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V: Bonds 16: Immunization

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Chapter 16: Immunization © Oltheten & Waspi 2012 Immunization Concept The sad story of John Q. Investor How to immunize a portfolio

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Chapter 16: Immunization © Oltheten & Waspi 2012 Bond Portfolio A portfolio of bonds is at risk from a change in market yields

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Chapter 16: Immunization Price/Reinvestment Risk Yields Increase Bond Prices Fall Reinvest at a higher rate Yields Decrease Bond Prices Rise Reinvest at a lower rate © Oltheten & Waspi 2012

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Chapter 16: Immunization © Oltheten & Waspi 2012 Immunization A portfolio of bonds is immunized if it is structured such that, for any change in yields, the price effect and the reinvestment effect cancel each other out. Duration Assets = Duration Liabilities

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Chapter 16: Immunization The sad story of John Q. Investor © Oltheten & Waspi 2012

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Chapter 16: Immunization © Oltheten & Waspi 2012 Client Requires $1,000,000 after two years Requires that the fund invest in only annual coupon bonds Three year 8% bonds Rolling one year 7% bonds Both come in denominations of $1000. Market Yield is 10%

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Chapter 16: Immunization Environment © Oltheten & Waspi 2012 8% bond $80$1080 $80 P=$ 7% bond $1070 P=$ 7% bond $1070 Liability $0 $1,000,000NPV=$ 7% bond $1070 P=$? ?

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Chapter 16: Immunization © Oltheten & Waspi 2012 John Q. Investor The firm for which he works has a Money Market account that pays the current rate (10%) John was not permitted to put the $826,446.28 in the firm’s Money Market account. Why not?

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Chapter 16: Immunization © Oltheten & Waspi 2012 John Q. Investor Calculates NPV of $1,000,000 @ 10% $826,446.28 Purchases 870 8% bonds at $826,726.20 © Oltheten & Waspi 2012

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Chapter 16: Immunization Back of the Napkin Calculations … © Oltheten & Waspi 2012

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Chapter 16: Immunization © Oltheten & Waspi 2012 If Yields stay at 10% … YEAR 0If Yields = Purchase 870 8% bonds YEAR 110% 870 * $80 coupons YEAR 210% $69,600 after one year in the money market account 870 * $80 coupons Sale of 870 8% bonds with 1 year to maturity at $981.82 per $1000 bond TOTAL PORTFOLIO After Trust disbursement:

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Chapter 16: Immunization © Oltheten & Waspi 2012 If Yields decrease to 9% YEAR 0If Yields Purchase 870 8% bonds YEAR 19% 870 * $80 coupons YEAR 29% $69,600 after one year in the money market account 870 * $80 coupons Sale of 870 8% bonds with 1 year to maturity at $990.83 per $1000 bond TOTAL PORTFOLIO After Trust disbursement:

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Chapter 16: Immunization © Oltheten & Waspi 2012 If Yields increase to 11% YEAR 0If Yields Purchase 870 8% bonds YEAR 111% 870 * $80 coupons YEAR 211% $69,600 after one year in the money market account 870 * $80 coupons Sale of 870 8% bonds with 1 year to maturity at $972.97 per $1000 bond TOTAL PORTFOLIO After Trust disbursement:

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Chapter 16: Immunization © Oltheten & Waspi 2012 Back of the Napkin Calculations … YEAR 0If Yields If Yields =If Yields Purchase 870 8% bonds-$826,726.20 -$826,726.200 YEAR 19%10%11% 870 * $80 coupons$69,600.00 YEAR 29%10%11% $69,600 after one year in the money market account $75,864.00$76,560.00$77,256.00 870 * $80 coupons$69,600.00 Sale of 870 8% bonds with 1 year to maturity 870 * 990.83 = $862,022.10 870 * 981.82 = $854,183.40 870 * 972.97 = $846,483.90 TOTAL PORTFOLIO $1,007,486.10$1,000,343.40$993,339.90 After Trust disbursement:$7,486.10$343.40- $6,660.10

