Fi8000 Valuation of Financial Assets pot

1
Fi8000Fi8000
Valuation ofValuation of
Financial AssetsFinancial Assets
Fall Semester 2009Fall Semester 2009
Dr. Isabel Dr. Isabel TkatchTkatch
Assistant Professor of FinanceAssistant Professor of Finance
Debt instrumentsDebt instruments
☺☺Types of bondsTypes of bonds
☺☺Ratings of bonds (default risk)Ratings of bonds (default risk)
☺☺
Spot and forward interest rateSpot and forward interest rate
☺☺
Spot

and

forward

interest

rateSpot

and

forward

interest

rate
☺☺The yield curveThe yield curve

☺☺DurationDuration
Bond CharacteristicsBond Characteristics
☺☺ A bond is a security issued to the lender (buyer) by the A bond is a security issued to the lender (buyer) by the
borrower (seller) for some amount of cash.borrower (seller) for some amount of cash.
☺☺ The bond obligates the issuer to make specified The bond obligates the issuer to make specified
payments of interest and principal to the lender, on payments of interest and principal to the lender, on
specified dates.specified dates.
☺☺ The typical The typical coupon bondcoupon bond obligates the issuer to make obligates the issuer to make
coupon payments, which are determined by the coupon payments, which are determined by the coupon coupon
raterate as a percentage of the as a percentage of the par valuepar value ((face valueface value). ).
When the bond matures, the issuer repays the par value.When the bond matures, the issuer repays the par value.
☺☺ ZeroZero coupon bondscoupon bonds are issued at discount (sold for a are issued at discount (sold for a
price below par value), make no coupon payments and price below par value), make no coupon payments and
pay the par value at the maturity date.pay the par value at the maturity date.
Bond Pricing Bond Pricing ExamplesExamples
☺☺The par value of a riskThe par value of a risk free zero coupon bond is free zero coupon bond is
$100. If the continuously compounded risk$100. If the continuously compounded risk free free
rate is 4% per annum and the bond matures in rate is 4% per annum and the bond matures in
three months, what is the price of the bond three months, what is the price of the bond
today?today?
today?today?
☺☺A risky bond with par value of $1,000 has an A risky bond with par value of $1,000 has an
annual coupon rate of 8% with semiannual annual coupon rate of 8% with semiannual
installments. If the bond matures 10 year from installments. If the bond matures 10 year from
now and the risknow and the risk adjusted cost of capital is 10% adjusted cost of capital is 10%
per annum compounded semiannually, what is per annum compounded semiannually, what is
the price of the bond today?the price of the bond today?
Yield to Maturity Yield to Maturity ExamplesExamples
☺☺What is the What is the yield to maturityyield to maturity (annual, (annual,
compounded semiannually) of the risky couponcompounded semiannually) of the risky coupon

bond, if it is selling at $1,200?bond, if it is selling at $1,200?
☺☺What is the What is the ex
p
ected
y
ield to maturit
y
ex
p
ected
y
ield to maturit
y
of the of the
py ypy y
risky couponrisky coupon bond, if we are certain that the bond, if we are certain that the
issuer is able to make all coupon payments but issuer is able to make all coupon payments but
we are uncertain about his ability to pay the par we are uncertain about his ability to pay the par
value. We believe that he will pay it all with value. We believe that he will pay it all with
probability 0.6, pay only $800 with probability probability 0.6, pay only $800 with probability
0.35 and won’t be able to pay at all with 0.35 and won’t be able to pay at all with
probability 0.05.probability 0.05.
Default Risk and Bond RatingDefault Risk and Bond Rating
☺☺ Although bonds generally promise a fixed flow of Although bonds generally promise a fixed flow of
income, in most cases this cashincome, in most cases this cash flow stream is uncertain flow stream is uncertain
since the issuer may default on his obligation.since the issuer may default on his obligation.
☺☺ US government bonds are usually treated as free of US government bonds are usually treated as free of
default (credit) risk. Corporate and municipal bonds aredefault (credit) risk. Corporate and municipal bonds are
default

(credit)

risk.

Corporate

and

municipal

bonds

are

default

(credit)

risk.

