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Annuity – finite series of equal payments that occur at regular intervals◦ If the first payment occurs at the end of the

period, it is called an ordinary annuity◦ If the first payment occurs at the beginning of

the period, it is called an annuity due Perpetuity – infinite series of equal

payments

Annuities and Perpetuities Defined

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If the payment stream lasts forever, it is called a perpetuity.

The PV of perpetuity is calculated by dividing the cash payment by the interest rate:

PV=C/r

Important: first payment starts a period from now. What happens if it starts today?

Perpetuities – Basic Formulas

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What would you pay to own a guaranteed income of $1,000 per year to be received forever, if interest rates are 4%?

Example (perpetuity)

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Assume interest rates are 10%.You wish to buy a car making three

installments: $4,000 a year from now, $4,000 two years from now, $4,000 three years from now.

How much money would you have to place in an account today to generate this stream of cash flows?

Example (annuity)

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PV of an Annuity: the Long Method◦You would need to place $9,947.41 in an

account today to generate the desired cash flows:

$4,000 / (1.10) = $3,636.36$4,000 / (1.10)2 = $3,305.79

$9,947.41

PV today: 0 1 2 3

-$4,000 -$4,000 -4,000

$4,000 / (1.10)3 = $3,005.26

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Can you think of a way to solve the previous problem using the perpetuity formula?

The relation between annuity and perpetuity

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Perpetuity: PV = C / r Annuities:

Annuities and Perpetuities – Basic Formulas

r

rCFV

rr

CPV

t

t

1)1(

)1(1

1

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◦Assume interest rates are 4.3884%.◦You have just won a lottery and must

choose between the following two options: Receive a cheque for $150,000 today. Receive $10,000 a year for the next 25

years.

KEY QUESTIONS FOR YOU:

Which option gives you the biggest “winnings”

How should you tackle this kind of problem?

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Suppose you win the Publishers Clearinghouse $10 million sweepstakes. The money is paid in equal annual installments of $333,333.33 over 30 years. If the appropriate discount rate is 5%, how much is the sweepstakes actually worth today?

Annuity – Sweepstakes Example

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You know the payment amount for a loan and you want to know how much was borrowed. Do you compute a present value or a future value?

You want to receive 5,000 per month in retirement. If you can earn .75% per month and you expect to need the income for 25 years, how much do you need to have in your account at retirement?

Questions:

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What is the difference in value between a 50-year bond and the perpetual bond offering $100 cash flows when interest rates are 10%?

The difference in value between a long Annuity and a Perpetuity

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Suppose you want to borrow $20,000 for a new car. You can borrow at 8% per year, compounded monthly (8/12 = .66667% per month). If you take a 4 year loan, what is your monthly payment?

Finding the Payment

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You ran a little short on your spring break vacation, so you put $1,000 on your credit card. You can only afford to make the minimum payment of $20 per month. The interest rate on the credit card is 1.5 percent per month. How long will you need to pay off the $1,000?

Finding the Number of Payments –

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Suppose you borrow $2,000 at 5% and you are going to make annual payments of $734.42. How long before you pay off the loan?

Finding the Number of Payments – Another Example

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Suppose you borrow $10,000 from your parents to buy a car. You agree to pay $207.58 per month for 60 months. What is the monthly interest rate?

Finding the Rate

Microsoft Office Excel Worksheet

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Trial and Error Process◦ Choose an interest rate and compute the PV of the

payments based on this rate◦ Compare the computed PV with the actual loan

amount◦ If the computed PV > loan amount, then the

interest rate is too low◦ If the computed PV < loan amount, then the

interest rate is too high◦ Adjust the rate and repeat the process until the

computed PV and the loan amount are equal

Annuity – Finding the Rate Without a Financial Calculator

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You can use the PMT or CF key on a financial calculator for the equal payment

The sign convention still holds

In Excel:

Annuities and the Calculator

Microsoft Office Excel Worksheet

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You want to receive $5,000 per month for the next 5 years. How much would you need to deposit today if you can earn .75% per month?

Suppose you have $200,000 to deposit and can earn .75% per month.◦ How many months could you receive the $5,000

payment?◦ How much could you receive every month for 5

years?

Some more questions:

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Suppose you begin saving for your retirement by depositing $2,000 per year in an IRA. If the interest rate is 7.5%, how much will you have in 40 years?

