Business F723

Fixed Income Analysis

Week 5

Liability Funding and Immunization

2

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

3

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

4

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

5

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

6

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

7

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

8

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 surpluse.g. $10 million assets, duration 10 $9.2 million liabilities, duration 15 what happens with a 1% decrease in YTM?

9

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

10

FASB 115

AccountClassification

Accountingmethod

Will AffectSurplus

Will AffectEarnings

Held toMaturity

AmortizedCost

No No

Available forsale

MarketValue

Yes No

Tradingaccount

MarketValue

Yes Yes

11

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

12

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 A B CFace value $1,000 $1,000 $1,000Coupon rate 10% 12% 7.5%Maturity 8 years 14 years 20 yearsPrice $1,116.52 $1,333.26 $950.52

13

The 8-year, 10% Bond

YTM CouponsInterest on

interestFuture price

of bondEnding

ValueTotal

Return0% 800 0 1000 1800.00 6.06%2% 800 62.89 1000 1862.89 6.50%4% 800 131.96 1000 1931.96 6.97%6% 800 207.84 1000 2007.84 7.47%8% 800 291.23 1000 2091.23 8.00%

10% 800 382.87 1000 2182.87 8.56%12% 800 483.63 1000 2283.63 9.15%14% 800 594.40 1000 2394.40 9.77%16% 800 716.21 1000 2516.21 10.42%18% 800 850.17 1000 2650.17 11.10%20% 800 997.49 1000 2797.49 11.82%

14

The 8-year, 10% Bond

0

500

1000

1500

2000

2500

3000

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

15

The 20-year, 7.5% Bond

Face Value purchased 1,174.65$

YTM CouponsInterest on

interestFuture price

of bondEnding

ValueTotal

Return Liability0% 704.79$ 0 $2,231.83 2936.62 12.46% 2091.232% 704.79$ 55.41 $1,860.87 2621.07 10.96% 2091.234% 704.79$ 116.26 $1,563.45 2384.50 9.71% 2091.236% 704.79$ 183.11 $1,323.85 2211.75 8.73% 2091.238% 704.79$ 256.57 $1,129.87 2091.23 8.00% 2091.23

10% 704.79$ 337.31 $972.04 2014.14 7.51% 2091.2312% 704.79$ 426.07 $842.95 1973.81 7.25% 2091.2314% 704.79$ 523.66 $736.79 1965.24 7.19% 2091.2316% 704.79$ 630.97 $649.03 1984.79 7.32% 2091.2318% 704.79$ 748.99 $576.05 2029.83 7.61% 2091.2320% 704.79$ 878.77 $515.03 2098.59 8.05% 2091.23

16

The 20-year, 7.5% Bond

0

500

1000

1500

2000

2500

3000

3500

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

17

The 14-year, 12% Bond

Face Value purchased 837.44$

YTM CouponsInterest on

interestFuture price

of bondEnding

ValueTotal

Return0% 803.94$ 0 $1,440.40 2244.34 8.92%2% 803.94$ 63.20 $1,308.71 2175.86 8.52%4% 803.94$ 132.61 $1,191.69 2128.25 8.23%6% 803.94$ 208.87 $1,087.52 2100.33 8.06%8% 803.94$ 292.66 $994.63 2091.23 8.00%

10% 803.94$ 384.76 $911.66 2100.37 8.06%12% 803.94$ 486.01 $837.44 2127.39 8.22%14% 803.94$ 597.33 $770.92 2172.20 8.49%16% 803.94$ 719.74 $711.22 2234.91 8.87%18% 803.94$ 854.36 $657.54 2315.84 9.33%20% 803.94$ 1002.40 $609.20 2415.54 9.88%

18

The 14-year, 12% Bond

2050

2100

2150

2200

2250

2300

2350

2400

2450

0% 2% 4% 6% 8% 10% 12% 14% 16% 18% 20%

19

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

20

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

21

Multiple Bonds

Duration 5.827 7.998 10.425 8.00052.74% 47.26% 100.00%

YTMFV A, per $1000 FV

FV C, per $1174.65 FV Portfolio Bond B Liability

0% $1,800.00 $2,936.62 $2,337.12 $2,244.34 2091.232% $1,862.89 $2,621.07 $2,221.17 $2,175.86 2091.234% $1,931.96 $2,384.50 $2,145.81 $2,128.25 2091.236% $2,007.84 $2,211.75 $2,104.20 $2,100.33 2091.238% $2,091.23 $2,091.23 $2,091.23 $2,091.23 2091.23

