Topic Overview
Lesson 3.1
3.1.1 Definition of an Annuity
3.1.2 Present Value of an
Ordinary Annuity
3.1.3 Present Value of an
Annuity Due
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 2
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3.1.1 Present value of an annuity
 An annuity is similar to a perpetuity, in that
it is a constant stream of equal cash flows,
except that it only occurs for a fixed period
of time
ORDINARY ANNUITY
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 3
0 2
C
1
C C
3
C
4
 Once again, note that the first cash flow
occurs at the end of the first period of the
annuity – this is called an ordinary annuity
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3.1.1 Present value of an annuity
 If the cash flows occur at the beginning of
each period, as shown in the diagram
below, this is called an annuity due
ANNUITY DUE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 4
0 2
C
1
C C
3
C
4
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3.1.2 Present value of an annuity
 It would be quite possible to calculate the
present value of an annuity by repeatedly
using the formula for the present value of a
single cash flow, and then summing the
present value of each cash flow; i.e.
PRESENT VALUE OF AN ORDINARY ANNUITY
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 5
but it can be shown that this can be
simplified to a single formula
     
1 2
2 …
1 1 1
n
n
PV C C C
r r r
   
  
(4.3)
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3.1.2 Present value of an annuity
 The formula for the present value of an
ordinary annuity is:
PRESENT VALUE OF AN ORDINARY ANNUITY
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 6
where C = the constant cash flow (with the
first cash flow occurring one
period in the future)
r = the interest rate per period
n = the number of periods
 
1 1 1
1 n PV C
r r
 
    
    
(4.5)
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3.1.3 Present value of an annuity
 The formula for the present value of an
annuity due is:
PRESENT VALUE OF AN ANNUITY DUE
where C = the constant cash flow (with the
first cash flow occurring
immediately)
r = the interest rate per period
n = the total number of payments
  1
1 1 1
1 n PV C C
r r 
 
     
    
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 7
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3.1.4 Future value of an annuity
 The formula for the future value of an annuity
(the value of the cash flow stream at the end
of the annuity) is:
FUTURE VALUE OF AN ORDINARY ANNUITY
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 8
where C = the constant cash flow (first cash
flow one period in the future)
r = the interest rate per period
n = the number of periods
11  1 n FV C r
r
    (4.6)
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Topic Overview
Lesson 3.2
3.2.1 Solving for the Rate of
Return
3.2.2 Solving for the Annuity
Cash Flow
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 9
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3.2.1 Solving for other variables
 If you have a formula with one unknown
variable, and that variable only appears
once, rearranging the formula should allow
you to solve for that variable
 E.g. we can derive Formula 3.2 from 3.1:
REARRANGING FORMULAS
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 10
 
Substituting PV for C
 
1
1
n
n
FV PV r
PV FV
r
  
 

Dividing both sides of
the equation by (1 + r)n
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3.2.1 Solving for other variables
 We can take the formula that relates the
present and future value, and rearrange the
equation to get a formula for the rate of
return that allows the present value to grow
to the future value over n periods:
SOLVING FOR THE RATE OF RETURN
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 11
     
 
1/
1
n r FV
 1  PV n FV PV r 
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3.2.2 Solving for other variables
 If we know the rate of return, the number of
periods and the present value of an annuity,
we can then rearrange the PV formula to
solve for the cash flow
SOLVING FOR THE ANNUITY CASH FLOW
12
 
 
    
    
1 1 1
1 n PV C
r r 
 

 
  
    
1 1 1
1 n
C PV
r r
(4.9)
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics
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3.2.2 Solving for other variables
 Similarly, if we know the rate of return, the
number of periods and the future value of an
annuity, we can then rearrange the FV
formula to solve for the cash flow
SOLVING FOR THE ANNUITY CASH FLOW
13
  11  1 n FV C r
r 
   
