fi9-interestrisk

7/18/2019 fi9-interestrisk

1/23

Interest Rate Risk

Interest rate changes have significant effects on many

financial firms net income, asset value, liability value and

equity value (net difference between assets and liabilities).

Three Traditional Ways to Measure Interest ate is!

". e#ricing $a# % focuses on net interest income changes.

&. Maturity $a# % focuses on equity value changes % ignorescash flow timing.

'. uration $a# % focuses on equity value including cash

flow timing.

uration $a# is the most com#lete and #recise measure.


2/23

Repricing GapThe re#ricing ga# is the dollar value of the difference

between the boo! values of assets and liabilities with a

certain range of maturity (called a buc!et).

*te#s to +alculate the e#ricing $a# and +umulative $a#

". ist the firms assets and liabilities by buc!et.

&. e#ricing $a# - (assets % liabilities) by buc!et.

'. +umulative $a# - sum of e#ricing $a#s.

The effect of interest rate changes on a firms net income is

II - ($a#)

where II is the annualized change in net interest incomeand is the annualinterest rate change.


3/23

Repricing Gap ExampleTime /eriod 0ssets iabilities $a# +m. $a#

" day &1 '1 %"1 %"1

" day % ' months '1 21 %"1 %&1

' % 3 months 41 56 %"6 %'6

3 % "& months 71 41 &1 %"6" % 6 years 21 '1 "1 %6

8ver 6 years "1 6 6 1

ote9 emand de#osits are e:cluded from liabilities because

the interest rates #aid (;ero) do not change.


4/23

Weaknesses of Repricing Gap

". It ignores mar!et value changes of assets and liabilities.

&. 0ggregation of assets and liabilities can be misleading

when their distributions within a buc!et differ.

'. unoff #roblems % some assets or liabilities may mature

#artially or com#letely before the stated maturity date

% e.g., '1 year mortgages seldom last '1 years.

2. unoffs may be sensitive to interest rate changes.

6. Ignores the effect of off%balance%sheet items.

*ee *M >oldings "1< ('?&111) @dgar filing for e:am#le.


5/23

Example: Chap 8 - Pro! "

+onsider the following balance sheet.

+ash "1 8vernight e#os "41

" mon, 4.16A Tbill 46 4%yr 5.66A *ub. eb. "61

' mon, 4.&6A Tbill 46

&%yr, 4.6A Tnote 61

5%yr, 5.73A Tnote "11

6%yr, 5.&A, muni &6

(reset % 3 months) @quity "6

Total 0ssets ''6 Total iab. B @quity ''6

a. '1 day re#ricing ga# - 46 % "41 - %76

7" day re#ricing ga# - (46 B 46) % "41 - %&1

&%yr re#ricing ga# - (46 B 46 B 61 B &6) % "41 - 66


6/23

b. '1 day im#act of a .6A rise or a .46A dro# in all rates.

II - (%76 million) (.116) - %246,111.

II - (%76 million) (%.1146) - 4"&,611

c. 0ssume one%year runoffs of C"1 million for &%yr Tnote and

C&1 million for 5%year Tnote.

"%yr re#ricing ga# - (46 B 46 B "1 B &1 B &6) % "41 - '6

d. edo #art b.

II - ('6 million) (.116) - "46,111.

II - ('6 million) (%.1146) - %&3&,611


7/23

#aturit$ Gap #odelThe Maturity $a# measures the difference between a firms

weighted average asset maturity (M0) and weighted average

liability maturity (M).

Maturity $a# - (M0% M)

M0- W0"M0"B W0&M0&B W0'M0'B D B W0nM0n

M- W"M"B W&M&B W'M'B D B WnMn

W0i- (mar!et value of asset i)?(mar!et value of total assets).

Wi- (mar!et value of liability E)?(mar!et value of total liab.)

M0iis the maturity of asset i.M is the maturit of liabilit .


8/23

#aturit$ Gap and the Effect of

Interest Rates on E%uit$ &alueWhen (M0% M) F 1then an increase (decrease) in interest

rates is e:#ected to decrease (increase) a financial

firms equity.

When (M0% M) G 1then an increase (decrease) in interestrates is e:#ected to increase (decrease) a financial

firms equity.

@quity - 0ssets % iabilities

or in change form,

@quity - 0ssets % iabilities

@ uit 0ssets and iabilities are measured in mar!et value.


