Simple Interest Formula I = PRT

I = PRT I = interest earned (amount of money the bank pays you) P = Principle amount invested or borrowed. R = Interest Rate usually given as a percent (must changed to decimal before plugging it into formula) T = Time (must be measured in years) or converted to years by dividing by 12 months

Converting Change % to decimal 1) 12% 2) 5% 3) 2 ½ % 4) 8.5% Change from decimal to % 5) .098 6) .455 Answers 1) .12 2) .05 3) .025 4) .085 5) 9.8% 6) 45.5% Move 2 places to left & drop % sign Move 2 places to right & add % sign

I = PRT Solve for one of variables: Solving for I Plug in numbers for P, R, & T. Then multiply Solving for other variables Plug in what you know. Multiply the numbers that are on same side then divide by that answer.

Interest paid by bank is unknown Principle (invested) 1. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of $5,000 is invested at 4.5%, what is the total simple interest accumulated in the checking account after 2 years. I = PRT I= I=$450 Interest paid by bank is unknown Principle (invested) Rate changed to decimal Time is 2 years Multiply (5,000) (.045) (2)

Interest paid by bank is unknown Principle (invested) 2. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of $7,000 is invested at 7.5%, what is the total simple interest accumulated in the checking account after 3 years. I = PRT I= I=$1575 Interest paid by bank is unknown Principle (invested) Rate changed to decimal Time is 3 years Multiply (7,000) (.075) (3)

Principle (invested) is unknown Rate changed to decimal Time is 1 year 3. When invested at an annual interest rate of 6% an account earned $180.00 of simple interest in one year. How much money was originally invested in account? I = PRT 180= 180 = .06P .06 .06 3,000 = P Interest paid by bank Principle (invested) is unknown Rate changed to decimal Time is 1 year Multiply Divide P (.06) (1)

Principle (invested) is unknown Rate changed to decimal Time is 1 year 4. When invested at an annual interest rate of 7% an account earned $581.00 of simple interest in one year. How much money was originally invested in account? I = PRT 581= 581 = .07P .07 .07 $8,300 =P Interest paid by bank Principle (invested) is unknown Rate changed to decimal Time is 1 year Multiply Divide P (.07) (1)

5. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of $7,000 accumulate $910 of interest in the account after 2 years, what was the annual simple interest rate on the savings account? I = PRT 910= 910 = (7,000)(2)R 910 = 14,000 R 14,000 14,000 0.065 = R 6.5% = R Interest paid by bank Principle (invested) Rate is unknown Time is 2 years Regroup & Multiply Divide Change to % (7,000) (R) (2)

6. A savings account is set up so that the simple interest earned on the investment is moved into a separate account at the end of each year. If an investment of $2,000 accumulate $360 of interest in the account after 4 years, what was the annual simple interest rate on the savings account? I = PRT 360= 360 = (2,000)(4)R 360 = 8,000 R 8,000 8,000 0.045 = R 4.5% = R Interest paid by bank Principle (invested) Rate is unknown Time is 4 years Regroup & Multiply Divide Change to % (2,000) (R) (4)

Interest paid by bank Principle (invested) Rate is unknown 7. Sylvia bought a 6-month $1900 certificate of deposit. At the end of 6 months, she received a $209 simple interest. What rate of interest did the certificate pay? Interest paid by bank Principle (invested) Rate is unknown Time is 6 months (divide by 12) Regroup & Multiply Divide Change to % I=PRT 209= 209=(1900)(6/12)R 209=950R 950 0.22 = R 22% = R 1900 (R) (6/12)

Interest paid by bank - Unknown Principle (invested) Rate is .045 8. An investment earns 4.5% simple interest in one year. If the money is withdrawn before the year is up, the interest is prorated so that a proportional amount of the interest is paid out. If $2400 is invested, what is the total amount that can be withdrawn when the account is closed out after 2 months? Interest paid by bank - Unknown Principle (invested) Rate is .045 Time is 2 months (divide by 12) Multiply Now, since the money is being withdrawn, add the interest to the principal. I=PRT I= (2400) (.045) (2/12) I=$18 $18 + $2400 = $2418 $2418 will be withdrawn

Compound Interest

An Example Suppose that you were going to invest $5000 in an IRA earning interest at an annual rate of 5.5% How would you determine the amount of interest you’ve made on your investment after one year?

