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BUS 1300: Principles of Finance
Principles of Finance
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BUS 1300: Principles of Finance
Time Value of Money:Multi-Period Cash Flows
Overview of Class
• What happens if we have multiple periods
between cash flows?
• How do we handle multiple payments?
• Practical examples
2
Perpetuities
• A level perpetuity represents a set of equal payments that last indefinitely (i.e. to
infinity)
• How do we find the PV of an infinite series of cash flows?!?!?!
• The easiest way to think about it is to ask the following question. How much do I
need to invest today in order to pay myself $100 a year forever if the first payment
starts a year from today and interest rates are expected to remain constant at 10%
per year?
– If I invest $1,000 today, the investment will be worth $1,000(1.1), or $1,100
next year. I can pay myself $100, reinvest the $1,000, and do this each year,
thereby generating a cash flow of $100 per year forever. Since this is the same
pattern as the cash flows I am valuing, the PV of the perpetuity is $1,000 since
identical assets with identical cash flows and risks must sell for the same value
• PV0 = c/r
where c is the first cash flow in year one. The cash flow and the interest rate
must remain constant
6
In class problem #2
• Suppose Vanderbilt is considering endowing a chair. If the
holder of the chair is to receive $30,000 per year from the
endowment (beginning one year from now), how large a
donation is required if interest rates are expected to remain at
10%?
PV0 = $30,000/.10 = $300,000
7
Growing Perpetuities
• The present value of this set of cash flows is
• where C1 is the cash flow at date 1, g is the constant growth rate in cash
flows, and r is the constant opportunity cost of capital
• If the university now expects the salaries of the chair holder to increase by
5% per year, how much is the required endowment?
PV0 = $30,000/(.10-.05) = $600,000
8
gr
C
−
= 10 PV
C1 C1(1+g) C1(1+g)
2 C1(1+g)
3 …
Payment ↑ ↑ ↑ ↑ …
date 0 1 2 3 4 …
WARNINGS!
• (1) r>g. Perpetuity formula uses long run equilibrium values, which are not likely to
hold over the early years of a project
• (2) The present value is based on the assumption that the first payment occurs one
period after the date of valuation. Are you getting a break when the first car
payment is due one month after you drive the car off the lot?
• Consider a perpetuity that makes its first payment today of $2.00. The payment
grows at 4% per year and r=10%. What is the PV of this cash flow?
• PV0 = $2.00 + $2.00(1.04)/(.10-.04) = $36.67
• Note that the first payment in year one incorporates the growth in the cash flow
from $2.00 in year 0 to $2(1.04) or $2.08 in year 1
• Another way to solve this is to think of yourself as standing at time t-1 and using
the payment of $2 at t=0 (present) as the first payment the carry the amount
forward to today (hint: play close attention to the subscripts):
– Step1: PVt-1 = 2 / (0.10 – 0.04) = $33.33
– Step 2: PV0 = PVt-1 * (1 + 0.10) = $36.67
9
Perpetuity in the future
• You start looking into retirement plans and you’re convinced
that you’re going to live forever. You have the opportunity to
purchase a perpetuity that will pay $10k forever and starts
paying in 60 years. The discount rate is 5%. How much are you
willing to pay for this perpetuity?
59 60 61 62 63 ….
|----------------|----------------|--------------|----------------|
10,000 10,000 10,000 10,000 ….
The words forever indicate that it’s a perpetuity. There is no mention of a growth rate so we can safely assume it is a
level perpetuity (i.e. not a growing annuity). The problem is only tricky because those cash flows don’t start for another
60 years. We can think of this as a two step process. First, let’s condense the cash flows of the perpetuity into a single
value. Once we’ve accomplished that, we can then take that single cash flow and move it through time.
Value (perpetuity) = C / r = 10,000 / 0.05 = $200, 000
Thus, the perpetuity’s value is worth $200,000. But at what point in time is it worth $200,000. The perpetuity formula
assumes that the first payment is made one year from today. So if the first payment is at time 60, using the formula
gives us the value at time t=59. Once we know that, we can apply the simple time Present Value Formula.
PV = Ct / (1+r)
t = 200,000 / (1.05)^59 = $11,242.46
That says that we need $11k today to be able to fund the perpetuity that starts paying in 60 years time.
10
Mixing Perpetuities & Single Cash Flows
What happens when there are single cash flows
and a perpetuity?
