Download Chapter 8

Chapter 8
Mathematics of Finance:
An Introduction to Basic
Concepts and Calculations
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-1
Learning Objectives
• Differentiate between simple and compound
interest rate calculations
• Differentiate between nominal and effective
interest rate calculations
• Calculate present and future values of cash flows
• Calculate the yield of a security
• Calculate the present value of an annuity
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-2
Chapter Organisation
8.1 Simple Interest
–
–
–
–
Simple interest accumulation
Present value
Yields
Holding period yield
8.2 Compound Interest
–
–
–
–
–
Compound interest accumulation (future value)
Present value
Present value of an annuity
Accumulated value of an annuity (future value)
Effective rates of interest
8.3 Summary
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-3
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
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-4
8.1
Simple Interest (cont.)
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-5
8.1
Simple Interest (cont.)
• Simple interest is interest paid on the original
principal amount borrowed or invested
– The principal is the initial, or outstanding, amount
borrowed or invested
– With simple interest, interest is not paid on previous
interest
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-6
Simple interest accumulation
• The amount of interest paid on debt, or earned on a deposit
is
d
I  A
i
365
(8.1)
where
• A is the principal
• d is the duration of the loan, expressed as the number of
interest payment periods (usually one year)
• i is the interest rate, expressed as a decimal
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-7
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:
I = A  d/365  i
= 10 000  365/365  0.08
= $800
– Example 2: Had the same loan been for two years then
the total amount of interest paid would be:
I = 10 000  730/365  0.08
= $1600
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-8
Simple interest accumulation (cont.)
– The market convention (common practice occurring in a
particular financial market) for the number of days in the
year is 365 in Australia and 360 in the US and the
euromarkets
– 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:
I = 10 000  90/365  0.08
= $197.26
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-9
Simple interest accumulation (cont.)
– The final amount payable (S) on the borrowing is the sum
of the principal plus the interest amount
– Alternatively, the final amount payable can be calculated
in a single equation:
S=A+I
= A + (A  n  i)
S = A[1 + (n  i)]
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
(8.2)
8-10
Simple interest accumulation (cont.)
• The final amounts payable in the three previous
examples are:
– Example 1a:
S = 10 000 [1 + ( 1  0.08)]
= 10 800
– Example 2a:
S = 10 000 [1 + ( 2  0.08)]
= 11 600
– Example 3a:
S = 10 000 [1 + ( 90/365  0.08)]
= 10 197.26
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-11
Present value with simple interest
• 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
– The yield is the effective rate of return received
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-12
Present value with simple interest (cont.)
• Equation for calculating the present value of a
future amount is a re-arrangement of Equation
8.2
A
S
[1  ( n  1)]
8.3
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-13
Present value with simple interest (cont.)
– Example 5: A company discounts (sells) a commercial bill
with a face value of $500 000, 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 value. The
investor who buys the bill will get back the face value at
the maturity date. The price of the bill will be:
$500 000
$500 000
Price 

1  (180/365  0.0875)] (1 0.04315)
 $479 317.14
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-14
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 Treasury
note (T-note)
365  face value
365  (yield/100  days to maturity)
365  $500 000

365  (0.0875  180)
 $479 317.14
Price 
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8.4
8-15
Present value with simple interest (cont.)
– Example 6: What price per $100 of face value would a
funds manager be prepared to pay to purchase 180-day
T-notes if the current yield on these instruments was
7.35% per annum?
365  $100
Price 
365  (0.0735  180)
$36 500

