1. Interest Rate Gap. Duration
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1 . Ineres Rae Gap. Duraion Mauriy Gap Problem. Mauriy Gap A bank invess $00 million in 3-year, 0% fixed rae bonds (assume hese are all asses) In he same ime, i issuses $90 million in -year, 0% percen fixed rae bonds (assume hese are all liabiliies). Show he marke value of equiy assuming ineres raes change in he range 8-7 percen. (b) Is i he ineres rae risk eliminaed when mauriy of he issued bonds is 3 years? Soluion Asses Liabiliies Mauriy 3 Coupon rae 0% 0% Face value YTM Asses Liabiliies Equiy Equiy 8% 05,5 9,67 3,49 3,49 9% 0,53 90,83,7,7 0% 00,00 90,00 0,00 0,00 % 97,56 89,9 8,37 -,63 % 95,0 88,39 6,80-3,0 3% 9,9 87,6 5,3-4,69 4% 90,7 86,84 3,87-6,3 5% 88,58 86,09,50-7,50 6% 86,5 85,34,8-8,8 7% 84,53 84,6-0,08-0,08 (b) Asses Liabiliies Mauriy 3 3 Mauriy 0% 0% Face value YTM Asses Liabiliies Equiy Equiy 8% 05,5 94,64 0,5 0,5 9% 0,53 9,8 0,5 0,5 0% 00,00 90,00 0,00 0,00 % 97,56 87,80 9,76-0,4 % 95,0 85,68 9,5-0,48 3% 9,9 83,6 9,9-0,7 4% 90,7 8,64 9,07-0,93 5% 88,58 79,73 8,86 -,4 6% 86,5 77,87 8,65 -,35 7% 84,53 76,08 8,45 -,55
2 Mauriy gap is he difference beween he weighed average mauriies of asses and liabiliies. I may be greaer han, equal o or less han zero. A rise in ineres raes reduces he marke values of asses and liabiliies. If he mauriy of asses is longer han mauriy of liabiliies (mauriy gap is greaer han zero), he marke value of asses falls by more han he marke value of liabiliies. The equiy of he bank declines. Mauriy maching does no perfecly immunize or proec agains ineres rae risk. Mauriy gap is an incomplee measure of ineres rae risk. Duraion gap is much more accurae risk measure. Duraion Gap Duraion is he average life of an asse, or more exacly, he weighed average ime o mauriy using he relaive presen values of he cash flows as weighs. Duraion is measured in years. The modified duraion is a measure of he ineres sensiiviy of an asse s price. The larger is he modified duraion, he price of he asse is more sensiive. You can calculae he modified duraion (insrumen wih annual cash flows) using he general formula: () D = ( + i) T = CF ( + i) W EXCEL FUNCTIONS DURATION(selemen, mauriy, coupon, yld, frequency, basis) Reurns he annual duraion of a securiy wih periodic ineres paymens. Duraion is defined as he weighed average of he presen value of he cash flows, and is used as a measure of a bond price's response o changes in yield. MDURATION(selemen, mauriy, coupon, yld, frequency, basis) Reurns he modified Macauley duraion for a securiy wih an assumed par value of $00. MDURATION = DURATION / ( + marke yield/coupon paymens per year) Selemen is he securiy's selemen dae, expressed as a serial dae number. Mauriy is he securiy's mauriy dae, expressed as a serial dae number. Coupon is he securiy's annual coupon rae. Yld is he securiy's annual yield. Frequency is he number of coupon paymens per year. For annual paymens, frequency = ; for semiannual, frequency = ; for quarerly, frequency = 4. Basis is he ype of day coun basis o use. Basis Day coun basis 0 or omied US (NASD) 30/360 Acual/acual Acual/360 3 Acual/365 4 European 30/360
