Lecture 5 - Bond Portfolio Management - IRRM - Immunization and ALM
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Transcript of 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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43.1110001.005.1
12
12
:onImmunizati
4.76:ueequity valinLoss
43.118001.005.1
15
67.610001.0
05.1
7
dA
yearsA
LDD
dL
dA
LA