bond primer slides 2007
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Derivatives: A Primer on Bonds
First Part: Fixed Income Securities
Bond Prices and Yields
Term Structure of Interest Rates
Second Part: TSOIR
Term Structure of Interest Rates
Interest Rate Risk & Bond Portfolio Management
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Bond Prices and Yields
Time value of money and bond pricing
Time to maturity and risk
Yield to maturity
vs. yield to call
vs. realized compound yield
Determinants of YTM
risk, maturity, holding period, etc.
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Bond Pricing
Equation:
P= PV(annuity) + PV(final payment)
=
Example: Ct= $40; Par = $1,000; disc. rate = 4%; T=60
)1()1(1 r
Par
r
couponT
T
tt
000,1$06.95$94.904$)04.01(
000,1$
)04.01(
40$60
60
1
t
tP
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Prices vs. Yields
P yield
intuition
convexity
BKM6 Fig. 14.3; ; BKM4 Fig. 14.6
intuition: yield P price impact
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Measuring Rates of Return on Bonds
Standard measure: YTM
Problems
callable bonds: YTM vs. yield to call
default risk: YTM vs. yield to expected default
reinvestment rate of coupons
YTM vs. realized compound yield
Determinants of the YTM
risk, maturity, holding period, etc.
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Measuring Rates of Return on Bonds 2
Yield To Maturity definition
discount rate such that NPV=0 interpretation
(geometric) average return to maturity
Example: Ct= $40; Par = $1,000; T=60; sells at par
%4)1(
000,1$
)1(
40$000,1
60
60
1
yyy tt
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Measuring Rates of Return on Bonds 3
Yield To Call definition
discount rates.t.NPV=0, with TC = earliest call date deep discount bonds vs. premium bonds
BKM6 Fig. 14.4; ; BKM4 Fig. 14.7
Example: Ct= $40, semi; Par = $900; T=60; P = $1,025;callable in 10 years (TC=20), call price = $1,000
%4)1(
000,1$
)1(
40$025,1
20
20
1
ytcytcytct
t
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Measuring Rates of Return on Bonds 4
Yield To Default definition
discount rates.t.NPV=0, with TD= expected default date
default premium and business cycle
economic difficulties and flight to quality
Example: Ct= $50, semi; Par = $1,000; T=10; P = $200;expected to default in 2 years (TC=4), recover $150
)1(
150$
)1(
50$200
4
4
1 ytdytdtt
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Measuring Rates of Return on Bonds 5
Coupon reinvestment rate YTM assumption: average
problem: not often true
solution: realized compound yield forecast future reinvestment rates
compute future value (BKM6 Fig.14.5; BKM4 Fig.14.9)
compute the yield (rcy) such that NPV = 0
practical?
need to forecast reinvestment rates
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Bond Prices over Time
Discount bonds vs. premium bonds coupon rate < market interest rates
built-in capital gain (discount bond)
coupon rate > market interest rates
built-in capital loss (premium bond)
Behavior of prices over time
BKM6 Fig. 14.6; BKM4 Fig. 14.10
Tax treatment capital gains vs. interest income
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Discount Bonds
OID vs. par bonds original issue discount (OID) bonds
less common
coupon need not be 0
par bonds
most common
Zeroes what? mostly Treasury strips
how? certificates of accrual, growth receipts, ...
annual price increase = 1-year disc. factor
(BKM6 Fig. 14.7; BKM4 Fig. 14.11)
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OID tax treatment -- Discount Bonds 2
Idea for zeroes
built-in appreciation = implicit interest schedule
tax the schedule as interest, yearly
tax the remaining price change as capital gain or loss
Other OID bonds
same idea
taxable interest = coupon + computed schedule
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OID tax treatment -- Discount Bonds 3
Example
30-year zero; issued at $57.31; Par = $1,000
compute YTM:
1styear taxable interest
%10)1(
000,1$31.57$
30
y
y
73.5$31.57$04.63$%)101(
000,1$
%)101(
000,1$3029
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OID tax treatment -- Discount Bonds 4
Example (continued)
interests on 30-year bonds fall to 9.9%
capital gain
tax treatment: taxable interest= $5.73; capital gain
41.7$31.57$72.64$)1.01(
000,1$
)099.01(
000,1$3029
68.1$04.63$72.64$)1.01(
000,1$
)099.01(
000,1$2929
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Term Structure of Interest Rates
Basic question link between YTM and maturity
Bootstrapping short rates from strips forward rates and expected future short rates
Recovering short rates from coupon bonds
Interpreting the term structure
does the term structure contain information?
