InInterestterest RRateate RiRisksk M Modelingodeling$1,000 face value and a 10% annual coupon rate, but different maturities, as shown in Table 9.1. Bond # Face value Maturity Annual

Interest Rate Risk ModelingInterest Rate Risk ModelingThe Fixed Income Valuation Course

Sanjay K. Nawalkhaj yGloria M. SotoNatalia A. Beliaeva

• Interest Rate Risk Modeling : The Fixed Income Valuation Course. Sanjay K. Nawalkha, Gloria M. Soto, N li K B li 200 Wil FiNatalia K. Beliaeva, 2005, Wiley Finance. – Chapter 9 :

Key Rate Durations with VaR Analysis

• Goals:– Use the key rate durations to hedge against the

h i fi it b f k i t t tchanges in a finite number of key interest rates.– Use the VaR (value at risk) analysis to estimate the

maximum loss at a given level of confidencemaximum loss at a given level of confidence.– Know the limitations of the key rate model.

2

Chapter 9 : Key Rate Durations with VaR Analysis

• Introduction

• Key Rate Changes• Key Rate Changes

• Key Rate Durations and Convexities

• Risk Measurement and Management

• Key Rate Durations and Value At Risk Analysis

• Limitations of the Key Rate Modely

3


• Introduction






4

Introduction• The key rate durations hedge against the changes in the

finite number of key interest rates that proxy for thefinite number of key interest rates that proxy for the shape changes in the entire term structure.

• The key rate duration model describes the shifts in the term structure as a discrete vector representing the p gchanges in the key zero-coupon rates of various maturities.

• Key rate durations are defined as the sensitivity of the f ffportfolio value to the given key rates at different points

along the term structure.

5

Introduction• It does not require a stationary covariance structure of

interest rate changesinterest rate changes.

• The model allows for any number of key rates therefore• The model allows for any number of key rates, therefore, interest rate risk can be modeled and hedged to a high degree of accuracy.g y

• The number of key rates durations to be used and the ycorresponding choice of key rates remain quite arbitrary under the key rate model.

• It gives only the linear exposures to the key rates.

6


• Introduction






7

Key Rate Changes

• Any smooth change in the term structure of zero coupon• Any smooth change in the term structure of zero-coupon yields can be represented as a vector of changes in a number of properly chosen key rates:number of properly chosen key rates:

( )= Δ Δ ΔK1 2︵︶, ︵︶, ︵︶︵9.1 ︶mTSIR shift y t y t y twhere is the zero-coupon rate for term and

define the set of m key rates.

︵︶iy t it1 2︵︶ , ︵︶ , . . . , ︵︶my t y t y t y

• The shift in the term structure is approximated by a

1 2︵︶ , ︵︶ , , ︵︶my y y

The shift in the term structure is approximated by a piecewise linear function of the changes in the m key rate.

8

Key Rate Changes

• The changes in all other interest rates are approximated• The changes in all other interest rates are approximated by linear interpolation of the changes in the adjacent key rates.rates.

• The linear interpolation is performed in two steps:The linear interpolation is performed in two steps:– Step one:

Define the linear contribution made by the︵︶s t tDefine the linear contribution made by the change in the i th key rate, , to the change in a given zero-coupon rate .

︵, ︶is t tΔ ︵︶y tｉ

Δ ︵︶y t– Step two:

Add up the linear contributions for i=1,2,…,m, to ︵, ︶is t t

9

pobtain change in the given zero-coupon rate . Δ ︵︶y t

Key Rate Changes• The linear interpolation is performed in two steps:

Step one as:– Step one as: ⎧Δ <⎪

−⎪Δ ≤ ≤⎨

1 1

2

︵︶

︵︶︵︶

y t t tt ts t t y t t t t= Δ ≤ ≤⎨ −⎪

⎪ >⎩

21 1 1 22 1

2︵, ︶︵︶

0

s t t y t t t tt t

t t

−⎧ <⎪ −⎪

10 it tt t −

−−

+

⎪Δ ≤ ≤⎪ −⎪= ⎨

−⎪Δ

1 11

1

︵︶

︵, ︶︵9.2 ︶

︵︶i

i i ii i

ii

t ty t t t tt ts t tt tt t t t

　　　　　+

++

+

⎪Δ ≤ ≤⎪ −⎪

>⎪⎩

1 11

1

︵︶

0

ii i i

i i

i

y t t t tt t

t t

10

+⎩=

1for 2,3,..., -1, and

i

i m

Key Rate Changes– Step one as (continued):

−⎧ <⎪

−⎪Δ⎨

11

0︵︶︵︶

m

m

t tt tt t t t t t− −

−

⎪= Δ ≤ ≤⎨ −⎪⎪Δ >⎩

1 11

︵, ︶︵︶

︵︶

mm m m m

m m

m m

s t t y t t t tt t

y t t t

St t

⎩ m m

– Step two as:

Δ = + + +1 2︵︶︵, ︶︵, ︶... ︵, ︶︵9.3 ︶my t s t t s t t s t t1 2︵︶︵, ︶︵, ︶︵, ︶︵︶my

11

Key Rate Changes• Figure 9.1 shows the magnitudes of under three

cases when i=1 i=j (for any given value of j=2 3 m-1)

︵, ︶is t tcases, when i=1, i=j (for any given value of j=2,3,…,m 1), and i=m, consistent with equation 9.2.

