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CHAPTER 6

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Level Cash Flows: Annuities and Perpetuities

Quick Links

Multiple Cash Flows

Cash Flows That Grow at a Constant Rate

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Future Value of Multiple Cash Flows

Solving future value problems with multiple cash flows.

1. Draw timeline to ascertain each cash flow is placed in correct time period.

2. Calculate future value of each cash flow for its time period.

3. Add up the future values.

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Exhibit 6.1: Future Value of

Two Cash Flows

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Three Cash Flows

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Multiple Cash Flows

 _{Many business situations call for computing}

present value of a series of expected future cash flows.

 _{Determining market value of security.}

 _{Deciding whether to make capital investment}

 _{Process similar to determining future value of}

multiple cash flows.

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_{Next, calculate present value of each cash flow}

using equation 5.4 from the previous chapter. Present Value of Multiple Cash Flows

 _{Finally, add up all present values.}

 _{First, prepare timeline to identify magnitude and}

timing of cash flows.

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Exhibit 6.3: Present Value of

Three Cash Flows

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The Value of a Gift to the

University



Suppose that you made a gift to your

university, pledging $1,000 per year for four

years and $3,000 for the fifth year, for a

total of $7,000. After making the first three

payments, you decide to pay off the final

two payments of your pledge because your

financial situation has improved. How much

should you pay to the university if the

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Buying a Used Car

 For a student—or anyone else—buying a used car

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The Investment

Decision

 _{You are thinking of buying a business, and your investment adviser} presents you with two possibilities. Both businesses are priced at $60,000, and you have only $60,000 to invest. She has provided you with the following annual and total cash flows for each business,

along with the present value of the cash flows discounted at 10 percent:

 Cash flow ($ thousands)

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Annuities and Perpetuities

 _{Individual investors may make constant payments on}

home or car loans, or invest fixed amount year after year saving for retirement.

 _{Many situations exist where businesses and}

individuals would face either receiving or paying constant amount for a length of period.

Level Cash Flows

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 _{Annuity: any financial contract calling for equally}

spaced level cash flows over finite number of periods.

Annuities and Perpetuities

 _{Perpetuity: contract calling for level cash flow}

payments to continue forever.

 _{Ordinary annuities: constant cash flows occurring}

at end of each period.

Level Cash Flows

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Present Value of an Annuity

 _{Can calculate present value of annuity same way}

present value of multiple cash flows is calculated.

 _{Instead, simplify equation 5.4 in chapter 5 to}

obtain annuity factor.

 _{Results in equation 6.1 that can be used to}

calculate the annuity’s present value.

Level Cash Flows

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Exhibit 6.3: Present Value of

Three Cash Flows

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(6.1) i i) (1 1 1 CF i factor) value Present (1 CF annuity an for factor value Present CF PVA n n                         

Level Cash Flows

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Level Cash Flows

A financial contract pays $2,000 at the end of each year for three years and the appropriate discount rate is 8% percent? What is the present value of these cash flows?

Present Value of Annuity example

n 3

1 1

Present value factor = = = 0.7938 (1+i) (1+0.08)

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Finding Monthly or Yearly Payments Example

Level Cash Flows

You have just purchased a $450,000 condominium. You were able to put $50,000 down and obtain a 30-year

fixed rate mortgage at 6.125 percent for the balance. What are your monthly payments?

n 360

Monthly interest rate = 6.125 % / 12 months = 0.51042 %

1 1

Present value factor = 0.1599589 (1+i) (1.0051042)

1 - Present value factor PV annuity factor =

i

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Preparing a Loan Amortization Schedule

 _{Amortization: the way the borrowed amount}

(principal) is paid down over life of loan.

 _{Monthly loan payment is structured so each}

month portion of principal is paid off; at time loan matures, it is entirely paid off.

Level Cash Flows

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Exhibit 6.5: Amortization

Table for a 5-Yr, $10K Loan

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 _{Amortized loan: each loan payment contains}

some payment of principal and an interest payment.

Preparing a Loan Amortization Schedule

 _{Loan amortization schedule is a table showing:}  _{loan balance at beginning and end of each}

period.

 _{payment made during that period.}

Level Cash Flows

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Future Value of an Annuity

 _{Future value annuity calculations usually involve}

finding what a savings or investment activity is worth at some future point.

_{E.g. saving periodically for vacation, car,}

house, or retirement.

 _{We can derive the future value annuity equation}

from the present value annuity equation (equation 6.1). This results in equation 6.2.