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Chapter 16: Immunization © Oltheten & Waspi 2012 Back of the Napkin Calculations … YEAR 0If Yields If Yields =If Yields Purchase 870 8% bonds-$826,726.20 -$826,726.200 YEAR 19%10%11% 870 * $80 coupons$69,600.00 YEAR 29%10%11% $69,600 after one year in the money market account - $696.00 $76,560.00 + $696.00 870 * $80 coupons$69,600.00 Sale of 870 8% bonds with 1 year to maturity 870 * + $9.01 = + $7,838.70 870 * 981.82 = $854,183.40 870 * - $8.85 = - $7,699.50 TOTAL PORTFOLIO $1,007,486.10$1,000,343.40$993,339.90 After Trust disbursement:$7,486.10$343.40- $6,660.10

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Chapter 16: Immunization Duration of the Liability © Oltheten & Waspi 2012 TCash FlowNPVNPV/PT*NPV/P 1$0$0.000%0.000 2$1,000,000.00$826,446.28100%2.000 Present Value = $826,446.28100%2.000 years

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Chapter 16: Immunization Duration of the Assets © Oltheten & Waspi 2012 TCash FlowNPVNPV/PT*NPV/P 1$80$72.737.654%0.07654 2$80$66.126.958%0.13915 3$1080$811.4285.388%2.56167 Present Value = $950.26100%2.78 years TCash FlowNPVNPV/PT*NPV/P 1$1070$972.73100.000%1.00000 Present Value = $972.73100%1.00 years

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Chapter 16: Immunization © Oltheten & Waspi 2012 Duration Assets = Duration Liabilities Liability: 2.00 years Assets:2.78 years (Three year 8% bond) 1.00 year (One year 7% bond) Can we combine these assets in such a way that the portfolio duration equals 2 years?

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Chapter 16: Immunization © Oltheten & Waspi 2012 Immunization Let x = % of the portfolio invested in 8% bonds 1-x = % of the portfolio invested in 7% bonds Duration Assets = Duration Liabilities

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Chapter 16: Immunization © Oltheten & Waspi 2012 The Immunized Portfolio 8% bonds: 0.5618 * $826,446.28 = 488.6 bonds $950.26 7% bonds: 0.4382 * $826,446.28 = 372.3 bonds $972.73 We must begin with at least $826,446.28

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Chapter 16: Immunization © Oltheten & Waspi 2012 The Immunized Portfolio Purchase 489 8% bonds and 372 7% bonds for a total invested portfolio of $826,532.70 All cash (Coupons and matured bonds) received at the end of year 1 will be reinvested in new one year 7% bonds. © Oltheten & Waspi 2012

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Chapter 16: Immunization The Immunized Portfolio – year 0 © Oltheten & Waspi 2012 YEAR 0If Yields If Yields =If Yields Purchase 489 8% bonds @ $950.26 -$464,677.14 Purchase 372 7% bonds @ $972.73 -$361,855.56 TOTAL PORTFOLIO -$826,532.70

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Chapter 16: Immunization © Oltheten & Waspi 2012 The Immunized Portfolio – year 1 YEAR 19%10%11% 489 * $80 coupons$39,120.00 372 7% bonds mature @ $1070$398,040.00 Total Cash Received$437,160.00 Price of the new 7% bonds/$1000$981.65$972.73$963.96 Number of 7% bonds to purchase445449453 Purchase of 7% bonds-$436,834.25-$436,755.77-436,673.88 Residual cash$325.75$404.23$486.12

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Chapter 16: Immunization © Oltheten & Waspi 2012 The Immunized Portfolio – year 2 YEAR 19%10%11% Number of 7% bonds maturing445449453 7% bonds mature @ $1070$476,150.00$480,430.00$484,710.00 489 * $80 coupons$39,120.00 Price of 8% bonds with 1 year to maturity per $1000 $990.83$981.82$972.97 Sale of the 489 8% bonds$484,515.87$480,109.98$475,782.33 Residual cash + interest $325.75 $29.32 $404.23 $40.42 $486.12 $53.47 Total Portfolio Cash$1,000,140.94$1,000,104.63$1,000,151.92 After Trust Disbursement$140.94$104.63$151.92

