We have to work with money every day. While balancing your checkbook or calculating your monthly expenditures on espresso requires only arithmetic, when we start saving, planning for retirement, or need a loan, we need more mathematics. Show
Simple InterestDiscussing interest starts with the principal, or amount your account starts with. This could be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is calculated as a percent of the principal. For example, if you borrowed $100 from a friend and agree to repay it with 5% interest, then the amount of interest you would pay would just be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the original principal plus the interest. Simple One-time InterestI = P0r Example 1A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much interest will you earn?
One-time simple interest is only common for extremely short-term loans. For longer term loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In that case, interest would be earned regularly. For example, bonds are essentially a loan made to the bond issuer (a company or government) by you, the bond holder. In return for the loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which time the issuer pays back the original bond value. Example 2 Suppose your city is building a new park, and issues bonds to raise the money to build it. You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much interest will you earn? Simple Interest over TimeI = P0rt a APR—Annual Percentage Rate Interest rates are usually given as an annual percentage rate (APR)—the total interest that will be paid in the year. If the interest is paid in smaller time increments, the APR will be divided up. Example 3 Treasury Notes (T-notes) are bonds
issued by the federal government to cover its expenses. Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a maturity in 4 years. How much interest will you earn?
Try it Now 1A loan company charges $30 interest for a one month loan of $500. Find the annual interest rate they are charging. Show Answer I = $30 of interest P0 = $500 principal r = unknown t = 1 month Using I = P0rt, we get 30 = 500r1. Solving, we get r = 0.06, or 6%. Since the time was monthly, this is the monthly interest. The annual rate would be 12 times this: 72% interest. Compound InterestNote from Professor Pinegar: I studied on whether to write this at the beginning or the end. I may not use the same variables in finance as the authors of these sections. Below are the formulas I generally use.As you view the formulas below, let me tell you a little about some differences you might see.
Source: Kevin Pinegar With simple interest, we were assuming that we pocketed the interest when we received it. In a standard bank account, any interest we earn is automatically added to our balance, and we earn interest on that interest in future years. This reinvestment of interest is called compounding. Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly. How will our money grow? The 3% interest is an annual percentage rate (APR)—the total interest to be paid during the year. Since interest is being paid monthly, each month, we will earn 3%12=0.25%\displaystyle\frac{{{3}\%}}{{12}}={0.25}\% per month.
To find an equation to represent this, if Pm =P0(1+rk)m\displaystyle{P}_{{m}}={P}_{{0}}{\left({1}+\frac{{r}}{{k}}\right)}^{{m}} In this formula: Compound InterestPN is the balance in the account after N years. Example 4 A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a higher interest rate, but you cannot access your investment for a specified length of time. Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much will you have in the account
after 20 years?
Let us compare the amount of money earned from compounding against the amount you would earn from simple interest
As you can see, over a long period of time, compounding makes a large difference in the account balance. You may recognize this as the difference between linear growth and exponential growth. Evaluating Exponents on the Calculator When we need to calculate something like 5 Example 5 You know that you will need $40,000 for your child's education in 18 years. If your account earns 4% compounded quarterly, how much would you need to deposit now to reach your goal?
So you would need to deposit $19,539.84 now to have $40,000 in 18 years. Rounding It is important to be very careful about rounding when calculating things with exponents. In general, you want to keep as many decimals during calculations as you can. Be sure to Example 6To see why not over-rounding is so important, suppose you were investing $1000 at 5% interest compounded monthly for 30 years.
If we first compute
If you're working in a bank, of course you wouldn't round at all. For our purposes, the answer we got by rounding to 0.00417, three significant digits, is close enough—$5 off of $4500 isn't too bad. Certainly keeping that fourth decimal place wouldn't have hurt. Using Your CalculatorIn many cases, you can avoid rounding completely by how you enter things in your calculator. For example, in the example above, we needed to calculate P30=1000(1+0.0512)12×30\displaystyle{P}_{{30}}={1000}{\left({1}+\frac{{0.05}}{{12}}\right)}^{{{12}\times{30}}} We can quickly calculate 12 × 30 = 360, giving P30=1000(1+0.0512)360\displaystyle{P}_{{30}}={1000}{\left({1}+\frac{{0.05}}{{12}}\right)}^{{{360}}} Now we can use the calculator.
