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Lecture 3

ADM 3351 Lecture Notes - Lecture 3: The Monthly, Financial Instrument, Zero-Coupon Bond


Department
Administration
Course Code
ADM 3351
Professor
Chen Guo
Lecture
3

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CHAPTER 2
PRICING OF BONDS
REVIEW OF TIME VALUE OF MONEY
Money has time value because of the opportunity to invest it at some interest
rate.
Future Value
The future value of any sum of money invested today is: Pn = P0 (1 + r)n,
where n = number of periods, Pn = future value n periods from now (in
dollars), P0 = original principal (in dollars), r = interest rate per period (in
decimal form), and the expression (1 + r)n represents the future value of $1
invested today for n periods at a compounding rate of r.
Example: You deposit $100 with a bank for one year. The bank promises to
pay you the principle amount back in one year plus $8 interest. Your annual
return is
08.01
100
108
1
00
0
P
P
P
PP
rnn
or 8% per year.
You may interpret that the annual interest rate is 8%, with which the future
value of your deposit can be found by the compounding formula as
108)08.01(100)1( 11
01 rPP
If you extend your deposit and the interest with the bank for another year,
your future value will be
116)08.1(100)1( 22
02 rPP
Compounding Assumption and APR Quote
Consider the above example again. The bank may tell you that it is going to
invest and reinvest your deposit semiannually, i.e., twice a year. However,
there is no reason why you should care how the bank is going to spend the
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money that you lend to the bank. All you care is that the bank will pay you
the principle back plus $8 interest at the end of the year.
The compounding story or assumption of the bank may be expressed as
follows:
108)1( 12
01
s
rPP
The relationship implies that
03923.108.11 s
r
Hence, the semiannual interest rate is 0.03923 or 3.923% per half-year.
The controversy is that the bank is allowed, by the government regulation, to
quote the annual interest rate in the form of Average Percentage Rate or
Annual Percentage Rate (APR) as
%846.7%923.322 s
rAPR
per year
which appears lower than the actual annual interest rate of 8%.
The bank may also argue that your deposit will be compounded every
month, i.e.,
108)1( 112
01
m
rPP
The monthly rate can be solved as
or 0.6434% per month.
The bank can quote the APR based on the monthly compounding
assumption as
%7208.7%6434.01212 m
rAPR
per year,
which is even lower than the APR based on the semiannual compounding
assumption.
There is a popular formula that relates the Effective Annual Rate (EAR) and
an APR as
m
m
APR
EAR
11
where m is the number of compounding per year.
This formula could be misleading because it creates an impression as if one
can arbitrarily divide an APR quote by any m of his choice or at his
convenience.
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The government regulation stipulates that the APR quote must clearly
indicates its compounding assumption, i.e., how it is derived from the first
place. It implies that the APR can only be and must be divided by one and
only one m that is consistent with its compounding assumption.
For example, when you see a quote 8% without indicating its compounding
assumption, you have to ask for it. You cannot make up a compounding
assumption out of your convenience, because you not the one who quotes
the interest rate. If the compounding assumption is clearly indicated, you
certainly cannot set the divisor m at other value than the one that is
consistent with the compounding assumption.
If it is EAR, m=1.
If it is APR based on semiannual compounding, m=2; it cannot be anything
else. The only valid interpretation of the quote is that the semiannual rate is
rs = 8%/2=4%, because the APR is originally derived as 2rs. Once you
realize that the only divisor is m=2, you can find the EAR by the above
formula as
0816.1)04.1(
2
08.0
11 2
2
EAR
, i.e., EAR=8.16%.
The relationship between the interest rate per period and the EAR can be
expressed as
R
dbwsmmqs errrrrrr 36526241242 )1()1()1()1()1()1(1
,
where R is the annualized continuous compounding interest rate.
3
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