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Financial Analysis
Class #3: Annuities and Perpetuities
Ana Elisa Pereira
August 2, 2018
Class #3: Annuities and Perpetuities Financial Analysis
Stock and Bond Valuation
• Present value is the most important concept when it comes to
valuing investments of all types, including stocks and bonds
• These investments rarely have just two or three payments
• Stocks may pay dividends forever
• Some bonds have monthly payments for 30 years
• Sometimes it is not necessary to compute a present value with over
360 terms
• Some shortcut formulas can help us compute present values if...
• the project has a particular set of cash flows
• and the discount rate is constant
Class #3: Annuities and Perpetuities Financial Analysis
Stock and Bond Valuation
• Present value is the most important concept when it comes to
valuing investments of all types, including stocks and bonds
• These investments rarely have just two or three payments
• Stocks may pay dividends forever
• Some bonds have monthly payments for 30 years
• Sometimes it is not necessary to compute a present value with over
360 terms
• Some shortcut formulas can help us compute present values if...
• the project has a particular set of cash flows
• and the discount rate is constant
Class #3: Annuities and Perpetuities Financial Analysis
Stock and Bond Valuation
• Present value is the most important concept when it comes to
valuing investments of all types, including stocks and bonds
• These investments rarely have just two or three payments
• Stocks may pay dividends forever
• Some bonds have monthly payments for 30 years
• Sometimes it is not necessary to compute a present value with over
360 terms
• Some shortcut formulas can help us compute present values if...
• the project has a particular set of cash flows
• and the discount rate is constant
Class #3: Annuities and Perpetuities Financial Analysis
Perpetuities
• A simple perpetuity is a project with a stream of constant cash flows
that repeat forever
• If the cost of capital (discount rate) is constant and the payments
are constant or grow at a constant rate, we can use perpetuities
formulas to compute present values
• An example: At a constant interest rate of 10% per year, how
much would you have to invest today to receive $2 each year,
starting next year, forever?
Class #3: Annuities and Perpetuities Financial Analysis
Perpetuities
• A simple perpetuity is a project with a stream of constant cash flows
that repeat forever
• If the cost of capital (discount rate) is constant and the payments
are constant or grow at a constant rate, we can use perpetuities
formulas to compute present values
• An example: At a constant interest rate of 10% per year, how
much would you have to invest today to receive $2 each year,
starting next year, forever?
Class #3: Annuities and Perpetuities Financial Analysis
Perpetuities
• A simple perpetuity is a project with a stream of constant cash flows
that repeat forever
• If the cost of capital (discount rate) is constant and the payments
are constant or grow at a constant rate, we can use perpetuities
formulas to compute present values
• An example: At a constant interest rate of 10% per year, how
much would you have to invest today to receive $2 each year,
starting next year, forever?
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity
• The present values of the future payments:
• Height of bars expresses that nominal cash flows are the same
• Their widths indicate that present values decrease
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity
• The present values of the future payments:
• Height of bars expresses that nominal cash flows are the same
• Their widths indicate that present values decrease
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity
• If we sum the first 50 terms, we get $19.83
• If we sum the first 100, we get $ 19.9986
• Mathematically, the sum converges to $20 sharp
But how do we compute this infinte sum of (discounted) payments?
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity
• If we sum the first 50 terms, we get $19.83
• If we sum the first 100, we get $ 19.9986
• Mathematically, the sum converges to $20 sharp
But how do we compute this infinte sum of (discounted) payments?
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity
• If we sum the first 50 terms, we get $19.83
• If we sum the first 100, we get $ 19.9986
• Mathematically, the sum converges to $20 sharp
But how do we compute this infinte sum of (discounted) payments?
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity formula
• The simple perpetuity formula is:
PV0 =
C1
r
which is the result of the sum
PV0 =
C1
1 + r +
C2
(1 + r)2
+ ... + Ct
(1 + r)t
+ ...
with Ct = C1 for all t, and r is the constant discount rate
• The time subscript “1” in the formula is to remind us that the first
cash flow occurs not now, but next period
• Applying it to our example, we get
PV0 =
$2
0.1 = $20
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity formula
• The simple perpetuity formula is:
PV0 =
C1
r
which is the result of the sum
PV0 =
C1
1 + r +
C2
(1 + r)2
+ ... + Ct
(1 + r)t
+ ...
with Ct = C1 for all t, and r is the constant discount rate
• The time subscript “1” in the formula is to remind us that the first
cash flow occurs not now, but next period
• Applying it to our example, we get
PV0 =
$2
0.1 = $20
Class #3: Annuities and Perpetuities Financial Analysis
The simple perpetuity formula
• The simple perpetuity formula is:
PV0 =
C1
r
which is the result of the sum
PV0 =
C1
1 + r +
C2
(1 + r)2
+ ... + Ct
(1 + r)t
+ ...
with Ct = C1 for all t, and r is the constant discount rate
• The time subscript “1” in the formula is to remind us that the first
cash flow occurs not now, but next period
• Applying it to our example, we get
PV0 =
$2
0.1 = $20
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
• What if the cash flows increase over time?
