V: Bonds 16: Immunization

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

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

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

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

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

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%

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

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, in the firm’s Money Market account.  Why not?

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

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

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 $ per $1000 bond TOTAL PORTFOLIO After Trust disbursement:

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 $ per $1000 bond TOTAL PORTFOLIO After Trust disbursement:

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 $ per $1000 bond TOTAL PORTFOLIO After Trust disbursement:

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

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

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

Chapter 16: Immunization Duration of the Assets © Oltheten & Waspi 2012 TCash FlowNPVNPV/PT*NPV/P 1$80$ % $80$ % $1080$ % Present Value = $ %2.78 years TCash FlowNPVNPV/PT*NPV/P 1$1070$ % Present Value = $ %1.00 years

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?

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

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

Chapter 16: Immunization © Oltheten & Waspi 2012 The Immunized Portfolio  Purchase 489 8% bonds and 372 7% bonds for a total invested portfolio of $826,  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

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

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

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

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

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

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

Chapter 16: Immunization : Duration of the Liability © Oltheten & Waspi 2012 TCash FlowNPVNPV/PT*NPV/P 2$250,000$240, $250,000$230, $250,000$221, $250,000$213, P = $906, semis = yrs

Bonds IV

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

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

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

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

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

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

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

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

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

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