9/23

Example: Ch 8! '( - )ond

Instead of #ortgage+ounty Han! has the following Halance sheet9

+ash C&1 emand e#osits C"11

"6%yr, "1A oan "31 6%yr, 3A + Halloon &"1

'1%yr, 5A Hond '11 &1%yr, 4A ebenture "&1

@quity 61

Total 0ssets 251 Total iab. 0nd @q. 251

a. What is the Maturity $a#=

M0- 1(&1) B "6("31) B '1('11)J?251 - &'.46

M- 1("11) B 6(&"1) B &1("&1)J?251 - 5.1&

M$0/ - &'.46 % 5.1& - "6.4' years


10/23

b. What is the ga# if all interest rates rise by "A=

oan Kalue - "3/K0 "6,.""J B "31/K "6,.""J - "25.27

Hond Kalue - &2 /K0 '1,.17J B '11/K '1,.17J - &37.15

M0- 1(&1) B "6("25.27) B '1(&37.15)J?2'4.3 - &'.6'

+ Kalue - "&.3/K0 6,.14J B &"1/K 6,.14J - &1".'7

ebenture Kalue - 5.2/K0 &1,.15J B "&1/K &1,.15J - "15.&&M- 1("11) B 6(&1".'7) B &1("15.&&)J?217.3" - 4.77

M$0/ - &'.6' % 4.77 - "6.62

c. Mar!et Kalue of @quity falls by && to &5 (2'4.3 % 217.3").


11/23

*uration Gap #odel

uration is a better measure of asset or liability interest rate

ris! than maturity. The duration formula is

- time weight : (discount cash flows)?(Hond /rice)

- duration

+Lt - cash flow in time #eriod t

- yield to maturity (interest rate) #er #eriodT - maturity in #eriods % usually semi%annual

*

C+ t

,C+

,

t

t

t

t

t

t

= +

+

=

=

"

"

"

"

( )

( )

( )


12/23

0 *horter Way to +alculate a

+ou#on HondNs uration

where Tis the number of #ayments % for a thirty

year bond with semi%annual cou#ons T - 31

cis the cou#on rate #er #eriod % for a "&A cou#on #aid semi%annually, c - .13.

is the yield to maturity #er #eriod % for a

7A yield with semi%annual cou#ons - .126

*,

,

, - c ,

c , ,-

=+

+ +

+ +

( ) ( ) ( )

I( ) J

" "

" "


13/23

@O0M/@9 '1 year treasury bond % "&A cou#on (#aid

semi%annually) % 7A yield

- &1.54 semi%annual #eriods or "1.22 annual #eriods

ote9 ield and interest rate are used interchangeably here

because a bonds Pinterest rateQ is called its Pyield.Q

* = + + + + +

( . ).

I( . ) (. . )JI. I( . ) J . J

" 126126

" 126 31 13 12613 " 126 " 12631


14/23

Rsing uration to @stimate

Hond /rice +hangeInterest rate changes affect the value of #romised #aymentsand the value of additional income from reinvested

#ayments. uration measures both effects.

uration is the elasticity (from economics) of the asset orliability #rice with res#ect to a yield change.

Lor a bond #aying semi%annual cou#ons9

n - the new semi%annual yield

o - the old semi%annual yield - duration in semi%annual #eriods

)"()(

)"()"(A

o

on

,

,,x*,

,x*P+

=+

+=


15/23

@O0M/@9 '1 yr Treasury

"&A cou#on (#aid semiannually)

uration - &1.54 semi%annual #eriods

8ld yield - 7A annual % ew ield - 5.6A annual

- .16 - 6A


16/23

*uration Gap

*imilar to the Maturity $a#, uration $a# measures the

difference between a firms weighted average asset uration

(0) and weighted average liability uration ().

uration $a# - (0% )

0- W0"0"B W0&0&B W0'0'B D B W0n0n

- W""B W&&B W''B D B Wnn

W0i- (mar!et value of asset i)?(mar!et value of total assets).

Wi- (mar!et value of liability E)?(mar!et value of total liab.)

0iis the duration of asset i.


17/23

*uration and the Effect of

Interest Rates on E%uit$ &alue0 more #recise measure of the effect of an interest ratechange on a financial firms equity value is9

@quity - %0% !J0(n% o)?(" B o)

where !-?0 and 0% !J is the leverage%adEusted

uration $a#, hereafter referred to as Eust the uration $a#.

