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Lecture 4: Financial Math
& Cash Flow Valuation II
C. L. Mattoli
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
1
Intro




Last week we looked at the concept of time
value of money.
The basic idea that you should have carried
away is that any money has different values
at different times.
Last week, we began slowly and considered
the case of only one cash flow.
There are many situations, investments and
securities that promise to pay a bunch of
cash flows in different years.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
2
Intro



In this lecture, we will continue our
discussion and analysis of time value, and
we will apply those concepts to these
more complicated situations.
We already have developed the basic
equation for time valuation.
In this lecture we will use that equation to
form more complicated aggregate
equations to value aggregated cash flows.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
3
Intro


We will also study some more theory and
discuss the additional assumptions that we
must make for this next step in time
valuation.
This will form a major basis for everything
else that we shall study from here on, so it
is crucial that you understand and can use
these concepts and equations.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
4
Rate of return: Interlude



A rate of return is a specific example of a
growth rate.
A growth rate is simply the percentage
change (%Δ) of anything, A: %ΔA = (A1 –
A0)/A0.
In finance time value, the growth rate is over
time, and we usually annualize it and call it
the annual rate of return: r = [(FV – PV)/PV]/n
where FV is PV’s value in the future, and we
divide by n because the growth term is a
HPR.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
5
Rate of return: Interlude




From that definition of return, the PV/FV
equation follows by solving for PV or FV: FV
= PV(1 + nr)
It answers the question how will an amount
grow through simple interest.
In compound interest, we implicitly assume
that intermediate cash flows are reinvested.
Thus, under compound interest we must
reinvest any cash flow that we get from the
investment, during our holding period (HP).
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
6
Rate of return: Interlude


The simple example is a savings account
at a bank where you are paid periodic
interest on the money, and you leave the
whole thing in the account.
In that case, your money is automatically
reinvested. In other cases, it is an implicit
assumption, as we shall discuss toward
the end of the lecture: the reinvestment
assumption.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
7
Rate of return: Interlude



The further questions are why will it grow,
and why do we need it to grow?
It will grow because we rent it out to other
people as debt or equity for their business.
We want it to grow because we give up
consumption; inflation erodes buying
power; and we want to earn money and
stay ahead of inflation when we do our
deferred spending.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
8
PV/FV for Multiple CF’s
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
9
Intro




PV/FV is one of the most important
concepts in finance, and, you will have to
admit, it does not seem too complicated.
The next step is to be able to find FV/PV
for a stream of cash flows, instead of just
one.
The step is simple, but it involves some
technical issues.
We begin by looking at FV, as we did in
the case of single cash flows, but first …
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
10
Time lines: a Pictorial device



A time line is a diagrammatic representation of cash
flows, either received or paid, or both.
It gives you a way to picture the process.
PV
FV1
FV2
0
1
2
This diagram relates a present value to two future
values. The FV’s could also represent payments in
future years.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
11
FV for multiple CF’s


Suppose that we make a $100 deposit, right
now, to a bank account earning 10%/yr., a
second $100 deposit, a year from now, and
we keep the money in bank til 5 years from
now.
Then, we can break it down into FV1 =
$100(1+10%)5 = $161.05, for the first deposit,
and FV2 = $100(1+10%)4 = $146.41, for the
second, since it will be in bank for only 4
years.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
12
FV for multiple CF’s



The total FV is just FV1+FV2 = $307.46, five
years from now.
An example of a situation in which payments
are made into an investment account year
after year, is contributions to a retirement
fund. Then, the future value would be the
money that you have at the end of the
extended period of years for your retirement.
We can make a general formula for cash
flows invested in year n, and held in the
account til year m.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
13
FV for multiple CF’s



For each cash flow that you put into an
investment account, n years from now, and
keep in the account til m years from now will
earn return for m – n years.
So, the future value of each of those cash
flows will be FVn(1 + r)(m-n).
Then, your total future value will be the sum
of those valued in year m, and we can write a
complicated-looking equation, but not really
complicated, as follows …
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
14
FV for multiple CF’s


The general formula for cash flows, CFm,
invested in year m, and held in the account til
year n as:
The symbol, , is used to denote the sum of
the objects to its right, indexed by m, over the
specified range of m, in this case, m = 0, 1,2,
…,n.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
15
FV for multiple CF’s