1 1 1 n
C FV
r
r
(4.9)
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics
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Topic Overview
Lesson 3.3
3.3.1 Interest Rates
3.3.2 Effective Annual Rate
3.3.3 Real and Nominal Interest
Rates
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3.3.1 Interest rates
 Interest rates can be quoted in different ways,
and we need to know how to interpret them
and convert between them
 So far we have mainly concentrated on the
interest rate per compounding period – the
discount rate (r) used in most TVM formulas
 Banks usually quote an annual interest rate
(the APR, or Annual Percentage Rate), and
they should also tell you how often interest is
compounded
ANNUAL PERCENTAGE RATE (APR)
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 15
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3.3.1 Interest rates
 As we’ve seen, to find the interest rate per
compounding period, we divide the APR by
the number of compounding periods per year
 The formula for finding the interest rate per
compounding period is:
INTEREST RATE PER COMPOUNDING PERIOD
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 16
r APR
m

where m = the number of compounding
periods per year
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3.3.2 Interest rates
 The problem with the APR is that it ignores
the effect of compounding
 For example, consider
these interest rate quotes:
EFFECTIVE ANNUAL RATE (EAR)
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 17
10.8% p.a. compounded
annually
10.5% p.a. compounded
quarterly
10% p.a. compounded
daily
 They have different
APRs and different
compounding periods
 This makes comparisons between rates with
different compounding periods difficult
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3.3.2 Interest rates
 If you need to compare alternative financing
arrangements, the best method is to convert
the quoted rate, or APR, to an Effective
Annual Rate (EAR)
 The EAR tells you how much you are
effectively paying or receiving, taking into
account the effect of compounding
 The EAR is the annually-compounded APR
that would leave you in the same position in
terms of interest paid or received
EFFECTIVE ANNUAL RATE (EAR)
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 18
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3.3.2 Interest rates
 The formula for Effective Annual Rate is:
EFFECTIVE ANNUAL RATE (EAR)
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 19
where EAR = The Effective Annual Rate
APR = The Annual Percentage Rate
m = the number of compounding
periods per year
1 1
m EAR APR
m
      
 
(5.3)
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3.3.3 Interest rates
 So far we have restricted our discussion to the
nominal interest rate, which is the quoted or
observable interest rate paid or received
 Because we normally experience inflation,
which reduces the purchasing power of
dollars over time, the nominal interest rate
does not represent the increase in purchasing
power from an investment
 The rate that measures the increase in
purchasing power is the real interest rate
REAL AND NOMINAL INTEREST RATES
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 20
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3.3.3 Interest rates
 The relationship between the nominal and
real interest rate is as follows:
REAL AND NOMINAL INTEREST RATES
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 21
 The second formula above is a convenient
approximation when interest rates are low
1 1 1  n r
n r
r r π
r r π
   
