Business F723
Fixed Income Analysis
Week 5
Liability Funding and Immunization
Institutional Investors
• Investment strategies and planning horizon
dictated by nature of their liabilities
• Liability funding strategy to set cash flows
from assets = cash flow to liabilities
• Basic principal; to minimize risk, set the
duration of assets = duration of liabilities
• Bad example: US Savings & Loan collapse
2
Types of Institutions
• Depository Institutions; in the Spread
Banking business, make money on spread
between assets and liabilities (banks, ins.)
• Pension funds; try to cover defined benefits
at minimum cost
• Mutual funds and others; no fixed liabilities,
try to generate maximum return
3
Types of Liability
• Type 1; certain in time and amount; GIC
• Type 2; certain in amount but not time
– Life insurance policy
• Type 3; certain time in but not amount
• Type 4; certain in neither time nor amount
– Auto insurance policy
4
Liquidity Concerns
• If cash flows to liabilities are uncertain,
liquidity becomes a serious concern
– GIC: early withdrawal with penalty
– Life insurance; cash surrender or loan value
– Mutual funds; net disposals
5
Asset/Liability Management
• Two primary goals of financial institutions
– Earn a reasonable return on investment
– Maintain a surplus of assets over liabilities, also
called Surplus Management
• Trade-off between risk and return
• Risk must be measured for both assets and
liabilities
6
Types of Surplus
• Economic Surplus; present value of assets
in excess of the present value of liabilities
• Accounting Surplus; as specified by GAAP
• Regulatory Surplus; as specified by various
regulatory bodies charged with protecting
the stakeholders in various institutions
7
Economic Surplus
• Best from a finance standpoint
• Surplus = PV Assets - PV Liabilities
• If duration of the assets and liabilities are
not the same, a change in interest rates can
change the value of the surplus
e.g. $10 million assets, duration 10
$9.2 million liabilities, duration 15
what happens with a 1% decrease in YTM?
8
Accounting Surplus
• Financial reporting according to GAAP,
FASB 115 in USA
– Amortized cost (book value)
– Market value
– Lower of cost or market
• Which method is allowed depends on what
the institution intends
9
FASB 115
Account
Accounting Will Affect
Classification method
Surplus
Will Affect
Earnings
Held to
Maturity
No
No
Available for Market
sale
Value
Yes
No
Trading
account
Yes
Yes
Amortized
Cost
Market
Value
10
Regulatory Surplus
• Uses Regulatory Accounting Principals
(RAP)
• no overall guiding rules, each jurisdiction
and regulatory body is free to determine the
rules that the financial institution must
follow for reporting to the regulatory body
11
Immunization
• Defined by F. M. Reddington in 1952
– The investment of the assets in such a way that
the existing business is immune to a general
change in the rate of interest
• For funding a single liability, consider 3
bonds and a liability of $2,091.23 due in 8
years
Bond
Face value
Coupon rate
Maturity
Price
A
$1,000
10%
8 years
$1,116.52
B
$1,000
12%
14 years
$1,333.26
C
$1,000
7.5%
20 years
$950.52
12
The 8-year, 10% Bond
YTM Coupons
0%
800
2%
800
4%
800
6%
800
8%
800
10%
800
12%
800
14%
800
16%
800
18%
800
20%
800
Interest on Future price
interest
of bond
0
1000
62.89
1000
131.96
1000
207.84
1000
291.23
1000
382.87
1000
483.63
1000
594.40
1000
716.21
1000
850.17
1000
997.49
1000
Ending
Value
1800.00
1862.89
1931.96
2007.84
2091.23
2182.87
2283.63
2394.40
2516.21
2650.17
2797.49
Total
Return
6.06%
6.50%
6.97%
7.47%
8.00%
8.56%
9.15%
9.77%
10.42%
11.10%
11.82%
13
The 8-year, 10% Bond
3000
2500
2000
1500
1000
500
0
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
14
The 20-year, 7.5% Bond
Face Value purchased
YTM
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
Coupons
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 704.79
$ 1,174.65
Interest on Future price
interest
of bond
0
$2,231.83
55.41
$1,860.87
116.26
$1,563.45
183.11
$1,323.85
256.57
$1,129.87
337.31
$972.04
426.07
$842.95
523.66
$736.79
630.97
$649.03
748.99
$576.05
878.77
$515.03
Ending
Value
2936.62
2621.07
2384.50
2211.75
2091.23
2014.14
1973.81
1965.24
1984.79
2029.83
2098.59
Total
Return
12.46%
10.96%
9.71%
8.73%
8.00%
7.51%
7.25%
7.19%
7.32%
7.61%
8.05%
Liability
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
15
The 20-year, 7.5% Bond
3500
3000
2500
2000
1500
1000
500
0
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
16
The 14-year, 12% Bond
Face Value purchased
YTM
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
Coupons
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$ 803.94
$
837.44
Interest on Future price
interest
of bond
0
$1,440.40
63.20
$1,308.71
132.61
$1,191.69
208.87
$1,087.52
292.66
$994.63
384.76
$911.66
486.01
$837.44
597.33
$770.92
719.74
$711.22
854.36
$657.54
1002.40
$609.20
Ending
Value
2244.34
2175.86
2128.25
2100.33
2091.23
2100.37
2127.39
2172.20
2234.91
2315.84
2415.54
Total
Return
8.92%
8.52%
8.23%
8.06%
8.00%
8.06%
8.22%
8.49%
8.87%
9.33%
9.88%
17
The 14-year, 12% Bond
2450
2400
2350
2300
2250
2200
2150
2100
2050
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
18
Why?
