6
Learning 22/03/2011
AFF1300
Lecture Objectives
Understand and carry out simple interest
calculations to determine:
Mathematics of Finance
accumulated amount
present value
yields
holding period yield
accumulated amount (future value)
present value
present value of an annuity
accumulated value of an annuity (future value)
effective rate of interest
Understand and carry out compound
interest calculations to determine:
Please attempt all questions and make sure to bring a
calculator for this lecture
8-1
8-2
8.1 Simple Interest (cont.)
8.1 Simple Interest
Introduction
Focus is on the mathematical techniques for
calculating the cost of borrowing and the return
earned on an investment
Table 8.1 defines the symbols of various formulae
Although symbols vary between textbooks,
formulae are consistent
8-3
8.1 Simple Interest (cont.)
8-4
Simple interest accumulation
The amount of interest paid on debt, or earned on a
deposit is:
Simple interest is interest paid on the
original principal amount borrowed or
invested
I = A
d
i
365
(8.1)
where:
The principal is the initial, or outstanding,
amount borrowed or invested
A
d
With simple interest, interest is not paid on
previous interest
i
is the principal
is the duration of the loan, expressed as the
number of interest payment periods (usually
one year)
is the interest rate, expressed as a decimal
8-5
8-6
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Simple interest accumulation (cont.)
Simple interest accumulation (cont.)
Example 1: If $10 000 is borrowed for one year,
and simple interest of 8% per annum is charged,
the total amount of interest paid on the loan would
be:
The market convention (common practice occurring
in a particular financial market) is for the number of
days in the year to be 365 in Australia and 360 in
the US and the euromarkets
I = A d/365 i
= 10 000 365/365 0.08
365/365
= $800
Example 3: If the amount is borrowed at the same
rate of interest but for a 90-day term, the total
amount of interest paid would be:
Example 2: Had the same loan been for two years
the total amount of interest paid would be:
I = 10 000 90/365 0.08
= $197.26
I = 10 000 730/365 0.08
= $1600
8-7
Simple interest accumulation (cont.)
Simple interest accumulation (cont.)
The final amount payable (S) on the borrowing
is the sum of the principal plus the interest
amount. [We have to pay what we borrowed]
The final amounts payable in the three
previous examples are:
Example 1a:
S = 10 000 [1 + ( 1 0.08)]
= 10 800
Alternatively the
Alternatively, the final amount payable can be
calculated in a single equation:
S=A+I
= A + (A n i)
S = A[1 + (n i)]
8-8
Example 2a:
S = 10 000 [1 + ( 2 0.08)]
= 11 600
Example 3a:
S = 10 000 [1 + ( 90/365 0.08)]
= 10 197.26
(8.2)
8-9
8-10
Present value with simple interest
(cont.)
Present value with simple interest
Equation for calculating the present value of a
future amount is a re-arrangement of Equation 8.2
The present value is the current value of a
future cash flow, or series of cash flows,
discounted by the required rate of return
Alternatively, the present value of an amount
of money is the necessary amount invested
today to yield a particular value in the future
A=
S
(8.3)
[1 + ( n i)]
The yield is the effective rate of return received
8-11
8-12
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Present value with simple interest (cont.)
Example: A company discounts (sells) a
commercial bill with a face value of $100 000, a
term to maturity of 180 days, and a yield of 8.75%
per annum. How much will the company raise on
the issue?
(Commercial bills are discussed in Chapter 9.)
Briefly, a bill is a security issued by a company to
raise funds. A bill is a discount security, i.e. it is
issued with a face value payable at a date in the
future but in order to raise the funds today the
company sells the bill today for less than the face
ll th bill
th
th
value. The investor who buys the bill will receive
face value at the maturity date. The price of the
bill will be:
Present value with simple interest
(cont.)
Equation 8.3 may be rewritten to facilitate
its application to calculating the price (i.e.
present value) of another discount security,
the
the Treasury note (T-note)
365 face value
365 + (yield/100 days to maturity)
365 $100 000
=
365 + (0.0875 180)
= $95,863.49
Price =
Try now
(8.4)
8-14
8-13
Present value with simple interest
(cont.)
