Duration, convexity and portfolio immunization

Transcription

1 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess Duraio, covexiy ad porfolio immuizaio Some priciples of bods prices As is kow, a bod s price is give by: C C C F + r r( + ( + + F If we derive ha wih respec o is parameers, C, F, r ad or, parially, we ca examie how is price is affeced. A aleraive mehod is o examie i usig umerical values. Amog he priciples, we ca ame he followig: (i) The higher he r, he lower is, because: C F Example (i): F,000, C 00, r 0.0,...0.0, 0 (ιι) The bod prices chage over ime, if r C F, because: Cl( + Fl( + ( C rf)l( + γ) r( + r( +

2 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess Example (ii): F,000, C 00, r(a) 0.0, r(b) 0.08, r(c) 0., 0, 5, 0, 5, 0 The followig able shows he bod prices for various & r r 0.0,000,000,000,000,000 r 0.08,000,079.85,34.9,7.8,96.36 r 0., This meas ha premium bods fall over ime, while discou bods icrease. Boh will reach F a 0. (iii) For similar ieres rae chages, he shor bods are less volaile compared o log bods If a corporaio issues loger bods, i will pay he give coupos per period, irrespecively if he ieres rae icreases (ad herefore he corporaio will pay less) or falls (ad cosequely he corporaio will pay more). I addiio, if he ieres rae falls, i would be beer if i could issue ew oes a lower coupos. Tha would be easier if i had shor bods o replace ad raher expesive if i had loger bods o replace. This ca explai why he shor bods are, relaively, risk free. Example (iii): F,000, C 00, r 0.0,..., 0.0,,..., 0 The followig hree-dimesioal graph shows clearly ha he shorer bods are less volaile ha he loger oes, for all chages i yield o mauriy.

3 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess (iv) The more freque he coupo paymes, he lower he volailiy if he yield chages This is due o he fac ha he ivesor who receives C ofe, she ca save ha o a higher ieres rae, if i icreases. Such a opio does o exis if C is paid seldom. A similar argume applies if he ieres rae declies. Example (iv): F,000, C A 00 (per yea, C B 00 (every secod yea,, r 0.05, 0.0, 0.0. The followig able shows i. r 0.05 r 0.0 r 0.0 Volailiy Α,09.97, % Β,097.5, % (v) There exis a liear (posiive) relaioship bewee a bods price ad is coupo, as for similar bods wih same ime o mauriy, because: C r r( + > 0 Example (v): F,000, C 0,...,60, r 0.0, 0 C 0 ( + 0.) 0. 0.( + 0.) , ad, if C ( + 0.) This implies ha C. (vi) The %-chage of he large coupo bod is lower ha of he small coupo bod. Example (vi): F 00, C A 0, C B, 5, r 0.08, 0.0, 0.. 3

4 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess r 0.08 r 0.0 r 0. Volailiy Α % Β % Why? The aswer is foud i he bods duraio. Duraio From he priciple (i) above, i is clear ha whe he ieres rae icreases (decreases) he price of a bod decreases (icreases), oher higs beig uchaged. This egaive relaioship hough is more complicaed. O oe had, he price of a bod falls as he ieres rae icreases, because he coupo paymes are worh less. O he oher had, from priciple (vi) we ca argue ha hese paymes ca be reivesed a he higher ieres rae. Thus, a some poi, hese wo effecs will cacel each oher. The ime where hese effecs cacel each oher is called a bod s duraio (Du. There are various mehods o defie ad esimae he duraio. Below we will defie i wih wo aleraive (ad ideical) formulas. Boh defiiios are plausible if coupos are cosa ad are paid a specific periods (ofe per year or per half yea. Duraio as elasiciy We showed above ha: C F () We ca express ha derivaive as elasiciy, by muliplyig i wih: ( +. The produc becomes he bod s duraio. 4

5 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess + ( C F ( ( C + F Dur () This formula shows he perceage chage of a bod s price caused by a perceage chage of ieres rae, ad is measured i years. Corary from he bod s legal life o mauriy, i.e., he duraio shows he bod s ecoomic life. The ecoomic life is always shorer ha he legal life for all o zero-coupo bods. The ecoomic ad legal lives coicide for he zero-coupo bods, because boh such bods pay oly pricipal a mauriy day. Example: F,000, C 30, r 0.0, 4. Is price is,095.0, ad -3, r Therefore, is duraio is: Dur ( 3,375.75) years.,095.0 If isead, C 0, is price is 683.0, ad -,483.68, r implyig duraio of: Dur (,483.68). 4 years Duraio as average ime of all cash flows uil mauriy Accordig o his mehod, he duraio is defied as: Dur w, where, w discoued cash flows. ( F + C ), i.e., weighs based o For he same example we obai he same duraio as he able below shows. 5

6 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess eriod F + C (F+C)(.) - w (3)/ Dur()*(4) () () (3) (4) (5) , Sum, A impora advaage wih duraio is ha i provides he same iformaio as hree variables (r, C ad ). Isead of checkig all of hem, whe bod prices are compared, i is sufficie o compare heir duraio. A disadvaage wih duraio hough is ha i requires horizoal yield curve, i.e. shor ad log ieres raes a he same level. Duraio ad volailiy The duraio ca be used o esimae a bod s volailiy (σ). The formula () ca be rewrie as: ( + r ) Dur Dur (3) ( + r ) Tha shows how a bod s price chages, if he ieres rae chages (give is Dur ad ). For he same example, we ca esimae is volailiy if he ieres rae icreases, or falls wih 0.5 perceage uis , (.0), i.e. a ew price of, ,078., ad, 3.39( ) 0.005, , i.e. a ew price of, (.0) 6.87,.97.,.97,078. Thus he bod s volailiy is: σ ,095. 6

