Mathematical methods for finance (preparatory course) Simple numerical examples on bond basics

Transcription

1 Mahemaical mehods for finance (preparaory course) Simple numerical examples on bond basics

2 . Yield o mauriy for a zero coupon bond = = 92 days (=0.252 yrs) Face value = 00 r % 2

3 2. Diry price vs. clean price Bond wih an annual 6% coupon (wo 3% semiannual paymens) aymen daes: April s, Ocober s Dae: May s ( monh afer las coupon paymen) Clean price = Accrued ineres = 3 /6 = 0.50 Diry price = =

4 3. Yield o mauriy The yield such ha he presen value of fuure cashflows equals he diry price of he bond n ( YTM) F F represen fuure cashflows a mauriy is he diry price of he bond 4

5 3. Yield o mauriy (con d) Example Dae 0/0/20XX Clean price Coupon 6 Accrued in Las coupon paymen 0/0/20XX Diry price ( x) ( x) 3 ( x) Cashflows aymen daes 0/04/20XX+ 0/0/20XX+ 0/04/20XX+2 0/0/20XX+2 0/04/20XX+3 CF mauriy (yrs) Sum CF presen value ( x) Solving for x we obain ha x (Yield o mauriy) = 3.800% 2 03 ( x) ( 3.80%) ( 3.80%) 3 ( 3.80%).5 3 ( 3.80%) 2 03 ( 3.80%) 2.5 lease noe ha he ha maers is he DIRTY price (no he clean) 5

6 3. Yield o mauriy (con d) Example 2 Valuaion dae differen from coupon dae Dae 0//20XX Clean price Coupon 6 Accrued in Las coupon paymen 0/0/20XX Diry price Cashflows aymen daes 0/04/20XX+ 0/0/20XX+ 0/04/20XX+2 0/0/20XX+2 0/04/20XX+3 CF mauriy (days) CF mauriy (yrs) Sum CF presen value ( 3.90%) ( 3.90%) ( 3.90%) ( 3.90%) ( 3.90%) Yield o mauriy = 3.900% 6

7 4. Ineres rae ris: price vs reinvesmen ris Ineres componen Coupons Realized H reurn Capial gain/loss roceedings from reinvesmen urchase price Face value or sale price For a bond wih a fixed coupon, he uncerainy on oal ex pos reurn derives from: - proceedings from coupon reinvesmen (reinvesmen ris) - sale price if he bond is sold before mauriy (price ris) Noe: opposie effec of up/down rae moves on he wo componens 7

8 5. Effecs of YTM changes 5.. YTM increases vs YTM decreases Example: ideal 4-yr bond wih annual 5% coupons iniial YTM = 5% (hence bond quoes a par, ie 00) YTM = 5% = 00 YTM = 4 % = D%=+3.63% YTM = 6 % = D%=-3.47% The effec of idenical YTM changes is asymmeric 8

9 5. Effecs of YTM changes 5.2. The level of iniial YTM YTM = 8% = YTM = 7 % = D%=+3.5% YTM = 3% = YTM = 9 % = D%=-3.36% The effec of YTM changes is sronger if iniial YTM is lower YTM = 2 % =.42 D%=+3.7% YTM = 4 % = D%=-3.54% 9

10 5. Effecs of YTM changes 5.3. Differen coupon raes YTM = 5% (iniial level) YTM YTM=4% YTM YTM = 6% 4-yr bond zero coupon 4-yr bond 3% coupon 4-yr bond 5% coupon = = D% = % = 92.9 = D% = % = 00 = D% = % = 79.2 D% = % = D% = % = D% = % Lower coupon higher reacion o idenical YTM changes 0

11 5. Effecs of YTM changes 5.4. Differen mauriies YTM = 5% (iniial level) YTM YTM=4% YTM YTM = 6% 2-yr bond 5% coupon 4-yr bond 5% coupon 6-yr bond 5% coupon = 00 = 0.89 D% = +.89% = 00 = D% = % = 00 = D% = % = 98.7 D% = -.83% = D% = % = D% = % Longer ime o mauriy higher reacion o idenical YTM changes

12 5. Effecs of YTM changes Summary Direcion of YTM changes rice iniial YTM level (-) change is lined o coupon rae (-) remaining ime o mauriy (+) All hese effecs can be capured by undersanding he wo ey indicaors of ineres rae ris: Duraion (and modified duraion) Convexiy 2

13 6. Duraion Duraion is he weighed average of he ime o mauriy of all bond cashflows in which weighs are represened by he presen value of each cashflow divided by he diry price of he bond D n Time o mauriy of each cashflow ( F YTM) resen value of each cashflow Sum of all cashflows presen values = diry price 3

14 6. Duraion Mauriy 4 yrs Annual coupon: 5 % YTM: 4.80 % Diry price: Sum Cashflows V CF V CF / (%) 4.74% 4.52% 4.3% 86.43% 00% [V CF /] Duraion Inuiively, duraion is higher when: YTM is lower (greaer weigh for higher mauriies) Coupon is lower (greaer weigh for higher mauriies) Time o mauriy is greaer 4

