CHAPTER 8 5 ANNUITIES PRESENT VALUE Definitions Definitions

Definitions (cont’d) Present Value: The principal that would have to be invested now to

Steps to Find the Present Value 1) Draw a timeline 2) Consider each payment

Example 2 (cont’d 2) A=Rx. N A = 1502. 94 x 240 A =

Example 3 Sarah wants to buy a house 5 years from now for $150,

Example 3 (cont’d) Use the Formula: PV = A/(1+i)ⁿ Then, substitute in the numbers

Example 3 (cont’d 2) Therefore, Sarah should invest $112, 088. 73 today to earn

Key Ideas q The present value of an annuity is the value of the

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CHAPTER 8. 5 ANNUITIES: PRESENT VALUE

Definitions

Definitions (cont’d) Present Value: The principal that would have to be invested now to get a specific future value in a certain amount of time; PV is used for present value instead of P, since P is used for principal. Simple Ordinary Annuity: a payment or deposit that is equal to the compounding period and made at the end of the interval.

Formulas

Formulas (cont’d)

Steps to Find the Present Value 1) Draw a timeline 2) Consider each payment separately, using the present value formula 3) Put each payment into a geometric sequence 4) Use the geometric series formula to calculate the sum of the previous sequence. or Substitute the terms in the present value formula

Example 1

Example 1 (cont’d)

Example 2

Example 2 (cont’d)

Example 2 (cont’d 2) A=Rx. N A = 1502. 94 x 240 A = $ 360 706. 60 To find the interest, subtract the present value from the total amount. I = A – PV = $ 360 706. 60 - $ 200 000 = $ 160 706. 60 Therefore, over the length of 20 years for the loan, Victoria will have to pay $ 160 706. 60 in interest.

Example 3 Sarah wants to buy a house 5 years from now for $150, 000. The interest is 6%/a compounded annually, how much should she invest today to earn 150, 000 in 5 years. Given: A= 150, 000 i=0. 06 n=5

Example 3 (cont’d) Use the Formula: PV = A/(1+i)ⁿ Then, substitute in the numbers = 150, 000/(1 + 0. 06)5 = 150, 000/(1. 338225578) = 112, 088. 73

Example 3 (cont’d 2) Therefore, Sarah should invest $112, 088. 73 today to earn $150, 000 in 5 years.

Key Ideas q The present value of an annuity is the value of the annuity at the beginning of the term. It is the sum of all present values of the payments and can be written as the geometric series PV= R x (1+i)^-1+ R x (1+i)^-2+ (1+i)^-3…+R x (1+i)^-n PV= Present value R= Regular payment i= Interest rate per compounding period, expressed as a decimal n= Number of compounding periods q q The formula for the sum of a geometric series can be used to determine the present value of an annuity The formula for the present value of an annuity is: PV= R x (1 -(1+i)^-n/i) PV= Present value R= Regular payment each compounding period i= Interest rate per compounding period, expressed as a decimal n= Number of compounding periods

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