• Duration of the 3 bonds (Macauly’s, modified
duration would be similar since YTM is constant)
– The 8-year, 10% Bond = 5.827
– The 14-year, 12% Bond = 7.998
– The 20-year, 7.5% Bond = 10.425
• A bond with a duration equal to the duration
of the liability will have offsetting price and
reinvestment risks
19
Multiple Bonds
• Set portfolio duration equal to duration of
obligation
• That gives 52.74% of the funds in bond A
and 47.26% of the funds in bond C
20
Multiple Bonds
Duration
YTM
0%
2%
4%
6%
8%
10%
12%
14%
16%
18%
20%
5.827
52.74%
FV A, per
FV C, per
$1000 FV $1174.65 FV
$1,800.00
$2,936.62
$1,862.89
$2,621.07
$1,931.96
$2,384.50
$2,007.84
$2,211.75
$2,091.23
$2,091.23
$2,182.87
$2,014.14
$2,283.63
$1,973.81
$2,394.40
$1,965.24
$2,516.21
$1,984.79
$2,650.17
$2,029.83
$2,797.49
$2,098.59
7.998
Portfolio
$2,337.12
$2,221.17
$2,145.81
$2,104.20
$2,091.23
$2,103.14
$2,137.22
$2,191.60
$2,265.09
$2,357.02
$2,467.22
10.425
8.000
47.26% 100.00%
Bond B
$2,244.34
$2,175.86
$2,128.25
$2,100.33
$2,091.23
$2,100.37
$2,127.39
$2,172.20
$2,234.91
$2,315.84
$2,415.54
Liability
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
2091.23
21
Portfolio
$2,500
$2,450
$2,400
$2,350
$2,300
$2,250
$2,200
$2,150
$2,100
$2,050
0%
2%
4%
6%
Portfolio
8%
10%
Bond B
12%
14%
16%
18%
Liability
22
Rebalancing a Portfolio
• What is the duration of an investment in the
8-year, 10% Bond, six months later, if the
YTM is now 7.5%?
• Note: duration of cash = zero
FV
CR
T
YTM
Price
$1,000.00
12%
13.5
7.5%
$1,377.94
Duration
Cash
7.942241
$60.00
Duration
7.61084
23
Rebalancing Considerations
• As time passes and interest rates change, the
duration of an immunized portfolio can drift
away from the target duration
• Buying and selling bonds can bring the
duration back to the target, but will give rise
to transaction costs
• Frequent rebalancing can be expensive, but
it will reduce the risk from duration drift
24
Immunization Risk
• Since duration measures the approximate
change in price for a parallel change in the
yield curve, duration matching leaves some
risk in an immunized portfolio
• Fong and Vasicek developed a measure of
immunization risk
2
2
2
CFn n  H 
CF1 1  H  CF2 2  H 
Risk 

 ... 
