6.2 Calculating PV & FV

Annuity Due

where payments or receipts occur at the beginning of each period (i.e., the first payment is today at t = 0).

Annuity

stream of equal cash flows that occurs at equal intervals over a given period. the size of the periodic cash flow is defined by the payment (PMT) variable on your calculator.

Ordinary Annuity

the most common type of annuity. It is characterized by cash flows that occur at the end of each compounding period.

What is the future value of an ordinary annuity that pays $200 per year at the end of each of the next three years, given the investment is expected to earn a 10% rate of return?

N = 3; I/Y = 10; PMT = -200; CPT → FV = $662.00

Future Value Factor / Future Value Interest Factor

(1+I/Y)^N

Ways to calculate FV annuity due

1) Set calculator to BGN mode 2) calculate the FV of an ordinary annuity, and simply multiply the resulting FV by [1 + periodic compounding rate (I/Y)].

Kodon Corporation issues preferred stock that will pay $4.50 per year in annual dividends beginning next year and plans to follow this dividend policy forever. Given an 8% rate of return, what is the value of Kodon's preferred stock today?

4.50 / .08 = $56.25

Assume the Kodon preferred stock in the preceding examples is scheduled to pay its first dividend in four years, and is non-cumulative (i.e., does not pay any dividends for the first three years). Given an 8% required rate of return, what is the value of Kodon's preferred stock today?

4.50 / .08 = $56.25 56.25 ------- = $44.65 (1.08)^3

What is the PV of an annuity that pays $200 per year at the end of each of the next three years, given a 10% discount rate?

N = 3; I/Y = 10; PMT = -200; FV = 0; CPT → PV = $497.37

Interest Rate in Discounting Process

Discount Rate, Opportunity Cost, Required Rate of Return, Cost of Capital

Future Value

IE: Compound Value. The amount to which a current deposit will grow over time when it is placed in an account paying compound interest. r = i / y. n = number of compounding periods.

Given a discount rate of 10%, calculate the PV of a $200 cash flow that will be received in two years.

N = 2; I/Y = 10; FV = 200; CPT → PV = −$165.29 (ignore the sign)

Calculate the FV of a $200 investment at the end of two years if it earns an annually compounded rate of return of 10%

N = 2; I/Y = 10; PV = -200; CPT → FV = $242

What is the future value of an annuity that pays $200 per year at the beginning of each of the next three years, commencing today, if the cash flows can be invested at an annual rate of 10%?

N = 3; I/Y = 10; PMT = -200; CPT → FV = $728.20

Given a discount rate of 10%, what is the present value of an annuity that makes $200 payments at the beginning of each of the next three years, starting today?

N = 3; I/Y = 10; PMT = -200; CPT → PVAD = $547.11

A bond will make coupon interest payments of 70 euros (7% of its face value) at the end of each year and will also pay its face value of 1,000 euros at maturity in six years. If the appropriate discount rate is 8%, what is the present value of the bond's promised cash flows?

N = 6; PMT = 70; I/Y = 8; FV = 1,000; CPT PV = -953.77

What is the present value of four $100 end-of-year payments if the first payment is to be received three years from today and the appropriate rate of return is 9%?

Step 1: Find the present value of the annuity as of the end of year 2 (PV2). N = 4; I/Y = 9; PMT = -100; FV = 0; CPT → PV = PV2 = $323.97 Step 2: Find the present value of PV2.. N = 2; I/Y = 9; PMT = 0; FV = -323.97; CPT → PV = PV0 = $272.68

Present Value

Today's value of a cash flow that is to be received at some point in the future. Process for finding FV is known as discounting. I = I/Y; n = number of compounding periods.

Ways to calculate PV annuity due

With an annuity due, there is one less discounting period since the first cash flow occurs at t = 0 and thus is already its PV. This implies that, all else equal, the PV of an annuity due will be greater than the PV of an ordinary annuity. Ways to Calculate 1) Set calculator to BGN 2) Calculate the PV of an ordinary annuity and multiply by [1 + periodic compounding rate (I/Y)].

Perpetuity

a financial instrument that pays a fixed amount of money at set intervals over an infinite period of time. In essence, a perpetuity is a perpetual annuity. Most preferred stocks are examples of perpetuities since they promise fixed interest or dividend payments forever. Without going into all the excruciating mathematical details, the discount factor for a perpetuity is just one divided by the appropriate rate of return (i.e., 1/r). Given this, we can compute the PV of a perpetuity. PVperpetuity = PMT / IY