# Financial Mathematics

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1 Financial Mathatics A ractical Guid for Actuaris and othr Businss rofssionals B Chris Ruckan, FSA & Jo Francis, FSA, CFA ublishd b B rofssional Education Solutions to practic qustions Chaptr 7 Solution 7. Th bond pas annual coupons of 8 pr \$00 of par valu: Th pric of th bond pr \$00 of par is: t 00 CFt 8a Using this pric in th forula for odifid duration, w hav: t ( Ia) ModD In ordr to calculat th prsnt valu of th incrasing annuit-idiat, w first calculat th prsnt valu of an annuit-du: 4 (.05) a (.05) % (.05) ( Ia) % Using th prsnt valu of th incrasing annuit-idiat, w hav: ( Ia) ModD

2 Intrst Thor for Actuarial Studnts Solutions to practic qustions Chaptr 7 Solution 7. W ar not givn th copounding frqunc of th ild, but it is not ndd to answr th qustion. Using th rlationship btwn pric and odifid duration, w hav: ( )( ModD) (0.00)(7.0) 0.70% Whn th ild incrass to 6.0%, th pric dclins b 0.70%: 4.7( ) \$3.95 Solution 7.3 Sinc thr is onl on cash flow, w can calculat th odifid duration without calculating th pric of th bond: t 40 0(, 500)(.0) 0 ModD (, 500)(.0) (.0).0 Solution 7.4 Th bond pas siannual coupons of \$4 pr \$00 of par valu, and th siannual ild is 5%: and Th pric of th bond pr 00 of par is: t 00 CFt 4a 4(3.5460) Using th pric in th forula for odifid duration, w hav: t ModD ( Ia).05 (8.6488) Solution 7.5 Th Macaula duration of a zro-coupon bond is qual to its aturit: t tcf t 5, 000 MacD ,000 B rofssional Education

3 Solutions to practic qustions Chaptr 7 robabilit Thor for Actuarial Studnts Solution 7.6 Th odifid duration is th Macaula duration dividd b on plus th priodic ild: MacD 5 ModD Solution 7.7 Th odifid duration is : MacD ModD Using th rlationship btwn pric and odifid duration, w hav: ( )( ModD) ( )(9.8844) 0.494% If th ild falls to 8.95%, thn th pric incrass b approxiatl 0.494%: 8.( ) \$8.5 Solution 7.8 Sinc th coupon rat is qual to th bond s ild, th bond is pricd at par, and th Macaula duration is thrfor qual to th prsnt valu of a 30-ar annuit-du: 60 ( ) () () (.03) MacD a a.03 a (.03) 4.59 n Th odifid duration is : MacD 4.59 ModD Solution 7.9 Th onthl ffctiv ild is Th aount of th onthl pant is dnotd blow as t. Notic that w do not nd to dtrin th aount of th onthl pant to calculat th duration. Th first onthl cash flow occurs at ti t, th scond onthl cash flow occurs at ti t, and so on: t t t 80 t tcf t (.0) (.0) (.0) MacD t t t 80 (.0) (.0) (.0) t ( Ia) ( Ia) 5, % 80 % 5.44 t a a % 80 % 80 B rofssional Education 3

4 Intrst Thor for Actuarial Studnts Solutions to practic qustions Chaptr 7 Solution 7.0 Th ffctiv duration is 8.8: EffD 8.8 0( ) 7.79()(0.005) Solution 7. Th ffctiv duration is: EffD 4.0 0( ) (0.666)()(0.000) Solution 7. Th ffctiv convxit is 0.74: ( ) (0.666) EffC 0.74 ( ) (0.00) (0.666) 0 0 Solution 7.3 Sinc th ild incrass, th prcntag chang in pric is ngativ: ( ) ( )(Duration) (Convxit) (0.0075) (0.0075)(4.0) (0.74) % Th nw pric is \$98.4: Nw pric ( ) Solution 7.4 Th gnral xprssion for th pric of a prptuit-idiat paing \$ at th nd of ach ar is: ( ) whr: annual ffctiv ild It is a sipl attr to find th first and scond drivativs of th pric of th prptuit-idiat: '( ) ''( ) 3 4 B rofssional Education