Thus, for example, with n = 2 and m = 0 to 2,
we have, written out in full
= CF0(1+r)(2-0) + CF1(1+r)(2-1) +CF2(1+r)(2-2)
= CF0(1+r)2 + CF1(1+r)1 +CF2

Get used to this notation because it will be
used throughout the course.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
16
FV for multiple CF’s




The formula looks complicated, but you
should, instead, focus on the meaning of FV
of a cash flow.
FVn = PV(1 + r)n means that PV is invested
for n years at r rate of return.
When we look at multiple cash flows, we
have to figure out how long each one is
invested, which can be tricky.
Again, we can make it clearer, if we use time
lines.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
17
2-dimensional time lines and FV



The use of time lines can be really helpful in
looking at situations of PV and FV, and you
should use them to organize your thoughts
and equations.
For FV, we use a 2-dimensional time line, as
shown below, to help you better understand
what is happening with FV of multiple cash
flows.
Basically, each deposit grows from the time
it is put in the investment account til the
end of the investment horizon.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
18
Two-dimensional time line
Cash Flows, C, are received at the times shown, and the grey cells
represent the years that each cash flow receives interest. FV(5 years
of deposits, Cm , start year0) = C0(1+k)4 + C1(1+k)3+ C2(1+k)2 + C3(1+k)1 +
C4(1+k)0
0
1
2
3
4
Totals
C0 C0(1 + k) C0(1 + k)2 C0(1 + k)3
C0(1 + k)4
C0(1 + k)4
C1(1 + k)2
C1(1 + k)3
C1(1 + k)3
C2(1 + k)2
C2(1 + k)2
C1 C1(1 + k)
C2 C2(1 + k)
C3 C3(1 + k)
C4
C3(1 + k)
C4
Sum
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
19
MCF FV example



Let us look at a detailed example of FV of
MCF’s.
Assume that you deposit into an investment
account $100, now; $200, in 1 year; and
$300, 2 years from now. How much will you
have in the account, 3 years from now, if you
earn 10%/year, on your investment?
Note: the first deposit will earn interest for 3
years, the second will earn interest for 2
years, and the third will earn interest for 1
year.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
20
MCF FV example

Thus, the total that you will have in the
account at the end of 3 years will be the
sum of: $100(1 + 10%)3 = $133.10,
$200(1 + 10%)2 = $242, and $300(1 +
10%) = $330, for a total of $705.10.
 That also means that, over the period,
you have earned $705.1 –
($100+$200+$300) = $105.10 in total
interest income.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
21
Annuities: a special MCF case



0
A special case of multiple cash flows is
annuity payments.
An annuity is a stream of cash flows, A,
that are all equal and evenly spaced in
time, e.g., yearly, quarterly, monthly, etc.
We show an n-year annuity in a time line,
here.
$A
$A
$A
$A
1
2
3
4
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
$A
…
n
22
Annuities: a special MCF case



There are annuities sold by insurance
companies.
More importantly, this type of even cash
flow is actually useful, also, for valuing
certain other types of securities, which we
shall see in the next lecture.
By tradition, an ordinary annuity CF
stream begins one period in the future,
as is shown in the previous slide.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
23
Annuities: a special MCF case

Happily, there is a compact equation for the
future value of an annuity, FVA.


 (1  k ) n  1 
FVA  A 

k



This type of equation, with compounding is of
the class called geometric equations, and the
method that leads to the compact equation is
actually quite simple.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
24
PV of Multiple CF’s
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
25
PV of MCF’s


Since we can find PV for any future CF,
and PV is right now, time = n = 0, then, if we
have PV’s for a bunch of future CF’s, the
PV of the sum of the CF’s is the sum of all
of the PV’s of those CF’s.
The general formula for a stream of CF’s,
CFi, discounted at rate, k, is given by
n
PV  
i0
n
CF i
(1  k)
i
  CF i (1  k)
i
i0
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
26
PV of MCF’s




Notice that we have begun our summation at
i = 0. which is more general than the usual
equation that you might see.
However, some investment instruments, for
example, a so-called annuity due, begin
right now.
Another example is the value of a stock that
you buy today and it just happens to be
paying a dividend, today.
In the equation, CF0/(1+k)0 = CF0 since
anything to the “power” 0 is equal to 1.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
27
MCF PV example