Corporate

and

municipal

bonds

are

considered risky.considered risky.
☺☺ Providers of bond quality rating:Providers of bond quality rating:
☺☺ Moody’s Investor ServicesMoody’s Investor Services
☺☺ Standard and Poor’s CorporationStandard and Poor’s Corporation
☺☺ Duff & PhelpsDuff & Phelps
☺☺ Fitch Investor ServiceFitch Investor Service
2
Default Risk and Bond RatingDefault Risk and Bond Rating
☺☺AAA (Aaa) is the top rating.AAA (Aaa) is the top rating.
☺☺Bonds rated BBB (Baa) and above are Bonds rated BBB (Baa) and above are
considered considered investmentinvestment grade bondsgrade bonds
☺☺Bonds rated lower than BBB are considered Bonds rated lower than BBB are considered
ltilti
dd
jkbdjkbd
specu
l
a
ti
vespecu
l
a
ti
ve gra
d
egra
d
e or or
j
un

k

b
on
d
s
j
un
k

b
on
d
s
☺☺Risky bonds offer a riskRisky bonds offer a risk premium. The greater premium. The greater
the default risk the higher the the default risk the higher the default riskdefault risk
premiumpremium
☺☺The The yield spreadyield spread is the difference between the is the difference between the
yield to maturity of high and lower grade bond. yield to maturity of high and lower grade bond.
Estimation of Default RiskEstimation of Default Risk
☺☺The determinants of the The determinants of the bond default riskbond default risk (the (the
probability of bankruptcy) and probability of bankruptcy) and debt quality ratingsdebt quality ratings
are based on measures of financial stability:are based on measures of financial stability:
☺☺Ratios of earnings to fixed costs;Ratios of earnings to fixed costs;
☺☺Levera
g
e ratios
;
Levera
g

e ratios
;
g;g;
☺☺Liquidity ratios;Liquidity ratios;
☺☺Profitability measures;Profitability measures;
☺☺CashCash flow to debt ratios.flow to debt ratios.
☺☺A complimentary measure is the A complimentary measure is the transition matrixtransition matrix ––
estimates the probability of a change in the rating of estimates the probability of a change in the rating of
the bond.the bond.
The TermThe Term Structure of Interest RatesStructure of Interest Rates
☺☺ The The short interest rateshort interest rate is the interest rate is the interest rate
for a given time interval (say one year, for a given time interval (say one year,
which does not have to start today).which does not have to start today).
☺☺ The The yield to maturityyield to maturity ((spot ratespot rate) is the ) is the
internal rate of return (say annual) of a zero internal rate of return (say annual) of a zero
coupon bond, that prevails today and coupon bond, that prevails today and
corresponds to the maturity of the bond.corresponds to the maturity of the bond.
ExampleExample
In our previous calculations we’ve assumed In our previous calculations we’ve assumed
that all the that all the short interest ratesshort interest rates are equal. Let are equal. Let
us assume the following:us assume the following:
Year (date)Year (date)
ShortShort
For the time intervalFor the time interval
Year

(date)Year

(date)
Short

Short

Interest rateInterest rate
For

the

time

intervalFor

the

time

interval
00rr
11
= 8%= 8% t = 0 to t = 1t = 0 to t = 1
11rr
22
= 10%= 10% t = 1 to t = 2t = 1 to t = 2
22rr
33
= 11%= 11% t = 2 to t = 3t = 2 to t = 3
33rr
44
= 11%= 11% t = 3 to t = 4t = 3 to t = 4
ExampleExample

What is the price of the 1, 2, 3 and 4 years What is the price of the 1, 2, 3 and 4 years
zerozero coupon bonds paying $1,000 at coupon bonds paying $1,000 at
maturity?maturity?
Maturit
y
Maturit
y
ZeroZero Coupon Bond PriceCoupon Bond Price
11 $1,000/1.08 = $925.93$1,000/1.08 = $925.93
22 $1,000/(1.08*1.10) = $841.75$1,000/(1.08*1.10) = $841.75
33 $1,000/(1.08*1.10*1.11) = $758.33$1,000/(1.08*1.10*1.11) = $758.33
44 $1,000/(1.08*1.10*1.11$1,000/(1.08*1.10*1.11
22
) = $683.18) = $683.18
ExampleExample
What is the yieldWhat is the yield toto maturity of the 1, 2, 3 and maturity of the 1, 2, 3 and
4 years zero4 years zero coupon bonds paying $1,000 at coupon bonds paying $1,000 at
maturity?maturity?
Mt itMt it
PiPi
Yi ld t M t itYi ld t M t it
M
a
t
ur
ityM
a
t
ur
ity

