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Lecture

ACTSC445 Lecture Notes - Reinvestment Risk, Actuarial Science, Pension


Department
Actuarial Science
Course Code
ACTSC445
Professor
Jiahua Chen

Page:
of 3
ACTSC 445: Asset-Liability Management – Fall 2008
Department of Statistics and Actuarial Science, University of Waterloo
Unit 7 – Dedicated Bond Portfolio: A Case Study
References (recommended readings): Chap. 48 of Fabozzi.
In this unit, we use an example based on a pension plan to illustrate the use of a dedicated bond
portfolio for cash flow matching.
The liabilities in this example are taken to be the expected benefit payouts to a closed block of current
retirees, and a 35-year schedule is used.
A few notes on liabilities:
Liability obligations must be projected accurately.
Sometimes, terminated vested participants (former employees who are vested in the pension plan
and are entitled to benefit payouts commencing sometime in the future) are also included, and
also active partcipants over the age of 50. When this is the case, various mortality, termination,
and benefit assumptions must be reviewed periodically.
The following table is an excerpt of the schedule of expected benefit payouts for this example (Exhibit
48-1 in Fabozzi).
Retired lives liabilities
Retired lives liabilities plus terminated vested
Year Dollar Payout Dollar Payout
1992 1,250,000 2,000,000
1993 15,000,000 24,000,000
1994 14,916,015 24,519,000
1995 13,445,985 25,021,000
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.
..
.
..
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2027 2,123,504 10,982,000
2028 1,337,297 9,869,000
Total 283,758,000
Once we have the liability schedule, the next step is to identify the portfolio’s constraints. A typical
constraint is to force the portfolio to contain only bonds with low credit risk. Also, mortgages are
usually undesirable because of the prepayment uncertainty.
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The following table gives the constraints used for this example (Exhibit 48-2 of Fabozzi).
Minimum Maximum
Quality (Credit Rating)
Treasury 20% 100%
AAA 0 100%
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.
..
.
..
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BBB 0 0
Sector
Treasury 20% 100%
Industrial 0 30%
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.
..
.
..
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Concentration
Max. in one issue 10%
Min. in one issuer 10%
Call Constraints on Corporate
Securities Noncallable only
Lot Size
Conditional minimum $2,000,000 (par)
Increment $1,000,000 (par)
Maximum Unlimited
Note: it is typical to not enforce a perfect match of the liability schedule. Because of that, we can have
cash flows from the portfolio that need to be reinvested. Assumptions for the reinvestment rate are
usually pretty conservative (3% in the example).
Once the constraints are identified, and an assumption for the reinvestment rate is made, the goal is to
construct a portfolio that matches the liability schedule at the least cost possible. Methods like linear
programming or (better) integer programming can be used for that.
The following table describes the chosen dedicated portfolio (Exhibit 48-3 from Fabozzi).
Par S&P Security Coupon Maturity Price Yield Duration Market Val.
($000) ($000)
5,000 GOV United States 5.000 12/31/1993 101.266 3.849 1.08 5,154
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..
.
..
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..
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3,000 AAA Southern RY CO 8.350 12/15/1999 105.411 7.356 5.30 3,263
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..
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..
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1,500 GOV Resolution FDG 0.000 10/15/2026 6.520 8.211 42.28 98
139,000 AA+ 6.942 16.0 years 86.213 7.835 7.37 123,160
Based on this portfolio and the liability schedule, we can analyse the cash flows arising from this
matching (Exhibit 48-4 in Fabozzi).
Period Beginning Maturing Coupon Reinvestment Cash Flow Liability Ending
Ending Balance Principal Income Income Available Schedule Balance
12/31/1992 0 0 2,740 8 2,748 1,250 1,498
12/31/1993 1,498 5,000 9,661 176 16,335 15,000 1,335
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..
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..
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12/31/2028 4,068 0 0 123 4,189 1,337 2,852
Total 139,000 139,794 7817 283,758
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A few things to note about this example:
1. The cash flow match is almost perfect, so the reinvestment risk is not very important.
2. The yield to maturity on the chosen portfolio is 7.835%.
3. The actuarial investment rate assumption used to compute the present value of the liabilities is
5%, for a value of $159,818,000. However, the chosen portfolio with its yield of 7.835% manages
to pay out these liabilities (as long as all coupon and principal payments are done it a timely
and punctual way) for an initial price of only $123,160,000. Based on this, the actuary has the
flexibility of increasing the assumed rate from 5% to possibly 7.835%, which ultimately may end
up reducing the current contribution requirements. More precisely, we have (Exhibit 48-5 in
Fabozzi)
Yield Rate Dollar Amount
1. Total liabilities - $283,758,000
2. PV of total liabilities at 5.00% 159,818,000
3. Portfolio cost (market value) at 7.83% 123,160,000
4. Potential savings (23) - 36,658,000
Percent savings (4/2) 22.94% -
Percent savings (4/3) 29.76%
Finally, instead of opting for passive management, a dedicated bond portfolio can be reoptimized over
time. The reoptimization should be executed through a dealer firm in order to guarantee any take-out.
For example, we may have (Exhibit 48-6 in Fabozzi)
Market Value Average Take-out
Original dedicated portfolio $100,000 AA+ -
Reoptimized dedicated portfolio $ 99,400 AA+ $600
(marked to market 1 year later)
Takeouts generated from such strategies can then be used to reposition the portfolio according to some
objectives (e.g., increase quality, shorten duration, etc.), through active management.
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