Fixed income price and concept of duration

Fixed income price and bond price elasticities experts use the concept of duration bond elasticity defined ratio of the percent change in price to be expected with  a percent change in gross yield is called duration while considering duration and its various uses and interpretations and its limitations and misuses keep in mind that duration is always a number such as 1.5 that is associated with fixed income securities and its measures various attributes of those securities including their price sensitivity to yield changes.
As calculation for duration keep in mind duration is that  duration for a given fixed income security is a single number.This number has two apparently completely different interpretation of the same mathematical expression there is two interpretation of duration.
1)Weighted average payment  with the cash payment of security.
2)Number is a bond price elasticity with respect to change in yield.
Macaulay’s defined duration

Duration=d1= 1/P0(1cF/1+i+2cF/(1+i)2_ _ _ _ _ _+McF/(1+i)m+MF/(1+i)m

Macualey’s duration -1)as the weighted average payment date of the cash flow from the bond where the weight are the present value of the cash payment at the yield to maturity by the payment dates 1,2_ _ _ _ _ _up to M
2)Second interpretation of duration -This interpretation of duration is the elasticity of bond prices with respect to changes in gross yields
Elasticity can be defined as

d1=dpdi(1+i)p=dpp/di(1+i)=percent change ppercent change(1+i)                                

This duration number measures the interest rate sensitivity of the bond.
Calculation of duration
There is two method of duration calculation
1)Using macaulay’s definition
2)Calculating two bond prices at two different yield and dividing the percent bond price change by the percent gross yield change.
Calculation according to Macaulay’s formula
Each coupan payment date must be multiplied by the present value of the associated coupan payment or payment of principal .A four year U.S. treasury is used the 8.5 percent coupan in july 2000.
Example 8.5 percent U.S.treasury note maturing july 2000 priced  percent of 22 power of 113/32 face value on july 13 1995,to yield 4.71 percent to maturity.

                                                                     Macaulay’s formula

Date of payment Payment period Cash amount Present value Present value times payment period
1/2000 0.5 42.5 41.53 20.77
7/2000 1 42.5 40.59 40.59
1/2001 1.5 42.5 39.67 59.50
7/2001 2 42.5 38.77 77.53
1/2002 2.5 42.5 37.8 94.71
7/2002 3 42.5 37.02 111.07
1/2003 3.5 42.5 36.18 126.64
7/2003 4 1042.4 867.34 3469.37
Column total 1138.99 4000.17

Macaulay’s Duration =3.51 =4000.17/1138.99
                                                   B.Ratio of percent changes in prices

Price at 4.71 percent yield to maturity =$1138.99
Price at 5.71 percent yield to maturity =$1098.68

d1  =($1138.99$1098.68)/$1138.99/.01/1.0471=.03539/.00955 =3.706



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