Δy (t 1)

Δy (t j )

Shi

ft Δy (t m)

t(1) t(2) t(j–1) t(j) t(j+1) t(m–1) t(m)

Term

12

Figure 9.1 Linear contributions of the key rates shifts

Key Rate Changes• Figure 9.2 shows the magnitudes of under all m

cases (i e i=1 2 m) consistent with equation 9 2

︵, ︶is t tcases (i.e., i=1,2,…,m) consistent with equation 9.2.

Δy (t 1)

Δy (t )

hift

Δy (t 5)Δy (t 1)

Δy (t 2)Δy (t 3)

Δy (t 4)

Δy (t )...

Sh Δy (t m-1) Δy (t m)

t(1) t(2) t(3) t(4) t(5) ... t(m–1) t(m)

Term

13

Figure 9.2 Collection of the linear contributions of the key rate shifts

Key Rate Changes• Figure 9.3

The sum of the key rate shifts along the maturity rangeThe sum of the key rate shifts along the maturity range leads to a piecewise linear approximation for the shift in the term structure. This approximation given by equationthe term structure. This approximation given by equation 9.3. final curve

eres

t rat

eIn

te

estimated curve

t(1) t(2) t(3) t(4) t(5) ... t(m–1) t(m)

Term

initial curve

14

Term

Figure 9.3 The term structure shift


• Introduction






15

Key Rate Durations and Convexities

• This section is going to derive key rate durations and convexitiesconvexities….

• assuming the cash flows from a bond portfolio are fixed and the maturities of the cash flows coincide with theand the maturities of the cash flows coincide with the maturities of the chosen key rates.

16



– Key Rate Durations

– Key Rate ConvexitiesKey Rate Convexities

17

Key Rate Durations and ConvexitiesKey Rate Durations

• The set of key rate shifts can be used to evaluate the change in the price of any fixed-income security.

• Infinitesimal and instantaneous shift in a specific key rate,, results in an instantaneous price change given Δ ︵︶y tｉ

as : (continued)︵︶︵︶ 9.4 ︶i i

P KRD i y tPΔ

= − ⋅ Δ （

18


where is the i th key rate duration, defined as the (negative) percentage change in the price resulting

︵︶KRD i

from the change in the i th key rate:

1 P∂1︵︶︵9.5 ︶

︵︶iPKRD i

P y t∂

= −∂

　　　　

• The total price change due to all key rate changes is given as the sum of price changes resulting from individual key rate changes:individual key rate changes:

1 2 ... ︵9.6 ︶P P P PΔ = Δ + Δ + + Δ

19

1 2 ... ︵9.6 ︶mP P P PΔ Δ + Δ + + Δ


• The set of KRDs forms a vector of m risk measures, representing the first-order price sensitivities of the securities to the m key rates:

︵1 ︶︵2 ︶︵︶︵9 7 ︶KRD KRD KRD KRD⎡ ⎤︵1 ︶︵2 ︶︵︶︵9.7 ︶KRD KRD KRD KRD m⎡ ⎤= ⎣ ⎦K

• The total percentage change in price due to an infinitesimal shift in the term structure can be obtained as the sum of the effect of each key rate shift on thethe sum of the effect of each key rate shift on the security price.

20


• We can get the total percentage change in price by substituting equation 9.4 into equation 9.6, as follows:

︵︶︵︶︵9.8 ︶

m

iP KRD i y tΔ = − ⋅ Δ∑

1

︵︶︵︶︵︶ii

yP =∑

or using a matrix notation:

︵9.9 ︶

TP KRD yPΔ

= − ⋅ Δ

21



– Key Rate Durations

– Key Rate ConvexitiesKey Rate Convexities

22

Key Rate Durations and ConvexitiesKey Rate Convexities

• When the shift in the term structure is not infinitesimal, the previous framework must be extended to account for the second-order nonlinear effects of the key rate shifts.

• They are given as the key rate convexities and are d fi ddefined as:

21 P∂21

︵, ︶︵ , ︶︵9.10 ︶

︵︶︵︶i jPKRC i j KRC j i

P y t y t∂

= =∂ ∂

23


• The key rate durations and convexities of a portfolio can be obtained as the weighted average of the key rate duration and convexities of the portfolio.

• The percentage change in the price of a security can be i t d b T l i i i th kapproximated by Taylor series expansion using the key

rate durations and convexities as follows:

1 1 1

1

︵︶︵︶︵ , ︶︵︶︵︶︵9 . 1 2 ︶

2

m m m

i i ji i j

P KRD i y t KRC i j y t y tP = = =Δ

≈ − ⋅ Δ + ⋅ Δ ⋅ Δ∑ ∑∑

24


• When the term structure exhibits a parallel shift, all key rates shift by the same amount and equation 9.12 can be written as:

21P D CONΔ Δ Δ 2 ︵9 . 1 4 ︶

2D y CON y

P≈ − Δ + Δ 　　

=

= ∑1

︵︶, and m

iD KRD i

= =

= ∑∑1 1 ︵

, ︶

m m

i jCON KRC i j

25

Key Rate Durations and ConvexitiesKey Rate Convexities• Example 9.1

Consider five bonds 1,2,3,4, and 5, all of which have a $1,000 face value and a 10% annual coupon rate, but different maturities, as shown in Table 9.1.