• The growing perpetuity formula allows us to compute present
values in a similar way
• as long as payments grow at a constant rate g ...
• and that this rate is smaller than the interest rate, g < r
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
• What if the cash flows increase over time?
• The growing perpetuity formula allows us to compute present
values in a similar way
• as long as payments grow at a constant rate g ...
• and that this rate is smaller than the interest rate, g < r
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
• Back to our example...
Imagine a perpetuity that pays $2 next year, grows at 5% per year, and
faces a cost of capital of 10%
• The first 30 discounted terms add up to $30.09
• The first 100 discounted terms add up to $39.64
• The first 200 discounted terms add up to $39.98
• Mathematically, the sum converges to $40
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
• Back to our example...
Imagine a perpetuity that pays $2 next year, grows at 5% per year, and
faces a cost of capital of 10%
• The first 30 discounted terms add up to $30.09
• The first 100 discounted terms add up to $39.64
• The first 200 discounted terms add up to $39.98
• Mathematically, the sum converges to $40
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
• Back to our example...
Imagine a perpetuity that pays $2 next year, grows at 5% per year, and
faces a cost of capital of 10%
• The first 30 discounted terms add up to $30.09
• The first 100 discounted terms add up to $39.64
• The first 200 discounted terms add up to $39.98
• Mathematically, the sum converges to $40
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
• The present values of thefuture payments:
• Height of bars expresses that nominal cash flows grow over time
• Their widths indicate that present values decrease
• the area of each bar is smaller than the previous one
• this is important for the sum to be finite
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
• The present values of the future payments:
• Height of bars expresses that nominal cash flows grow over time
• Their widths indicate that present values decrease
• the area of each bar is smaller than the previous one
• this is important for the sum to be finite
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity formula
• A stream of cash flows growing at rate g each period and
discounted at a rate r is worth
PV0 =
C1
r − g
• This is the growing perpetuity formula
• Important to note that:
• The first cash flow, C1, occurs next period (period 1)
• The second cash flow C2 = C1 (1 + g) occurs two periods from now...
and so on, forever
• g can be negative
• The formula only makes sense for r > g
Class #3: Annuities and Perpetuities Financial Analysis
The growing perpetuity formula
• A stream of cash flows growing at rate g each period and
discounted at a rate r is worth
PV0 =
C1
r − g
• This is the growing perpetuity formula
• Important to note that:
• The first cash flow, C1, occurs next period (period 1)
• The second cash flow C2 = C1 (1 + g) occurs two periods from now...
and so on, forever
• g can be negative
• The formula only makes sense for r > g
Class #3: Annuities and Perpetuities Financial Analysis
Application: Stock valuation with a Gordon Growth Model
• Perpetuities assume constant interest rate and eternal payments...
• So of course they are rarely correct
• However, they provide a very useful approximation!
• Consider a stable business with profits estimated in $1 million next
year
• Its profits are expected to grow at 2% per year
• The firm faces a cost of capital of 8%
• How much should that firm be worth, more or less?
Class #3: Annuities and Perpetuities Financial Analysis
Application: Stock valuation with a Gordon Growth Model
• Perpetuities assume constant interest rate and eternal payments...
• So of course they are rarely correct
• However, they provide a very useful approximation!
• Consider a stable business with profits estimated in $1 million next
year
• Its profits are expected to grow at 2% per year
• The firm faces a cost of capital of 8%
• How much should that firm be worth, more or less?
Class #3: Annuities and Perpetuities Financial Analysis
Application: Stock valuation with a Gordon Growth Model
• Perpetuities assume constant interest rate and eternal payments...
• So of course they are rarely correct
• However, they provide a very useful approximation!
• Consider a stable business with profits estimated in $1 million next
year
• Its profits are expected to grow at 2% per year
• The firm faces a cost of capital of 8%
• How much should that firm be worth, more or less?
Class #3: Annuities and Perpetuities Financial Analysis
Application: Stock valuation with a Gordon Growth Model
• Perpetuities assume constant interest rate and eternal payments...