Assume that we will get payments: $100, one
year from now, $300, 3 years from now, and
$500, 6 years from now, and our required rate of
return (RRR) is 10%.
Then, the PV is PV = $100(1+10%)-1 +
$300(1+10%)-3 + $500(1+10%)-6 = $90.91 +
$225.39 + $282.24 = $598.54.
What that means is: if someone offers you a
right to receive those payments, and you want to
earn 10% on your money over the years, then,
you should pay, at most, $598.54 for that right.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
28
MCF PV example


That is the true use of PV: to find out
what you should pay for an investment,
like stocks, bonds, real estate, or any
business investment (intrinsic value).
You get future cash flows for giving out
money, in the present, and you want to
know how much to pay, if you want to
earn a certain rate of return on
investment.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
29
The Reinvestment assumption in PV



There is one technical issue in PV MCF,
which is sometimes conceptually difficult for
students to grasp, but it is not too difficult to
understand, if you try.
If you will receive just one future CF, FV, and
you want to earn a certain APR, k,
compounded annually on your initial
investment for n years, then, you should pay
PV = FV/(1+k)n.
In that way FV = PV(1+k)n , so FV is PV
getting interest, k, compounded over n years.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
30
The Reinvestment assumption in PV


Indeed, compounding is all about the
situation in which you continue to reinvest
the interest, return, or cash flows that you
get during each interest period (yearly,
monthly, etc.).
Thus, there is already a reinvestment
assumption, in the case that we have only
one initial cash flow, PV, that grows into
FV by compounding, or reinvesting the
intermediate period returns.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
31
The Reinvestment assumption in PV




Now, consider two CF’s, $10, paid 1 year
from now, and $110, paid 2 years from
now.
Assume that your RRR is 10%, then, PV =
$10/(1+10%) + $110/(1+10%) = $100.
So, you invest $100 and get $110+$10 =
$120 over the 2 years.
However, if you take PV(1+k)n =
$100(1+10%)2 = $121, not $120.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
32
The Reinvestment assumption in PV



We said that we wanted to earn 10% on our
initial investment, and we discounted the
future CF’s to get PV, the investment we
should make to get our RRR, but it appears
that we didn’t get that.
The problem is, actually, that everything
needs to be valued at the same point in
time, in finance, when adding values.
However, we added $10 in year 2 to $110
in year 3 to get $120 total, but values in year
2 and 3 are not on an equal footing.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
33
The Reinvestment assumption in PV



We have to make an adjustment: value year 2
CF, in year 3, by pushing it forward as FV($10 in
year 2) = $10(1+10%) = $11, in year 3.
Then, we will have $11+$110 = $121, and we
will have truly earned 10% per year
compounded for 2 years on our initial investment
of $100.
What it means is that you have to reinvest
intermediate (in the middle of your holding
period) CF’s at the same interest rate and let
them also earn that rate over the life of the
investment.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
34
The Reinvestment assumption in PV



That is commonly known as the
reinvestment assumption in PV MCF’s.
We said that PV and FV are just flip sides of
the same concept, and the reinvestment
assumption in the case of compounding and
MCF’s is what makes FV and PV of MCF’s
correspond, in the same way as PV and FV,
in the case of only one CF.
It is a simple matter of taking PV(1+k)n =
CF1(1+k)n-1 + CF2(1+k)n-2 + … + CFn-1(1+k)
+CFn = FV.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
35
The Reinvestment assumption in PV



That equation is, then, just the FV of all of
the CF’s valued at the end of the holding
period.
Moreover, it is not unreasonable to require
that cash flows coming over the years
should be reinvested. It would be stupid
not to reinvest, if we were investing, in the
first place.
Thus, the reinvestment assumption is not
only necessary, but it also makes good
business investment sense.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
36
The real point: one time



We usually talk about PV, the value
right now, or some FV.
The real point is that, in finance, we
realize that money has a time value.
Because of that, if we are to value
things and we want to compare their
values, then, they all have to be
valued at the same time.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
37
The real point: one time



We could value them all, 3 years from
now, for example, if we wanted to.
Moreover, if we had all CF’s valued 3
years from now, we could just add them
up, and discount the sum back to now,
PV, by dividing that summed number by
the PV factor for 3 years from now.
Or we could push them forward into
another future year.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
38
The real point: one time