• This is common when we forecast company cash flows, but at some point we
assume a slower, steady growth rate
• Cash flow in Year 1: $50
• Cash flow in Year 2: $100
• Cash flow in Year 3: $150
• Every year thereafter, cash flows will grow at 4% forever
• How much is this string of cash flows worth today if the discount rate is 8%?
See next slide for explanation
11
Mixing Perpetuities & Single Cash Flows
• Cash flow in Year 1: $50
• Cash flow in Year 2: $100
• Cash flow in Year 3: $150
• Every year thereafter, cash flows will grow at 4% forever
• How much is this string of cash flows worth today if the discount rate is 8%?
We can deal with the single cash flows individually then add the perpetuity with
growth
Present value = 50/1.081 + 100/1.082 + 150/1.083 + [(150*1.04)/(.08-.04]/1.083
Present value = $3,347.05
Note, we treat the individual cash flows just like before. When dealing with the
perpetuity with growth, we account for the year 4 cash flow by growing the year cash
flow by the growth rate. Because the first cash flow of the growing perpetuity occurs
in year 4, we discount back by 3 periods. Recall the application of the perpetuity
formula assumes the first cash flow is one period from today, so use the formula gives
us the value at year 3, so we have to discount it back by an additional 3 periods.
12
Level Annuities
• Annuities are a set of equal cash flows that are received up to and
including a terminal date T. The generalize present value formula is
• Level annuities have the following characteristics
– Equal cash flows such that C1 = C2 = C3 … = CT
– Discount rate stays constant over the life of the annuity. Note that there is no
time subscript on the discount rate
– Cash flows have a defined terminal date T years from the date the annuity is
valued. This is in contrast to a level perpetuity whose cash flows are infinite
13
)r+(1
C
+ ... +
)r+(1
C
+
)r+(1
C
+
r+1
C
= PV T
T
3
3
2
21
0
Annuity Valuation
• Rather than make T individual calculations to arrive at the sum of the
present value of the cash flows for an annuity, we can derive a single
formula that uses C, r and T as inputs. The following represents an
expanded discussion of pages 28-29
• The goal is to find the sum of the present value of all cash flows beginning
at date 1 and ending at date T.
1 2 3 T-1 T T+1 T+2 ...
C C C C C 0 0
• As a starting point, we know that the present value of cash flows that
begin at date 1 and extend to infinity is C/r.
1 2 3 T-1 T T+1 T+2 ...
C C C C C C C
14
• However, C/r is too large relative to the present value of the
annuity.
• The cash flows for an annuity stop after year T, whereas the
perpetuity has cash flows that extend to infinity. Thus, the
perpetuity value of C/r overstates the value of the annuity by
the present value of all the cash flows from years T+1 to
infinity
• Thus, we can value the annuity by first valuing the level
perpetuity whose cash flows extend to infinity, and then
subtract the present value of the cash flows from years T+1 to
infinity, leaving us with all the cash flows from years 1 through
year T
15
Intuition for this approach
You go to Chile’s for lunch and they have two
offers:
1) Entrée, side, and a drink ($15)
2) Entrée and a drink ($10)
What does that imply about the price of a side?
Well given the only difference between the two is the side, you might infer
that the value of the side is worth $5. If it wasn’t then you could recreate the
meal at different prices. Relating this analogy, we are trying to isolate a series
of payments (our side) by utilizing our knowledge of the prices for two
different perpetuities (offers 1 and 2).
16
• Define P1 as the level perpetuity whose cash flows
begin in year 1, and P2 as the level perpetuity whose
cash flows begin in year T+1. By subtracting the cash
flows of P2 from P1, the remaining cash flows will
begin in year 1 and end in year T
17
1 2 3 T-1 T T+1 T+2 ...
P1 C C C C C C C
P2 0 0 0 0 0 C C
P1 – P2 = A C C C C C 0 0
• What is the present value of P2, defined as PV0(P2)
• Where along the timeline for P2 does it have the
value C/r?
• Recall that the present value is calculated one period
prior to the first payment. Since the first cash flow is
at date T+1, P2 must have value C/r at date T. This
value then needs to be discounted back T periods for
find PV0(P2)
18
Annuity Formula Derivation
19
• PV0(P1) =
• PV0(P2) =
• PV(A) = - =
r
C
)r+(1
r
C
T
1
r
C
)r+(1
r
C
T
1
rrr
C
T
+
−
)1(
11
The second term is adjustment to perpetuity formula
Why go through the derivation?
