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

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Presentation on theme: "V: Bonds 16: Immunization. Chapter 16: Immunization © Oltheten & Waspi 2012 Immunization  Concept  The sad story of John Q. Investor  How to immunize."— Presentation transcript:

1 V: Bonds 16: Immunization

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

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

4 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

5 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

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

7 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%

8 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 $1070 P=$? ?

9 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?

10 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

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

12 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:

13 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:

14 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:

15 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

16 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

17 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

18 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

19 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?

20 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

21 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

22 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

23 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

24 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

25 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

26 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

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

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

29 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

30 Bonds IV

31 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

32 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

33 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

34 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

35 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

36 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

37 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

38 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

39 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

40 Return

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