378.23
 $96.50
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-16
Calculations of yields
– In the previous examples the return on the instrument or
yield was given
– However, in other situations it is necessary to calculate
the yield on an instrument (or cost of borrowing)
i = 365 x I
d
A
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
(8.5)
8-17
Calculations of yields (cont.)
– Example 7: What is the yield (rate of return) earned on a
deposit of $50 000 with a maturity value of $50 975 in 93
days? That is, this potential investment has a principal (A)
of $50 000, interest (I) of $975 and an interest period (d)
of 93 days.
i = 365 x
93
$975
$50 000
= 0.07653
= 7.65%
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-18
Holding period yield (HPY)
• 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
 The investor’s cash flow position has unexpectedly changed
and cash is needed
 A better rate of return can be earned in an alternative
investment
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-19
Holding period yield (HPY) (cont.)
– 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. Tnote
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-20
Holding period yield (HPY) (cont.)
• The HPY will be
– Greater than the yield to maturity when the market yield
declines from the yield at purchase, i.e. interest rates
have decreased and the price of the 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
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-21
Chapter Organisation
8.1 Simple Interest
–
–
–
–
Simple interest accumulation
Present value
Yields
Holding period yield
8.2 Compound Interest
–
–
–
–
–
Compound interest accumulation (future value)
Present value
Present value of an annuity
Accumulated value of an annuity (future value)
Effective rates of interest
8.3 Summary
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-22
8.2
Compound Interest
• Compound interest (unlike simple interest) is paid
on
– The initial principal
– Accumulated previous interest entitlements
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-23
Compound interest accumulation (future
value) (cont.)
• 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)
• This method can be simplified using the general
form of the compounding interest formula
S = A(1 + i)n
(8.6)
• Applying Equation 8.6 to Example 10
S = 5000 (1 + 0.15)3 = $7604.38
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-24
Compound interest accumulation (future
value) (cont.)
• On many investments and loans, interest will
accumulate more frequently than once a year, e.g.
daily, monthly, quarterly, etc.
– Thus, it is necessary to recognise the effect of the
compounding frequency on the inputs i and n in Equation
8.6
– If interest had accumulated monthly on the previous loan,
then
i = 0.15/12 = 0.0125 and n = 3  12 = 36
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-25
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:
I = 12.00 % p.a. / 2
= 0.06
and:
n = 4  2 = 8 periods
so:
S = 8000(1 + 0.06)8
= 8000(1.593848)
= $12 750.78
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-26
Present value with compound interest
– 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
(8.7a)
(1 + i)n
A = S(1 + i)-n
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
(8.7b)
8-27
Present value with compound interest
(cont.)
– Example 12: What is the present value of $18 500
received at the end of three years if funds could presently
be invested at 7.25% per annum, compounded annually?
Using Equation 8.7a:
A=
S
(1 + i)n
=
$18 500
(1 + 0.0725)3
=
$18 500
1.233650
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
= $14 996.15
8-28
Present value of an ordinary annuity
• An annuity is a series of periodic cash flows of the
same amount
– Ordinary annuity—series of periodic cash flows occur at
end of each period (equation 8.8)
A = C [ 1 – (1 + i)-n ]
i
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
(8.8)
8-29
Present value of an ordinary annuity (cont.)
– Example 14: The present value of an annuity of $200,
received at the end of every three months for ten years,
where the required rate of return is 6.00 per cent per
annum, compounded quarterly, would be:
C = $200
i = 6.00%/4 = 1.50% or 0.015
n = 4 x 10 = 40
Therefore
A = $200 [ 1 – (1 + 0.015)-40 ]
0.015
= $200 [ 29.915 845 2 ]
= $5983.17
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-30
Present value of an annuity due
• Annuity due—cash flows occur at the beginning of
each period (Equation 8.9)
A = C [ 1 – (1 + i)-n ] (1 + i)
i
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PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
(8.9)
8-31
Present value of an annuity due (cont.)
– Example 15: The present value of an annuity of $200,
received at the beginning of every three months for ten
years, where the required rate of return is 6.00 per cent
per annum, compounded quarterly, would be:
C = $200
i = 6.00%/4 = 1.50% or 0.015
n = 4 x 10 = 40
Therefore
A = $200 [ 1 – (1 + 0.015)-40 ]
0.015
= $200 [ 29.915 845 2 ](1.015)
= $6072.92
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-32
Present value of a Treasury bond
– Equation 8.10 is used to calculate the price (or present
value) of a Treasury bond
A = C [ 1 – (1 + i)-n ] + S(1 + i)-n
(8.10)
i
– Example 16 in the textbook illustrates the application of
Equation 8.10
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-33
Accumulated value of an annuity (future
value)
• The accumulated (or future) value of an annuity is
given by Equation 8.11
S = C [ (1 + i)n - 1 ]
(8.11)
i
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-34
Accumulated value of an annuity (future
value) (cont.)
– Example 17: A university student is planning to invest the
sum of $200 per month for the next three years in order
to accumulate sufficient funds to pay for a trip overseas
once she has graduated. Current rates of return are 6 per
cent per annum, compounding monthly. How much will
the student have available when she graduates?
C = $200
i = 6.00%/12 = 0.50% or 0.005
n = 3 x 12 = 36
Therefore
S = $200 [ (1 + 0.005)36 - 1 ]
0.005
= $200 [ 39.3361 ]
= $7867.22
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-35
Effective rates of interest
• 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
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-36
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?
A = $8000
i = 12%/2 = 6% or 0.06
n=4x2 = 8
Therefore
S = $8000 (1 + 0.06)8
= $8000 (1.06)8
= $12 750.78
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-37
Effective rates of interest (cont.)
– Example 18b: What would the maturity value of the same
deposit be if interest was compounded annually, rather
than semi-annually as in Example 18a?
S = $8000 (1.12)4
= $12 588.15
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-38
Effective rates of interest (cont.)
• The formula for converting a nominal rate into an
effective rate is
ie = (1 + i/m )m – 1
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-39
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?
(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%
(c) ie = (1 + 0.10/12)12 – 1
= (1.008333)12 – 1
= 0.1047 or 10.47%
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-40
Chapter Organisation
8.1 Simple Interest
–
–
–
–
Simple interest accumulation
Present value
Yields
Holding period yield
8.2 Compound Interest
–
–
–
–
–
Compound interest accumulation (future value)
Present value
Present value of an annuity
Accumulated value of an annuity (future value)
Effective rates of interest
8.3 Summary
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-41
8.3
Summary
• 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
• The present value and future value of an
investment or loan can be calculated using either
simple or compound interest
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-42
8.3
Summary (cont.)
• 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
Copyright  2007 McGraw-Hill Australia Pty Ltd
PPTs t/a McGrath’s Financial Institutions, Instruments and Markets 5e by Viney
Slides prepared by Anthony Stanger
8-43