V: Bonds
16: Immunization
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
© Oltheten & Waspi 2012
Price/Reinvestment Risk
 Yields Increase
 Bond Prices Fall
 Reinvest at a higher
rate
Chapter 16: Immunization
 Yields Decrease
 Bond Prices Rise
 Reinvest at a lower
rate
© 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
© Oltheten & Waspi 2012
The sad story of John Q. Investor
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 $1000.
 Market Yield is 10%
Chapter 16: Immunization
© Oltheten & Waspi 2012
Environment
Liability
NPV=$
$0
$1,000,000
$80
$80
8% bond
P=$
$1080
7% bond
P=$
Chapter 16: Immunization
$1070
7% bond
P=$?
?
$1070
7% bond
P=$?
?
$1070
© 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?
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
Chapter 16: Immunization
© Oltheten & Waspi 2012
© Oltheten & Waspi 2012
Back of the Napkin Calculations
…
Chapter 16: Immunization
© Oltheten & Waspi 2012
If Yields stay at 10% …
YEAR 0
Purchase 870 8% bonds
YEAR 1
870 * $80 coupons
YEAR 2
$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
If Yields =
10%
10%
TOTAL PORTFOLIO
After Trust disbursement:
Chapter 16: Immunization
© Oltheten & Waspi 2012
If Yields decrease to 9%
YEAR 0
Purchase 870 8% bonds
YEAR 1
870 * $80 coupons
YEAR 2
$69,600 after one year in the money market account
If Yields 
9%
9%
870 * $80 coupons
Sale of 870 8% bonds with 1 year to maturity at $990.83
per $1000 bond
TOTAL PORTFOLIO
After Trust disbursement:
Chapter 16: Immunization
© Oltheten & Waspi 2012
If Yields increase to 11%
YEAR 0
Purchase 870 8% bonds
YEAR 1
870 * $80 coupons
YEAR 2
$69,600 after one year in the money market account
If Yields 
11%
11%
870 * $80 coupons
Sale of 870 8% bonds with 1 year to maturity at
$972.97 per $1000 bond
TOTAL PORTFOLIO
After Trust disbursement:
Chapter 16: Immunization
© Oltheten & Waspi 2012
Back of the Napkin Calculations …
YEAR 0
Purchase 870 8% bonds
YEAR 1
870 * $80 coupons
YEAR 2
$69,600 after one year in the
money market account
870 * $80 coupons
Sale of 870 8% bonds with 1
year to maturity
TOTAL PORTFOLIO
After Trust disbursement:
Chapter 16: Immunization
If Yields 
-$826,726.20
9%
$69,600.00
If Yields =
-$826,726.20
10%
$69,600.00
If Yields 
-$826,726.200
11%
$69,600.00
9%
10%
11%
$75,864.00
$69,600.00
870 * 990.83 =
$862,022.10
$1,007,486.10
$76,560.00
$69,600.00
870 * 981.82 =
$854,183.40
$1,000,343.40
$77,256.00
$69,600.00
870 * 972.97 =
$846,483.90
$993,339.90
$7,486.10
$343.40
- $6,660.10
© Oltheten & Waspi 2012
Back of the Napkin Calculations …
YEAR 0
Purchase 870 8% bonds
YEAR 1
870 * $80 coupons
YEAR 2
$69,600 after one year in the
money market account
870 * $80 coupons
Sale of 870 8% bonds with 1
year to maturity
TOTAL PORTFOLIO
After Trust disbursement:
Chapter 16: Immunization
If Yields 
-$826,726.20
9%
$69,600.00
If Yields =
-$826,726.20
10%
$69,600.00
If Yields 
-$826,726.200
11%
$69,600.00
9%
10%
11%
- $696.00
$69,600.00
870 * + $9.01 =
+ $7,838.70
$76,560.00
$69,600.00
870 * 981.82 =
$854,183.40
+ $696.00
$69,600.00
870 * - $8.85 =
- $7,699.50
$1,007,486.10
$1,000,343.40
$993,339.90
$7,486.10
$343.40
- $6,660.10
© Oltheten & Waspi 2012
Duration of the Liability
T
1
2
Cash Flow
$0
$1,000,000.00
NPV
$0.00
$826,446.28
Present Value = $826,446.28
Chapter 16: Immunization
NPV/P
T*NPV/P
0% 0.000
100% 2.000
100% 2.000 years
© Oltheten & Waspi 2012
Duration of the Assets
T
Cash Flow
NPV
NPV/P
T*NPV/P
1
2
$80
$80
$72.73
$66.12
7.654%
6.958%
0.07654
0.13915
3
$1080
$811.42
85.388%
2.56167
Present Value = $950.26
100%
2.78 years
T
Cash Flow
NPV
NPV/P
1
$1070
$972.73
100.000%
1.00000
Present Value = $972.73
100%
1.00 years
Chapter 16: Immunization
T*NPV/P
© 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:
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
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.