2
1 y
1  y 
1  y n
25
Immunization Risk
• Calculate the
immunization risk for
the 12%, 14-year bond
at a YTM of 8%,
given a horizon of 8
years
T
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
(n-h)^2
225
196
169
144
121
100
81
64
49
36
25
16
9
4
1
0
1
4
9
16
25
36
49
64
81
100
121
144
CF
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
60
1060
term
12,981
10,873
9,014
7,386
5,967
4,742
3,693
2,806
2,066
1,459
974
600
324
139
33
31
118
256
438
658
911
1,193
1,498
1,823
2,164
2,518
50,902
125,568
26
Immunization Risk
• Calculate the
immunization risk for
the portfolio at a YTM
of 8%, given a horizon
of 8 years
T
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
(n-h)^2
225
196
169
144
121
100
81
64
49
36
25
16
9
4
1
0
1
4
9
16
25
36
49
64
81
100
121
144
169
196
225
256
289
324
361
400
441
484
529
576
CF A
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
26.37
553.77
CF C
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
575.96
Portfolio
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
47.19
574.59
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
20.82
575.96
Term
10,209
8,551
7,090
5,808
4,693
3,729
2,905
2,207
1,625
1,148
766
472
255
109
26
11
41
89
152
228
316
414
520
633
751
874
1,000
1,128
1,258
1,389
1,519
1,649
1,778
1,904
2,029
2,151
2,270
2,386
69,100
143,180
27
Zero-Coupon Bonds
• From the immunization risk measure (or
just by intuition) we can see that a pure
discount bond maturing on the date of the
obligation has no immunization risk
• Unfortunately, in practice, the zero-coupon
bond has a lower yield than coupon bonds
• This leads to another risk/return trade-off
28
Credit Risk and Target Yield
• If one or more of the bonds in the portfolio
defaults or suffers a downgrade, the target yield
may not be realized
• The tactic to minimize this risk is to restrict the
allowable bonds to those with a level of credit risk
with which the client is comfortable
• Similar to the zero-coupon problem this brings up
another risk/return trade-off
29
Call Risk
• If any of the bonds in the portfolio are
callable, this will increase the risk that the
target value will not be reached
• Restricting bonds to those which are not
callable or are trading at a deep discount
will reduce the level of call risk… and the
expected return on your investment
30
Building the Portfolio
• After deciding on the allowable bonds,
build a portfolio that matches the duration
of the obligation
• Mathematical tools can be used to minimize
the objective function, which is often the
immunization risk measure
• Alternatively, this can be done by matching
both duration and convexity
31
Contingent Immunization
• A strategy where a safety net return is lower
than that currently available is acceptable
• This allows the fund manager to pursue an
active trading strategy to seek higher yields
• If the portfolio value drops to a point where
there is no safety cushion, then the strategy
will change to immunization
32
Multiple Liabilities
• If there are multiple liabilities in the future,
the duration matching condition is still valid
but extra conditions must also be satisfied
• The distribution of durations of the assets
must be wider than that of the liabilities
• The present value of the portfolio must
equal the present value of the liabilities
33
Multiple Liabilities
• As with single liability immunization, the
portfolio is only protected against parallel
shifts in the yield curve
• Fong and Vasicek’s immunization risk
measure can be used in this case too
• Immunization strategies for one type of
non-parallel shift can increase the risk from
a different non-parallel shift
34
Cash Flow Matching
• Multiple liabilities can be hedged by
creating a portfolio where the cash inflows
are equal to the required cash outflows
• This can be done by matching the final
required cash flow to that of a bond’s final
payment, the next to last payment can be
covered by the previous bond’s coupon plus
the final payment of another bond, etc.
35
Cash Flow Matching Example
• You are required to pay $2m every six
months for the next 3 years
• Construct a bond portfolio to fund this
obligation
Time
1
2
3
4
5
6
Face
Coupon rate
Liability
2,000,000
2,000,000
2,000,000
2,000,000
2,000,000
2,000,000
Bond A
1,913,876
9.0%
Bond B
1,858,132
6.0%
86,124
86,124
86,124
86,124
86,124
2,000,000
55,744
55,744
55,744
55,744
1,913,876
Bond C
Bond D
1,752,954
1,689,595
12.0%
7.5%
Cash Flow
105,177
63,360
105,177
63,360
105,177
1,752,954
1,858,132
Bond F
1,624,610
8.0%
Bond G
1,554,651
9.0%
Portfolio
64,984
1,689,595
1,624,610
2,000,000
2,000,000
2,000,000
2,000,000
2,000,000
2,000,000
36
Combining Active and
Immunization Strategies
• A mixed strategy of actively managing part
of the portfolio and actively managing the
rest of the portfolio
Im. target rate - minimum acceptable return
Im. target rate - expected worst case active return
8%  4%
Active componant 
 57.14%
8%  1%
Active componant 
37