債券定價(jià)原理的證明

1、C(1 r)2CFV(11) FV(1 _r)n (1 r)nC (7 r (1 r)n)_r)nTheorem 1: Bond prices move in versely to the yield to maturity of bonds.示導(dǎo)得:dP drFVFV1r1r(1 r)n r n (1 r)n)FV(1FV n22nn 1r (1 r)(1 r)(1 r) r n、 FV nn(1 r)FV nm(1 r)2n 1r (1 r)n 1r n (1 r)2£)r2(1 r)n 1即證明分子部分小于 0:(1 r)r n(1、n 1/ar) (1r rn)(1r)(1r

2、)n(1r rn)(1r)(1C：rC；r2(1r rn)(1r)(1nrIII)(1r rn)(1nrr nr2 III)(r)0所以有：dP0，即債券價(jià)格與到期收攵益率反向變動(dòng)。dr式子第二項(xiàng)小于因此要證明第一項(xiàng)小于0即可,0,I)Theorem 2. For a given change in yield to maturityfrom the nominal yield,the changeFV cFVr (1 r )n (1 r )n令 f (n)FV(1 r)nFV cr (1 r )nin bond prices are greater, the Ion ger is the t

3、erm to maturity.(1)如r變小，由原理1可知債券價(jià)格P會(huì)上升，此時(shí)債券價(jià)格變動(dòng)數(shù)值為:1P FV c (-r1FVn )nFVr (1 r)(1 r )當(dāng)c r時(shí)，P FV，當(dāng)r發(fā)生變動(dòng)時(shí)，P的大小與n的關(guān)系體現(xiàn)在對(duì)其求導(dǎo)得：f (n) FV r(1 r)上式的符號(hào)由c r決定，前面假設(shè)r變小，所以c r 0，因此f (n) 0。即n越大， P增加得越多。(2)如r變大，由原理1可知債券價(jià)格 P會(huì)下降，此時(shí)債券價(jià)格變動(dòng)數(shù)值為:P FV FV c (- r1_)FV r (1 r )(1 r )P的大小與n的關(guān)系體現(xiàn)在FV cr (1 r )nFV(1亍同樣令f(n)FV cFV

4、r (1 r )n (1 r )n對(duì)其求導(dǎo)得：f(n) FV (r c) ln(1 r )? r(1 r)由于前面假設(shè)r變大，所以rc 0,因此 f (n)0。即n越大，P增加得越多。Theorem 3: The percentagecha nge describedin theorem 2 in creasesat a diminishingrate as the time to maturity in creases.隨著時(shí)間的增加，P的數(shù)值越來越大，但增加的部分越來越少。即求其二階導(dǎo)。(1)如 r 變小， 此時(shí) f (n) FV (仝) ln(1 r ? >0, r (1 r )f

5、 (n)FV 1)r (1 r )如r變大，此時(shí)f(n) FV (匚) ln(1 r ? r (1 r )n2f (n)FVIn (1 r )(1 r )n因此原理3成立。Theorem 4: Price moveme nts result ing from equal absolute (or, what is the same, from equal proport ion ate) in creases and decreases in the yield to maturity areasymmetric; that is, a decrease in yield to maturit

6、y raises bond prices more than the same in crease in yield to maturity lowers prices.收益率相同的增加或減少時(shí)，債券價(jià)格的變動(dòng)是不對(duì)稱的。 收益率的減少導(dǎo)致的債券價(jià)格增 加超過收益率的增加導(dǎo)致的債券價(jià)格的減少。由原理dP1已知o，所以我們要證明drd2P dr2FVc(丄(1 r) r n)22 一 、n 1 丿FVFV n 2 n 1 丿 /r r (1 r) (1 r)2 (n 1)(1 r)r (1 r nr)(2 3r nr) FV n (n 1)c _r3 mFV3n 2n 2r (1 r)(1 r)

7、c 2(1 r)n 2 (n 1)(1 r)r (1 r nr)(2 3r nr) FV n (n 1) r33n 2r (1 r)上式的符號(hào)由分子決定，即FVc 2(1 r)n 2(n 1)(1r)r(1 r n r )(2 3r nr)FV n (n1) r3決定。這個(gè)式子中，取后-項(xiàng)大于0,因此只要部分大于0即可，該部分為：2(1n 2r)(n1)(1r)r (1rnr)(2 3r nr)2(1r)2(1 nrn(n 1) 2 r22 2)(nr nr r r ) (25r3nr 4nr2C222、3r nr)(24r 2nr3nr22r2 n2 2、r )2(2 4r 2nr 3nr2

8、r2n2r2)0d2P因此有： 70drTheorem 5: The higher is the coup on carried by the bond, the smaller will be the percentage price fluctuations for a given change in the yield to maturity except for one year securities and con sols.息票率越高，債券價(jià)格變化的百分比越小。設(shè)f(c)dPdrn 1FV c (1 r) r n (1 r)r2(1 r)n 1FV nn 1(1 r)故有:f(c)

9、 FV (1(1 r)n12n 1r (1 r)前面已經(jīng)證明(1 r) r n (1 r)n 10,故f (c)0因此，息票率越高，債券價(jià)格變化的幅度越小。設(shè)原來債券價(jià)格為 PFV c(1 1 -)r r (1 r)_FV(1n，后來債券收益率下降，則r)價(jià)格變?yōu)镻 FV cn)FV(1 r)n債券價(jià)格變動(dòng)百分比為:1。FV(r ) (1 r 廠(1 r)n11FVFV c (n)nr r (1 r) (1 r)n(1 1 、 1c (n)nr (r ) (1 r )(1 r )1(1 1 ) 1c (r r (1 r)n) (1 r)n令 f(c)1，f1(11，k1，k2(1 r )r(111，上式可化簡(jiǎn)為:r)f(c)