P
r
i
ce
P
r
i
ce
Yi
e
ld

t
o
M
a
t
ur
it
y
Yi
e
ld

t
o
M
a
t
ur

it
y
11 $925.93$925.93 yy
11
= 8.000%= 8.000%
22 $841.75$841.75 yy
22
= 8.995%= 8.995%
33 $758.33$758.33 yy
33
= 9.660%= 9.660%
44 $683.18$683.18 yy
44
= 9.993%= 9.993%
3
The TermThe Term Structure of Interest RatesStructure of Interest Rates
The price of the zeroThe price of the zero coupon bond is calculated coupon bond is calculated
using the using the short interest ratesshort interest rates (r(r
tt
, t = 1,2…,T). For , t = 1,2…,T). For
a bond that matures in T years there may be up to a bond that matures in T years there may be up to
T different short rates.T different short rates.
Price = FV / [(1+rPrice = FV / [(1+r
11
)(1+r)(1+r
22
)…(1+r)…(1+r
TT
)])]
The The yieldyield toto maturitymaturity (y(y

TT
) of the zero) of the zero coupon coupon
bond that matures in T years, is the internal rate of bond that matures in T years, is the internal rate of
return of the bond cash flow stream.return of the bond cash flow stream.
Price = FV / (1+yPrice = FV / (1+y
TT
))
TT
The TermThe Term Structure of Interest RatesStructure of Interest Rates
The price of the zeroThe price of the zero coupon bond paying $1,000 coupon bond paying $1,000
in 3 years is calculated using the short term rates:in 3 years is calculated using the short term rates:
Price = $1,000 / [1.08*1.10*1.11] = Price = $1,000 / [1.08*1.10*1.11] = $758.33$758.33
The The yieldyield toto maturitymaturity (y(y
33
) of the zero) of the zero coupon coupon
bond that matures in 3 years solves the equationbond that matures in 3 years solves the equation
$758.33$758.33 = $1,000 / (1+y= $1,000 / (1+y
33
))
33
yy
33
= = 9.660%.9.660%.
The TermThe Term Structure of Interest RatesStructure of Interest Rates
Thus the Thus the yieldsyields are in fact are in fact geometric geometric
averagesaverages of the of the short interest ratesshort interest rates in in
each periodeach period
(1+y(1+y
))
TT

=(1+r=(1+r
)(1+r)(1+r
)(1+r)(1+r
))
(1+y(1+y
TT
))
TT
=

(1+r=

(1+r
11
)(1+r)(1+r
22
)
…
(1+r)
…
(1+r
TT
))
(1+y(1+y
TT
) = [(1+r) = [(1+r
11
)(1+r)(1+r
22
)…(1+r)…(1+r

TT
)])]
(1/T)(1/T)
The The yield curveyield curve is a graph of bond yieldis a graph of bond yield toto
maturity as a function of timematurity as a function of time toto maturity.maturity.
The Yield Curve (Example)The Yield Curve (Example)
YTM
9.660%
9.993%
Time to Maturity
24
8.000%
8.995%
13
The TermThe Term Structure of Interest RatesStructure of Interest Rates
If we assume that all the If we assume that all the short interest ratesshort interest rates ((rr
tt
, t , t
= 1, 2…,T) are equal, then all the = 1, 2…,T) are equal, then all the yieldsyields ((yy
TT
) of ) of
zerozero coupon bonds with different maturities (T = 1, coupon bonds with different maturities (T = 1,
2…) are also equal and the yield curve is flat.2…) are also equal and the yield curve is flat.
AA
flatflat
yield curve is associated with an expectedyield curve is associated with an expected
A

A

flatflat
yield

curve

is

associated

with

an

expected

yield

curve

is

associated

with

an

expected

constant interest rates in the future;constant interest rates in the future;