The previous steps were assuming you have a "one operation at a time" calculator; a more advanced calculator will often allow you to type in the entire expression to be evaluated. If you have a calculator like this, you will probably just need to enter: David Lippman, Math in Society,
"Finance," licensed under a CC BY-SA 3.0 license. Developing Financial Intuition Rarely is it the case these days that you invest $100 of your money at, say, 5% per year and get
Interest paid on original balance only: constant rate of growth
Interest paid on overall balance: constant percentage growth $265.33$100=2.65\displaystyle\frac{{\${265.33}}}{{\${100}}}={2.65} 2.65−1=1.65=165%\displaystyle{2.65}-{1}={1.65}={165}\% gain $200100−1=1.00=100%\displaystyle\frac{{\${200}}}{{100}}-{1}={1.00}={100}\% gain Simple Interest Balance Formula If interest is paid according to a simple interest schedule and we define Example 1Verify that the 20-year balance for a $100 investment at 5% yearly interest is $200 by using the simple interest balance formula. Solution We have that Building a Compound Interest Formula For compound interest the idea is fairly simple. Recall that growth by a percentage is called Compound Interest Balance Formula If interest is paid according to a compound interest schedule, where interest is paid on the Example 2Confirm that if you invest $100 for 20 years at an annual interest rate of 5% compounded annually, that you will have a balance of $253.33. Solution We have
Note: Weeks and days vary depending on year. For ease of use, we ignore this detail. Example 3A bank offers you a nominal annual rate of 5% compounded monthly. You invest $100 and plan on keeping it invested for 20 years. Calculate your balance after 20 years. Then, compare this to the value found in example 2 based on annual compounding and comment on the effect of compounding periods. Solution We have that i =.0512 months≈.00417or.417%\displaystyle{i}=\frac{{.05}}{{{12}\ {m}{o}{n}{t}{h}{s}}}\approx{.00417}{\quad\text{or}\quad}{.417}\% per month. We calculate Effect of Compounding Frequency on Accumulated Balance (Future Value), As the frequency of compounding interest increases, so does the accumulated balance.
We can see that, while the balance is slightly larger than that of the previous compounding period, the differences become quite small as the frequency increases more and more. Example 4Is 12% given annually the same thing as 1% given monthly? Why or why not? Solution Suppose a person deposits Annual Percentage Yield So, if 12% once is not the same as 1% 12 times, what percentage 112.68100=1.1268 \displaystyle\frac{{112.68}}{{100}}={1.1268} This means that the overall growth was 12.68%, a percentage larger than 12. Recall that the rate of 12% is called the nominal annual rate. The rate that you APY=1(1+rn)n×1−1=(1+rn)n−1\displaystyle{A}{P}{Y}={1}{\left({1}+\frac{{r}}{{n}}\right)}^{{{n}\times{1}}}-{1}={\left({1}+\frac{{r}}{{n}}\right)}^{{{n}}}-{1} Thus, APY =(1+rn)n−1\displaystyle{A}{P}{Y}={\left({1}+\frac{{r}}{{n}}\right)}^{{{n}}}-{1} Alternatives to the formula?Absolutely! If the amount invested is different than $1, calculate what it will become in one year. Take the year-end amount, divide it by the original, and subtract 1. Example 5Let's say you invest $325 at 10% compounded semi-annually (twice a year) for 5 years. What is the APY? Solution Since we want the
In my opinion, it is much easier to understand and remember the intuitive approach on the right. Needless to say, you'll get the same answer. Milos Podmanik, By the Numbers, "Compound Interest and Exponential Growth," licensed under a
CC BY-NC-SA 3.0 license. What is compound amount on Rs 2500 at the rate of 8% for 2 years?∴ Amount will be Rs. 3025 and Interest will be Rs. 525 If Compounded Annually.
What is the compound interest on 2500 for 2 years at the rate of interest of 4% per annum?= 2704 - 2500 = Rs. 204 C.I. - S.I.
What is the compound interest on Rs 2500 for 2?2500 for 2 years at 12% p.a.? Compound interest in first year on a sum=Simple interest. So compound interest after 1 year=2500*1*12/100=Rs 300. For the next year Sum=2500+300=2800.
What is the compound interest on Rupees 3000 at 6% for 2 years?Detailed Solution. ∴ The compound interest for 2 years is Rs. 1320.
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