10% $2,182.87 $2,014.14 $2,103.14 $2,100.37 2091.2312% $2,283.63 $1,973.81 $2,137.22 $2,127.39 2091.2314% $2,394.40 $1,965.24 $2,191.60 $2,172.20 2091.2316% $2,516.21 $1,984.79 $2,265.09 $2,234.91 2091.2318% $2,650.17 $2,029.83 $2,357.02 $2,315.84 2091.2320% $2,797.49 $2,098.59 $2,467.22 $2,415.54 2091.23

22

Portfolio

$2,050

$2,100

$2,150

$2,200

$2,250

$2,300

$2,350

$2,400

$2,450

$2,500

0% 2% 4% 6% 8% 10% 12% 14% 16% 18%

Portfolio Bond B Liability

23

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 $1,000.00 Duration 7.942241CR 12% Cash $60.00T 13.5YTM 7.5% Duration 7.61084Price $1,377.94

24

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

25

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

nn

y

HnCF

y

HCF

y

HCFRisk

1

...1

2

1

1 2

2

22

21

26

Immunization Risk

• Calculate the immunization risk for the 12%, 14-year bond at a YTM of 8%, given a horizon of 8 years

T (n-h)^2 CF term1 225 60 12,981 2 196 60 10,873 3 169 60 9,014 4 144 60 7,386 5 121 60 5,967 6 100 60 4,742 7 81 60 3,693 8 64 60 2,806 9 49 60 2,066

10 36 60 1,459 11 25 60 974 12 16 60 600 13 9 60 324 14 4 60 139 15 1 60 33 16 0 60 - 17 1 60 31 18 4 60 118 19 9 60 256 20 16 60 438 21 25 60 658 22 36 60 911 23 49 60 1,193 24 64 60 1,498 25 81 60 1,823 26 100 60 2,164 27 121 60 2,518 28 144 1060 50,902

125,568

27

Immunization Risk

• Calculate the immunization risk for the portfolio at a YTM of 8%, given a horizon of 8 years

T (n-h)^2 CF A CF C Portfolio Term1 225 26.37 20.82 47.19 10,209 2 196 26.37 20.82 47.19 8,551 3 169 26.37 20.82 47.19 7,090 4 144 26.37 20.82 47.19 5,808 5 121 26.37 20.82 47.19 4,693 6 100 26.37 20.82 47.19 3,729 7 81 26.37 20.82 47.19 2,905 8 64 26.37 20.82 47.19 2,207 9 49 26.37 20.82 47.19 1,625

10 36 26.37 20.82 47.19 1,148 11 25 26.37 20.82 47.19 766 12 16 26.37 20.82 47.19 472 13 9 26.37 20.82 47.19 255 14 4 26.37 20.82 47.19 109 15 1 26.37 20.82 47.19 26 16 0 553.77 20.82 574.59 - 17 1 20.82 20.82 11 18 4 20.82 20.82 41 19 9 20.82 20.82 89 20 16 20.82 20.82 152 21 25 20.82 20.82 228 22 36 20.82 20.82 316 23 49 20.82 20.82 414 24 64 20.82 20.82 520 25 81 20.82 20.82 633 26 100 20.82 20.82 751 27 121 20.82 20.82 874 28 144 20.82 20.82 1,000 29 169 20.82 20.82 1,128 30 196 20.82 20.82 1,258 31 225 20.82 20.82 1,389 32 256 20.82 20.82 1,519 33 289 20.82 20.82 1,649 34 324 20.82 20.82 1,778 35 361 20.82 20.82 1,904 36 400 20.82 20.82 2,029 37 441 20.82 20.82 2,151 38 484 20.82 20.82 2,270 39 529 20.82 20.82 2,386 40 576 575.96 575.96 69,100

143,180

28

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

29

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

30

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

31

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

32

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

33

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

34

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

35

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.

36

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

Bond A Bond B Bond C Bond D Bond F Bond G PortfolioFace 1,913,876 1,858,132 1,752,954 1,689,595 1,624,610 1,554,651 Coupon rate 9.0% 6.0% 12.0% 7.5% 8.0% 9.0%

Time Liability1 2,000,000 86,124 55,744 105,177 63,360 64,984 1,624,610 2,000,000 2 2,000,000 86,124 55,744 105,177 63,360 1,689,595 2,000,000 3 2,000,000 86,124 55,744 105,177 1,752,954 2,000,000 4 2,000,000 86,124 55,744 1,858,132 2,000,000 5 2,000,000 86,124 1,913,876 2,000,000 6 2,000,000 2,000,000 2,000,000

Cash Flow

37

Combining Active and Immunization Strategies

• A mixed strategy of actively managing part of the portfolio and actively managing the rest of the portfolio

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return active case worst expected - rate target Im.

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