An An upward slopingupward sloping yield curve is associated with yield curve is associated with
an expected increase in the future interest rates;an expected increase in the future interest rates;
A A downward slopingdownward sloping yield curve is associated yield curve is associated
with an expected decrease in the future interest with an expected decrease in the future interest
rates.rates.
The Forward Interest RateThe Forward Interest Rate
☺☺ The The yield to maturityyield to maturity ((spot ratespot rate) is the internal ) is the internal
rate of return of a zero coupon bond, that rate of return of a zero coupon bond, that
prevails today and corresponds to the maturity prevails today and corresponds to the maturity
of the bond.of the bond.
☺☺
TheThe
forward interest rateforward interest rate
is the rate of return ais the rate of return a
☺☺
The

The

forward

interest

rateforward

interest

rate
is

the

rate

of

return

a

is

the

rate

of

return

a

borrower will pay the lender, for a specific loan, borrower will pay the lender, for a specific loan,
taken at a specific date in the future, for a taken at a specific date in the future, for a
specific time period. If the principal and the specific time period. If the principal and the
interest are paid at the end of the period, this interest are paid at the end of the period, this
loan is equivalent to a forward zero coupon loan is equivalent to a forward zero coupon
bond.bond.
4
The Forward Interest RateThe Forward Interest Rate

Suppose the price of 1Suppose the price of 1 year maturity zeroyear maturity zero coupon coupon
bond with face value $1,000 is $925.93, and the bond with face value $1,000 is $925.93, and the
price of the 2price of the 2 year zeroyear zero coupon bond with $1,000 coupon bond with $1,000
face value is $841.68. face value is $841.68.
If there is no opportunity to make arbitrage profits, If there is no opportunity to make arbitrage profits,
what is the 1what is the 1 year forward interest rate for the year forward interest rate for the
second year?second year?
How will you construct a synthetic 1How will you construct a synthetic 1 year forward year forward
zerozero coupon bond (loan of $1,000) that coupon bond (loan of $1,000) that
commences at t = 1 and matures at t = 2?commences at t = 1 and matures at t = 2?
The Forward Interest RateThe Forward Interest Rate
If there is no opportunity to make arbitrage profits, If there is no opportunity to make arbitrage profits,
the 1the 1 year forward interest rate for the second year year forward interest rate for the second year
must be the solution of the following equation:must be the solution of the following equation:
(1+y(1+y
))
22
=(1+y=(1+y
)(1+f)(1+f
))
(1+y(1+y
22
))
22
=

(1+y=

(1+y
11

)(1+f)(1+f
22
)
,
)
,
wherewhere
yy
TT
= yield to maturity of a T= yield to maturity of a T year zeroyear zero coupon bondcoupon bond
ff
tt
= = 11 year forward rate for year tyear forward rate for year t
The Forward Interest RateThe Forward Interest Rate
In our example, In our example, yy
11
= 8% and y= 8% and y
22
= 9%. Thus,= 9%. Thus,
(1+0.09)(1+0.09)
22
= (1+0.08)(1+f= (1+0.08)(1+f
22
))
ff
= 0 1001 = 10 01%= 0 1001 = 10 01%
ff
22
=

0
.
1001

=

10
.
01%
.
=

0
.
1001

=

10
.
01%
.
Constructing the loan (borrowing):Constructing the loan (borrowing):
1. Time t = 0 CF should be zero;1. Time t = 0 CF should be zero;
2. Time t = 1 CF should be +$1,000;2. Time t = 1 CF should be +$1,000;
3. Time t = 2 CF should be 3. Time t = 2 CF should be $1,000(1+f$1,000(1+f
22
) = ) = $1,100.1.$1,100.1.
The Forward Interest RateThe Forward Interest Rate
Constructing the loan:Constructing the loan:

we would like to borrow $1,000 a year from now we would like to borrow $1,000 a year from now
for a forward interest rate of 10.01%.for a forward interest rate of 10.01%.
1.1.
(
#3
)
CF
(
#3
)
CF
00
= $925.93 but it should be zero. We offset that = $925.93 but it should be zero. We offset that
()()
00
cash flow if we buy the 1cash flow if we buy the 1 year zero coupon bond for year zero coupon bond for
$925.93. That is, if we buy $925.93/$925.93 = 1 units of $925.93. That is, if we buy $925.93/$925.93 = 1 units of
the 1the 1 year zero coupon bond;year zero coupon bond;
2.2. (#1) CF(#1) CF
11
should be equal to $1,000;should be equal to $1,000;
3.3. (#2) CF(#2) CF
22
= = $,1000*1.1001 = $,1000*1.1001 = $1,100.1. We generate that $1,100.1. We generate that
cash flow if we sell 1.1001 of the 2cash flow if we sell 1.1001 of the 2 year zeroyear zero coupon coupon
bond for 1.1001* $841.68 = $925.93.bond for 1.1001* $841.68 = $925.93.
Bond Price SensitivityBond Price Sensitivity
☺☺ Bond prices and yields are inversely related.Bond prices and yields are inversely related.
☺☺ Prices of Prices of longlong term bondsterm bonds tend to be more tend to be more
sensitive to changes in the interest rate sensitive to changes in the interest rate

(required rate of return / cost of capital) than (required rate of return / cost of capital) than
those of shortthose of short

term bonds (compare two zeroterm bonds (compare two zero
those

of

shortthose

of

short
term

bonds

(compare

two

zero

term

bonds

(compare

two

zero

coupon bonds with different maturities).coupon bonds with different maturities).
☺☺ Prices of Prices of high couponhigh coupon raterate bondsbonds are less are less
sensitive to changes in interest rates than sensitive to changes in interest rates than
prices of low couponprices of low coupon rate bonds (compare a rate bonds (compare a
zerozero coupon bond and a couponcoupon bond and a coupon paying bond paying bond
of the same maturity).of the same maturity).
DurationDuration
The observed bond price properties suggest that the The observed bond price properties suggest that the
timingtiming and and magnitudemagnitude of of all cash flowsall cash flows affect bond affect bond
prices, not only timeprices, not only time toto maturity. maturity. Macaulay’s durationMacaulay’s duration is a is a
measure that summarizes the timing and magnitude effects measure that summarizes the timing and magnitude effects
of all
p
romised cash flows.of all
p
romised cash flows.
pp
()
1
Cash flow weight:
/1

Macauley's Duration:

t
t
t

T
t
t
CF y
w
B
ondPrice
Dtw
=
+
=
=⋅
∑
5
ExampleExample
Calculate the duration of the following bonds:Calculate the duration of the following bonds:
1.1. 8% coupon bond; $1,000 par value; 8% coupon bond; $1,000 par value;
semiannual installments; Two years to semiannual installments; Two years to
maturity; The annual discount rate ismaturity; The annual discount rate is
maturity;

The

annual

discount

rate

is

maturity;

The

annual

discount

rate

is

10%, compounded semi10%, compounded semi annually.annually.
2.2. ZeroZero coupon bond; $1,000 par value; coupon bond; $1,000 par value;
Two year to maturity; The annual Two year to maturity; The annual
discount rate is 10%, compounded semidiscount rate is 10%, compounded semi
annually.annually.
Properties of the DurationProperties of the Duration
☺☺ The The duration of a zeroduration of a zero coupon bond coupon bond
equals its time to maturity;equals its time to maturity;
☺☺ Holding maturity and par value constant, Holding maturity and par value constant,
the bond
’
sthe bond
’
s
duration is lowerduration is lower
when thewhen the
the

bond s

the

bond s

duration

is

lowerduration

is

lower
when

the

when

the

coupon rate is highercoupon rate is higher;;
☺☺ Holding couponHolding coupon rate and par value rate and par value
constant, the bond’s constant, the bond’s duration generally duration generally
increasesincreases with its with its time to maturitytime to maturity
Macaulay’s DurationMacaulay’s Duration
Bond price (p) changes as the bond’s yield Bond price (p) changes as the bond’s yield

to maturity (y) changes. We can show that to maturity (y) changes. We can show that
the proportional price change is equal to the the proportional price change is equal to the
proportional change in the yield times theproportional change in the yield times the
proportional