• So of course they are rarely correct
• However, they provide a very useful approximation!
• Consider a stable business with profits estimated in $1 million next
year
• Its profits are expected to grow at 2% per year
• The firm faces a cost of capital of 8%
• How much should that firm be worth, more or less?
Class #3: Annuities and Perpetuities Financial Analysis
Applying perpetuities
• The firm will probably not exist forever, so the growing perpetuity
formula indicates this firm should be worth no more than
Business value = PV0 =
$1mi
0.08− 0.02 = $16, 666, 667
• Of course, in real life there are significant uncertainties
• next year’s profits may be different
• firm may grow at different rate in the future
• cost of capital may change
• $16.7 million should be considered a useful approximation, not an
exact number
Class #3: Annuities and Perpetuities Financial Analysis
Applying perpetuities
• The firm will probably not exist forever, so the growing perpetuity
formula indicates this firm should be worth no more than
Business value = PV0 =
$1mi
0.08− 0.02 = $16, 666, 667
• Of course, in real life there are significant uncertainties
• next year’s profits may be different
• firm may grow at different rate in the future
• cost of capital may change
• $16.7 million should be considered a useful approximation, not an
exact number
Class #3: Annuities and Perpetuities Financial Analysis
Applying perpetuities
• The firm will probably not exist forever, so the growing perpetuity
formula indicates this firm should be worth no more than
Business value = PV0 =
$1mi
0.08− 0.02 = $16, 666, 667
• Of course, in real life there are significant uncertainties
• next year’s profits may be different
• firm may grow at different rate in the future
• cost of capital may change
• $16.7 million should be considered a useful approximation, not an
exact number
Class #3: Annuities and Perpetuities Financial Analysis
Applying perpetuities
• Growing perpetuities are sometimes directly applied to the stock
market
• If you expect dividends to grow by, say, g = 5% forever, you believe
the appropriate discount rate is 10%, and if you expect dividends to
be $10 per share next year, then you would think that the stock
price today should be
Stock price P today = $1010%− 5% = $200
• In this context, the growing perpetuity model is called the Gordon
Growth Model:
Stock price P today = Dividends D next yearr − g
Class #3: Annuities and Perpetuities Financial Analysis
Applying perpetuities
• Growing perpetuities are sometimes directly applied to the stock
market
• If you expect dividends to grow by, say, g = 5% forever, you believe
the appropriate discount rate is 10%, and if you expect dividends to
be $10 per share next year, then you would think that the stock
price today should be
Stock price P today = $1010%− 5% = $200
• In this context, the growing perpetuity model is called the Gordon
Growth Model:
Stock price P today = Dividends D next yearr − g
Class #3: Annuities and Perpetuities Financial Analysis
Quick exercise
1. An ’Americano’ coffee in size ’tall’ at Starbucks costs $1.75. How
much should cost a coffee plan that allows you to have an
Americano every day forever? Suppose your kids, grandkids, etc, will
inherit the Starbucks plan, and that the appropriate discount rate is
0.02% per day.
With constant price, we have
PV0 =
1.75
0.0002 = $8, 750
Class #3: Annuities and Perpetuities Financial Analysis
Quick exercise
1. An ’Americano’ coffee in size ’tall’ at Starbucks costs $1.75. How
much should cost a coffee plan that allows you to have an
Americano every day forever? Suppose your kids, grandkids, etc, will
inherit the Starbucks plan, and that the appropriate discount rate is
0.02% per day.
With constant price, we have
PV0 =
1.75
0.0002 = $8, 750
Class #3: Annuities and Perpetuities Financial Analysis
Annuities
• The second type of cash flow stream for which we have a simple PV
computation is the annuity
• Unlike a perpetuity, payments stop after T periods
• An annuity is a series of payments of the same amount for T years
If the interest rate is 10% per period, what is the value of the annuity
that pays $5 per period for 10 periods?
• One could simple sum the present value of all cash flows
• But there is a quick (and thus easier) way to do it
Class #3: Annuities and Perpetuities Financial Analysis
Annuities
• The second type of cash flow stream for which we have a simple PV
computation is the annuity
• Unlike a perpetuity, payments stop after T periods
• An annuity is a series of payments of the same amount for T years
If the interest rate is 10% per period, what is the value of the annuity
that pays$5 per period for 10 periods?