15 7. Duraion and he effecs of ineres rae changes The effec of a change in he yield o mauriy equal o DYTM is given by Modified Duraion (MD) D D DYTM YTM Example: if D=4.20 and YTM=5%... hen a +% in YTM (5% 6%) implies a 4% reducion in he bond s price D D YTM DYTM (6% 5%) 4 % 4% 5

16 7. Duraion and he effecs of ineres rae changes roof. Firs-order Taylor-McLaurin approximaion f ( x0 h) f ( x0) f '( x0) h can be applied o he price of he bond which is a funcion of he YTM: change in diry price (D) F n ( YTM) ( YTM0 DYTM) ( YTM0) d dytm DYTM rae shoc 6

17 7. Duraion and he effecs of ineres rae changes Duraion! 7 D YTM YTM F YTM YTM F YTM YTM F YTM F dytm d YTM F YTM F From n n n n n n ) ( ) ( ) ( ) )( ( ) ( ) (

18 7. Duraion and he effecs of ineres rae changes d D DYTM dytm D D DYTM YTM D D DYTM YTM bu d dytm ( finally!) YTM D Key assumpion: same change in discouning rae for all cashflows (as hey are all discouned a YTM) 8

19 8. Using duraion o esimae D Mauriy 4 yrs Annual coupon: 5 % YTM: 4.80 % Diry price: Sum Cashflows V CF V CF / (%) 4.74% 4.52% 4.3% 86.43% 00% [V CF /] Duraion Modified Duraion D YTM %

20 8. Using duraion o esimae D A. Effecs of a % increase D D DYTM 3.554% 3.554% YTM D 3.554% ' 97.3 B. Effecs of a % decrease Exac ' D D DYTM 3.554% 3.554% YTM D 3.554% ' Exac ' Noe: rue prices are underesimaed in boh cases 20

21 9. Using duraion and convexiy o esimae D The formula D D DYTM YTM which is based on a firs-order (linear) approximaion can be refined o a second-order approximaion as D D YTM where C is convexiy DYTM C Noe: convexiy effec is always posiive 2 n C YTM 2 DYTM F 2 ) ( YTM 2 2

22 9. Using duraion and convexiy o esimae D Example: calculaing convexiy 5% annual coupon - Time o mauriy 4 yrs 6% YTM Sum Cashflows V CF V % =V CF / 4.89% 4.6% 4.35% 86.6% V% (+ 2 )V % rice Duraion Convexiy 22

23 9. Using duraion and convexiy o esimae D Example: effec of a % ineres rae increase Duraion only Duraion + convexiy D D D 3.78 DYTM % 3.507% YTM.06 D YTM C DYTM 2 ( YTM % (%) ) 2 DYTM % 0.080% 3.427% Acual price change (repricing) ' D acual ' %

24 0. A few exra noes on convexiy Srongly lined o mauriy ( squared ) C=D+D 2 +dispersion of cash flows (i.e. given duraion, bonds, or porfolios, wih greaer cashflows dispersion have greaer convexiy) Is i always beer for a bond porfolio manager o have higher convexiy? 24

25 . Numerical examples abou he erm srucure: (a) YTM, zero and forward raes Year Yield* Discoun facor for year Forward rae (, +) Zero coupon rae Discoun facor Annual forward rae 3% 3,000% 0,97 2 3,20% 3,203% 0,939 3,407% 3 3,25% 3,254% 0,908 3,355% 4 3,26% 3,263% 0,879 3,292% 5 3,27% 3,273% 0,85 3,33% * assuming one annual coupon f S ( i ) i S i S 25

26 . Numerical examples abou he erm srucure: (b) boosrapping Zero coupon raes 0,5,50%,80%,5,95% 2 2,05% (.50%) 0.5 We also now ha a coupon bond wih mauriy = 2.5 years semiannual coupon =.25 has YTM = 2.5% and herefore =00.96 How can we derive he 2.5yrs zero coupon rae?.25 (.80%).25 (.95%).5.25 ( 2.05%) ( x) Known zero raes Solving for x we obain 2.2% (=2.5 yrs zero coupon rae) Unnown erm 26

27 27

28 2,5 6-monh USD Libor Spo and Forward as of 3/08/206 2,5 0,5 0 Spo Forward 28

29 2. Immunizaion and Babcoc's formula Inuiion: finding a holding period such ha he wo componens of reurn variabiliy can offse Realized H reurn Ineres componen Capial gain/loss Coupons roceedings from reinvesmen urchase price Face value or sale price 29

30 2. Immunizaion heorem and Babcoc's formula Babcoc's formula Y p =Y 0 + D Y H where: Y p is he yield over he holding period; Y 0 is he porfolio YTM before raes shoc; D is duraion; H is he holding period; DY is he change (parallel shif) in YTM 30