5 Solutions to practic qustions Chaptr 7 robabilit Thor for Actuarial Studnts Now w can driv gnral xprssions for th odifid duration and convxit of th prptuit: '( ) ModD ( ) ''( ) Convxit ( ) Intrstingl, th odifid duration of a prptuit-idiat paing \$ at th nd of ach ar is qual to its pric: ric 0.0 ModD 0.0 Convxit Solution 7.5 Whn thr is just on cash flow, th currnt ild of th bond is not ndd in ordr to calculat Macaula duration and Macaula convxit: δ t tcf 5 t δ 5CF5 MacD 5 5δ CF5 δ t tcf 5δ t 5CF5 MacC 5 5δ CF5 For zro-coupon bonds, th Macaula convxit is th squar of th Macaula duration. Solution 7.6 Sinc th ild incrass, th prcntag chang in pric is ngativ: ( ) ( )(Duration) (Convxit) (0.0063) (0.0063)5.35 (39.9) % Solution 7.7 Th prsnt valu of th assts and th prsnt valu of th liabilitis ar: VA 7, , , , 000 VL 55, Th first iunization condition, that th prsnt valu of th assts ust qual th prsnt valu of th liabilitis, is satisfid. B rofssional Education 5

6 Intrst Thor for Actuarial Studnts Solutions to practic qustions Chaptr 7 Th Macaula duration of th assts and th Macaula duration of th liabilitis ar: MacDA 0.5(5) 0.5(5) 0 MacD 0 L Th scond iunization condition, that th duration of th assts b qual to th duration of th liabilitis, is satisfid. In Solution 7.5, w saw that th Macaula convxit of a zro-coupon bond is qual to th squar of its Macaula duration. Thrfor, th Macaula convxitis ar: Convxit of 5-ar bond 5 5 Convxit of 5-ar bond 5 5 MacCA 0.5(5) 0.5(5) 5 MacCL 0 00 Th third iunization condition, that th convxit of th assts ust b gratr than th convxit of th liabilitis is satisfid. Th copan s position is thrfor iunizd. Solution 7.8 Th total aount invstd ust qual th prsnt valu of th liabilit: 00,000 V L 56, Th Macaula duration of th liabilit is 5: MacD L 5 If x % of th \$56,74.69 is invstd in th 4-ar bond and ( x%) is invstd in th 0-ar bond, w us iunization condition # to solv for x %: MacD A MacD L (%)4 x ( x%)0 5 6% x 5 x% 83.33% and x% 6.67% Chcking to s if iunization condition #3 is satisfid: MacC MacC L A (4) 0.667(0) 30 Sinc th convxit of th assts is gratr than th convxit of th liabilitis, th copan s position is iunizd. To stablish its asst portfolio, th copan invsts \$47,85.58 in th 4-ar bond and \$9,457. in th 0-ar bond: , , , , B rofssional Education

7 Solutions to practic qustions Chaptr 7 robabilit Thor for Actuarial Studnts Solution 7.9 Modifid duration and ffctiv duration ar calculatd blow: t ModD 8 (7.9) 3(07.80).07 3 (.07) (.07) (.07) 0.37(.07) EffD.3 ( ) (0.37)()(0.0) 0 Th ratio of odifid duration to ffctiv duration is.67: Solution 7.0 Modifid duration is th drivativ of pric with rspct to th ild, dividd b pric: '( ) 700 ModD 7 ( ) 00 W ar not told th copounding frqunc of th 8% ild. Whn w ar not givn th copounding frqunc of a ild, it is custoar to assu that it has th sa copounding frqunc as th coupons. Whn, as in this cas, w ar also not givn th copounding frqunc of th coupons, it is custoar to assu that th ild is an annual ffctiv ild: MacD MacD ModD 7 MacD 7(.08) Th Macaula duration is B rofssional Education 7

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