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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 = $300 If 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

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**13.1 Compound Interest Period Interest Credited**

Times Credited per year Rate per compounding period Annual year 1 R Semiannual 6 months 2 Quarterly quarter 4 Monthly month 12

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13.1 Compound Interest Compound interest formula: M = the compound amount or future value P = principal i = interest rate per period of compounding n = number of periods I = interest earned

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13.1 Compound Interest Time Value of Money – with interest of 5% compounded annually. 2000 2010 2020

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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:

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

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13.1 Compound Interest Example: (continued) quarterly: monthly:

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**13.2 Daily and Continuous Compounding**

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

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**13.2 Daily and Continuous Compounding**

Time Value of Money – with 5% interest compounded continuously. 2000 2010 2020

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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.

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**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.

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**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.

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**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

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13.3 Finding Time and Rate Given a principal of $12,000 with a compound amount of $17, 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

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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.

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13.3 Finding Time and Rate Example:Given an investment of $13200, compound amount of $ 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 = … so i = 7%.

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**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.

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**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:

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**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)

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**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.

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**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?

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**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)

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

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

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**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.

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**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

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**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:

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**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?

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**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.

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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.)

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**14.3 Sinking Funds Amount of a sinking fund payment:**

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

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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.

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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.

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

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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

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15.1 Open-End Credit Example: Find the finance charge for an average daily balance of $ with monthly interest rate of 1.4% finance charge

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15.1 Open-End Credit Example: Find the interest for the following account with monthly interest rate of 1.5% Previous balance $412.48 November 5 Billing date November 18 Payment $150 November 30 Dinner and play $84.50

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**15.1 Open-End Credit Example(continued)**

Average balance = 30 = $341.56 Finance charge = .015 = $5.12 Balance at end = = $352.10 Date # days until chg balance (2)(3) November 5 13 $412.48 November 18 12 $262.48 November 30 5 $346.98 1734.9 December 5 30 (total days)

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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

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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

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15.2 Installment Loans Example: Given the following data, find the finance charge and the total installment cost Total installment cost Finance charge Amount Financed Down Payment Cash Price # of payments Amount of payment $650 $125 $775 24 $32

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15.2 Installment Loans Example: Given the following data, find the annual percentage rate using table from table 15.2 # payments = 12, APR is approximately 13% Amount Financed Finance Charge # of payments $345 $24.62 12

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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

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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) Principal Interest Time in days Partial payments $5800 10% 120 $2500 on day 60

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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):

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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

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**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:

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**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?

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**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?

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Chapter 9: Mathematics of Finance

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