求教！有关修正久期(modified duration)

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<p>求教!</p><p>Modified duration等于　麦考利duration/(1+YTM/n)  </p><p>解释是为了消除一年多次计息的误差？</p><p>为什么呢？</p><p>请问能否给出数学推导呢？或者直观说下原因来帮助理解也非常感谢！</p>

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关键词：Duration Modified ration ATION ratio 麦考利 数学

修正久期就是债券价格对实际YTM的一阶导数。

P: price of bond

D: Macaulay duration

Dm: modified duration

r: yield

Ci: i'th coupon

In case of continuously compounded yield, [from the definition of duration] 

<1>     (dP/dr)*(1/P)= -D

then, for a small change in yield,

<2>     dP/P= -D*dr+O((dr)^2),

where the second term on RHS takes in account in convexity.

However, the above relation is not valid anymore in the case of yearly compounded yield. 

Here, we can simply prove that the Dm fits into<1> and <2>:

define Dm=D/(1+r/n), 

assume the simplest case, which n=1,

P=sum[Ci/((1+r)^i)],

where i is the number of years after the starting date Ci is paid. 

By the definition of duration, 

D=(1/P)*sum{[Ci/((1+r)^i)]*i},

then, take the derivative of D w.r.t. r and multiply by (1+r)/P, gives

(dP/dr)*[(1+r)/P]= -D,

or,

<1'>     (dP/dr)*(1/P)= -Dm,

and hence,

<2'>     dP/P= -Dm*dr+O((dr)^2,

which are valid for yearly compounded yield.

跪谢。。。

他求一阶导 是不是跟自由度有什么关系啊？