Panne(Author)
- 2020(Publication Date)
- Routledge(Publisher)
...There are only two cash flows, one at the beginning and one at the end. It is obvious that money put on Compound Interest accumulates faster than on simple interest, because summing the cashflow now results in $469, which is $69 more than for simple interest. Algebraically, it is easily seen that the principal is multiplied every year by the factor 1 + r, so that, if an amount A is put on Compound Interest, with an annual interest rate of r, its value after t years will be A (1 + r) t. This formula represents the future value of the amount A after t periods of Compound Interest investment at rate r. Panel 5.4 Cashflow, Principal, and Interest for 5 -Year Bond with Compound Interest Panel 5.5 Compound Interest Results Panel 5.5 gives the future values for A = $ 1,000, and a number of values for t and r, while Panel 5.6 gives these values in graphical form. Cell B4 contains the formula = A ★ (1 + B $ 3) ^ $ A 4 which is copied to B4:F9. Note the effect of higher interest rates for longer horizons. For r = 2–3% it takes 30 years to double the original amount. For r = 5%, it takes 15 years, for r = 10%, 7 years, and for r = 15%, only 5 years. Long-term interest rates have historically varied from 3% to 15%, and with an average of around 10% over the last 10 years. But the average inflation rate over the last 10 years has been 5%, so that the real interest rate has been around 5%. Short-term and long-term interest rates may be different. For example, a 1-year GIC may give a 6% interest, a 5-year GIC 7%, while a 10-year government bond may give 8%. The following is based on the interest rate being the same for all terms. Panel 5.6 The Effect of Compound Interest The future value formula is, of course, valid for all kinds of constant percentage growth, such as long-term economic growth expressed in the growth of real Gross Domestic Product (GDP) or income per capita, inflation, population growth, resource use, and environmental degradation, to name just a few...
eBook - ePubFixed Income Securities
Concepts and Applications
- Sunil Kumar Parameswaran(Author)
- 2019(Publication Date)
- De Gruyter(Publisher)
...In other words, the earlier that one starts investing, the greater will be the return. Example 1.5. Jesus was born 2019 years ago. Assume that an investment of $1 was made in that year in a bank which has been paying 1% interest per annum since then, compounded annually. What will be the accumulated balance at the end of 2019? 1 × (1.01) 2019 = $ 530, 705, 596 Thus the terminal amount will be in excess of 530 million. Properties of Simple and Compound Interest 1. If N = 1 (that is, an investment is made for one year) then both the simple and the Compound Interest techniques will give the same accumulated value. 2. If N < 1 (that is, the investment is made for less than a year), the accumulated value using simple interest will be higher. That is (1 + r N) > (1 + r) N if N < 1 A lot of people are not aware of this. Thus, if a bank were to quote an interest rate of r % per annum compounded annually, and you were to deposit for nine months, the payoff would be greater if the bank were to compute the interest on a simple interest basis. Using similar logic, if a bank were to quote a rate of r % per annum with semiannual compounding, and you were to deposit for three months, the payoff would be greater if the bank were to use simple interest. 3. If N > 1 (that is, the investment is made for more than a year), the accumulated value using Compound Interest will always be greater. That is (1 + r N) < (1 + r) N if N > 1 Simple interest is usually used for short-term transactions, that is, for investments for a period of one year or less. Consequently, simple interest is the norm for money market calculations. 1 However, in the case of capital market securities, that is, medium to long-term debt securities and equities, we use the Compound Interest principle. Simple interest is at times used as an approximation for Compound Interest over fractional periods. Example 1.6. Take the case of Andrew Gordon who has deposited $10,000 with ABC Bank for five years and six months...
eBook - ePub- Arsen Melkumian(Author)
- 2012(Publication Date)
- Routledge(Publisher)
...Further, the total payment due on a simple interest loan is The simple interest method does not charge interest on interest and is not widely used in practice. 10.2 Compound Interest The vast majority of lenders require the borrower to pay Compound Interest on a loan. Further, most deposit accounts at banks and other financial institutions pay Compound Interest on the balance in the account. The future value FV of an asset earning Compound Interest is given by where r is the interest rate and t is the number of periods (typically years) an asset is held for. For example, suppose $2000 is lent for eight years at 5 percent per year, compounded annually. Then, the lump sum the borrower must pay at the end of eight years is Thus, the Compound Interest on the loan is Increased competition between financial institutions has forced many depositary institutions to Compound Interest more frequently than once per annum, with some depositary institutions compounding interest monthly or even daily. Now, let us say you hold an $8000 deposit in a bank that offers 12 percent interest rate compounded monthly. At the end of five years your account balance will be The general formula for the future value of an asset that pays interest of r percent per annum, compounded n times per year, is given by To summarize, future value (FV) is the amount of money that an investment made today will become at some future date, given a certain fixed interest rate. E XAMPLE 10.1 Suppose you deposit $2000 in a bank that offers a 5 percent interest rate on time deposits (compounded annually). In seven years the $2000 will grow to In other words, the future value of $2000 in seven years is approximately $2814.20 if the interest rate is 5 percent. E XAMPLE 10.2 Let us say you hold an $8000 deposit in an account that earns interest at the annual rate of 12 percent...
eBook - ePubShares Made Simple
A Beginner's Guide to Sharemarket Success
- Roger Kinsky(Author)
- 2010(Publication Date)
- Wiley(Publisher)
...Chapter 3: The power of compounding In this chapter I discuss the power of compounding, and lay the groundwork for a reliable method that you can use for long-term wealth accumulation with shares. Compounding returns Compounding kicks in when you make a profit on an investment and you reinvest that profit. In the case of an investment in an interest-bearing account, this is known as Compound Interest. On the other hand, if you don’t reinvest your profits this is known as simple interest. Tip Compounding is very important and really kicks in over the long term. If you’re trying to make a short-term gain (for example, by short-term share trading), you don’t need to consider compounding as it’s not significant in the short term. The longer the term of the investment, the more benefit you’ll derive from compounding. Compounding works because by reinvesting your profits you’re adding to your investment capital. If you reinvest your profits this year, then next year you’ll make more profit because you’re getting a return on the capital reinvested in addition to the capital you originally invested. In effect, next year you’re making a profit on the profit you made this year. Tip To make compounding work, you need to reinvest profits and not withdraw them. It’s a human temptation to want to spend profits, but if you want to really grow your wealth over the long term you must resist the impulse to spend in the short term. Example 1 You invest $100 000 in a portfolio of shares. Your portfolio makes a profit of 10% in the first year and you reinvest your profit (that is, you don’t sell any shares to realise the gains)...