We just need to have them all valued in
the same year.
In that manner, for example, if we are
offered different cash flows in different
future years, we can compare them, in
the present, by looking at their present
values.
A simple example of that is consider two
choices: $100, now, or $110, a year from
now. Which is a better offer?
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
39
The real point: one time




Answer: if you can make more than 10% on your
money by investing it for a year, you will have
more than $110.
Thus, which is better, in this case, depends on
your opportunity rate of return, which you can
also use to find out the PV of the $110, a year
from now.
The other thing that we can do with the scheme
is to compare values of the same cash flow at
different times.
Then, we are looking at the question: what is or
was the growth rate of that cash flow over time,
and we call that the rate of return.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
40
Annuity PV


Again, we can discuss annuities, which are
just the same amount of CF paid for a
number of periods, evenly spaced in time.
Thus, if the first payment will come in 1 year
from now and the last comes n years from
now, we have:
PV = A/(1+r)1 + A/(1+r)2 + … + A/(1+r)n
n
 
i0
n
A
(1  k)
i
 A
(1  k)
i
i0
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
41
PV annuities

Fortunately, again, geometric math allows us
to get a compact reduced equation for the PV
of even, evenly spaced in time, CF’s, A, for n
years, present value of annuity: PVA.


 1  (1  k)  n 
PVA  A 

k


PVA
$A
$A
$A
$A
0
1
2
3
4
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
$A
…
n
42
PV annuities


The other fortunate thing is that annuities
can be found either pure or as part of the
CF streams from actual investment
instruments.
In addition to annuities sold by financial
companies, coupon bond debt securities
pay a coupon interest payment that is the
same dollar amount, period after period, in
addition to the final payment of principal.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
43
PV annuities


So, part of the valuation of a coupon
bond will involve the PVA of coupons.
The other part will be the discounted value
of the final payment.
By having a simplified formula we will have
a simple way to figure out the price that we
should pay for those future even cash
flows, PV, given our own RRR =k.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
44
PV annuities




Consider an example. A lottery contest
promises to pay a $5,000,000 prize.
That sum will actually be paid out over 20
years with equally-divided annual
payments.
Thus, the prize will pay you $250,000 a
year annuity payments for 20 years.
Not a bad annual income, but are you
really getting $5,000,000?
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
45
PV annuities



As students of finance who understand that
money has time value, we are not that easily
fooled.
If our opportunity rate for investment is
6%, then we know that the prize is actually
worth PVA, right now.
Putting in the numbers, we find

 1  (1  6%)
PVA  $250,000 
6%


 20
  $2,867,480


.31
Just over half the advertized value.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
46
Annuity language.




As we mentioned earlier, annuity payments that
begin one year from now are traditionally called
ordinary annuities.
If the CF’s begin right now, time = 0, we call it an
annuity due.
If payments begin, m years from now, and
payments are made for n years, it is called an nyear deferred annuity with deferment m – 1
years after ordinary.
We call an infinite annuity payment stream that
begins 1 year from now a (ordinary) perpetuity.
(C) 2008-2009 Red Hill Capital Corp.,
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47
Annuity relationships


Then, for example, we can use our PVA
ordinary to get the right values for the others.
An n-year annuity due is like a payment of
A, right now, and n-1 payments in future
years, so we can write


 1  (1  k)  ( n -1) 
PVAD  A  A 

k


PVA+A
0
1
A
A
2
A
3
A
4
A
5
A
6
A
A
7 …… n – 1 .... n
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
48
Annuity Relations



The ordinary annuity equation just gives a
value of a bunch of cash flows, one year
before the first cash flow.
Although we can use the equation to find PV,
if the first CF is 1 year from now, the
equation doesn’t really know what time it
is.
Equations just have rules and inputs. Put n,
A, and k, into the PVA, and it values the n
CF’s one year before they began.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
49
Annuity Relations




Then, for example, we could even use
ordinary PVA to get annuity due, in another
way.
Take PVA for n payments, beginning right
now, and it will give a value one year
before now (t= -1).
Next push that forward from a year ago to
now as PVA(1+k).
Thus, another way to write PV annuity due is
PVAD = PVA(1+k).
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
50
Annuity Relations




If you take the time, you can see that the two
equations for annuity due are the same in the
end (they have to be).
We can also use this n-period annuity
equation to value an infinite number of equal
payments.
Interestingly, the value is finite, not infinite.
To value a perpetuity payment stream that
begins one period from now, we can use
PVA, again.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
51
Annuity Relations