• Example of no arbitrage: if we can create the same
cash flows using two different products then the
must have the same price.
• What happens if you can buy a side at Chile’s for $4?
• A very simple look into the financial engineering that
is dominating finance
20
Generalized annuities
• General formula for annuity
– n periods, payment C, constant interest rate r:
• WARNING: Once again, the first payment is at date
1 not at date 0
• How about the NPV for our factory?
21
740,886$
)14.1(14.
1
14.
1
$170,000PV
100
=
−= $210,0 0 $611,879.58
4
Growing annuity
• What happens if the annuity is set to grow at
a constant rate?
• Can go through same derivation as before
since we know the present value of a growing
perpetuity.
• PV(Growing Annuity) =
−
1 −
1+
1+
22
Annuities example #3
If someone wanted to sell you a financial product that pays $100
a month for the next 5 years, what is the maximum you would be
willing to pay for the product? Assume the monthly interest rate
is 1%. What is the maximum you would pay if the payment grew
at a rate of 0.5%?
This is an annuity (fixed payment, finite time)
5 years of monthly payments = 5 * 12 = 60 payments
total
PV = 100 * [(1/0.1) – (1/ (0.01 * (1.01)60)]=$4,495.50
Growing perpetuity
PV = (100/(0.01 – 0.005))* [1 – (1.005/1.001)60]=$5,150.52
23
Annuity in the future
Suppose you want to buy an annuity today that will start paying
$1,000/year in 5 years. What would you pay today for an annuity
that pays out every year for 10 years with the first payment in 5
years? Assume an annual interest rate of 8%.
We’re back in a world where we have cash flows that occur at some point in the future. The problem tells us
everything we need (including that this is an annuity), but we need to use a timeline so we understand how to
get it into today’s dollars.
5 6 7 …. 14 15
|----------------|----------------|--------------|----------------|
1,000 1,000 1,000 … 1,000 1,000
Let’s take it as a two step approach again. First let’s find the value of the annuity.
Value (Annuity) = C * [(1/r) – (1/(r*(1+r)^T))] = 1,000 * [(1/0.08) – (1/(0.08*(1+0.08)^10))] = $6,710.08
Note, the value of T in this equation is capturing the number of payments, so we use 10. Again, the formula for
the annuity assumes that the first payment is being made one year from today. Thus when we apply this
formula it is giving us the value as of (5-1)=4. We converted the string of cash flows into a single cash flow so
we can return to the present value formula.
PV = Ct / (1+r)t = $6,710.08 / (1.08)^4 = $4,932.11
This value represents how much money you would have (or need to spend) to create an annuity that pays
$1,000 a year for 10 years, with the first payment starting 5 years from today. The most frequently missed
areas are: 1) why we use 10 in the value of the annuity formula and 2) why we use 4 when discounting that
value back. Make sure you are comfortable with these. 24
In class problem #4
• (BMA, Chapter 2, question 19) As a winner of a breakfast cereal
competition, you can choose one of the following prizes:
(a) $100,000 now
(b) $180,000 at the end of five years
(c) $11,400 a year forever beginning one year from now
(d) $19,000 for each of 10 years with the first payment in one year
(e) $6,500 next year and increasing thereafter by 5% a year
forever
If the interest rate is 12%, which is the most valuable prize?
(a) $100,000
(b) $180,000/(1.12)5 = $102,137
(c) $11,400/.12 = $95,000
(d)
(e) $6,500/(.12-.05) = $92,857
25
In class problem #5
• (a) The Whitt’s are planning for the college education of their newborn son, Dim.
Mr. and Mrs. Whitt estimate that college expenses will run $50,000 per year when
their son (hopefully) reaches college in 18 years. The annual interest rate is
expected to remain at 4% during the next few decades. How much money must
they deposit in the bank each year (beginning on Dim's 1'st birthday and ending
when he turns 17 so that their son will be supported through 4 years of college?
Assume the first tuition payment is due on Dim's 18'th birthday.
0 1 2 3 4 ... 16 17 18 19 20 21
--------------------------------------------------------
x x x x x x -50K -50K -50K -50K
• Let’s first determine the PV of the cash outflows. The annuity formula assumes that the first
payment is made one year after the date the annuity is valued. Therefore, the level annuity
formula produces the total discounted value of his education as of Dim's 17th birthday since
the first tuition payment is made on his 18th birthday.
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