 
where rn = nominal interest rate
rr = real interest rate
π = the expected inflation rate
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Topic Overview
Lesson 3.4
3.4.1 The Term Structure of
Interest Rates
3.4.2 Interest Rate Determination
3.4.3 The Pure Expectations
Theory
3.4.4 The Liquidity Premium
Theory
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 22
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3.4.1 Interest rates
 Until now we have treated “the interest rate”
as a single variable, but interest rates usually
vary across different financial assets
 Interest rates may reflect different variables,
such as credit risk, but the most important
variable is the term to maturity
 The relationship between interest rates and
term to maturity – the pattern of different rates
applying to different maturities – is referred to
as the term structure of interest rates
INTEREST RATE “STRUCTURE “
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 23
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3.4.1 Interest rates
 The term structure of interest rates can be
represented graphically, via a yield curve
YIELD CURVES
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 24
The yield curve
shows the interest
rate, or yield
Interest rate
Maturity
Short-term Medium-term Long-term
, that
is currently
available for
otherwise identical
securities with
different maturities
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3.4.1 Interest rates
 Yield curves can take on a variety of shapes.
Commonly observed shapes include:
YIELD CURVES
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 25
An upward
sloping yield
curve is referred
to as “normal”
because it is the
most commonly
observed shape
Positive or “normal”
Interest rate
Maturity
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3.4.1 Interest rates
 Yield curves can take on a variety of shapes.
Commonly observed shapes include:
YIELD CURVES
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 26
When there is
little or no
difference
between shortand
long-term
rates, the yield
curve is referred
to as flat
Interest rate
Maturity
Flat
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3.4.1 Interest rates
 Yield curves can take on a variety of shapes.
Commonly observed shapes include:
YIELD CURVES
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 27
A downward
sloping yield
curve is referred
to as “inverse”
because it is the
opposite of a
“normal” curve
Negative or “inverse”
Interest rate
Maturity
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3.4.1 Interest rates
 Yield curves can take on a variety of shapes.
Commonly observed shapes include:
YIELD CURVES
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 28
A humped yield
curve, or other
non-standard
shape, usually
occurs when
there are varying
expectations
about future rates
Interest rate
Maturity
Humped
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3.4.2 Interest rates
 Interest rates are set by the RBA, which uses
the cash rate to implement monetary policy
INTEREST RATE DETERMINATION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 29
 The cash rate is a very short term rate, but
as a result of market forces, changes in the
cash rate flow on to longer-term rates
Monetary policy Cash rate
The rate charged on
overnight loans between
financial institutions
The main tool used to
influence economic growth,
inflation and unemployment
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3.4.3 The pure expectations theory
 Long-term rates, and therefore the term
structure of interest rates and the shape of
the yield curve, are largely determined by
investors’ expectations regarding future
interest rates
 This is based on the assumption that capital
markets are perfect and that investors are
indifferent between investments with
differing maturities – they will always choose
the investment providing the highest return
PURE EXPECTATIONS THEORY ASSUMPTIONS
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 30
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3.4.3 The pure expectations theory
Lecture Example 2.28 A
If short-term rates and long-term rates are equal
(say, 5% p.a.) but you expect short-term rates to
go up in the future (to 6%), would you prefer…
PURE EXPECTATIONS THEORY PREDICTION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 31
A
B
A single long-term
investment?
A series of successive
short-term investments?
 If short-term rates and long-term rates are
equal (say, 5% p.a.) but you expect shortterm
rates to go up in the future (to 6%),
would you prefer a single long-term
investment or a series of successive short
term investments?
 You would prefer a series of short-term
investments, because when you “roll over”
each investment to the next one, you can
take advantage of higher interest rates
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3.4.3 The pure expectations theory
Lecture Example 2.28 B
If you (and everyone else as smart as you)
prefers to invest short-term rather than longterm,
but there are still people who want to
borrow long-term, what will happen to long-term
interest rates?
PURE EXPECTATIONS THEORY PREDICTION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 32
A
C
Increase
Remain the same
B Decrease
 If you (and everyone else as smart as you)
prefers to invest short-term rather than
long-term, but there are still people who
want to borrow long-term, what will happen
to long-term interest rates?
 Long-term rates would increase until they
are high enough to attract investors, at
which point short-term investments do not
provide higher returns
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3.4.3 The pure expectations theory
 As a result of this process, we would expect
current long-term rates to be the geometric
average of expected future short-term rates
 If this were not the case, arbitrage
opportunities would exist – arbitrageurs
could borrow at the point on the yield curve
where rates are lowest and invest where
they are highest, and the resulting market
forces would restore an equilibrium in which
the above prediction holds
PURE EXPECTATIONS THEORY PREDICTION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 33