change

in

the

yield

times

the

proportional

change

in

the

yield

times

the

duration.duration.
(1 )
(1 )
Py
D
Py
ΔΔ+
=− ⋅
+
Modified DurationModified Duration
Practitioners commonly use the modified Practitioners commonly use the modified
duration measure duration measure D*=D/(1+y),D*=D/(1+y), which can be which can be
presented as a measure of the bond price presented as a measure of the bond price
sensitivity to changes in the interest ratesensitivity to changes in the interest rate
sensitivity

to

changes

in

the

interest

rate
.
sensitivity

to

changes

in

the

interest

rate
.
*
P
D
y
P
Δ
=
−⋅Δ
ExampleExample
Calculate the percentage price change for the following Calculate the percentage price change for the following
bonds, if the semibonds, if the semi annual interest rate increases from 5% to annual interest rate increases from 5% to
5.01%:5.01%:
1.1. 8% coupon bond; $1,000 par value; semiannual 8% coupon bond; $1,000 par value; semiannual
installments; Two years to maturity; The annual installments; Two years to maturity; The annual
d
i
scou

nt r
a
t
e
i
s
1
0%,

co
m
pou
n
ded

se
mi
d
i
scou
nt r
a
t
e
i
s
1
0%,

co

m
pou
n
ded

se
mi
a
nn
ua
ll
y
.
a
nn
ua
ll
y
.
d scou a e s 0%, co pou ded sed scou a e s 0%, co pou ded se
auayauay
2.2. ZeroZero coupon bond; $1,000 par value; Two year to coupon bond; $1,000 par value; Two year to
maturity; The annual discount rate is 10%, maturity; The annual discount rate is 10%,
compounded semicompounded semi annually.annually.
3.3. A zeroA zero coupon bond with the same duration as the 8% coupon bond with the same duration as the 8%
coupon bond (1.8852 years or 3.7704 6coupon bond (1.8852 years or 3.7704 6 months months
periods. The modified duration is 3.7704/1.05 = 3.591 periods. The modified duration is 3.7704/1.05 = 3.591
66 months periods).months periods).
ExampleExample
The percentage price change for the following bonds as a The percentage price change for the following bonds as a

result of an increase in the interest rate (from 5% to result of an increase in the interest rate (from 5% to
5.01%):5.01%):
1.1. ∆P/P = ∆P/P = D*·∆y = D*·∆y = (3.7704/1.05)·0.01% = (3.7704/1.05)·0.01% = 0.03591%0.03591%
2.2. ∆P/P = ∆P/P = D*·∆y = D*·∆y = (4. /1.05)·0.01% = (4. /1.05)·0.01% = 0.03810%0.03810%
3.3. ∆P/P = ∆P/P = D*·∆y = D*·∆y = (3.7704/1.05)·0.01% = (3.7704/1.05)·0.01% = 0.03591%0.03591%
Note that:Note that:
When two bonds have the same duration (not time to When two bonds have the same duration (not time to
maturity) they also have the same price sensitivity to maturity) they also have the same price sensitivity to
changes in the interest rate: 1 vs. 3.changes in the interest rate: 1 vs. 3.
When the duration (not timeWhen the duration (not time toto maturity) is higher for maturity) is higher for
one of the bonds then the price sensitivity of that bond one of the bonds then the price sensitivity of that bond
is also high: 1 vs. 2; 3 vs. 2.is also high: 1 vs. 2; 3 vs. 2.
6
The Use of DurationThe Use of Duration
☺☺ It is a simple It is a simple summary statisticsummary statistic of the effective of the effective
average maturity of the bond (or portfolio of average maturity of the bond (or portfolio of
fixed income instruments);fixed income instruments);
☺☺
Duration can be presented as aDuration can be presented as a
measure ofmeasure of
☺☺
Duration

can

be

presented

as

a

Duration

can

be

presented

as

a

measure

of

measure

of

bond (portfolio) price sensitivitybond (portfolio) price sensitivity to changes to changes
in the interest rate (cost of capital);in the interest rate (cost of capital);
☺☺ Duration is an essential Duration is an essential tool in portfolio tool in portfolio
immunization:immunization: hedging interest rate risk.hedging interest rate risk.
Uses of Interest Rate HedgesUses of Interest Rate Hedges
☺☺ Owners of fixedOwners of fixed income portfolios protecting income portfolios protecting
against a rise in ratesagainst a rise in rates