certainty vs. uncertainty
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Terminology
Term structure = yield curve (BKM6 Fig. 15.1)
= plot of the YTM as a function of bond maturity
= plot of the spot rate by time-to-maturity
Short rate vs. spot rate
1-period rate vs. multi-period yield
spot rate = current rate appropriate to discount a
cash-flow of a given maturity
BKM6 Figure 15.3; BKM4 Figure 14.3
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Extracting Info re:Short Interest Rates
From zeroes non-linear regression analysis
bootstrapping
From coupon bonds
system of equations
regression analysis (no measurement errors)
Certainty vs. uncertainty
forward rate vs. expected future (spot) short rate
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Bootstrapping Fwd Rates from Zeroes
Forward rate break-even rate BKM Fig. 15.4
equates the payoffs of roll-over and LT strategies
Uncertainty no guarantee that forward = expected future spot
General formula
f1= YTM1 and 1)1(
)1(1
1
n
n
n
nn
YTM
YTMf
)1()1()1( 11 nn
n
n
n fYTMYTM
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Bootstrapping Fwd from Zeroes 2
Data
BKMTable 15.2 & Fig. 15.1
4 bonds, all zeroes (reimbursable at par of $1,000)
T Price YTM
1 $925.93 8%
2 $841.75 8.995% 3 $758.33 9.66%
4 $683.18 9.993%
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Bootstrapping Fwd Rates from Zeroes 3
Forward interest rate for year 1
Forward interest rate for year 2
%8
)11(
000,1$
)11(
000,1$93.925$ 11
yf
yf
)2
1(
93.925$
)2
1%)(81(
000,1$
)2
1)(1
1(
000,1$
275.841$
ffffP
%102
)2
1(
93.925$75.841$
f
f
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Bootstrapping Fwd Rates from Zeroes 4
Short rate for years 3 and 4
keep applying the method
you findf3= 11% =f4
General Formula
f1= YTM1
1
1)1(
)1(1
n
n
n
nn
YTM
YTMf
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Yield, Maturity and Period Return
Data
2 bonds, both zeroes (reimbursable at par of $1,000)
T Price YTM
1 $925.93 8%
2 $841.75 8.995%
Question investor has 1-period horizon; no uncertainty
does bond 2 (higher YTM) dominate bond 1?
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Yield, Maturity and Period Return 2
Answer: Nope
Bond 1 HPR:
Bond 2 HPR:
f2= 10%
price in 1 year =Par/(1+f2) = $ 909.09
capital gain at year-1 end =
%81
)1
1(
000,1$
93.925$
93.925$000,1$
HPR
y
%875.841$
75.841$09.909$1
HPR
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Fwd Rate & Expected Future Short Rate
Interpreting the term structure
Short perspective
liquidity preference theory (investors)
liquidity premium theory (issuer)
Expectations hypothesis
Long perspectiveMarket Segmentation vs. Preferred Habitat
Examples
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Fwd Rate & Exp. Future Short Rate 2
Short perspective
liquidity preference theory (short investors)
investors need to be induced to buy LT securities
example: 1-year zero at 8% vs. 2-year zero at 8.995%
liquidity premium theory (issuer)
issuers prefer to lock in interest rates
f2E[r2]
f2E[r2] + risk premium
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Fwd Rate & Exp. Future Short Rate 3
Long perspective
long investors wish to lock in rates
roll over a 1-year zero at 8% or lock in viaa 2-year zero at 8.995%
E[r2]f2
f2E[r
2] - risk premium
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Fwd Rate & Exp. Future Short Rate 4
Expectation Hypothesis risk premium = 0 and E[r2]f2
idea: arbitrage
Market segmentation theory idea: clienteles
ST and LT bonds are not substitutes
reasonable?