7 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess Volailiy ca also be defied as: σ Dur (4) ( + Tha formula provides of course he same volailiy because, σ % (.). If we derive (4) wih respec o Dur, we obai: V volailiy. > 0, i.e., he loger he duraio, he higher he riciples of Duraio (i) < 0, ad, > 0 Thus, he duraio falls as ime passes (which is obvious), a dimiishig rae (i.e., i is covex). The bod porfolios mus herefore be rebalaced, if oe wishes o keep a give duraio. (ii) < 0, ad, > 0 r r The duraio falls a higher ieres raes, a dimiishig rae, because he mos rece coupo paymes are discoued by higher ieres raes ad coribue herefore o a lower weigh. I our example, if r chages from 0.0 o 0.5, he duraio falls from 3.39 o approximaely 3.3 years. (iii) > 0, ad, < 0 Log bods have obviously loger duraio, eve if he duraio rae declies. Tha depeds o he fac ha duraio is iflueced by oher parameers as well ad o oly o he ime o mauriy. (iv) < 0, ad, > 0 C C 7

8 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess The duraio falls for higher coupo paymes, a dimiishig rae, because he higher he coupo, he higher he weigh ha he firs coupos receive. We showed previously ha zero-coupo bods have he loges duraio. Thus, if oe wishes o keep a log duraio, should seek for bods payig low coupos. Covexiy Whe he duraio is expressed as elasiciy, i would lead o rough approximaios for large chages i he ieres raes. Addiioal measures are herefore eeded, o ake io cosideraio boh he curvaure ad he sesiiviy of ieres rae chages. Such a measure which is ofe used is he covexiy (Co). The secod derivae of price wih respec o ieres rae is give from (5): ( + ) C + + ( + ) F + 0 (5) As wih he case of duraio, we ca also express his derivaive as ( + elasiciy, by muliplyig i wih. The oucome, see below, is covexiy. ( ) + r ( + ) C ( + ) C + ( + ) F + + ( + ) F + Co (6) Sice covexiy ad duraio are relaed posiively o each oher, he same priciples apply as for duraio. For isace higher ieres raes or coupo paymes would lead o lower covexiy, while loger bods would lead o higher covexiy. Higher covexiy should be preferred o lower covexiy, if wo bods have he same duraio, as he followig figure shows. If he 8

9 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess ieres rae icreases, he bod s price wih he higher covexiy will fall less. Similarly, if he ieres rae declies he same bod will icrease more, compared o he bod wih lower covexiy. Bods wih similar duraio bu differe covexiies High Low 0 T r0 r Immuizaio of ieres rae chages Very ofe, pesio fuds ad isurace compaies seek afer bods wih loger duraio, because heir liabiliies (obligaio paymes) ake place i he fuure. O he oher had, bods wih log mauriies are more volaile i ieres rae chages. They mus herefore fid a bod porfolio ha immuizes (euralises) hese effecs. Obviously, a bod porfolio s fuure value depeds o he yield curve ha prevails uil he mauriy day. If he fuure value of liabiliies is equal o he fuure value of bod porfolio a every poi of ime, he obligaios are fully immuized, o maer if he yield curve shifs upward or dowward. A simple immuizaio sraegy o achieve ha is o equae he bod porfolio s duraio (ad covexiy) wih he liabiliies duraio (ad covexiy). Le us examie his sraegy, wih he help of duraio oly. 9

10 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess Example: Compay Alfa ows a machie ha res i o Bea, over a period of 8 years. Bea pays yearly re paymes (coupos) of m. SEK; r %. Alfa s V 8 milllio (.) 9.94 millio V ( ) Because, C millio. is duraio is: Dur - (-34.65) 3. 9 years (.) 34.65, Alfa ca secure his V 9,94 m. SEK, by issuig, for isace, wo bods, a zero-coupo for year (wih a V equal o X) ad a 6- year level coupo bod (wih a V equal o Y), so ha he weighed average duraio will be equal o 3.9 years. For simpliciy, assume ha he yield of he 6-year bod is also equal o%. Obviously, X + Y Ay liear combiaio of X, such as.9 m. ad Y, such as 8.03 m., mus herefore lead o he followig weighed average duraio of 3.9 years, i.e.,.9 Dur x Dur 9.94 y years. Sice Dur x, Alfa mus issue a 6-year level coupo bod Y, wih Dur y 4.6 years. Oher combiaios ha he above, for isace 5 % a -year bod (i.e.,485 m. SEK) ad 75 % a 6-year bod (i.e. 7,455 m. SEK.), would icrease Dur y o 4.86 years. If he ieres rae icreases o 3 %, from (3) we obai: D ( 3.9) ,. i.e. a ew V If he 6-year bod has he same duraio as before (4.6 years), ad he proporios remaied uchaged (i.e. o rebalace of porfolio akes place), hese bods, immuize almos perfecly. 0

11 //004 Chrisos apahrisodoulou, Mälardale Uiversiy, School of Busiess Aleraively, small chages are eeded o immuize perfecly. For isace, he ew proporios mus be: X (9.597 ) + Y (4.6) X X + (4.6) 3.9 X ad Y Noice ha hese proporios are almos equal o he previous oes, sice: Aoher possibiliy is of course o decrease he duraio, or o ry o reegoiae he corac ad re he machie o less ha 8 years.