Future Values for Annuities

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You are ready to buy a house and you have $20,000 for a down payment and closing costs. Closing costs are estimated to be 4% of the loan value. You have an annual salary of $36,000 and the bank is willing to allow your monthly mortgage payment to be equal to 28% of your monthly income. The interest rate on the loan is 6% per year with monthly compounding (.5% per month) for a 30-year fixed rate loan. How much money will the bank loan you? How much can you offer for the house?

Buying a House

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Example: the opportunity cost is 4%. I have a rent agreement for a house of $1000 monthly. What is the PV of renting the house for a year.

Annuity Due

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You are saving for a new house and you put $10,000 per year in an account paying 8%. The first payment is made today. How much will you have at the end of 3 years?

Annuity Due – Example 2

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Annuity Due Timeline

0 1 2 3

10000 10000 10000

32,464

35,016.12

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You want to have $1 million to use for retirement in 35 years. If you can earn 1% per month, how much do you need to deposit on a monthly basis if the first payment is made in one month?

What if the first payment is made today? You are considering preferred stock that

pays a quarterly dividend of $1.50. If your desired return is 3% per quarter, how much would you be willing to pay?

Questions

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You are offered the opportunity to put some money away for retirement. You will receive five annual payments of $25,000 each beginning in 40 years. How much would you be willing to invest today if you desire an interest rate of 12%?

Saving For Retirement

An unequal cash flow that we are able to value easily is the growing annuity – where the cash flow grows at a constant rate g.

Assume you own a Condo that currently generates 12,000 a year. You expect this to grow at a rate of 3%, while the discount rate is 8%. What is the value of the Condo?

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Growing Perpetuity

gr

CperpetuitygrowingPV

) (

The PV of a finite stream of cash flow that grows at a constant rate g.

Assume the cash flow from the condo is only yours for next 20 years. What is it value today?

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Growing Annuity

t

r

g

gr

CgrowingPV

1

11)annuity (

A real estate appraiser is assessing the value of a piece of land in Vancouver.

Currently the land is unoccupied but is zoned for commercial use. Plans

have been approved to build a five-story office building. Construction is

expected to start in 1 year and will take 2 years to complete, at a total cost

of $3 million. For simplicity, assume that the costs are paid in equal

amounts at the start of each construction year.

Suppose a constant annual cash flow of $400,000, net of all taxes and

operating cost, is expected at the end of each year of operation, and the

building lasts for 50 years. What is the maximum you would be willing to

pay for the land if the discount rate is 8 percent. Explain your answer.

If the cash flow from the tenants grows at 1.5% per annum, after the first year

of occupancy, recalculate the price you would be willing to pay for the land.

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Growing Annuity: Example 2

When Marilyn Monroe died, ex husband Joe DiMaggio vowed to place fresh flows on her grave every Sunday as long as he lived. A bunch of fresh flowers that the former baseball player thought appropriate for the star cost about $5 when she died in 1962. Based on actuarial tables, Joe could expect to live for 30 years after the actress died. Assuming the inflation per week is 0.0736%, that the stated interest per week is 0.1905%, and assuming 52 weeks in a year, what was the PV of this commitment?

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Growing Annuity: Example 3

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This is the actual rate paid (or received) after accounting for compounding that occurs during the year.

If you want to compare two alternative investments with different compounding periods you need to compute the EAR and use that for comparison.

Effective Annual Rate (EAR)

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This is the annual rate that is quoted by law By definition APR = period rate times the

number of periods per year Consequently, to get the period rate we

rearrange the APR equation:◦ Period rate = APR / number of periods per year

You should NEVER divide the effective rate by the number of periods per year – it will NOT give you the period rate

Annual Percentage Rate

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What is the APR if the monthly rate is .5%?◦ .5(12) = 6%

What is the APR if the semiannual rate is .5%?◦ .5(2) = 1%

What is the monthly rate if the APR is 12% with monthly compounding?◦ 12 / 12 = 1%

Computing APRs

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You ALWAYS need to make sure that the interest rate and the time period match.◦ If you are looking at annual periods, you need an

annual rate.◦ If you are looking at monthly periods, you need a

monthly rate. If you have an APR based on monthly

compounding, you have to use monthly periods for lump sums, or adjust the interest rate appropriately if you have payments other than monthly

Things to Remember

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Suppose you can earn 1% per month on $1 invested today.◦ What is the APR? 1(12) = 12%◦ How much are you effectively earning?

FV = 1(1.01)12 = 1.1268 Rate = (1.1268 – 1) / 1 = .1268 = 12.68%

Suppose if you put it in another account, you earn 3% per quarter.◦ What is the APR? 3(4) = 12%◦ How much are you effectively earning?

