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CHAPTER

6. CHAPTER. Finance. Chapter 6 p 1. 6.1 Interest. Section 6.1 p 2. 6.1 Interest. Definition. Simple Interest Simple interest is interest computed on the principal for the entire period it is borrowed. Section 6.1 p 3. 6.1 Interest. Theorem. Simple Interest Formula

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  1. 6 CHAPTER Finance Chapter 6 p1

  2. 6.1 Interest Section 6.1 p2

  3. 6.1 Interest Definition Simple Interest Simple interest is interest computed on the principal for the entire period it is borrowed. Section 6.1 p3

  4. 6.1 Interest Theorem Simple Interest Formula If a principal P is borrowed at a simple interest rate of R% per annum (where R% is expressed as the decimal r = R/100 for a period of t years, the interest I is I =Prt (1) Section 6.1 p4

  5. 6.1 Interest Theorem The amount A owed at the end of t years is the sum of the principal P borrowed and the interest I charged. That is, A =P +I =P +Prt =P(1 +rt) (2) Section 6.1 p5

  6. 6.1 Interest Theorem Discounted Loans Let r be the per annum rate of interest, t the time in years, and L the amount of the loan. Then the proceeds R is given by R =L −Lrt =L(1 −rt) (3) where Lrt is the discount, the interest deducted from the amount of the loan. Section 6.1 p6

  7. 6.2 Compound Interest Section 6.2 p7

  8. 6.2 Compound Interest Determine the Future Value of a Lump Sum of Money In working with problems involving interest, we use the term payment period as follows: Annually Once per year Semiannually Twice per year Quarterly 4 times per year Monthly 12 times per year Daily 365 times per year* If the interest due at the end of each payment period is added to the principal, so that the interest computed for the next payment period is based on this new amount of the old principal plus interest, then the interest is said to have been compounded. That is, compound interest is interest paid on the initial principal and previously earned interest. * Some banks use 360 times per year. Section 6.2 p8

  9. 6.2 Compound Interest Theorem Compound Interest Formula The amount A after t years due to a principal P invested at an annual interest rate r compounded n times per year is (1) The amount A is typically referred to as the future value of the account, while P is called the present value. Section 6.2 p9

  10. 6.2 Compound Interest Now suppose that the number n of times that the interest is compounded per year gets larger and larger. This is equivalent to h = n/r getting larger and larger. Table 2 (see next slide) illustrates what happens to the expression (1 + 1/h)has h takes on increasingly larger values; notice it is getting closer to a particular number, which we designate as e. Section 6.2 p10

  11. 6.2 Compound Interest TABLE 2 The bottom number in the right column is the number e correct to eight decimal places and is the same as the entry given for e on your calculator (if expressed correctly to eight decimal places). The number e is an irrational number, so its decimal equivalent is a nonrepeating, nonterminating decimal. Section 6.2 p11

  12. 6.2 Compound Interest Definition The number e is defined as the number that the expression (3) approaches as h gets larger and larger. Section 6.2 p12

  13. 6.2 Compound Interest Definition When interest at per annum rate, r, is compounded on a principal P so that the amount after 1 year is Per, we say the interest is compounded continuously. Section 6.2 p13

  14. 6.2 Compound Interest Theorem Continuous Compounding The amount A after t years due to a principal P invested at an annual interest rate r compounded continuously is A = Pert(4) Section 6.2 p14

  15. 6.2 Compound Interest When people engaged in finance speak of the “time value of money,” they are usually referring to the present value of money. The present value of A dollars to be received at a future date is the principal that you would need to invest now so that it will grow to A dollars in the specified time period. The present value of money to be received at a future date is always less than the amount to be received, since the amount to be received will equal the present value (money invested now) plus the interest accrued over the time period. Section 6.2 p15

  16. 6.2 Compound Interest Theorem Present Value Formulas The present value P of A dollars to be received after t years, assuming a per annum interest rate r compounded n times per year, is (5) If the interest is compounded continuously, P = Ae−rt(6) Section 6.2 p16

  17. 6.3 Annuities; Sinking Funds Section 6.3 p17

  18. 6.3 Annuities; Sinking Funds An annuity is a sequence of equal periodic deposits. The periodic deposits can be annual, semiannual, quarterly, monthly, or any other fixed length of time. When the deposits are made at the same time the interest is credited, the annuity is termed ordinary*. The amount of an annuity is the sum of all deposits made plus all interest accumulated. *We shall concern ourselves only with ordinary annuities in this book. Section 6.3 p18

  19. 6.3 Annuities; Sinking Funds Theorem Amount of an Annuity Suppose P is the deposit made at the end of each payment period for an annuity paying an interest rate of i per payment period. The amount A of the annuity after n deposits is (1) Section 6.3 p19

  20. 6.3 Annuities; Sinking Funds A person with a debt may decide to accumulate sufficient funds to pay off the debt by agreeing to set aside enough money each month (or quarter, or year) so that when the debt becomes payable, the money set aside each month plus the interest earned will equal the debt. The fund created by such a plan is called a sinking fund. Section 6.3 p20

  21. 6.4 Present Value of an Annuity; Amortization Section 6.4 p21

  22. 6.4 Present Value of an Annuity; Amortization The present value of an annuity is the sum of the present values of the withdrawals. In other words, the present value of an annuity is the amount of money needed now so that if it is invested at a rate of i per payment period, n equal dollar amounts can be withdrawn without any money left over. Section 6.4 p22