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Chapter 16: Immunization © Oltheten & Waspi 2012 The Immunized Portfolio – year 2 YEAR 29%10%11% Number of 7% bonds maturing-4449+4 7% bonds mature @ $1070 - $4,280.00 $480,430.00 + $4,280.00 489 * $80 coupons$39,120.00 Price of 8% bonds with 1 year to maturity per $1000 $990.83$981.82$972.97 Sale of the 489 8% bonds + $4,405.89 $480,109.98 - $4,327.65 Residual cash$325.75 - $11.10 $444.23 $40.42 $486.12 + $13.05 TOTAL CASH RECEIVED$1,000,140.94$1,000,104.63$1,000,151.92 After Trust Disbursement$140.94$104.63$151.92

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Chapter 16: Immunization © Oltheten & Waspi 2012 Question 16-4 © Oltheten & Waspi 2012

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Chapter 16: Immunization 16 -4: Liability © Oltheten & Waspi 2012 $250,000

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Chapter 16: Immunization 15 - 2: Duration of the Liability © Oltheten & Waspi 2012 TCash FlowNPVNPV/PT*NPV/P 2$250,000$240,292.20.26500.5301 4$250,000$230,961.36.25471.0190 6$250,000$221,992.85.24491.4691 8$250,000$213,372.59.23531.8828 P = $906,619.001.00004.9010 semis = 2.4505 yrs

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Bonds IV

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Chapter 16: Immunization © Oltheten & Waspi 2012 Calculator Techniques Use the memory keys to store intermediate numbers. Faster because you are not spending all your time writing down and typing in numbers Minimizes typos Minimizes rounding errors

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Chapter 16: Immunization © Oltheten & Waspi 2012 Calculate Net Present Values 250000 [÷] 1.02 [x 2 ][=][STO][2] Stores 240,292.1953 into memory slot 2 [÷] 1.02 [x 2 ][=][STO][4] Stores 230,961.3565 into memory slot 4 [÷] 1.02 [x 2 ][=][STO][6] Stores 221,992.8455 into memory slot 6 [÷] 1.02 [x 2 ][=][STO][8] Stores 213,372.5929 into memory slot 8

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Chapter 16: Immunization © Oltheten & Waspi 2012 Calculate Net Present Value [RCL][2] [+] [RCL][4] [+] [RCL][6] [+] [RCL][8] [=] Calculates the NPV = 906,618.99 [STO][0] Store the total into memory slot 0

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Chapter 16: Immunization © Oltheten & Waspi 2012 Calculate Duration [RCL][2] [÷] [RCL][0] [x] 2 [=] [STO][2] Calculates the duration of the first payment at 0.530084187 and stores it back in slot 2 [RCL][4] [÷] [RCL][0] [x] 4 [=] [STO][4] Calculates the duration of the second payment at 1.019000745 and stores it back in slot 4

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Chapter 16: Immunization © Oltheten & Waspi 2012 ….. continued [RCL][6] [÷] [RCL][0] [x] 6 [=] [STO][6] Calculates the duration of the third payment at 1.469147556 and stores it back in slot 6 [RCL][8] [÷] [RCL][0] [x] 8 [=] [STO][8] Calculates the duration of the first payment at 1.882798354 and stores it back in slot 8

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Chapter 16: Immunization © Oltheten & Waspi 2012 Calculate Duration [RCL][2] [+] [RCL][4] [+] [RCL][6] [+] [RCL][8] [=] Calculates the Duration in semi-annual periods = 4.901030842 [÷][2] Calculates duration = 2.45 years

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Chapter 16: Immunization © Oltheten & Waspi 2012 Calculate Convexity [RCL][2] [x] 3 [=][STO][2] Calculates the convexity component of the first payment at 1.590… and stores it back in slot 2 [RCL][4] [x] 5 [=][STO][4] Calculates the convexity component at 5.095… and stores it back in slot 4

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Chapter 16: Immunization © Oltheten & Waspi 2012 …. continued [RCL][6] [x] 7 [=][STO][6] Calculates the convexity component at 10.284… and stores it back in slot 6 [RCL][8] [x] 9 [=][STO][8] Calculates the convexity component at 16.945… and stores it back in slot 8

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Chapter 16: Immunization © Oltheten & Waspi 2012 Calculate Convexity [RCL][2] [+] [RCL][4] [+] [RCL][6] [+] [RCL][8] [=] Calculates the Convexity in semi-annual periods 2 = 33.914… [÷][4] Calculates convexity = 8.48 years 2

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