Chapter 16: Immunization
& Waspi 2012
© Oltheten©&Oltheten
Waspi 2012
The Immunized Portfolio – year 0
YEAR 0
Purchase 489 8% bonds @
$950.26
If Yields 
-$464,677.14
If Yields 
If Yields =
-$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
Chapter 16: Immunization
© Oltheten & Waspi 2012
The Immunized Portfolio – year 1
YEAR 1
489 * $80 coupons
9%
$39,120.00
10%
$39,120.00
11%
$39,120.00
372 7% bonds mature @ $1070
$398,040.00
$398,040.00
$398,040.00
Total Cash Received
$437,160.00
$437,160.00
$437,160.00
Price of the new 7% bonds/$1000
$981.65
$972.73
$963.96
Number of 7% bonds to purchase
445
449
453
-$436,834.25
-$436,755.77
-436,673.88
$325.75
$404.23
$486.12
Purchase of 7% bonds
Residual cash
Chapter 16: Immunization
© Oltheten & Waspi 2012
The Immunized Portfolio – year 2
YEAR 1
Number of 7% bonds maturing
7% bonds mature @ $1070
489 * $80 coupons
Price of 8% bonds with 1 year to
maturity per $1000
Sale of the 489 8% bonds
Residual cash
+ interest
Total Portfolio Cash
After Trust Disbursement
Chapter 16: Immunization
9%
10%
11%
445
449
453
$476,150.00
$480,430.00
$484,710.00
$39,120.00
$39,120.00
$39,120.00
$990.83
$981.82
$972.97
$484,515.87
$480,109.98
$475,782.33
$325.75
$29.32
$404.23
$40.42
$486.12
$53.47
$1,000,140.94
$1,000,104.63
$1,000,151.92
$140.94
$104.63
$151.92
© Oltheten & Waspi 2012
The Immunized Portfolio – year 2
YEAR 2
Number of 7% bonds maturing
9%
10%
11%
-4
449
+4
7% bonds mature @ $1070
- $4,280.00
$480,430.00
+ $4,280.00
489 * $80 coupons
$39,120.00
$39,120.00
$39,120.00
$990.83
$981.82
$972.97
+ $4,405.89
$480,109.98
- $4,327.65
$325.75
- $11.10
$444.23
$40.42
$486.12
+ $13.05
$1,000,140.94
$1,000,104.63
$1,000,151.92
$140.94
$104.63
$151.92
Price of 8% bonds with 1 year to
maturity per $1000
Sale of the 489 8% bonds
Residual cash
TOTAL CASH RECEIVED
After Trust Disbursement
Chapter 16: Immunization
© Oltheten & Waspi 2012
Question 16-4
Chapter 16: Immunization
© Oltheten & Waspi 2012
© Oltheten & Waspi 2012
16 -4: Liability
$250,000
Chapter 16: Immunization
$250,000
$250,000
$250,000
© Oltheten & Waspi 2012
15 - 2: Duration of the Liability
T
Cash Flow
NPV
NPV/P
T*NPV/P
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
Chapter 16: Immunization
1.0000 4.9010 semis
= 2.4505 yrs
© Oltheten & Waspi 2012
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
Chapter 16: Immunization
© Oltheten & Waspi 2012
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
 [÷] 1.02 [x2][=][STO][6]
 Stores 221,992.8455 into memory slot 6
 [÷] 1.02 [x2][=][STO][8]
 Stores 213,372.5929 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 = 906,618.99

 [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
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
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
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
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 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
Chapter 16: Immunization
© Oltheten & Waspi 2012
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
Chapter 16: Immunization

© Oltheten & Waspi 2012
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