# 13.1 Compound Interest - PowerPoint PPT Presentation

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13.1 Compound Interest. Simple interest – interest is paid only on the principal Compound interest – interest is paid on both principal and interest, compounded at regular intervals

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13.1 Compound Interest

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### 13.1 Compound Interest

• Simple interest – interest is paid only on the principal

• Compound interest – interest is paid on both principal and interest, compounded at regular intervals

• Example: a \$1000 principal paying 10% simple interest after 3 years pays .1  3  \$1000 = \$300If interest is compounded annually, it pays .1  \$1000 = \$100 the first year, .1  \$1100 = \$110 the second year and .1  \$1210 = \$121 the third year totaling \$100 + \$110 + \$121 = \$331 interest

### 13.1 Compound Interest

• Compound interest formula:M = the compound amount or future valueP = principali = interest rate per period of compoundingn = number of periodsI = interest earned

### 13.1 Compound Interest

• Time Value of Money – with interest of 5% compounded annually.

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### 13.1 Compound Interest

• Example: \$800 is invested at 7% for 6 years. Find the simple interest and the interest compounded annually Simple interest:Compound interest:

### 13.1 Compound Interest

• Example: \$32000 is invested at 10% for 2 years. Find the interest compounded yearly, semiannually, quarterly, and monthly yearly:semiannually:

### 13.1 Compound Interest

• Example: (continued) quarterly:monthly:

### 13.2 Daily and Continuous Compounding

• Daily compound interest formula: divide i by 365 and multiply n by 365

• Continuous compound interest formula:

### 13.2 Daily and Continuous Compounding

• Time Value of Money – with 5% interest compounded continuously.

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### 13.2 Daily and Continuous Compounding

• Example: Find the compound amount if \$2900 is deposited at 5% interest for 10 years if interest is compounded daily.

### 13.2 Daily and Continuous Compounding

• Example: Find the compound amount if \$1200 is deposited at 8% interest for 11 years if interest is compounded continuously.

### 13.2 Daily and Continuous Compounding – Early Withdrawal

• Early Withdrawal Penalty:

• If money is withdrawn within 3 months of the deposit, no interest will be paid on the money.

• If money is withdrawn after 3 months but before the end of the term, then 3 months is deducted from the time the account has been open and regular passbook interest is paid on the account.

### 13.2 Daily and Continuous Compounding – Early Withdrawal

• Example: Bob Kashir deposited \$6000 in a 4-year certificate of deposit paying 5% compounded daily. He withdrew the money 15 months later. The passbook rate at his bank is 3½ % compounded daily. Find his amount of interest.Bob receives 15-3 = 12 months of 3.5 % interest compounded daily

### 13.3 Finding Time and Rate

• Given a principal of \$12,000 with a compound amount of \$17,631.94 and interest rate of 8% compounded annually, what is the time period in years?From Appendix D table pg 805( i = 8%) we find that n = 5 years

### 13.3 Finding Time and Rate

• Example:Find the time to double your investment at 6%.If you try different values of n on your calculator, the value that comes closest to 2 is 12. Therefore the investment doubles in about 12 years.

### 13.3 Finding Time and Rate

• Example:Given an investment of \$13200, compound amount of \$22680.06 invested for 8 years, what is the interest rate if interest is compounded annually?From Appendix D table pg 803( i = 7%) we find that for n=8, column A = 1.71818… so i = 7%.

### 13.4 Present Value at Compound Interest

• Example:Given an amount needed (future value) of \$3300 in 4 years at an interest rate of 11% compounded annually, find the present value and the amount of interest earned.

### 13.4 Present Value at Compound Interest

• Example: Assume that money can be invested at 8% compounded quarterly. Which is larger, \$2500 now or \$3800 in 5 years?First find the present value of \$3800, then compare present values:

### 14.1 Amount (Future Value) of an Annuity

• Annuity – a sequence of equal payments

• Payment period – time between payments

• “Ordinary annuity” – payments at the end of the pay period

• “Annuity due” - payments at the beginning of the pay period

• “Simple annuity” – payment dates match the compounding period (all our annuities are simple)

### 14.1 Amount (Future Value) of an Annuity

• Amount of an annuity - S (future value) of n payments of R dollars for n periods at a rate of i per period:

• Use you calculator instead of using appendix D.

### 14.1 Amount (Future Value) of an Annuity

• Example: Sharon Stone deposits \$2000 at the end of each year in an account earning 10% compounded annually. Determine how much money she has after 25 years. How much interest did she earn?

### 14.1 Amount (Future Value) of an Annuity

• Example: For S = \$50,000, i = 7% compounded semi-annually with payments made at the end of each semi-annual period for 8 years, find the periodic payment (R)

### 14.1 Amount (Future Value) of an Annuity

• Example: For S = \$21,000, payments (R) of \$1500 at the end of each 6-month period i = 10% compounded semi-annually. Find the minimum number of payments to accumulate 21,000.Trying different values for n, the expression goes over 14 when n = 11 (Exact value = 4.20678716(1500)=\$21310.18)

### 14.1 Amount of an Annuity Due

• An annuity due is paid at the beginning of each period instead of at the end. It is essentially the same as an ordinary annuity that starts a period early but without the last payment.

