lecture 5 - bond portfolio management - irrm - immunization and alm

7/23/2019 Lecture 5 - Bond Portfolio Management - IRRM - Immunization and ALM

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IMMUNIZATION

ALM


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IMMUNIZATION

Immunization of a portfolio:

strategy such that the acquired value of the portfolio is

greater or equal to a given value, Bond portfolio have an assured return for a specific

time horizon irrespective of interest rate changes

Given value: acquired value of the initial portfolio

with the initial yield to maturity.


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ALM

ALM: net asset (assetliabilities) isprotected against interest rates

fluctuations.


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1. PORTFOLIO IMMUNIZATION

1.1. Decomposition of portfolio risk

Reinvestment risk

Coupons are reinvested as a rate unknown Return of an investment depends on the

reinvestment rate.

Portfolio value risk

Prices of bonds vary with interest rates.

If interest rate changes, effective return isdifferent of yield (expected return).


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Steps to compute effective

returns Compute the acquired value, at the

market rate, of coupons at the horizon

date,

Compute the expected price at thehorizon date,

Compute effective return


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Example

Bond B1 (5000, 10%, 4 years, Duration:3.5 years)

Market rate: 8%, so bond price is5331.

Just after you buy bond B1 market rates goesto 7%, 9 %.

What is the effective return of yourinvestment:

Is sold just after the second coupon,

Is kept until maturity,

Is kept during the time of duration.


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Example (solution)differences between gain and losses

Market rate 7% Market rate 9%

2 yearsAcquired interest

Bond priceTOTAL

Effective Return

1 0355 271,206 306,20

8.76%

1 0455 087.966 132.96

7.26%

4 yearsAcquired interest

Bond priceTOTAL

Effective Return

2 219.975 000

7 219.97

7.88%

2 286.565 000

7 286.56

8.125%3,5 years

Acquired interestBond price

TOTALEffective Return

1 662,765317.056 979.81

8%

1 711.225 268.046 979.26

8%


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Comments

Two different effects of rate variations.

If interest rate increases (decreases)

An increase (decrease) of reinvestment ofcoupons,

Decrease (increase) in the selling value of

the bond.At duration time, this two effects

compensate exactly.


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1.2. Immunization

IMMUNIZATION THEOREM: Toimmunize a portfolio duration must be

maintained equal to the remainingmaturity of the portfolio (Fisher et Weil,1971).


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Example

You want to secure an amount of$1851=1000*1.08^8 in 8 years.

The current yield to maturity is 8 %. Compare the 2 strategies:

Buy a bond of maturity 8 years,

Buy a bond of duration 8 years (maturity 10years).

At the end of Year 4 the yield go from 8% to6%.

Give the value at the end of year 8 with the


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Example (solution)

Year CF

Reinvestt

Rate

End

Value CF

Reinvestt

Rate End Value

1 80 0,08 80,00 80 0,08 80,00

2 80 0,08 166,40 80 0,08 166,40

3 80 0,08 259,71 80 0,08 259,714 80 0,08 360,49 80 0,08 360,49

5 80 0,06 462,12 80 0,06 462,12

6 80 0,06 569,85 80 0,06 569,85

7 80 0,06 684,04 80 0,06 684,04

8 1080 0,06 1805,08 1117 0,06 1841,75

80 0,06

1080 0,062

06.1

1080

06.1

8080

80+80*

1.08


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Immunization is a complexstrategy

As with the passage of time, duration of theportfolio decrease, the portfolio must be

rebalanced periodically (see case study 1). The portfolio must also be rebalanced when

interest rate changes.

To avoid the change of shape of the termstructure, you must use bullet portfolio(bonds with maturity 7 to 9 years toimmunize at 8 years).


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1.3. Contingent Immunization

Contingent immunization consists of identifyingboth:

the available immunization target rate,

a lower safety net level return which a client would

be minimally satisfied or a minimum required rateof return.

The manager:

Pursues an active strategy until the availablepotential return is driven down to the safety netlevel,

Completely immunizes the portfolio and lock in thesafety net return.


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Example

An investor is willing to accept a 6%return over a 5-years investment

horizon. It is the safety net. Initialportfolio:$ 100 millions

The possible immunized rate of return

is 7.5%. The difference between 7.5% and 6%

is called the cushion spread.

This cushion permit active


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Example (follow)

Compute the required terminal value (at the safetynet rate) of the portfolio assuming semi-annualcompounding.

What is the value of assets required now (at the

current available return) to obtain the requiredterminal value? Deduce the safety margin in dollars.