Level Cash Flows

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Future Value of an Annuity Equation (6.2) i 1 i) (1 CF i 1 -factor value Future CF annuity an for factor value Future CF FVA n n         _ _      

Level Cash Flows

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4-Yr Annuity

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Level Cash Flows: Annuities

and Perpetuities



Finding the Interest Rate

◦

The present value of an annuity equation

can be used to find the interest rate or

discount rate for an annuity

◦

To determine the rate-of-return for an

annuity, solve the equation for

i

◦

Using a calculator is easier than a

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Perpetuities

 _{A perpetuity is constant stream of cash flows that}

goes on for infinite period.

 _{In stock markets, preferred stock issues are}

considered to be perpetuities, with issuer paying a constant dividend to holders.

 _{Equation for present value of a perpetuity can be}

derived from present value of an annuity equation with n tending to infinity.

Level Cash Flows

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Perpetuities CF i 0) (1 CF i i) (1 1 1 CF annuity an for factor value Present CF PVA                        

Level Cash Flows

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Perpetuities - Example

Level Cash Flows

Suppose you decided to endow a chair in finance. The goal of the chair is to provide the chair holder with $100,000 of additional financial support per

year forever. If the rate of interest is 8 percent, how much money will you have to give the university

foundation to provide the desired level of support?

PVA CF $100,000

= = = $1,250,000 i 0.08

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Exhibit 6.7: Ordinary

Annuity versus Annuity Due

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Annuity Due

 _{Annuity transformation method shows relationship}

between ordinary annuity and annuity due.

 _{Each period’s cash flow thus earns extra period}

of interest compared to ordinary annuity.

 _{Present or future value of annuity due is}

always higher than that of ordinary annuity.

Level Cash Flows

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Annuity Due Example

The value of the annuity due shown in Exhibit 6.7B is:

Level Cash Flows

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 _{In addition to constant cash flow streams, one}

may have to deal with cash flows that grow at a constant rate over time.

 _{These cash-flow streams called growing annuities}

or growing perpetuities.

Cash Flows That Grow at a

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Growing Annuity

 _{Business may need to compute value of multiyear}

product or service contracts with cash flows that increase each year at constant rate.

 _{These are called growing annuities.}

 _{Example of growing annuity: valuation of growing}

business whose cash flows increase every year at constant rate.

Cash Flows That Grow at a

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Growing Annuity

 _{Use this equation to value the present value of}

growing annuity (equation 6.5) when the growth rate is less than discount rate.





1

i

(6.5)

g

1

g

-i

CF

PVA

n 1 n

































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Growing Annuity Example

Cash Flows That Grow at a

Constant Rate

A coffee shop will be in business for 50-years. It produced $300,000 this year and the discount rate used by similar businesses is 15 percent. The cash flows will grow at 2.5 percent per year. What is the estimated value of the coffee shop?

1

50

CF = $300,000 (1 0.025) = $307,500

$307,500 1.025

 

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Growing Perpetuity

_{When cash flow stream features constant growing}

annuity forever.

 _{Can be derived from equation 6.5 when n tends}

to infinity and results in the following equation:

Cash Flows That Grow at a

Constant Rate

1

CF

PVA = (6.6) i - g

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Growing Perpetuity Example

Constant Rate

0

1 CF ×(1+g)

CF

PVA = _ =

Your account reports that a firm’s cash flow last year was $450,000 and the appropriate discount

rate for the club is 18 percent. You expect the firm’s cash flows to increase by 5 percent per year and

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 _{Interest rates can be quoted in financial markets}

in variety of ways.

 _{Most common quote, especially for a loan, is}

annual percentage rate (APR).

 _{APR represents simple interest accrued on loan}

or investment in a single period; annualized over a year by multiplying it by appropriate number of periods in a year.

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Calculating the Effective Annual Rate (EAR)

 _{Correct way to compute annualized rate is to}

reflect compounding that occurs; involves calculating effective annual rate (EAR).

 _{Effective annual interest rate (EAR) is defined as}

annual growth rate that takes compounding into account.

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Calculating the Effective Annual Rate (EAR)

EAR = (1 + Quoted rate/m)

m

– 1 (6.7)

m is the # of compounding periods during a year.

 _{EAR conversion formula accounts for number of}

compounding periods, thus effectively adjusts annualized interest rate for time value of money.

 _{EAR is the true cost of borrowing and lending.}

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Effective Annual Rate (EAR) Example

Effective Annual Interest Rate

Your credit card has an APR of 12 percent (1

percent per month). What is the effective annual interest rate?

EAR = (1 + 0.12/12)12 – 1

= (1.01)12 – 1

= 1.1268 – 1

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Examples: Calculating

EAR



Lender A: 10.40% compound monthly



Lender B: 10.90% compound annually



Lender C: 10.50% compound

quarterly