Annuities and Perpetuities Defined. 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
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PV=C/r
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?
-$4,000
-$4,000
-4,000
0
1
2
3
PV today:
$4,000 / (1.10) = $3,636.36
$4,000 / (1.10)2 = $3,305.79
$4,000 / (1.10)3 = $3,005.26
$9,947.41
The difference in value between a long Annuity and a Perpetuity
What is the difference in value between a 50-year bond and the perpetual bond offering $100 cash flows when interest rates are 10%?
In Excel:
0 1 2 3
10000 10000 10000
32,464
35,016.12
Growing Perpetuity
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?
Growing Annuity
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?
Growing Annuity: Example 2
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.
Growing Annuity: Example 3
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?
CompoundingNumber of timesEffective
periodcompoundedannual rate
Remember that the APR is the quoted rate
m is the number of compounding periods per year
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
Beginning Total Interest Principal Ending
Year Balance PaymentPaid Paid Balance
1$5,000$1,450$450$1,000$4,000
24,0001,3603601,0003,000
33,0001,2702701,0002,000
42,0001,1801801,0001,000
51,0001,090901,0000
Totals$6,350$1,350$5,000
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
24,164.541,285.46374.81910.653,253.88
33,253.881,285.46292.85992.612,261.27
42,261.271,285.46203.511,081.951,179.32
51,179.321,285.46106.141,179.320.00
Totals$6,427.30$1,427.31$5,000.00
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?