An Example How much money would you have in your IRA account? How much interest would you get after two years?

An Example How much money would you have in your IRA account after two years? What about 10 years?

Compound Interest Notice that the interest in our account was paid at regular intervals, in this case every year, while our money remained in the account. This is called compounding annually or one time per year.

Compound Interest Suppose that instead of collecting interest at the end of each year, we decided to collect interest at the end of each quarter, so our interest is paid four times each year. What would happen to our investment? Since our account has an interest rate of 5.5% annually, we need to adjust this rate so that we get interest on a quarterly basis. The quarterly rate is:

Compound Interest So for our IRA account of $5000 at the end of a year looks like: After 10 years, we have:

Compound Interest Formula P dollars invested at an annual rate r, compounded n times per year, has a value of A dollars after t years. Think of P as the present value, and F as the future value of the deposit.

Compound Interest Formula Notice that when we collected our interest more times during each year, i.e. we compounded more frequently, the amount of money in our account was actually greater than if only collected interest one time a year. What would happen to our money if we compounded a really large number of times?

Annuities Section 5.2

Introduction Let’s say you want to save money to go on a vacation, or you want to save money now for your baby’s college education. A strategy for saving a little bit of money in the present and having a big payoff in the future is called an annuity. An annuity is an account in which equal regular payments are made. There are two basic questions with annuities: Determine how much money will accumulate over time given that equal payments are made. Determine what periodic payments will be necessary to obtain a specific amount in a given time period.

Calculating short-term annuities Claire wants to take a nice vacation trip, so she begins setting aside $250 per month. If she deposits this money on the first of each month in a savings account that pays 6% interest compounded monthly, how much will she have at the end of 10 months? Claire’s first payment will earn 10 months interest. So F = 250(1 + .06/12)12(10/12). Note that the time t is 10/12. Therefore F = 250(1.005)10 = $262.79. Claire’s second payment will earn 9 months interest. Thus F = 250(1.005)9 = $261.48.

Table of future values Payment Future Value 1st 250(1.005)10 = $262.79 2nd 250(1.005)9 = $261.48 3rd 250(1.005)8 = $260.18 4th 250(1.005)7 = $258.88 5th 250(1.005)6 = $257.59 6th 250(1.005)5 = $256.31 7th 250(1.005)4 = $255.04 8th 250(1.005)3 = $253.77 9th 250(1.005)2 = $252.51 10th 250(1.005)1 = $251.25 Totaling up the future value column, we see that Claire has $2569.80 to use for her vacation. She earned $69.80 in interest.

Ordinary Annuity There are two types of annuity formulas. One formula is based on the payments being made at the end of the payment period. This called ordinary annuity. We will derive the ordinary annuity formula next.

Calculating Long Term Annuities The previous example reflects what actually happens to an annuity. The problem is what if the annuity is for 30 years. Future Value of the 1st payment for an ordinary annuity is F1 = PMT(1+r/n)m-1 The future value of the next to last payment is Fm-1 = PMT(1+r/n) The future value of the last payment is Fm = PMT. The total future value F = F1 + F2 + F3 + … + Fm-1 + Fm

Continuing the calculation of a long term annuity The future value is Eq1 Now multiply the equation above by (1+r/n) Eq2 Take Eq2 – Eq1 Note that m = nt. Simplifying gives the ordinary annuity future value formula

Formulas ANNUITY Formula

Example Find the future value of an ordinary annuity with a term of 25 years, payment period is monthly with payment size of $50. Annual interest is 6%. A = $34,649.70 Note: We only put in $15,000. This means that interest earned was $19,649.70!