PVP, present value (ordinary) perpetuity,
is just the limiting case of PVA, as n goes to
infinity.
Looking back at the PVA equation, as n  ,
the term (1+k)-n = 1/(1+k)n, goes to zero, so
from the PVA equation, we can write: PVP =
A/k.
Note also that we refer to this as an ordinary
perpetuity because even though infinity is
infinity, PVP gives a value for infinite
payments beginning 1 period from now.
(C) 2008-2009 Red Hill Capital Corp.,
Delaware, USA
52
Annuity Relations



If payments were, for example, to begin right
now, we would have to add a payment and
get the equivalent of annuity due for
perpetuity, or PVPD = A + PVP. It is an
important point to note.
For deferred annuities, we think in terms of
how many years after an ordinary annuity do
the payments begin.
So, if the first payment begins 4 years from
now, that is actually 4 – 1 = 3 years after an
ordinary annuity would begin.
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Annuity Relations



In general, we use the PVA for the right
number of payments.
That gives us a value one period
before the first payment.
You look at a time line to find the year
PVA gives a value, and you discount that
value back to time = 0 to get the present
value of a deferred annuity.
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Annuity Relations



Since it begins in year m, that will be m
– 1 years after an ordinary annuity.
PVA will give a value in year m – 1, one
year before the first payment, in year
m.
Thus PVA deferred to year m =
PVA/(1+k)m–1 to bring the PVA value in
year m – 1 back to time = 0, true PVA
deferred to year m.
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Annuity Relations
We show a simple pictorial example
description for annuity beginning 2
years from now, in the next slide.
 The value will be PVA deferred
payment year 2 = PVA(n-1
years)/(1+k) to bring the value back
one year to the present, t = 0.

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Deferred annuity pictures

1year deferred annuity using PVA
PVA
$A
$A
0
1
2
PVA defer 1
PVA(n-1)
$A
$A
$A
0
1
2
3
4
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$A
$A
n-1
$A
…
n
57
Installment loans



Another application of the annuity
concept is installment loans.
An installment loan is a loan that is paid
off in equal periodic installments:
annuity payments.
Thus, the value of the loan payments at
the time of borrowing is the PV of the
installment payments, discounted at the
interest rate that the banks wants to get
on the borrowings.
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Installment loans
PVA = amount borrowed, now.
 Typically, since we know how much
we want to borrow, and we know how
much interest the bank wants to
charge us, we can find the amount of
the installment payments, given the
length of time that we want to borrow
for.

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Installment example


Assume you want to borrow $10,000 for 5
years, and bank wants 8% interest on its
money.
We use the PVA equation, and solve for A:


 1  (1  k)  n 
A  PVA/ 

k



Thus,

 1  (1  8%)
A  $10,000/ 
8%

5
  $2,504.56


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Inside Installment Loans: Amortization



What is really happening in an installment
loan situation is this.
Each payment actually represents
interest owed on the outstanding loan
balance, plus some payment of
principal on the loan.
In that manner, each payment also
reduces the principal amount
outstanding.
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Inside Installment Loans: Amortization
Successive payments represent
less and less interest and more and
more principal payment, until the
last payment completely liquidates
the loan.
 We show a so-called loan
amortization schedule for the
example loan discussed, above, in
the slide, below.

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Inside Installment Loans: Amortization
The first year’s interest due to be paid
on $10,000 outstanding principal is
8%/yr x $10,000 = $800.
 The actual even annuity installment
payment that we figured is $2,504.56,
so that means that $1,704.56 =
$2,504.56 - $800 is applied to reduce
the principal to $10,000 – $1,704.56 =
$8,295.44.

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Loan Amortization Example


Over the life of the loan, total payments of
$12,522.82 are made.
Of that, we know the principal was $10,000, so
interest payments totaled $2,522.82.
Loan Amortization Table
Year
Payment
Interest
0
1
2
3
4
5 Totals
$2,504.56 $2,504.56 $2,504.56 $2,504.56 $2,504.56 $12,522.82
$800.00
$663.63
$516.36
$357.30
$185.52 $2,522.82
Principal
$1,704.56 $1,840.93 $1,988.20 $2,147.26 $2,319.04 $10,000.00
Principal
outstanding $10,000.00 $8,295.44 $6,454.51 $4,466.30 $2,319.04
$0.00
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Further notes on PVA