FV = 1(1.03)4 = 1.1255 Rate = (1.1255 – 1) / 1 = .1255 = 12.55%

Computing EARs - Example

Compounding Number of times Effective period compounded annual

rate Year 1 10.00000% Quarter 4

10.38129 Month 12

10.47131 Week 52

10.50648 Day 36510.51558 Hour 8,76010.51703 Minute 525,600

10.51709

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Compounding Periods for 10% APR

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EAR - Formula

1 m

APR 1 EAR

m

Remember that the APR is the quoted rate

m is the number of compounding periods per year

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You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use and why?

Decisions, Decisions II

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If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:

Computing APRs from EARs

1 - EAR) (1 m APR m

1

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Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay?

APR - Example

11.39%or

8655152113.1)12.1(12 12/1 APR

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Suppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments. The entire computer system costs $3,500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding. What is your monthly payment?

Computing Payments with APRs

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Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years?

Future Values with Monthly Compounding

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You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit?

Present Value with Daily Compounding

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Sometimes investments or loans are figured based on continuous compounding

EAR = eq – 1◦ The e is a special function on the calculator

normally denoted by ex

Example: What is the effective annual rate of 7% compounded continuously?◦ EAR = e.07 – 1 = .0725 or 7.25%

Continuous Compounding

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What is the definition of an APR? What is the effective annual rate? Which rate should you use to compare

alternative investments or loans? Which rate do you need to use in the time

value of money calculations?

Quick Quiz

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Treasury bills are excellent examples of pure discount loans. The principal amount is repaid at some future date, without any periodic interest payments.

If a T-bill promises to repay $10,000 in 12 months and the market interest rate is 7 percent, how much will the bill sell for in the market?

Pure Discount Loans

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Consider a 5-year, interest-only loan with a 7% interest rate. The principal amount is $10,000. Interest is paid annually.◦ What would the stream of cash flows be?

Years 1 – 4: Interest payments of .07(10,000) = 700 Year 5: Interest + principal = 10,700

This cash flow stream is similar to the cash flows on corporate bonds and we will talk about them in greater detail later.

Interest-Only Loan - Example

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Consider a $50,000, 10 year loan at 8% interest. The loan agreement requires the firm to pay $5,000 in principal each year plus interest for that year.

Click on the Excel icon to see the amortization table

Amortized Loan with Fixed Principal Payment - Example

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Each payment covers the interest expense plus reduces principal

Consider a 4 year loan with annual payments. The interest rate is 8% and the principal amount is $5,000.◦ What is the annual payment?

Click on the Excel icon to see the amortization table

Amortized Loan with Fixed Payment - Example

You take on a loan of $5,000 that matures in 5 years. What is your payment schedule if the loan bears a 9% interest and it is a

(a) Pure discount loan.(b) Interest only loan.(c) Fixed principal loan.(d) Fixed payment loan.

Question: Loans

Beginning Total Interest Principal Ending

Year Balance Payment Paid Paid Balance

1 $5,000 $1,450 $450 $1,000 $4,000 2 4,000 1,360 360 1,000 3,000 3 3,000 1,270 270 1,000 2,000 4 2,000 1,180 180 1,000 1,000 5 1,000 1,090 90 1,000 0Totals $6,350 $1,350 $5,000

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Amortization Schedule - Fixed Principal

Beginning Total Interest Principal Ending

Year Balance Payment Paid Paid Balance

1 $5,000.00 $1,285.46 $ 450.00 $ 835.46$4,164.54

2 4,164.54 1,285.46 374.81 910.65 3,253.88 3 3,253.88 1,285.46 292.85 992.61 2,261.27 4 2,261.27 1,285.46 203.51 1,081.95

1,179.32 5 1,179.32 1,285.46 106.14 1,179.32 0.00Totals $6,427.30 $1,427.31 $5,000.00

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Amortization Schedule - Fixed Payments

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There are web sites available that can easily prepare amortization tables

Click on the web surfer to check out the Bankrate.com site and work the following example

You have a loan of $25,000 and will repay the loan over 5 years at 8% interest.◦ What is your loan payment?◦ What does the amortization schedule look like?

Work the Web Example

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What is a pure discount loan? What is a good example of a pure discount loan?

What is an interest-only loan? What is a good example of an interest-only loan?

What is an amortized loan? What is a good example of an amortized loan?

Quick Quiz – Part VI

Your company will generate $55,000 in annual revenue each year for the next eight years from a new information database. The computer system needed to set up the database costs $250,000. If you can borrow the money to buy the computer system at 7.5% percent annual interest, can you afford the new system?

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Problem