Portfolio Dedication Principles Portfolio Immunization Principles Mathematical Programming Models

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LECTURE NOTES

Mathematical Modeling and its Application in Finance

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2

Stavros A. Zenios

Operations and Information Management Department

The Wharton School

University of Pennsylvania

Lecture 9:

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OUTLINE

n

Portfolio Dedication Principles

n

Portfolio Immunization Principles

n

Mathematical Programming Models

•

• Reading:Reading:

–

– S. Zenios, Financial Optimization, S. Zenios, Financial Optimization, pppp. 15. 15----2424

–

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4

Portfolio Dedication

Fixed income Asset/Liability management strategy: Fixed income Asset/Liability management strategy:

Example 1:

Example 1: Guaranteed Investment Contracts in the 70’s:Guaranteed Investment Contracts in the 70’s: Upward sloping yield curve

Upward sloping yield curve 3

3-- to 7-to 7-year maturity of year maturity of GICs GICs with low interest paymentswith low interest payments Proceeds reinvested in 10

Proceeds reinvested in 10-- to 30-to 30-year mortgages, public year mortgages, public bonds

bonds

What happened as interest rates rose in the late 1970s ? What happened as interest rates rose in the late 1970s ?

Example 2:

Example 2: Pension fund management may assume very conservative values Pension fund management may assume very conservative values for reinvestment opportunities. Result?

for reinvestment opportunities. Result?

Example 3: Court settlements. Example 3: Court settlements.

How to manage such risks? CASH

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Cashflow Matching

Time Time

Liabilities Liabilities Assets

Assets

No interest rate risk. No interest rate risk. Can we achieve

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6

Portfolio Dedication

Time Time

Liabilities Liabilities Assets

Assets

Reinvestment Reinvestment

Maturing assets + Proceeds from reinvestment = Liabilities, at e

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Portfolio Dedication

n

Determining the Liabilities for a Pension Fund

•

• Closed block of retirees:

Closed block of retirees:

–

– Very accurate estimateVery accurate estimate---benefits are known (+)benefits are known (+) –

– Only a small part of the fund’s liabilities (Only a small part of the fund’s liabilities (--))

–

– Declining stream of liabilities due to mortalityDeclining stream of liabilities due to mortality

•

• Retired

Retired

-

lives plus terminated vested participants

–

– Broader universe of participants (+)Broader universe of participants (+) –

– Vested benefits are known (+)Vested benefits are known (+)

•

• Anticipated retiree obligations

Anticipated retiree obligations

–

– Even broader universe (+)Even broader universe (+) –

– Benefits are not known with certainty (Benefits are not known with certainty (--))

–

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8

Portfolio Dedication

n

Setting constraints:

•

• Restrict investments to high

Restrict investments to high

-

quality bonds

•

• Restrictions by agency, sector, issuer (diversification of

Restrictions by agency, sector, issuer (diversification of

industry specific risk)

n

Example:

–

– QualityQuality

–

– Treasury 20 to 100%Treasury 20 to 100%

–

– Agency 0 to 100%Agency 0 to 100%

–

– AAAAAA 0 to 100%0 to 100%

–

– AA 0 to 100%AA 0 to 100%

–

– A 0 to 50%A 0 to 50%

–

– BBB 0BBB 0

–

– SectorSector

–

– Treasury 20 to 100%Treasury 20 to 100%

–

– Agency 0 to 100%Agency 0 to 100%

–

– Industrial 0 to 30%Industrial 0 to 30%

–

– Utility 0 to 30%Utility 0 to 30%

–

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Portfolio Dedication

n

Choosing the reinvestment

---

rollover

---

rate:

•

• Assuming high rollover rate will

Assuming high rollover rate will

prefund

liabilities if high

-

yield

securities are available with maturity before the liability date

s

•

• Assuming low reinvestment rates leads to tighter

Assuming low reinvestment rates leads to tighter

cashflow

matching, at the potential loss of portfolio yield

n

Example: Current actuarial practices allow 5 to 8

percent

n

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10

Portfolio Dedication: mathematical program

rate

nt

Reinvestme

:

period

at

security

from

generated

Cashflow

:

period

at

obligation

Liability

:

security

in

invested

Amount

:

security

of

(price)

Cost

:

ρ

t

j

cf

t

L

j

x

j

p

jt t

j j

Constraint: at each period t, the

Constraint: at each period t, the cashflow cashflow received from assets +received from assets + income from reinvestment = the liability obligation

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12

Example: Portfolio Dedication for a Pension Fund

Total liabilities:

$283,758,000

over 35 years (retired lives only)over 35 years (retired lives only) Reinvestment rate: 5%

Reinvestment rate: 5%

Present value of liabilities:

$159,818,000

Portfolio cost (market value):

$123,160,000

Portfolio yield: 7.83%

Savings (portfolio cost over present value of liabilities: 23%

Portfolio

Portfolio reoptimizationreoptimization. . Bond swaps.

Bond swaps.

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Portfolio Immunization

n

Broad fixed

-

income asset/liability management strategy

•

• Actuary F.M.

Actuary F.M.

Reddington

(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”

•

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14 ) 1 ( 1 1

)

1 (

duration

dollar

obtain the

we

yield

respect to

with

derivative

Taking

)

1 (

bond

of

Yield

:

period

at

bond

from

Cashflow

:

bond

of

lue)

present va

(or

Price

:

+ − = − =

∑

+

−

=

+

=

t T t j jt j t T t j jt j j jt j

r

tcf

d

r

cf

P

j

r

t

j

cf

j

P

Note: Dollar duration of a portfolio of bonds is additive Note: Dollar duration of a portfolio of bonds is additive

j N

j

p

d

x

d

∑

=

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Mathematical Program for

Portfolio Immunization

security

of

Duration

:

liability

of

lue

Present va

:

liability

of

Duration

:

security

of

Yield

:

security

in

invested

Amount

:

security

of

(price)

Cost

:

j

d

P

D

j

r

j

x

j

p

j L L j

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16

Mathematical Program for

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Understanding Portfolio Immunization:

Interpretation of duration

Effect of changing interest rates on asset value: Effect of changing interest rates on asset value:

Years Years Asset

Asset value value

100 100

9% bond in a 9% interest rate

9% bond in a 9% interest rate envenv..

30 years bond

Interest rates rise instantaneously to 15%

Market loss greater than Market loss greater than reinvestment gains

reinvestment gains Reinvestment gains

Reinvestment gains

greater than market loss greater than market loss

Duration Duration

6.79

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Understanding Portfolio Immunization:

Duration matched portfolios

Assume a target (liability) duration of 6.79 years Assume a target (liability) duration of 6.79 years

Time Time Barbell Portfolio

Barbell Portfolio

Time Time Bullet portfolio

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Yield of structured duration matched portfolios

2

2 55

Years Years Yield

Yield

Bullet structure Bullet structure

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20

What is the risk exposure of barbell and even

ladder portfolios?

Shape risk. Shape risk.

Add convexity constraints Add convexity constraints