3 Problem. Duraion Consider a Eurobond wih an annual coupon ha pays $000 in 5 years. Required Calculae he duraion and he modified of he bonds wih following characerisics: I. Coupon 0% and YTM 0%. II. Coupon 0% and YTM 5%. III. Coupon 0% and YTM 5%. (b) Show he sensiiviy analysis of he modified duraion on changes in coupon rae and YTM in he range -0%. Soluion Coupon rae 0% YTM 0% Year Cash Flow Disc. facor DCF Weigh Year * Weigh 00 0,909 90,9 0,0909 0, ,864 8,64 0,086 0, ,753 75,3 0,075 0, , ,30 0,0683 0, , ,0 0,6830 3,45 000,00,0000 4,699 Duraion is 4,7 years. Modified duraion is 4,7 * /(+0%) = 3,79. Coupon rae 0% YTM 5% Year Cash Flow Disc. facor DCF Weigh Year * Weigh 00 0, ,96 0,045 0, ,756 75,6 0,0908 0, , ,75 0,0790 0, ,578 57,8 0,0687 0, , ,89 0,6570 3,85 83,39,0000 4,089 Duraion is 4,08 years. Modified duraion is 4,08 * /(+5%) = 3,55. Coupon rae 0% YTM 5% Year Cash Flow Disc. facor DCF Weigh Year * Weigh 00 0,954 95,4 0,0783 0, , ,70 0,0746 0, , ,38 0,070 0, ,87 8,7 0,0676 0, , ,88 0,7085 3,545 6,47,0000 4,535 (b) Sensiiviy analysis Coupon D YTM D % 4,388 % 4,75 0% 3,7908 0% 3,7908 0% 3,450 0% 3,378 3
4 Problem 3. Modified Duraion. Sensiiviy Analysis Modified Duraion. Sensiiviy Analysis Annual cash flows Quarerly cash flows Mauriy year Coupon rae Coupon rae 0% % 5% 0% 0% 00% 0% % 5% 0% 0% 00% Yield o mauriy 0%,00,00,00,00,00,00,00,00 0,98 0,97 0,94 0,8 Y Y Y Y Y % 0,99 0,99 0,99 0,99 0,99 0,99,00 0,99 0,98 0,96 0,93 0,8 Y Y Y Y 5% 0,95 0,95 0,95 0,95 0,95 0,95 0,99 0,98 0,97 0,95 0,9 0,80 Y Y 0% 0,9 0,9 0,9 0,9 0,9 0,9 0,98 0,97 0,96 0,94 0,9 0,79 Y 0% 0,83 0,83 0,83 0,83 0,83 0,83 0,95 0,95 0,93 0,9 0,89 0,76 Y 00% 0,50 0,50 0,50 0,50 0,50 0,50 0,80 0,79 0,78 0,76 0,7 0,59 Mauriy 0 years Coupon rae Coupon rae 0% % 5% 0% 0% 00% 0% % 5% 0% 0% 00% Yield o mauriy 0% 0,00 9,59 8,50 7,75 7,00 5,9 0,00 9,56 8,38 7,56 6,75 5,57 Y Y Y Y Y % 9,90 9,47 8,34 7,58 6,83 5,76 9,98 9,50 8,7 7,44 6,6 5,46 Y Y Y Y 5% 9,5 9,00 7,7 6,9 6,8 5,0 9,88 9,8 7,83 6,94 6, 5,03 Y Y 0% 9,09 8,43 6,96 6,4 5,44 4,58 9,76 8,96 7, 6,8 5,48 4,5 Y 0% 8,33 7,3 5,54 4,77 4,9 3,58 9,5 8, 5,85 4,94 4,9 3,6 00% 5,00,35,07,03,0,00 8,00,09,0,0,00,00 Y Y Y Y Mauriy 00 years Coupon rae Coupon rae 0% % 5% 0% 0% 00% 0% % 5% 0% 0% 00% Yield o mauriy 0% 00,0 75,5 58,75 55,00 5,86 50,99 00,0 75,06 58,44 54,66 5,50 50,6 Y Y Y Y Y % 99,0 63,03 47,9 45,09 43,55 4,6 99,75 63,7 47,88 45,0 43,47 4,7 Y Y Y Y 5% 95,4,07 9,85 9,56 9,4 9,30 98,77,00 9,86 9,59 9,45 9,34 Y 0% 90,9 0,05 0,0 0,00 0,00 9,99 97,56 0,04 0,00 0,00 0,00 0,00 Y Y 0% 83,33 5,00 5,00 5,00 5,00 5,00 95,4 5,00 5,00 5,00 5,00 5,00 Y Y Y 00% 50,00,00,00,00,00,00 80,00,00,00,00,00,00 Y Do high yield bonds have high or low duraions? Do high coupon bonds have high or low duraions? Does duraion is more sensiive o yield or coupon rae? Do more frequen cash flows always reduce modified duraion? Imporan feaures of he modified duraion (True or False). Duraion increases wih he mauriy.. Duraion increases as yield decreases. 3. Duraion increases as he coupon rae decreases. Table. Modified duraion and convexiy of bonds Bonds Modified duraion Convexiy Coupon bonds T B( c[ + i]( [ + i) ] + it[ i c] ) T (annual D = B c[ + i] ( + i) cit + T+ ineress) Pi ( + i) C = 3 T+ Pi ( + i) Zero coupon T T( T + ) bonds D = ( + i ) C = ( + i) Consol bonds D = C = i i where: P - price, i annualized yield, c coupon rae, B face value, T - mauriy. [ ] [ ] [ ] [ ] i c i + i T T + 4
5 Duraion Model W () = - D i W Duraion and convexiy model P (3) = - D i + C( i) P Macaulay Duraion Model W i Dz (4) = Dz = i W + i + i Dz (5) D = + i Effecive duraion and effecive convexiy V- V+ (6) D = V0 y V- V+ V0 (7) C = V y 0 ( ) ( P/P) Error True relaionship dp g α = di α Error Ineres rae (i) 5