To find the number of payments, given,
PVA, A, and r, we can use the rule-ofthumb for doubling (from last lecture,
chapter 4), or we can use trial and error,
i.e., put in a guess, and see how close the
answer is, and try again.
The same applies to finding r, given PVA,
n, and A. Trial and error.
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PVA: the True Inverse of FVA



We want PV and FV to be inverse
operations of one another.
We saw, earlier, that there is an implicit
reinvestment assumption in the PV of
multiple CF’s to really get the compound
RRR that you want.
So assume that you pay PVA for an
annuity payment stream.
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PVA: the True Inverse of FVA



Then, take a factor of (1+k)n times PVA to
see what we get.
(1+k)n PVA = The future value of the
present value of n A payments = (1+k)nA[1
– (1+k)-n]/k = A[(1+k)n – 1]/k = FVA.
Thus, PVA and FVA are truly inverse
operations. If you pay PVA for an annuity
stream and you reinvest all of the
payments at the same discount rate, you
will have FVA at the end of the investment
horizon, in n years.
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Compounding more
than once a year
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Intro



There is no reason that we cannot compound
more than once a year.
Interest can be compounded semi-annually,
quarterly, monthly, weekly, daily, or
continuously, every nanosecond.
What we are looking at, mechanically, is, for
example, with semi-annual payments, half a
year’s interest gets paid into your account at
the end of 6 months, and you earn half a
year’s interest on the principal plus the first 6
month’s paid interest.
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Per Period Concepts.




To allow for compounding more than once
a year, we don’t need a new PV equation.
All we must do is to make some simple
adjustments to the way we use the
equations.
The equations don’t know that they are
using years for n or that we are using an
annual interest rate for k.
They are just equations with variables.
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Per Period Concepts.




We give meaning to the variables.
FV, for example, just takes a PV and
compounds a return on it n times.
Another way to use the equation is to
take r as a per period interest rate,
PPIR.
There is no a priori reason that n has to
represent years.
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Per Period Concepts.




N is just a set period, which could be years,
months, days, … , whatever.
We have to just a bit careful, here.
N and r are related by units, in the FV/PV
equation. N is a period of time, and r is the
interest rate that is earned over that period,
a PPIR
Then, for each period we compound
interest on a PV for n of those time-periods.
(or discount FV to PV) using the PPIR.
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Per Period Example




For example, assume that you are given an
annual interest rate, r, and interest is
compounded interest every month for 2
years.
First, find the PPIR = [r/yr.]/[12 months/year]
= a monthly rate of interest.
Then, compound that rate for 2 years x 12
months/year = 24 months (periods).
Take, for example, APR = 12%/year. Then,
PPIR = [12%/yr.]/[12 months/yr.] =
1%/month
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Per Period Example



Assume an initial principal (PV) of $20,000.
After 2 years you will have FV =
PV[1+(r/yr.)/(# periods/year)](n years x #
periods/year) = $20,000[1+12%/12]2years
x12months/year = $20,000(1+1%/month)24 months
= $25,394.69.
Compare that number to the number that
you would get, if you just compounded
interest once a year: FV =
$20,000(1+12%)2 = $25,088, which is
much less.
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Per Period Example


Thus, the effect of compounding more
and more times per year will be to get
more and more future value than with
less compounding.
In that regard, if you invest money, you
would like to have more compounding
per year, but if you borrow, you would
prefer less compounding per year.
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Annual Effective Rate



We have seen that the earning rate is
larger than it would be with annual
compounding when interest is
compounded more than once a year.
We want to some how quantify that
annual earning rate because we like to
compare annual rates.
So, assume that k = APR and m = # of
periods in a year.
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Annual Effective Rate


If we take PV and add compound
interest for one year (m periods), we will
have FV = PV(1+k/m)m at the end of a
year.
We can find the effective rate of
return over the year from our basic
equation for return that we have seen
since the first lecture: r = Income/initial
investment = (FV – PV)/PV.
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Annual Effective Rate



Moreover, it will be an annual return, in this
case, since we compound interest over m per
year periods = 1 year exactly.
Using the above equation for FV, we have
reffective annual rate of return = [PV(1+k/m)m –
PV]/PV = (1+k/m)m – 1 = reff.
All that the annual effective interest rate
(EAR) does is to give an equivalent annual
earning rate of return that resulted from a
nominal APR that is compounded more
than once a year.
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Annual Effective Rate