• To solve such a problem:

• Add 1 to the number of periods for the computation.

• After calculating the value for S, subtract the last payment.

### 14.1 Amount of an Annuity Due

• Example:Sharon Stone deposits \$500 at the beginning of each 3 months in an account earning 10% compounded quarterly. Determine how much money she has after 25 years

### 14.2 Present Value of an Annuity

• Present value of an annuity (A) made up of payments of R dollars for n periods at a rate of i per period:

### 14.2 Present Value of an Annuity

• Example: What lump sum deposited today would allow payments of \$2000/year for 7 years at 5% compounded annually?

### 14.2 Present Value of an Annuity

• Example: Kashundra Jones plans to make a lump sum deposit so that she can withdraw \$3,000 at the end of each quarter for 10 years. Find the lump sum if the money earns 10% per year compounded quarterly.

### 14.3 Sinking Funds

• Sinking fund – a fund set up to receive periodic payments.The purpose of this fund is to raise an amount of money at a future time.

• Bond – promise to pay an amount of money at a future time.(Sinking funds can be set up to cover the face value of bonds.)

### 14.3 Sinking Funds

• Amount of a sinking fund payment:

• Same formula as in section 14.1, except solved for the variable R.

### 14.3 Sinking Funds

• Example: 15 semiannual payments are made into a sinking fund at 7% compounded semiannually so that \$4850 will be present. Find the amount of each payment rounded to the nearest cent.

### 14.3 Sinking Funds

• Example: A retirement benefit of \$12,000 is to be paid every 6 months for 25 years at interest rate of 7% compounded semi-annually. Find (a) the present value to fund the end-of-period retirement benefit. ): (b) the end-of-period semi-annual payment needed to accumulate the value in part (a) assuming regular investments for 30 years in an account yielding 8% compounded semi-annually.

### 14.3 Sinking Funds

• Example(part b) – amount to save every 6 months for 30 years for this annuity

### 15.1 Open-End Credit

• Open-end credit – the customer keeps making payments until no outstanding balance is owed (e.g. charge cards such as MasterCard and Visa)

• Revolving charge account – a minimum amount must be paid …account might never be paid off

• Finance charges – charges beyond the cash price, also referred to as interest payment

• Over-the-limit fee – charged if you exceed your credit limit

### 15.1 Open-End Credit

• Example: Find the finance charge for an average daily balance of \$8431.10 with monthly interest rate of 1.4%finance charge

### 15.1 Open-End Credit

• Example: Find the interest for the following account with monthly interest rate of 1.5%

### 15.1 Open-End Credit

• Example(continued)

• Average balance = 10246.930 = \$341.56

• Finance charge = .015  341.56 = \$5.12

• Balance at end = 346.98 + 5.12 = \$352.10

### 15.2 Installment Loans

• A loan is “amortized” if both principal and interest are paid off by a sequence of periodic payments.For a house this is referred to as mortgage payments.

• Lenders are required to report finance charge (interest) and their annual percentage rate (APR)

• APR is the true effective annual interest rate for a loan

### 15.2 Installment Loans

• In order to find the APR for a loan paid in installments, the total installment cost, finance charge, and the amount financed are needed

• Total installment cost = Down payment + (amount of each payment  number of payments)

• Finance charge = total installment cost – cash price

• Amount financed = cash price – down payment

• Get:

• Use table 15.2 to get the APR

### 15.2 Installment Loans

• Example: Given the following data, find the finance charge and the total installment costTotal installment cost Finance charge

### 15.2 Installment Loans

• Example: Given the following data, find the annual percentage rate using table 15.2from table 15.2 # payments = 12, APR is approximately 13%

### 15.3 Early Payoffs of Loans

• United States rule for early payoff of loans:

• Find the simple interest due from the date the loan was made until the date the partial payment is made.

• Subtract this interest from the amount of the payment.

• Any difference is used to reduce the principal

• Treat additional partial payments the same way, finding interest on the unpaid balance

### 15.3 Early Payoffs of Loans

• Example: Given the following note, find the balance due on maturity and the total interest paid on the note.

• Find the simple interest for 60 days and subtract it from the payment.

• Subtract it from the payment:

• Reduce the principal by the amount from (2)

### 15.3 Early Payoffs of Loans

• Example(continued)… Interest due at maturity:Balance due on maturity (add reduced principal to interest):Total interest paid on the note (add interest paid to interest due at maturity):

### 15.3 Early Payoffs of Loans

• Rule of 78 (sum-of-the-balances method)Note (1+2+3+…+12) – sum of the month numbers adds up to 78 … used to derive the formula.U = unearned interest, F = finance charge, N = number of payments remaining, and P = total number of payments

### 15.4 Personal Property Loans

• From section 14.2, the present value of an annuity (A) made up of payments of R dollars for n periods at a rate of i per period:

### 15.4 Personal Property Loans

• A loan is made for \$3500 with an interest rate of 9% and payments made annually for 4 years. What is the payment amount?

### 15.4 Personal Property Loans

• A loan is made for \$4800 with an APR of 12% and payments made monthly for 24 months. What is the payment amount? What is the finance charge?