Preferred Habitat Theory investors do prefer some maturities
temptations exist
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Fwd Rate & Exp. Future Short Rate 5
In practice
liquidity preference + preferred habitat
hypotheses have the edge
Example
BKM Fig. 15.5
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Fwd Rate & Exp. Future Short Rate 6
Example 2
short term rates: r1r2 r3 10%
liquidity premium = constant 1% per year
YTM
%67.101%)111%)(111%)(101(1)1)(1)(1( 33 3213 ffry
%5.101%)1%101%)(101(1)1)(1( 212 fry
%1011 ry
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Measurement: Zeroes vs. Coupon Bonds
Zeroes
ideal
lack of data may exist (need zeroes for all maturities)
Coupon Bonds
plentiful
coupons and their reinvestment
low coupon rate vs. high coupon rate
short term rates they may have different YTM
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Short Rates, Coupons and YTM
Example
short rates are 8% & 10% for years 1 & 2; certainty
2-year bonds; Par = $1,000; coupon = 3% or 12%
Bond 1:
Bond 2:
%98.878.894$%)101%)(81(
030,1$
%)81(
30$
YTM
%94.887.053,1$%)101%)(81(
120,1$
%)81(
120$
YTM
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Measurements with Coupon Bonds 2
Example
2-year bonds; Par = $1,000; coupon = 3% or 12%
Prices: $894.78 (coupon = 3%); $1,053.87 (coupon = 12%)
Year-1 and Year-2 short rates
$ 894.78 = d1x 30 + d2x 1,030
$ 1,053.87 = d1x 120 + d2x 1,120
Solve the system: d2= 0.8417, d1= 0.9259
Conclude ...
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Measurements with Coupon Bonds 3
Example (continued)
%810.9259
1
1
11
11 rdr
%1018%)x0.8417(1
11
x)1(
12
21
2
rdr
r
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Measurements with Coupon Bonds 4
Practical problems pricing errors
taxes
are investors homogenous? investors can sell bonds prior to maturity
bonds can be called, put or converted
prices quotes can be stale
market liquidity
Estimation statistical approach
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Rising yield curves
Causes eithershort rates are expected to climb: E[rn]E[rn-1]
orthe liquidity premium is positive Fig. 15.5a
Interpretative assumptions
estimate the liquidity premium assume the liquidity premium is constant
empirical evidence
liquidity premium is not constant; past future?!
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Inverted yield curve
Easy interpretation if there is a liquidity premium
then inversion expectations of falling short rates why would interest rates fall?
inflation vs. real rates
inverted curve recession?
Example
current yield curve:The Economist
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Arbitrage Strategies
Question:
The YTM on 1-year-maturity zero coupon bonds is 5%
The YTM on 2-year-maturity zero coupon bonds is 6%.The YTM on 2-year-maturity coupon bonds with coupon rates of 12% (paid annually) is5.8%.
What arbitrage opportunity exists for an investment banking firm? What is the arbitragerofit?
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Arbitrage Strategies
Answer:
The price of the coupon bond, based on its YTM, is:
120 PA(5.8%, 2) + 1000 PF(5.8%, 2) = $1,113.99.
If the coupons were stripped and sold separately as zeros, then based on the YTM ozeros with maturities of one and two years, the coupon payments could be sold separately
for[120/1.05] + [1,120/1.06
2] = $1,111.08.
The arbitrage strategy is to:
buy zeros with face values of $120 and $1,120 and respective maturities of 1 and
2 yearssimultaneously sell the coupon bond.
The profit equals $2.91 on each bond.
