V: Bonds 16: Immunization. Chapter 16: Immunization © Oltheten & Waspi 2012 Immunization Concept ...

V: Bonds

16: Immunization

Chapter 16: Immunization © Oltheten & Waspi 2012

Immunization

Concept The sad story of John Q. Investor How to immunize a portfolio


Bond Portfolio

A portfolio of bonds is at risk from a change in market yields


Price/Reinvestment Risk

Yields Increase

Bond Prices Fall

Reinvest at a higher rate

Yields Decrease

Bond Prices Rise

Reinvest at a lower rate


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


The sad story of John Q. Investor


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 2012

8% bond

$80 $1080$80P=$

7% bond

$1070P=$

7% bond$1070

Liability

$0 $1,000,000NPV=$

7% bond

$1070

P=$??

P=$??


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?


John Q. Investor

Calculates NPV of $1,000,000 @ 10% $826,446.28

Purchases 870 8% bonds at $826,726.20



Back of the Napkin Calculations …



If Yields stay at 10% …

YEAR 0 If Yields =

Purchase 870 8% bonds

YEAR 1 10%870 * $80 coupons

YEAR 2 10%$69,600 after one year in the money market account870 * $80 couponsSale of 870 8% bonds with 1 year to maturity at $981.82 per $1000 bond

TOTAL PORTFOLIO

After Trust disbursement:


If Yields decrease to 9%

YEAR 0 If Yields Purchase 870 8% bondsYEAR 1 9%870 * $80 coupons

YEAR 2 9%$69,600 after one year in the money market account870 * $80 couponsSale of 870 8% bonds with 1 year to maturity at $990.83 per $1000 bond

TOTAL PORTFOLIO



If Yields increase to 11%

YEAR 0 If Yields Purchase 870 8% bondsYEAR 1 11%870 * $80 coupons

YEAR 2 11%$69,600 after one year in the money market account

870 * $80 couponsSale of 870 8% bonds with 1 year to maturity at $972.97 per $1000 bondTOTAL PORTFOLIO




YEAR 0 If Yields If Yields = If Yields Purchase 870 8% bonds -$826,726.20 -$826,726.20 -$826,726.200YEAR 1 9% 10% 11%870 * $80 coupons $69,600.00 $69,600.00 $69,600.00

YEAR 2 9% 10% 11%$69,600 after one year in the money market account $75,864.00 $76,560.00 $77,256.00870 * $80 coupons $69,600.00 $69,600.00 $69,600.00Sale 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



YEAR 0 If Yields If Yields = If Yields Purchase 870 8% bonds -$826,726.20 -$826,726.20 -$826,726.200YEAR 1 9% 10% 11%870 * $80 coupons $69,600.00 $69,600.00 $69,600.00

YEAR 2 9% 10% 11%$69,600 after one year in the money market account - $696.00 $76,560.00 + $696.00870 * $80 coupons $69,600.00 $69,600.00 $69,600.00Sale 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


Duration of the Liability


T Cash Flow NPV NPV/P T*NPV/P

1 $0 $0.00 0% 0.000

2 $1,000,000.00 $826,446.28 100% 2.000

Present Value = $826,446.28 100% 2.000 years


Duration of the Assets



1 $80 $72.73 7.654% 0.076542 $80 $66.12 6.958% 0.13915

3 $1080 $811.42 85.388% 2.56167

Present Value = $950.26 100% 2.78 years


1 $1070 $972.73 100.000% 1.00000

Present Value = $972.73 100% 1.00 years



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?


Immunization

Let x = % of the portfolio invested in 8% bonds1-x = % of the portfolio invested in 7% bonds



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


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.



The Immunized Portfolio – year 0


YEAR 0 If Yields If Yields = If Yields Purchase 489 8% bonds @ $950.26

-$464,677.14 -$464,677.14 -$464,677.14

Purchase 372 7% bonds @ $972.73

-$361,855.56 -$361,855.56 -$361,855.56

TOTAL PORTFOLIO -$826,532.70 -$826,532.70 -$826,532.70



YEAR 1 9% 10% 11%489 * $80 coupons $39,120.00 $39,120.00 $39,120.00372 7% bonds mature @ $1070 $398,040.00 $398,040.00 $398,040.00Total Cash Received $437,160.00 $437,160.00 $437,160.00Price of the new 7% bonds/$1000 $981.65 $972.73 $963.96

Number of 7% bonds to purchase 445 449 453Purchase of 7% bonds -$436,834.25 -$436,755.77 -436,673.88

Residual cash $325.75 $404.23 $486.12



YEAR 1 9% 10% 11%Number of 7% bonds maturing 445 449 4537% bonds mature @ $1070 $476,150.00 $480,430.00 $484,710.00489 * $80 coupons $39,120.00 $39,120.00 $39,120.00Price 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 $475,782.33Residual 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 $1,000,151.92After Trust Disbursement $140.94 $104.63 $151.92



YEAR 2 9% 10% 11%Number of 7% bonds maturing -4 449 +47% bonds mature @ $1070 - $4,280.00 $480,430.00 + $4,280.00489 * $80 coupons $39,120.00 $39,120.00 $39,120.00Price 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 - $4,327.65Residual cash $325.75

- $11.10$444.23$40.42

$486.12+ $13.05

TOTAL CASH RECEIVED $1,000,140.94 $1,000,104.63 $1,000,151.92After Trust Disbursement $140.94 $104.63 $151.92


Question 16-4



16 -4: Liability


$250,000$250,000$250,000 $250,000


15 - 2: Duration of the Liability



2 $250,000 $240,292.20 .2650 0.5301

4 $250,000 $230,961.36 .2547 1.0190

6 $250,000 $221,992.85 .2449 1.4691

8 $250,000 $213,372.59 .2353 1.8828

P = $906,619.00 1.0000 4.9010 semis= 2.4505 yrs

Bonds IV


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


Calculate Net Present Values

250000 [÷] 1.02 [x2][=][STO][2] Stores 240,292.1953 into memory slot 2

[÷] 1.02 [x2][=][STO][4] Stores 230,961.3565 into memory slot 4




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


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


….. 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


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


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


…. continued






Calculate Convexity

[RCL][2][+] [RCL][4][+] [RCL][6][+] [RCL][8] [=]

Calculates the Convexity in semi-annual periods2 = 33.914…

[÷][4] Calculates convexity = 8.48 years2

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