No Slide TitlePresent Value of a perpetuity or deferred perpetuity
LO 2.
LO 1.
Present Value and Future Value of an annuity whose payment size grows at a constant rate
Calculate the…
Learning Objectives
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A perpetuity is an annuity whose payments continue forever.
A $100,000 bequest is made to Seneca College to establish a perpetual bursary fund. If the college invests the funds to earn 6% compounded annually, the maximum amount that can be paid out on each anniversary of the bequest is …
$100,000 * 0.06 = $6,000
Ordinary Perpetuities
If more than this was to be paid out, a loss of principal would result.
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Present Value of:
If the payment interval equals the compounding interval, the perpetuity is an ordinary simple perpetuity
it is an ordinary general annuity
Otherwise…
12-*
What endowment is required to establish a perpetuity with an ongoing cost of $6,000 at the end of each month if interest is 6.0% compounded monthly in perpetuity?
= 6000 / (.06/12)
12-*
What monthly compounded nominal rate of return must an endowment of $1 million earn to fully fund a perpetuity with an ongoing cost of $4,000 at the end of each month?
i = 4000 / 1 000 000
= 0.004
= 0.4% per month
The required nominal rate of return is: 12 * 0.4% = 4.8% compounded monthly
PV = PMT/ i
12-*
What endowment is required to establish a perpetuity with an ongoing cost of $6,000 at the end of each month if interest is 6.0% compounded annually in perpetuity?
Since this is a general perpetuity, we need to determine c and i2
= .0833
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12-*
Since this is a general perpetuity, we need to determine c and i2
= .1667
= $ 211,743.26
What amount must be placed in a perpetual fund today if it earns 4.0% compounded semi-annually and monthly payments of $700 in perpetuity are to start 1 month from now?
PV
C =
12
12-*
What amount must be placed in a perpetual fund today if it earns 4.0% compounded semi-annually and monthly payments of $700 in perpetuity are to start 1 YEAR from now?
We have already determined the value at the beginning of the payments
PV = $ 211,743.26
This is the value 11 months from now
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Constant Growth Annuities
… Annuities in which the payments change by the same percentage from one payment to another
Let g = rate of growth in payment size between successive payments
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FV
= PMT
12-*
You intend to make RRSP contributions on Feb.28 of each year. You plan to contribute $2,000 in the first year and increase the contribution by 4% every year thereafter.
How much will you have in your RRSP at the time of your 20th contribution if the plan earns 7.5% compounded annually?
What will be the amount of your last contribution?
Constant Growth Annuities
Extract necessary data...
g = 4%
How much will you have in your RRSP at the time of your 20th contribution if the plan earns 7.5% compounded annually?
You intend to make RRSP contributions on Feb.28 of each year. You plan to contribute $2000 in the first year and increase the contribution by 4% every year thereafter.
Solve …
FV
= PMT
12-*
1.04
0.035
20
1.075
2000
20
2.1911
4.2479
2.0567
117,527.31
Solve …
Amount in the RRSP at the time of the 20th contribution
PMT =
(b) What will be the amount of your last contribution?
The final payment will be the Future Value of $2000 after 19 compoundings at 4%
= 2000( 1+ 0.04)19
= $4,213.70
You intend to make RRSP contributions on Feb.28 of each year. You plan to contribute $2000 in the first year and increase the contribution by 4% every year thereafter.
FV = PV(1 + i)n
Constant Growth Annuities
How much will it cost to purchase a 25-year ordinary annuity making semiannual payments that grow at the rate of 3% compounded semiannually?
The first payment is $10,000 and the funds used to purchase the annuity earn 5% compounded semiannually.
Solution
PMT =
Extract necessary data...
How much will it cost to purchase a 25-year ordinary annuity making semiannual payments that grow at the rate of 3% compounded semiannually? The first payment is $10,000 and the funds used to purchase the annuity earn 5% compounded semiannually.
Solve …
PV
= PMT
Cost of the annuity
How much will it cost to purchase a 25-year ordinary annuity making semiannual payments that grow at the rate of 3% compounded semiannually? The first payment is $10,000 and the funds used to purchase the annuity earn 5% compounded semiannually.
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