Assume the yield decreases to 5.6%, the marketvalue of the portfolio increases to $127.46 millions,give the new safety margin.

Same question if yield increases to 8.6% andportfolio value decreases to $88.23 millions.


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Example (solution)

Required terminal value is:

100(1.03)10 = $134.39 millions

Required asset at the current rate

X(1.0375)10 = $134.39

X = $93 millions Safety margin = 100 93 = $7 millions.


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Example (solution)

Asset value required to achieve therequired terminal value if current rate is

5.6%:X(1.028)10 = $134.39

X = $101.96 millions

New safety margin:

127.46 101.96 = $25.5 millions


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Example (solution)

Asset value required to achieve the required

terminal value if current rate is 8.6%:X(1.043)10 = $134.39

X = $88.21 millions

New safety margin: 88.23 88.21 = $0 million

At this yield level the immunization mode willbe triggered with an immunization target rateof 8.6%.

The yield at which the immunization modebecomes necessary is called the trigger point.


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2. ALM

Methods and strategies to measure andmanage risks due to the differences

between assets and liabilitiescharacteristics.

Difference in:

Amount, Date,

Risk (interest rate but also liquidity,

exchange rate risk)


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Objective of ALM

Protect equity (more generally surplus,a stock) = Value of assets value of

liabilities Protect an income (earnings, cash flow,

interest margin, a flow).


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2.1. Cash flow matching(CFM)

Construct the liabilities schedule.

Construct an asset portfolio that

reproduces the liabilities schedule.

Objective: minimize the investment inthe asset portfolio.


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Process for CFM

Method of matching by bonds of decreasing maturity

Step 0: construction of liabilities cash flows

Step 1: choose a bond with a maturity equal to thelast flow of liability, lets say, T. Compute the numberof bonds necessary to match this flow.

Step 2: compute the remaining liabilities cash flowsafter deducing cash flow of the first bond.

Step 3: choose a new bond with maturity T-1. Step 4: compute the remaining liabilities cash flows

after deducing cash flow of the second bond

And so on untill the shortest maturity of liabilities.


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Example

You must cover the following liabilitiesschedule.

Years Liabilities

1 1 000 000

2 1 000 000

3 1 000 000

4 1 000 000

5 1 000 000


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ExampleBond available

All bonds have a nominal of 100

maturity coupon Price

B1 5 3% 99.32

B2 4 5% 106.84

B3 3 4% 102.65

B4 2 6% 109.25

B5 1 3.50% 100.22


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Work to do

Give the asset portfolio structurematching liabilities. (cash flow

matching). Give the value of asset portfolio.


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t Liabilities O1 L1 O2 L2 O3 L3 O4 L4 O5 L5

1 1 000 000 29 127 970 873 46 230 924 643 35 564 889 079 50 328 838 751 838 764 -13

2 1 000 000 29 127 970 873 46 230 924 643 35 564 889 079 889 128 -49 -49

3 1 000 000 29 127 970 873 46 230 924 643 924 664 -21 -21 -21

4 1 000 000 29 127 970 873 970 830 43 43 43 43

5 1 000 0001 000027 -27 -27 -27 -27 -27

maturity coupon Price FTNber ofBonds

B1 5 3% 99.32 103 9709

B2 4 5% 106.84 105 9246

B3 3 4% 102.65 104 8891

B4 2 6% 109.25 106 8388

B5 1 3.50% 100.22 103.5 8104

Portfolio Value 4 593 373.55


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Cost/benefit of CFM Benefit:

Totally suppress risk,

Asset portfolio does not need to be rebalanced sotransaction costs are low.

Drawback: It is not always possible to construct an asset portfolio

perfectly reproducing liabilities schedule.

Less demanding method: To construct a portfolio such that acquired value of

Asset is greater than acquired value of liabilities.

Problem: reinvestment risk for cash flows of asset andliability.


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More Technical Aspects (Kocherlakota,Rosenbloom, Shiu, 1988)

0

)(,

sconstraint

jN

tLjtCjN

jPjNMin

j

j

Cash flow in t

of liability

Nber of bonds j

to buyPrice of

bond j

Cash flow generated

in t by bond j


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2.2. Duration matching

Objective: to protect the surplus of acompany.

LDAD

LDADED

LDEDAD

LA

LAE

LEA

0


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Example: life insurance company

Assets (investment on the market): 100

Duration: 7 years

Liabilities: 80 Duration: 15 years

Give the variation of equity value when

interest rate decreases from 5% to 4%.


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LDD

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lecture 5 - bond portfolio management - irrm - immunization and alm

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