Calculation of yields
Example: What price per $100 of face
value would a funds manager be prepared
to pay to purchase 180-day T-notes if the
current
current yield on these instruments was
6.35% per annum?
try now
In the previous examples, the return on the
instrument or yield was given (e.g. 7.35%)
However, in other situations it is necessary to
calculate the yield on an instrument (or cost of
borrowing)
i=
365
d
x
I
(8.5)
A
8-15
Calculation of yields (cont.)
Holding period yield (HPY)
Example : What is the yield (rate of return) earned
on a deposit of $100 000 with a maturity value of
$100 975 in 93 days? That is, this potential
investment has a principal (A) of $100 000, interest
(I) of $975 and an interest period (d) of 93 days.
i=
365
93
x
8-16
HPY is the yield on securities sold in the
secondary market prior to maturity
Short-term money market securities (e.g. T-notes)
may be sold prior to maturity because:
investment was intended as short-term management
of surplus cash held by investor
$975
$100 000
the investors cash flow position has unexpectedly
changed and cash is needed
= 0.03826
a better rate of return can be earned in an alternative
investment
= 3.83%
8-17
8-18
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Holding period yield (HPY) (cont.)
Holding period yield (HPY) (cont.)
The HPY will be:
The yield to maturity is the yield obtained by holding the
security to maturity
The HPY is likely to be different from the yield to
maturity
This is illustrated in Example 9 of the textbook with a
discount security using Equation 8.3 (8.4 can also be
used)
A discount security pays no interest but is sold today for
less than its face value, which is payable at maturity, e.g.
T-note
greater than the yield to maturity when the market
yield from declines the yield at purchase, i.e.
interest rates have decreased and the price of the
security
security increases
less than the yield to maturity when the market
yield increases from the yield at purchase, i.e.
interest rates have increased and the price of the
security decreases
8-19
8-20
Compound interest accumulation
(future value)
8.2 Compound Interest
When an amount is invested for only a small
number of periods it is possible to calculate
the compound interest payable in a relatively
cumbersome way (illustrated in Example 10
in the textbook)
Compound interest (unlike simple
interest)
interest) is paid on both:
Thi
This method can be simplified using the
th
general form of the compounding interest
formula
the initial principal; and
S = A(1 + i)n
the accumulated previous interest entitlements
(8.6)
Applying Equation 8.6 to Example 10
S = 5000 (1 + 0.15)3 = $7604.38
8-21
Compound interest accumulation
(future value) (cont.)
8-22
Compound interest accumulation
(future value) (cont.)
Example 11a: The effect of compounding can be
further understood by considering a similar deposit
of $8000 paying 12% per annum, but where
interest accumulates half-yearly for four years:
On many investments and loans, interest will
accumulate more frequently than once a
year, e.g. daily, monthly, quarterly
I = 12.00 % p.a. / 2
= 0.06
Thus, it is necessary to recognise the effect of the
compounding frequency on the inputs i and n in
Equation 8.6
and:
n = 4 2 = 8 periods
If interest had accumulated monthly on the
previous loan, then:
so:
S = 8000(1 + 0.06)8
= 8000(1.593848)
= $12 750.78
i = 0.15/12 = 0.0125 and n = 3 12 = 36
8-23
8-24
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Present value with compound interest
Present value with compound interest
(cont.)
The present value of a future amount is the
future value divided by the interest factor
(referred to as the discount factor) and is
expressed in equation form as:
A=
S
(1 + i)n
Example: What is the present value of
$16,500 received at the end of three years, if
funds could currently be invested at 7.25% per
annum, compounded annually? Using Equation
8.7a:
(8.7a)
A = S(1 + i)-n
A=
S
(1 + i)n
try now
(8.7b)
8-25
Present value of an ordinary annuity
(cont.)