• One could simple sum the present value of all cash flows
• But there is a quick (and thus easier) way to do it
Class #3: Annuities and Perpetuities Financial Analysis
Annuities
• The second type of cash flow stream for which we have a simple PV
computation is the annuity
• Unlike a perpetuity, payments stop after T periods
• An annuity is a series of payments of the same amount for T years
If the interest rate is 10% per period, what is the value of the annuity
that pays $5 per period for 10 periods?
• One could simple sum the present value of all cash flows
• But there is a quick (and thus easier) way to do it
Class #3: Annuities and Perpetuities Financial Analysis
The annuity formula
• A stream of constant equal cash flows beggining next period (time
1), lasting for T periods, and discounted at rate r , is worth
PV0 =
C1
r
[
1− 1
(1 + r)T
]
• This is the annuity formula.
• Where does it come from??
Class #3: Annuities and Perpetuities Financial Analysis
The annuity formula
• A stream of constant equal cash flows beggining next period (time
1), lasting for T periods, and discounted at rate r , is worth
PV0 =
C1
r
[
1− 1
(1 + r)T
]
• This is the annuity formula.
• Where does it come from??
Class #3: Annuities and Perpetuities Financial Analysis
The annuity as the subtraction of perpetuities
• Notice you can think of an annuity with T periods as
• An infinite series of payments starting next period...
• Minus another infinite series of payments starting at period T + 1!!
• That is,
Annuity = C1r︸︷︷︸
X
−
(C1
r
)
︸ ︷︷ ︸
Y
1
(1 + r)T︸ ︷︷ ︸
Z
• X = Perpetuity starting next period
• Y = Perpetuity starting at T + 1 (from the perspective of period T )
• Z = Discount factor to bring period T amounts to today’s dollars
• You should draw the cash flows to fix ideas!
Class #3: Annuities and Perpetuities Financial Analysis
The annuity as the subtraction of perpetuities
• Notice you can think of an annuity with T periods as
• An infinite series of payments starting next period...
• Minus another infinite series of payments starting at period T + 1!!
• That is,
Annuity = C1r︸︷︷︸
X
−
(C1
r
)
︸ ︷︷ ︸
Y
1
(1 + r)T︸ ︷︷ ︸
Z
• X = Perpetuity starting next period
• Y = Perpetuity starting at T + 1 (from the perspective of period T )
• Z = Discount factor to bring period T amounts to today’s dollars
• You should draw the cash flows to fix ideas!
Class #3: Annuities and Perpetuities Financial Analysis
The annuity as the subtraction of perpetuities
• Notice you can think of an annuity with T periods as
• An infinite series of payments starting next period...
• Minus another infinite series of payments starting at period T + 1!!
• That is,
Annuity = C1r︸︷︷︸
X
−
(C1
r
)
︸ ︷︷ ︸
Y
1
(1 + r)T︸ ︷︷ ︸
Z
• X = Perpetuity starting next period
• Y = Perpetuity starting at T + 1 (from the perspective of period T )
• Z = Discount factor to bring period T amounts to today’s dollars
• You should draw the cash flows to fix ideas!
Class #3: Annuities and Perpetuities Financial Analysis
The annuity formula
• Back to that question...
If the interest rate is 10% per period, what is the value of the annuity
that pays $5 per period for 10 periods?
PV0 =
C1
r
[
1− 1
(1 + r)T
]
= $50.1
[
1− 1
(1.1)10
]
= $30.72
Class #3: Annuities and Perpetuities Financial Analysis
The annuity formula
• Back to that question...
If the interest rate is 10% per period, what is the value of the annuity
that pays $5 per period for 10 periods?
PV0 =
C1
r
[
1− 1
(1 + r)T
]
= $50.1
[
1− 1
(1.1)10
]
= $30.72
Class #3: Annuities and Perpetuities Financial Analysis
Application: Fixed-rate mortgage payments
• The annuity formula is commonly used, since mortgage and several
loans have fixed payments
• A 30-year mortgage with monthly payments is a 360-payment
annuity
What you be your monthly payment if you signed a 30-year mortgage
loan for $500,000 at a quoted interest rate of 7.5% per annum?
• The true monthy interest rate is
r = 7.5%12 = 0.625%
(From the last class, remember the effective annual interest rate would
be
(
1 + 7.5%12
)12
− 1 = 8.08%)
Class #3: Annuities and Perpetuities Financial Analysis
Application: Fixed-rate mortgage payments
• The annuity formula is commonly used, since mortgage and several
loans have fixed payments
• A 30-year mortgage with monthly payments is a 360-payment
annuity
What you be your monthly payment if you signed a 30-year mortgage
loan for $500,000 at a quoted interest rate of 7.5% per annum?