  23. 6.4 Present Value of an Annuity; Amortization Theorem Present Value of an Annuity Suppose an annuity earns interest at the rate of i per payment period. If n withdrawals of $P are made at each payment period, the amount V required is (1) Here V is called the present value of the annuity. Section 6.4 p23

  24. 6.4 Present Value of an Annuity; Amortization A loan with a fixed rate of interest is said to be amortized if both principal and interest are paid by a sequence of equal payments made over equal periods of time. (The Latin word mort means “death.” Paying off a loan is regarded as “killing” it.) Section 6.4 p24

  25. 6.4 Present Value of an Annuity; Amortization Theorem Amortization The payment P required to pay off a loan of V dollars borrowed for n payment periods at a rate of interest i per payment period is (2) Section 6.4 p25

  26. 6.4 Present Value of an Annuity; Amortization Definition Face Amount (Face Value or Par Value) The face amount or denomination of a bond (normally $1000) is the amount paid to the bondholder at maturity. It is also the amount usually paid by the bondholder when the bond is originally issued. Section 6.4 p26

  27. 6.4 Present Value of an Annuity; Amortization Definition Nominal Interest (Coupon Rate) The nominal interest or coupon rate is the contractual interest paid on the bond. Section 6.4 p27

  28. 6.4 Present Value of an Annuity; Amortization Nominal interest is normally quoted as an annual percentage of the face amount. Nominal interest payments are conventionally made semiannually, so semiannual periods are used for compound interest calculations. Section 6.4 p28

  29. 6.4 Present Value of an Annuity; Amortization When the bond price is higher than the face amount, it is trading at a premium; when it is lower, it is trading at a discount. For example, a bond with a face amount of $1000 and a coupon rate of 8% may trade in the marketplace at a price of $1100, which means the true yield is less than 8%. To obtain the true interest rate of a bond, we view the bond as a combination of an annuity of semiannual interest payments plus a single future amount payable at maturity. The price of a bond is therefore the sum of the present value of the annuity of semiannual interest payments plus the present value of the single future payment at maturity. This present value is calculated by discounting at the true interest rate and assuming semiannual payment periods. Section 6.4 p29

  30. 6.5 Annuities and Amortization Using Recursive Sequences Section 6.5 p30

  31. 6.5 Annuities and Amortization Using Recursive Sequences Often, though, money is invested in equal amounts at periodic intervals. An annuity is a sequence of equal periodic deposits. The periodic deposits may be made annually, quarterly, monthly, or daily. When deposits are made at the same time that the interest is credited, the annuity is called ordinary. We will only deal with ordinary annuities here. The amount of an annuity is the sum of all deposits made plus all interest paid. Section 6.5 p31

  32. 6.5 Annuities and Amortization Using Recursive Sequences Theorem Annuity Formula If A0 = M represents the initial amount deposited in an annuity that earns a rate of r per annum compounded N times per year, and if P is the periodic deposit made at each payment period, then the amount An of the annuity after n deposits is given by the recursive sequence A0 = M, (1) *We use N to represent the number of times interest is compounded per annum instead of n, since n is the traditional symbol used with sequences to denote the term of the sequence. Section 6.5 p32

  33. 6.5 Annuities and Amortization Using Recursive Sequences Theorem Amortization Formula If $B is borrowed at an interest rate of r per annum compounded monthly, the balance An due after n monthly payments of $P is given by the recursive sequence (2) Formula (2) may be explained as follows: The initial loan balance is $B. The balance due after n payments will equal the balance due previously, plus the interest charged on that amount reduced by the periodic payment P. Section 6.5 p33

  34. 6 Summary Summary IMPORTANT FORMULAS Simple Interest Formula I = Prt Discounted Loans R = L – Lrt Compound Interest Formula A = Pert Amount of an Annuity Present Value of an Annuity Amortization Chapter 6 p34

  35. 6.Extra Multiple Choice Questions Select the best answerfor each of the followingmultiple choice questions. Section 6.MC p35

  36. 6.Extra Multiple Choice Questions • If $15,000 is deposited in an account that earns interest at a 5.25% annual rate, compounded monthly, find the amount in this account after seven years. • A. $21,610.07 • B. $21,460.80 • C. $21,644.44 • D. $6,644.44 Section 6.MC p36

  37. 6.Extra Multiple Choice Questions 2. If $235 is deposited at the end of each month for five years in an account earning interest at an annual interest rate of 7.15%, compounded monthly, find the amount in this account at the end of five years. A. $16,265.69 B. $15,108.15 C. $14,100.00 D. $16,889.81 Section 6.MC p37

  38. 6.Extra Multiple Choice Questions 3. If an automobile loan of $28,500 is to be repaid in equal monthly payments over five years, at an annual interest rate of 3.75%, compounded monthly, find the monthly payment for this loan. A. $521.66 B. $475.00 C. $432.60 D. $476.48 Section 6.MC p38

  39. 6.Extra Multiple Choice Questions 4. What is the principal amount of a 30-year, fixed-rate home loan, if the monthly mortgage payment for this loan is $1800 and the annual interest rate for this loan is 6.125%, compounded monthly? A. $312,700.30 B. $296,242.39 C. $288,013.44 D. $279,784.48 Section 6.MC p39

  40. 6.Extra Multiple Choice Questions Answers for the Multiple Choice Questions 1. C. $21,644.44 2. D. $16,889.81 3. A. $521.66 4. B. $296,242.39 Section 6.MC p40

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