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Fixed Income Portfolio Management
In general
bonds are securities just like other
use the CAPM
Bond Index Funds
Immunization
net worth immunization
contingent immunization
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Bond Index Funds
Idea
US indices
Solomon Bros. Broad Investment Grade (BIG) Lehman Bros. Aggregate
Merrill Lynch Domestic Master
composition
government, corporate, mortgage, Yankee bond maturities: more than 1 year
Canada: ScotiaMcLeod (esp. Universe Index)
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Bond Index Funds 2
Problems
lots of securities in each index
portfolio rebalancing
market liquidity
bonds are dropped (maturities, calls, defaults, )
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Bond Index Funds 3
Solution:
cellular approach
idea classify by maturity/risk/category/
compute percentages in each cell
match portfolio weights
effectiveness
average absolute tracking error = 2 to 16 b.p. / month
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Special risks for bond portfolios
cash-flow risk
call, default, sinking funds, early repayments,
solution: select high quality bonds
interest rate risk
bond prices are sensitive to YTM
solution
measure interest rate risk
immunize
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Interest Rate Risk
Equation: P= PV(annuity) + PV(final payment)
=
Yield sensitivity of bond Prices: P yield
Measure?
)1()1(1 r
Par
r
couponT
T
tt
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Interest Rate Risk 2
Determinants of a bonds yield sensitivity time to maturity
maturity sensitivity (concave function)
coupon rate
coupon sensitivity
discount bond vs. premium bond
zeroes have the highest sensitivity intuition: coupon bonds = average of zeroes
YTM initial YTM sensitivity
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Duration
Idea
maturity sensitivity
to measure a bonds yield sensitivity, measure its effective maturity
Measure
Macaulay duration:
1)1(
1
11
P
P
YTM
C
Pw
T
tt
tT
t
t)1.( YTMP
Cw
t
tt
t
T
t
wtD .1
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Duration 2
Duration = effective measure of elasticity
Proof
Modified duration
with
YTM
YTM
DP
P
1
)1(
.
YTMDP
P
.*
y
DD
1
*
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Duration 4
Interpretation 1
= average time until bond payment
Interpretation 2
% price change of coupon bond of a given duration
= % price change of zero with maturity = to duration
t
T
twtD .
1
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Duration 4
Example (BKM Table 15.3)
suppose YTM changes by 1 basis point (0.01%)
zero coupon bond with 1.8853 years to maturity
old price
new price
9623.831
05.1
10007706.3
6636.831
0501.1
1000
7706.3
YTM
YTMD
P
P
1
)1(.%0359.0
9623.831
9623.8316636.831
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Duration 5
Example: BKM4 Table 15.3
suppose YTM changes by 1 basis point (0.01%)
coupon bond
either compare the bonds price with YTM = 5.01%
relative to the bonds price with YTM = 5%
or simply compute the price change from the duration
%0359.005.1
%5%01.528853.1
1
)1(.
xx
YTM
YTMD
P
P
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Duration 6
Properties of duration (other things constant) zero coupon bond: duration = maturity
time to maturity
maturity duration exception: deep discount bonds
coupon rate
coupon duration
YTM YTM duration
exception: zeroes (unchanged)
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Duration 7
Properties of duration
duration of perpetuity =
less than infinity!
coupon bonds (annuities + zero)
see book
simplifies if par bond
y
yD
1
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Duration 8
Importance
simple measure
essential to implement portfolio immunization
measures interest rate sensitivity effectively
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Possible Caveats to Duration
1. Assumptions on term structure
Macaulay duration uses YTM
only valid for level changes in flat term structure
Fisher-Weil duration measure
T
tt
s
s
tT
t
t
r
Ct
PwtD
1
1
1 )1(
.1
.
T
ttt
T
t
tYTM
CtPwtD 11 )1(.
1.
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Possible Caveats to Duration 2
problems with the Fisher-Weil duration
assumes a parallel shift in term structure
need forecast of future interest rates bottom line: same problem as realized compound yield
Cox-Ingersoll-Ross duration
bottom line: lets keep Macaulay
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Possible Caveats to Duration 3
2. Convexity
Macaulay duration
first-order approximation:
small changes vs. large changes
duration = point estimate
for larger changes, an arc estimate is needed
solution: add convexity
)1(.* YTMDPP
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Possible Caveats to Duration 4
Convexity (continued)
second-order approximation:
2* ..21. YTMconvexityYTMD
PP
T
tt
t
YTM
Ctt
YTMPconvexity
1
2
2 )1().(
)1(
1
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Possible Caveats to Duration 5
Convexity: numerical example
P = Par = 1,000; T = 30 years; 8% annual coupon
computations give D*=11.26 years; convexity = 212.4 years
suppose YTM = 8% -> YTM = 10%
%52.2202.026.11.*
xYTMDP
P
%27.18..21. 2* YTMconvexityYTMD
PP
%85.18000,1
000,146.811
P
P
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Bottom Line on Duration
Very useful
But take it with a grain of salt for large changes
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Immunization
Why?