To eliminate the effect of interest rate changes on the value

of a firms equity (called immuni;ation), some havesuggested setting

Maturity $a# - (M0% M) - 1 or

uration $a# - (0% ) - 1.


18/23

0 more #recise way to Pimmuni;eQ equity value is by setting

0% !J - 1.

0 ty#ical situation is that the dollar amount of assets (0)

and liabilities () are given, then we select #articular assets

and liabilities with durations 0and so 0% !J - 1.

Lor solvent firms, we !now that (0 % ) - @ F 1 and ! G "

so that equity immuni;ation requires 0G .

Many financial firms have 0F ,which im#lies that they

are not immuni;ed.

To immuni;e equity as a #ercent of assets (@?0), setting0- is the #ro#er method.


19/23

Example: Ch! ". /0

The balance sheet of $otbuc!s Han! is

+ash '1 5A, &%yr e#osits &1

5.6A Led. Lunds &1 5.6A Led. Lunds 61

""A Lloat oan "16 7A @uro + "'1

"&A, 6%yr oan 36 @quity &1Total 0ssets &&1 Total iabilities &&1

a. Li:ed oan urationb. 0ssuming Lloating ate and Led Lunds have .'3 duration

0sset uration - '1(1) B 36(2.1') B "&6(.'3)J?&&1 - ".2

c. e#osits uration

1'.2

J"&.J")"&."("&.

)J"&."&(.6)"&."(

"&.

)"&."(6

=

++

++

+=*

7&6."J15.J")15."I(15I.

)J15.15(.&)15."I(

15.

)15."(

& =

++

++

+=*


20/23

d. 0ssuming the @uro + has .21" duration,

iab. uration - &1(".7&6) B "51(.21")J?&11 - .66'6

e. uration $a# - ".2 % (&11?&&1)(.66'6) - .57'5 years.

f. 0n "A increase in interest rates decreases equity by

@ - %.57'5(.1")S&&1 - %",733,'31

g. 0 decrease of .6A in interest rates increases equity by

@ - %.57'5(%.116)S&&1 - 75',"51

h. To eliminate the effects on equity, the ban! can increase

liability its duration to ".62 : (&11?&&1)(.66'6) - 1J,

decrease its asset duration to .61'& ".2 (&11?&&1)(:) - 1J,or some combination of the two.


21/23

Criticisms of *uration and

E%uit$ Immunization". 0s interest rates change, durations change, so one must

constantly rebalance assets and liabilities to !ee#

immuni;ed. Transactions costs may be large.

&. We have assumed all interest rates change by the same

amount but this is seldom true.

'. We have ignored default ris!. efault or #ayment

rescheduling can increase or decrease duration.

2. urations of floating rate instruments and demand

de#osits are unclear. Lor floating rate instruments we

usually assume duration equals the time to re#ricing.emand de#osits duration is assumed to be ;ero or small.


22/23

6. The most significant criticism is that duration is an

a##ro:imation and wor!s best for small changes in yields.

+onve:ity (+O) is a measure of the duration error when

yield changes are large. To get a better a##ro:imation to

#rice changes due to interest rate changes, one can adEust anearlier #rice change equation to9

&)(6.)"(

)(

)"(

)"(A

on

o

on

,,C1,

,,x*

,

,x*P +

+

=

+

+=

The change in equity value becomes9

@quity - %0% !J0(n% o)?(" B o)

B .6+O0% !+OJ0(n% o)&


23/23

Example of 2sing Con3exit$

>us!y Linancial has C"11 million of assets with a weighted

average duration of 5.6, a weighted average conve:ity of &11

and a yield of "1A. It also has C51 million of liabilities with

a weighted average duration of 3, a weighted average

conve:ity of 21 and a yield of "1A. If mar!et yields rise by& #ercentage #oints, what is the e:#ected change in >us!ys

equity value if conve:ity is ignored= >ow about if one

considers conve:ity=

@quity - %5.6 % .5(3)J"11(.1&)?(" B ."1) - %C3.4 MM

with conve:ity

@quity - %C3.4 B .6&11 % .5(21)J"11(.1&)&- %C'.&3 M

>ere ignoring conve:ity overestimates the negative change

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