Present value of an ordinary annuity
Example: The present value of an annuity of
$400, received at the end of every three
months for eight years, where the required rate
of return is 5 per cent per annum, compounded
quarterly, would be:
An annuity is a series of periodic cash
flows of the same amount
Ordinary annuityseries of periodic cash flows
Ordinary
occur at end of each period (Equation 8.8)
A = C [ 1 (1 + i)-n ]
8-26
C = $400
i = 5.00%/4 = 1.25% or 0.0125
n = 4 x 8 = 32
(8.8)
Therefore
i
Try now
8-27
8-28
Present value of an annuity due (cont.)
Present value of an annuity due
Example: The present value of an annuity of
$400, received at the beginning of every
three months for eight years, where the
required rate of return is 5per cent per annum,
compounded quarterly, would be:
Annuity duecash flows occur at the
beginning of each period (Equation 8.9)
A = C [ 1 (1 +
i)-n
] (1 + i)
C = $400
i = 5.00%/4 = 1.25% or 0.0125
n = 4 x 8 = 32
(8.9)
i
8-29
Try now
8-30
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22/03/2011
Accumulated value of an annuity (future
value)
Present value of a Treasury bond
Equation 8.10 is used to calculate the price (or
present value) of a corporate bond
A = C [ 1 (1 + i)-n ] + S(1 + i)-n
The accumulated (or future) value of an
annuity is given by Equation 8.11
(8.10)
i
S = C [ (1 + i)n - 1 ]
i
Example 16 in the textbook illustrates the
application of Equation 8.10
(8.11)
8-31
Accumulated value of an annuity (future
value) (cont.)
Effective rates of interest
Example: A university student is planning to
invest the sum of $100 per month for the next
four years in order to accumulate sufficient
funds to pay for a trip overseas once she has
graduated. Current rates of return are 7.2 per
cent per annum, compounding monthly. How
much will the student have available when she
graduates
graduates?
8-32
The nominal rate of interest is the annual
rate of interest, which does not take into
account the frequency of compounding
The effective rate of interest is the rate of
interest after taking into account the
frequency of compounding
Try now
8-33
Effective rates of interest (cont.)
8-34
Effective rates of interest (cont.)
Example 18a: A deposit of $8000 is made for four
years and will earn 12% per annum, with interest
compounding semi-annually. What will be the value
of the deposit at maturity?
Example 18b: What would the maturity value
of the same deposit be if interest was
compounded annually, rather than semiannually as in Example 18a?
A = $8000
i = 12%/2 = 6% or 0.06
n=4x2 = 8
Therefore:
S = $8000 (1.12)4
S = $8000 (1 + 0.06)8
= $12 588.15
= $8000 (1.06)8
= $12 750.78
8-35
8-36
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22/03/2011
Effective rates of interest (cont.)
Effective rates of interest (cont.)
Example 19: What is the effective rate of
interest if you are quoted:
(a) 10% per annum, compounded annually?
(b) 10% per annum, compounded semi-annually?
(c) 10% per annum, compounded monthly?
The formula for converting a nominal rate
into an effective rate is:
ie = (1 + i/m )m
1
(a) ie = (1 + 0.10/1)1 1
= (1.10)1 1
= 0.10 or 10%
(b) ie = (1 + 0.10/2)2 1
= (1.05)2 1
= 0.1025 or 10.25%
(8.12)
(c) ie = (1 + 0.10/12)12 1
= (1.008333)12 1
= 0.1047 or 10.47%
8-37
8.3 Summary
8-38
8.3 Summary (cont.)
Simple interest is interest paid on the original
principal amount borrowed or invested
Compound interest is paid on the initial
principal plus accumulated previous interest
entitlements
An annuity (ordinary or due) is a series of
periodic cash flows of the same amount,
of which both the present value and the
future value can be calculated
Unlike the nominal rate of interest, which
ignores the frequency of compounding,
the effective rate of interest takes into
account the frequency of compounding
The present value and future value of an
investment or loan can be calculated using
either simple or compound interest
8-39
8-40
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