This

**preview**shows pages 1-3. to view the full**14 pages of the document.**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

006434.0108.1

12

m

r

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.

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