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3.4.4 The liquidity premium theory
 The question then is…
LIQUIDITY PREMIUM THEORY ASSUMPTIONS
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 34
 Probably not
Why is an upwardsloping
yield curve
considered
“normal”?
Is it reasonable to assume
that, most of the time,
investors expect interest rates
to up rather than down?
In addition to investor
expectations, something else
is causing them to prefer
short‐term investments
As a result, those
wanting to borrow
long‐term have to
offer higher rates to
attract investors
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3.4.4 The liquidity premium theory
 The reason is that long-term investments are
riskier than short-term investments
 Small changes in interest rates cause a
much greater change in the value of a longterm
investment than a short-term
investment
 Because of this preference for short-term
investments, yields on long-term investments
will be somewhat higher than the predicted
yield based on purely on expectations
LIQUIDITY PREMIUM THEORY PREDICTION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 35
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3.4.4 The liquidity premium theory
 This phenomenon results in higher yields
for long-term securities, but has little effect
on short term securities
LIQUIDITY PREMIUM THEORY PREDICTION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 36
Interest rate
Maturity
“Pure expectations” yield curve
Observed
yield curve
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3.4.4 The liquidity premium theory
 The “upward bias” to what would otherwise be
a flat yield curve may help to explain why an
upward sloping yield is most commonly seen
LIQUIDITY PREMIUM THEORY PREDICTION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 37
Interest rate
Maturity
“Pure expectations” yield curve
Observed
yield curve
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3.4.4 The liquidity premium theory
 This hypothesis does not preclude an inverse
yield curve, but predicts that the yield curve
will be flatter (less negative) as a result
LIQUIDITY PREMIUM THEORY PREDICTION
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 38
Interest rate
Maturity
“Pure
expectations”
yield curve
Observed yield curve
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Topic 3 Appendices
Appendix 3.1 Financial calculator guide
Appendix 3.2 General calculator guide
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 39
La Trobe Business School
Although not required for this subject, many
students find a financial calculator easier to use
in many situations instead of formulas, or use it
to double check the results from using formulas
The following slides demonstrate how to solve
selected Lecture Examples from Topic 3 using a
financial calculator
Buttons shown are for the Texas Instruments
BAII Plus, although others are similar
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 40
Appendix 3.1
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Appendix 3.1
Lecture Example 3.1
What is the present value of $1 million per year
for the next 30 years (with the first payment in a
year’s time), if the interest rate is 8% p.a.?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 41
3 0
8 I/Y
1 PMT
CPT PV
N In this case the payment has been entered in
millions of dollars, so the answer will be in millions
of dollars ($11.26). To get the answer in dollars,
you can enter the payment as 1000000, or
multiply the answer by 1 million.
For Lecture Example 2.18, enter the number of
payments as 29 and then add $1 m to the answer.
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Appendix 3.1
Lecture Example 3.2
You are 25 years old, and have decided to save
$12,000 per year by making monthly deposits
into an account paying 12% p.a. compounding
monthly. How much will you have at age 65?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 42
4 8
1 I/Y
1 0
CPT FV
0 0 PMT
0 N Due to monthly
compounding, we are
entering the problem as if
it was $1000 per year at
1% p.a. for 480 years.
The answer is the same.
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Appendix 3.1
Lecture Example 3.3
An investment will cost you $1000 and will
return $2000 in 5 years. What is the annual
rate of return?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 43
5
1 0
2 0
CPT I/Y
N
0 0 PV
0 0 +/– FV
Either the present
value or the future
value needs to be
entered as a negative
number (using the +/‐
button) – otherwise
you will get an error
message
La Trobe Business School
Appendix 3.1
Lecture Example 3.4
What is the monthly repayment on a $250,000
mortgage loan, repayable over 20 years at an
interest rate of 12% p.a., compounding monthly?
FINANCIAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 44
2
1 I/Y
2 5
CPT PMT
4
0 0 0 0
0
PV
N
La Trobe Business School
Rounding errors
As discussed in Appendix 2.2, you need to avoid
or minimise rounding errors. The next two slides
show a couple of techniques to help you carry
out complicated calculations without writing down
intermediate answers and later entering them in
the calculator – which is almost always the
source of rounding errors.
If you would like to understand these techniques,
follow the steps in these slides carefully.
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 45
Appendix 3.2
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Lecture Example 3.5 – Using brackets
The calculation becomes 250000/(1/0.01(1-1/1.01^240)) and is entered as follows. If you are using the Chain system, you need 250000/(1/0.01(1-(1/(1.01^240))))

• i.e. you need more brackets.
  GENERAL CALCULATOR GUIDE
  FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 46
  Appendix 3.2
  1
  1 –
  yx 2
  ÷
  4 0 ) )
  2 5
  1
  0 0 0 0 ÷ (

0 . 0 1 × (

÷ 1 . 0 1
La Trobe Business School
Lecture Example 3.5 – Using the memory
If you find brackets tedious, you can minimise them
by storing intermediate answers in the calculator’s
memory. The following steps also reduce the use of
brackets by changing the order of some of the steps.
GENERAL CALCULATOR GUIDE
FIN1FOF Fundamentals of Finance – Topic 3 – Financial Mathematics 47