6 Problem 4. Modified Duraion and Convexiy A bond is currenly selling for $950 and has 5 years lef o mauriy and a par value of $000. The bond has a 0% coupon (payable annually). Calculae he YTM, he duraion, he modified duraion and he convexiy. (b) Calculae he impac of a +-%, +-5% changes in ineres raes on he price of he bond. Use duraion, duraion and convexiy approximaion and he exac price using bond valuaion. Soluion Year Cash Flow DCF Weigh Duraion Convexiy ,35 0,095 0,095 0, ,63 0,0859 0,78 0, ,75 0,0776 0,39 0, ,63 0,070 0,805, ,0 0,0634 0,368, ,39 0,0573 0,3435, ,4 0,057 0,36, ,40 0,0467 0,3739, , 0,04 0,3800 3, ,4 0,038 0,385 3, ,74 0,0345 0,379 3, ,58 0,03 0,3737 3, ,73 0,08 0,3657 4, ,5 0,054 0,3558 4, ,98 0,56 3,789 49,4870 YTM P Duraion Convexiy 0,68% 950,00,0000 8,06 83,336 Inernal rae of reurn (YTM) is 0,68%. Duraion is 8,0 years. Modified duraion is 8,0 * /(+0,7%) = 7,4. Convexiy is 83,34. Ad. CF CF ( + i) P( + i) P( + i) Relaive change in price P/P i i Marke Price P = - D i - D i + C( i) P Price Formula -5% 5,68% 48,07 37,05% 47,47% 50,3% -% 9,68% 04,54 7,4% 7,83% 7,85% 0% 0,68% 950,00 0,00% 0,00% 0,00% %,68% 883,39-7,4% -6,99% -7,0% 5% 5,68% 678,37-37,05% -6,63% -8,59% CF CF ( + ) CF ( i) + + P Maching he duraion of an asse o he invesor s arge horizon immunizes i agains ineres rae risk. Duraion can be calculaed for asses or liabiliies as a marke value weighed average of he individual duraions of all iems. The duraion gap is jus a difference beween he duraion of asse porfolio and he duraion of liabiliy porfolio. 6
7 (8) D A = w A D A + w A D A w An D An (9) D L = w L D L + w L D L w Ln D Ln (0) A = - DA A y () L = - D L L y () E = A - L = [- DA A + DL L] y E = - D - D k A (3) [ ] y k = L A - is a leverage measure. E = - D - D A (4) [ ] y A A L P Problem 5. Duraion Gap A bank invess $00 million in 3-year, 0% percen fixed rae bonds (assume hese are all asses). In he same ime, i issuses $90 million in 0% percen fixed rae bonds (hese are all liabiliies). Calculae he appropriae duraion of is liabiliies o mach he duraion of asses. Wha should be he mauriy of he issued bonds? (b) Show he marke value of equiy assuming ineres raes change in he range 8-7 percen. Is i he ineres rae risk compleely eliminaed? Soluion The duraion of he bank's asses is,5. Duraion gap should be equal o 0. D A - kd L = 0 Solving for he duraion of liabiliies, D L =,78. The mauriy of he issued bonds may be calculaed using ieraions. If he duraion gap is zero, he mauriy of he issued bonds will be 3,8. (b) Asses Liabiliies Mauriy 3 3,8 Coupon 0% 0% Face Value YTM Asses Liabiliies Equiy Equiy D A D L k Gap 8% 05,5 95,0 0,4 0,4,5,79 0,9 0,00 9% 0,53 9,46 0,07 0,07,5,78 0,9 0,00 0% 00,00 90,00 0,00 0,00,50,78 0,9 0,000 % 97,56 87,63 9,93-0,07,49,77 0,9-0,00 % 95,0 85,34 9,85-0,5,48,76 0,9-0,00 3% 9,9 83,4 9,78-0,,47,75 0,9-0,003 4% 90,7 8,0 9,70-0,30,46,74 0,9-0,004 5% 88,58 78,97 9,6-0,39,45,73 0,9-0,005 6% 86,5 76,99 9,53-0,47,44,7 0,9-0,006 7% 84,53 75,09 9,45-0,55,44,7 0,9-0,006 Duraion maching does no fully proec agains ineres rae risk. 7
8 Repricing Gap Repricing gap is he difference beween hose asses whose ineres raes will be repriced (ineres rae sensiive asses) and liabiliies whose ineres raes will be repriced (ineres rae sensiive liabiliies) over some fuure period. I is used o calculae ne ineres income change due o ineres rae changes a differen mauriy buckes. Problem 6. Repricing Gap Use he following Ineres Rae Risk Reporing Schedule o answer quesions a and b. Sensiive (cumulaive) Nonsensiive Toal wihin M wihin 3M wihin 6M wihin R Asses Cash Shor-erm insrumens Invesmen securiies Loans Toal asses Liabiliies and Capial Demand deposis Shor-erm deposis Passbook savings CDs Public and oher deposis Shor-erm borrowing Shareholder equiy Toal liabiliies and capial Calculae repricing gap and iners-sensiiviy raio for all periods. (b) Calculae he impac on ne ineres income if ineres raes change +-% for he four repricing gaps. Soluion wihin M wihin 3M wihin 6M wihin R Ineres-sensiiviy gap Ineres-sensiiviy raio,04 0,9 0,95 0,86 (b) Change in ne ineres income, when R+% Change in ne ineres income, when R-%
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