obligation to meet promises (pension funds)
protect future value of portfolio ratios, regulation, solvency (banks)
protect current net worth of institution
How? measure interest rate risk: duration
match duration of elements to be immunized
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Immunization
What? net worth immunization
match duration of assets and liabilities
target date immunization match inflows and outflows
immunize the net flows
Who?
insurance companies, pension funds target date immunization
banks net worth immunization
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Net Worth Immunization
Gap management
assets vs. liabilities
long term (mortgages, loans, )vs. short term (deposits, )
match duration of assets and liabilities
decrease duration of assets (ex.: ARM)
increase duration of liabilities (ex.: term deposits)
condition for success
portfolio duration = 0 (assets = liabilities)
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Target Date Immunization
Idea
Example: suppose interest rates fall
good for the pension fund
price risk
existing (fixed rate) assets increase in value
bad for the pension fund
reinvestment risk
PV of future liabilities increases
so more must be invested now
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Target Date Immunization 2
Solution
match duration of portfolio and funds horizon
single bondbond portfolio
duration of portfolio
= weighted average of components duration
condition: assets have equal yields
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Target Date Immunization 3
Question:
Pension funds pay lifetime annuities to recipients.
Firm expects to be in business indefinitely, its pension obligation perpetuity. Suppose, your pension fund must make perpetual payments of $2 million/year. The yield to maturity on all bonds is 16%.(a)duration of 5-year bonds with coupon rates of 12% (paid annually) is 4 years
duration of 20-year bonds with coupon rates of 6% (paid annually) is 11 years
how much of each of these coupon bonds (in market value) should you hold to bothfully fund and immunize your obligation?
(b)What will be the par value of your holdings in the 20-year coupon bond?
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Target Date Immunization 4
Answer:
(a)PV of the firms perpetual obligation = ($2 million/0.16) = $12.5 million. duration of this obligation = duration of a perpetuity = (1.16/0.16) = 7.25 years.
Denote by wthe weight on the 5-year maturity bond, which has duration of 4 years.Then,
wx 4 + (1 w) x 11 = 7.25, which implies that w= 0.5357. Therefore,
0.5357 x $12.5 = $6.7 million in the 5-year bond and 0.4643 x $12.5 = $5.8 million in the 20-year bond.
The total invested = $(6.7+5.8) million = $12.5 million, fully matching the fundingneeds.
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Target Date Immunization 5
Answer:
( b ) Price of the 20-year bond = 60 x PA(16%, 20) + 1000 x PF(16%, 20) = $407.11.
Therefore, the bond sells for 0.4071 times Par, and
Market value = Par value x 0.4071
=> $5.8 million = Par value x 0.4071
=> Par value = $14.25 million.
Another way to see this is to note that each bond with a par value of $1,000 sells for$407.11. If the total market value is $5.8 million, then you need to buy 14,250 bonds,which results in total ar value of $14,250,000.
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Dangers with Immunization
1. Portfolio rebalancing is needed
Time passes duration changes
bonds mature, sinking funds,
YTM changes duration changes
example: BKM4 Table 15.7
duration YTM5 8%
4.97 7%
5.02 9%
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Dangers with Immunization 2
2. Duration = nominal concept
immunization only for nominal liabilities
counter example
childrens tuition
why?
solution
do not immunize buy assets
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An Alternative? Cash-Flow Dedication
Buy zeroes
to match all liabilities
Problems difficult to get underpriced zeroes
zeroes not available for all maturities
ex.: perpetuity
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Contingent Immunization
Idea
try to beat the market
while limiting the downside risk
Procedure (BKM6 Fig. 16.10; BKM4 Fig. 15.6)
compute the PV of the obligation at current rates
assess available funds play the difference
immunize if trigger point is hit
top related