• The true monthy interest rate is
r = 7.5%12 = 0.625%
(From the last class, remember the effective annual interest rate would
be
(
1 + 7.5%12
)12
− 1 = 8.08%)
Class #3: Annuities and Perpetuities Financial Analysis
Application: Fixed-rate mortgage payments
• The annuity formula is commonly used, since mortgage and several
loans have fixed payments
• A 30-year mortgage with monthly payments is a 360-payment
annuity
What you be your monthly payment if you signed a 30-year mortgage
loan for $500,000 at a quoted interest rate of 7.5% per annum?
• The true monthy interest rate is
r = 7.5%12 = 0.625%
(From the last class, remember the effective annual interest rate would
be
(
1 + 7.5%12
)12
− 1 = 8.08%)
Class #3: Annuities and Perpetuities Financial Analysis
Application: Fixed-rate mortgage payments
• The annuity formula is commonly used, since mortgage and several
loans have fixed payments
• A 30-year mortgage with monthly payments is a 360-payment
annuity
What you be your monthly payment if you signed a 30-year mortgage
loan for $500,000 at a quoted interest rate of 7.5% per annum?
• The true monthy interest rate is
r = 7.5%12 = 0.625%
(From the last class, remember the effective annual interest rate would
be
(
1 + 7.5%12
)12
− 1 = 8.08%)
Class #3: Annuities and Perpetuities Financial Analysis
Application: Fixed-rate mortgage payments
• $500,000 is the PV0
• T = 360
• We want to compute C1
500, 000 = C10.625%
[
1− 1
(1 + 0.625%)360
]
≈ C1 × 143.018
C1 ≈ $3, 496.07
• You would have to pay $3,496.07 dollars every month starting next
month.
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• What if you have computed a present value assuming an infinite
stream of payments, but in reality there were only T payments?
• How far off were your computations?
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• What if you have computed a present value assuming an infinite
stream of payments, but in reality there were only T payments?
• How far off were your computations?
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• Imagine a firm is expected to have profits of $3 million forever,
starting next year
• The discount rate is 13% per year
• We can use perpetuities to estimate the value of the firm:
PV0 =
$3million
0.13 = $23.07million
• For how many years should the firm exist for the firm value
estimated using the annuity formula to be 95% of the value using
perpetuities?
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• Imagine a firm is expected to have profits of $3 million forever,
starting next year
• The discount rate is 13% per year
• We can use perpetuities to estimate the value of the firm:
PV0 =
$3million
0.13 = $23.07million
• For how many years should the firm exist for the firm value
estimated using the annuity formula to be 95% of the value using
perpetuities?
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• Imagine a firm is expected to have profits of $3 million forever,
starting next year
• The discount rate is 13% per year
• We can use perpetuities to estimate the value of the firm:
PV0 =
$3million
0.13 = $23.07million
• For how many years should the firm exist for the firm value
estimated using the annuityformula to be 95% of the value using
perpetuities?
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• Assuming the firm will exist for only T years, and not forever, we
have:
PV0 =
3
0.13
[
1− 11.13T
]
• For PV0 to be 95% of 30 million dollars, we would have
PV0 = 0.95×
3
0.13 =
3
0.13
[
1− 11.13T
]
0.95 =
[
1− 11.13T
]
=⇒ T = 24, 5 years
• It means that even if the firm only exists for about 25 years, we
would still have a pretty good approximation using perpetuities
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• Assuming the firm will exist for only T years, and not forever, we
have:
PV0 =
3
0.13
[
1− 11.13T
]
• For PV0 to be 95% of 30 million dollars, we would have
PV0 = 0.95×
3
0.13 =
3
0.13
[
1− 11.13T
]
0.95 =
[
1− 11.13T
]
=⇒ T = 24, 5 years
• It means that even if the firm only exists for about 25 years, we
would still have a pretty good approximation using perpetuities
Class #3: Annuities and Perpetuities Financial Analysis
Comparing perpetuities to annuities
• Assuming the firm will exist for only T years, and not forever, we
have:
PV0 =
3
0.13
[
1− 11.13T
]
• For PV0 to be 95% of 30 million dollars, we would have
PV0 = 0.95×
3
0.13 =
3
0.13
[
1− 11.13T
]
0.95 =
[
1− 11.13T
]
=⇒ T = 24, 5 years
• It means that even if the firm only exists for about 25 years, we
would still have a pretty good approximation using perpetuities
Class #3: Annuities and Perpetuities Financial Analysis