Investments, tenth edition

 Example  14.2 

Bond Pricing 

   

5

 Here is a quick derivation of the formula for the present value of an annuity. An annuity lasting  T  periods can be 

viewed as equivalent to a perpetuity whose first payment comes at the end of the current period  less  another per-

petuity whose first payment comes at the end of the ( T   1  1) 

 st 

  period. The immediate perpetuity net of the delayed 

perpetuity provides exactly  T  payments. We know that the value of a $1 per period perpetuity is $1/ r.   Therefore, 

the present value of the delayed perpetuity is $1/ r  discounted for  T  additional periods, or    

1

r

3

1

(1

1 r)

T

.  

The present value of the annuity is the present value of the first perpetuity minus the present value of the delayed 

perpetuity, or    

1

r

 

B1 2

1

(1

1 r)

T

R.   

   

6

 Here is a trap to avoid. You should not confuse the bond’s  coupon  rate, which determines the interest paid to the 

bondholder, with the market interest rate. Once a bond is issued, its coupon rate is fixed. When the  market   interest 

rate increases, investors discount any fixed payments at a higher discount rate, which implies that present values 

and bond prices fall.  

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P A R T   I V

 Fixed-Income 

Securities

   Bond prices are tedious to calculate without a spreadsheet or a financial calculator, 

but they are easy to calculate with either. Financial calculators designed with present and 

future value formulas already programmed can greatly simplify calculations of the sort we 

just encountered in Example 14.2. The basic financial calculator uses five keys that cor-

respond to the inputs for time-value-of-money problems such as bond pricing:

    1.   n  is the number of time periods. In the case of a bond,  n  equals the number of 

periods until the bond matures. If the bond makes semiannual payments,  n  is the 

number of half-year periods or, equivalently, the number of semiannual coupon 

payments. For example, if the bond has 10 years until maturity, you would enter 

20 for  n,  since each payment period is one-half year.  

   2.   i  is the interest rate per period, expressed as a percentage (not as a decimal). For 

example, if the interest rate is 6%, you would enter 6, not .06.  

   3.   PV  is the present value. Many calculators require that PV be entered as a negative 

number, in recognition of the fact that purchase of the bond is a cash  outflow,   while 

the receipt of coupon payments and face value are cash  inflows.   

   4.   FV  is the future value or face value of the bond. In general, FV is interpreted as 

a one-time future payment of a cash flow, which, for bonds, is the face (i.e., par) 

value.  

   5.   PMT  is the amount of any recurring payment. For coupon bonds, PMT is the cou-

pon payment; for zero-coupon bonds, PMT will be zero.    

 Given any four of these inputs, the calculator will solve for the fifth. We can illustrate with 

the bond in Example 14.2. 

 To find the bond’s price when the annual market interest rate is 8%, you would enter 

these inputs (in any order):   

    n    

  60   

   The bond has a maturity of 30 years, so it makes 60 semian-

nual payments.  

   i    

  4   

  The  semiannual  market interest rate is 4%.  

  FV   

  1,000   

   The bond will provide a one-time cash flow of $1,000 when 

it matures.  

  PMT   

  40   

  Each semiannual coupon payment is $40.   

 On most calculators, you now punch the “compute” key (labeled  COMP  or  CPT ) and 

then enter PV to obtain the bond price, that is the present value today of the bond’s 

cash flows. If you do this, you should find a value of  2 1,000. The negative sign signifies 

that while the investor receives cash flows from the bond, the price paid to  buy  the bond 

is a cash  out flow, or a negative cash flow. 

 If you want to find the value of the bond when the interest rate is 10% (the second 

part of Example 14.2), just enter 5% for the semiannual interest rate (type “5” and 

then “ i ”), and when you compute PV, you will find that it is  2 810.71.