☺☺ Corporations planning to issue debt securities Corporations planning to issue debt securities
protecting against a rise in ratesprotecting against a rise in rates
protecting

against

a

rise

in

ratesprotecting

against

a

rise

in

rates
☺☺ Investor hedging against a decline in rates for a Investor hedging against a decline in rates for a
planned future investmentplanned future investment
☺☺ Exposure for a fixedExposure for a fixed income portfolio is income portfolio is
proportional to modified durationproportional to modified duration
Hedging Interest Rate Risk: Textbook p. 802Hedging Interest Rate Risk: Textbook p. 802
Portfolio value = $10 millionPortfolio value = $10 million
Modified duration = 9 yearsModified duration = 9 years

If rates rise by 10 basis points (If rates rise by 10 basis points (bpbp) ) ΔΔy = y = ( .1% )( .1% )
Change in value = D*·∆y = ( 9 ) ( .1% ) = ( .9% ) or $90,000Change in value = D*·∆y = ( 9 ) ( .1% ) = ( .9% ) or $90,000
Price value of a basis point (PVBP) = Price value of a basis point (PVBP) =
$90,000 / 10 $90,000 / 10 bpbp = $9,000= $9,000
PVBP: PVBP: measures dollar value sensitivity to changes in measures dollar value sensitivity to changes in
interest ratesinterest rates
Hedging Interest Rate Risk: Text ExampleHedging Interest Rate Risk: Text Example
Hedging strategy: offsetting position in Treasury bonds Hedging strategy: offsetting position in Treasury bonds
futures. futures.
TT Bond futures contract calls for delivery of $100,000 par Bond futures contract calls for delivery of $100,000 par
value Tvalue T Bonds with 6% coupons and 20Bonds with 6% coupons and 20 years maturity.years maturity.
AssumptionsAssumptions::
Contract Modified duration = D* = 10 yearsContract Modified duration = D* = 10 years
Futures price = FFutures price = F
00
= $90 per $100 par value= $90 per $100 par value
(i.e., contract multiplier = 1,000)(i.e., contract multiplier = 1,000)
Hedging Interest Rate Risk: Text ExampleHedging Interest Rate Risk: Text Example
If rates rise by 10 basis points (If rates rise by 10 basis points (bpbp) ) ΔΔy = y = ( .1% )( .1% )
Change in value = D*·∆y = ( 10 ) ( .1% ) = ( 1% )Change in value = D*·∆y = ( 10 ) ( .1% ) = ( 1% )
Futures price change = Futures price change = ∆P ∆P == ( $90 ) ( 1% ) = $0.9( $90 ) ( 1% ) = $0.9
(
i.e.
,
from
$
90 to
$
89.10
)(

i.e.
,
from
$
90 to
$
89.10
)
(, $ $ )(, $ $ )
The gain on each short contract = 1,000 * $0.90 = $900The gain on each short contract = 1,000 * $0.90 = $900
Price value of a basis point (PVBP) = Price value of a basis point (PVBP) =
$900 / 10 $900 / 10 bpbp = $90= $90
Hedge Ratio: Text ExampleHedge Ratio: Text Example
H =
=
PVBP for the portfolio
PVBP for the hedge vehicle
$9,000
100
=
$90 per contract
=
100
contracts
100 T-Bond futures contract will serve to offset
the portfolio’s exposure to interest rate
fluctuations. The hedged position (long portfolio
+ short futures) has a PVBP of zero.
7
Practice ProblemsPractice Problems

BKM Ch. 14: 3, 4, 5, 8a, 9, 10, 14, 22BKM Ch. 14: 3, 4, 5, 8a, 9, 10, 14, 22
BKM Ch. 15:BKM Ch. 15:
Concept check: 8Concept check: 8 9;9;
End of chapter: 6, 14, CFA: 4, 10.End of chapter: 6, 14, CFA: 4, 10.
BKM Ch. 16: BKM Ch. 16:
Concept check: 1Concept check: 1 2;2;
End of chapter: 2End of chapter: 2 6, CFA: 3a6, CFA: 3a 3c.3c.