In other words: FV = (1+reff)PV for one
year. For more than 1 year, say n
years, FV = PV(1+reff)n = PV[(1+k/m)m]n
= PV(1+k/m)n x m.
The effective rate will always be higher
than the nominal interest rate, except
when the nominal rate is for the same
period as the compounding period.
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Matching CF periods to compounding
period



As a final technical note, we tell you that you
have to be careful to match compounding
periods with the period of cash flows.
For example, if CF’s come in quarters and
compounding is quarterly, use the quarterly
rate for making discounted CF’s.
If compounding is quarterly and CF’s come
only annually, you can compound 4 quarters,
find the annual effective rate and use that as
the discount factor for annual compounding.
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Bank Loans
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Types of Bank loans



Although you may not realize it, we have already
seen some templates for various types of bank
loans.
A pure discount loan is a loan that has one
balloon payment at maturity, so FV = PV(1+k)n.
It is like a LT bank loan version of the ST CB’s
we looked at earlier.
We saw installment loans with even periodic
payments. Principal borrowed = PVA of the
payments. Just let us point out that payments
might be monthly or quarterly, in practice, rather
than yearly.
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Types of Bank loans


A final type of loan is an interest-only loan,
whereby only interest payments are made
each intermediate year, with a final interest
payment plus principal amount at the end.
The principal remains unpaid til the end, so
interest is charged on the same principal
every period, so the interest payments will
form an annuity stream.
To value an interest-only loan, take PV =
kP[1 – (1+k)-n]/k + P/(1+k)n = P – P(1+k)-n +
P(1+k)-n = P, which is as it should be.
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Trailer
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Present or Future Value?


Present and future value are from the
same basic equation, but we must
understand what these concepts really
mean.
We calculate present value of future
cash flows because we want to know
the value of those future cash flows to
us now. Then, we can decide how
much to pay now for things like stocks,
bonds, and other investments, which are
always promises of money in the future.
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Present or Future Value?
We calculate future value of money
we have now or will get later to find
out how much money that we will
have at a future time.
 The basic difference is that present
value tells us how much something is
worth to us, now, while future value
tells us how many dollars we will
have at some point in the future but
not how much it is worth to us now.

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The learning experience: the use of logic
and pictures
Our use of logic and pictures to find
other annuity equations from the first
one for ordinary annuities is just an
example of how to attack PV and FV
problems, in general.
 We have the basic equations for a
one-payment cash flow. From it, we
construct general equations for
multiple cash flows.

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The learning experience: the use of logic
and pictures
Then, we derived specialized FV and
PV equations for an ordinary annuity,
using a simple trick.
 From those equations, we found
equations for other kinds of annuities,
using one- and two-dimensional time
lines, the logic of the situation, and
our understanding of the meanings of
the original ordinary annuity
equations.

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The learning experience: the use of logic
and pictures
In our future studies and in other
problems, we will be given a set of
cash flows, and we will be asked to
figure out an answer.
 Often, we will not be asked to figure
out PV or FV in the problem but part
of the problem will be, first, to figure
out which one we should calculate:
FV or PV.

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The learning experience: the use of logic
and pictures



After we figure out which one, FV or PV, is
called for in the problem, we set up a time line to
see more clearly what the situation is and if we
can use some of the simplified equations, like
PV or FV annuity, in the problem, maybe only for
part of it.
Then, we set up a final equation based on
everything that we can see about the situation.
The point is: don’t just memorize equations,
understand them, and then you will learn how to
use all of them for any situation.
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Exam Caliber Questions
1.
2.
An investment pays $50 per year for 100
years, except, in year 25, it pays $100, in
year 50, it pays $200, and in the final year, it
pays $500. If your RRR= 10%, how much
would you pay for this investment, now?
Explain the reinvestment assumption in the
multiple cash flow PV/FV theory.
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Tutorial Problems
Learning activity
● Attempt all of the chapter review
and self-test problems and the
critical thinking and concepts review
questions on pages 142 to 145.
Attempt questions and problems 1,
3, 4, 7, 12, 15, 19, 21, 31, 44.
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Next week



Monday April 6 is a holiday
Diane has informed me that Maurie has
informed her that we should do lectures in the
tutorials.
I will let you know by email what the rooms
will be next week.
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END
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