Bond # Face value Maturity Annual y(years) coupon

rate (%)1 $1 000 1 101 $1,000 1 102 $1,000 2 103 $1 000 3 103 $1,000 3 104 $1,000 4 105 $1,000 5 10

26

Table 9.1 Description of the bonds


– Assume that the one-, two-, three-, four-, and five-year continuously compounded zero-coupon rates define the set of five key rates and are give as:

︵1 ︶5% ︵2 ︶5.5% ︵3 ︶5.75%y y y= = =

︵4 ︶5.9% ︵5 ︶6%y y= =

– Consider the bond portfolio with a cash flow at time (for i=1, 2,…,N) given as:

CFｉit

︵︶1 ︵

9.15 ︶i iN

iy t t

i

CFPe ⋅=

= ∑ 　　　　

27

1i e=


– The first and second partial derivatives of the price with respect to the key rates are:

for all 1 2i iCF tP i N⋅∂ ︵︶2

for all 1,2,...

︵︶

0 f ll d ︵9 16 ︶

i i

i it y t

i

i Ny t e

P i j

⋅= − =∂

∂

22

0 for all , and ︵9.16 ︶

︵︶︵︶i ji j

y t y tCF tP

∂= ≠

∂ ∂

∂2 ︵︶ for all 1,2,...︵︶ i i

i it y t

i

CF tP i Ny t e ⋅

⋅∂= =

∂

28


– Key rate durations and convexities are defined as:

︵︶

1

︵︶ i ii i

t y tCF tKRD i

P e ⋅⋅

=︵, ︶0, , and 9.17 ︶

P e

KRC i j i j= ≠ （

2

︵, ︶0, , and 9.17 ︶

1

KRC i j i j

CF t

≠

⋅

（

︵︶1

︵, ︶ i ii i

t y tCF tKRC i i

P e ⋅⋅

=

29


– Using these formulas for the five bonds in Table 9.1, gives the results shown in Table 9.2.

Bond 1 Bond 2 Bond 3 Bond 4 Bond 5P i $1 046 35 $1 080 54 $1 110 42 $1 137 62 $1 162 74Price $1,046.35 $1,080.54 $1,110.42 $1,137.62 $1,162.74

KRD(1) 1.000 0.088 0.086 0.084 0.082KRD(2) 0.000 1.824 0.161 0.157 0.154KRD(3) 0 000 0 000 2 501 0 222 0 217KRD(3) 0.000 0.000 2.501 0.222 0.217KRD(4) 0.000 0.000 0.000 3.055 0.272KRD(5) 0.000 0.000 0.000 0.000 3.504

KRC(1 1) 1 000 0 088 0 086 0 084 0 082KRC(1,1) 1.000 0.088 0.086 0.084 0.082KRC(2,2) 0.000 3.648 0.323 0.315 0.308KRC(3,3) 0.000 0.000 7.503 0.666 0.651KRC(4 4) 0 000 0 000 0 000 12 219 1 087KRC(4,4) 0.000 0.000 0.000 12.219 1.087KRC(5,5) 0.000 0.000 0.000 0.000 17.521

D 1.000 1.912 2.748 3.518 4.229C 1.000 3.736 7.911 13.283 19.649

30

C 1.000 3.736 7.911 13.283 19.649

Table 9.2 Key rate durations and convexities for the five bonds


– Now consider a $10,000 portfolio with equal investment of $2,000 in each of the five bonds.

– The key rate duration measures of the portfolio are t dcomputed as:

︵1 ︶0.2 1 0.2 0.088 0.2 0.086 0.2 0.084 0.2 0.082 0.268PORTKRD = × + × + × + × + × =

︵2 ︶0.2 0 0.2 1.824 0.2 0.161 0.2 0.157 0.2 0.154 0.459

︵3 ︶0 2 0 0 2 0 0 2 2 501 0 2 0 222 0 2 0 217 0 588

PORTKRD

KRD

= × + × + × + × + × =

= × + × + × + × + × =︵3 ︶0.2 0 0.2 0 0.2 2.501 0.2 0.222 0.2 0.217 0.588

︵4 ︶0.2 0 0

PORT

PORT

KRD

KRD

× + × + × + × + ×

= × + .2 0 0.2 0 0.2 3.055 0.2 0.272 0.665× + × + × + × =

31

︵5 ︶0.2 0 0.2 0 0.2 0 0.2 0 0.2 3.504 0.701PORTKRD = × + × + × + × + × =


• Introduction






32

Risk Measurement and Management• Key rate durations give the risk profile of a fixed-income

securities across the whole term structuresecurities across the whole term structure.

• Figure 9 4 shows the typical key rate duration profile of• Figure 9.4 shows the typical key rate duration profile of a coupon-bearing bond.

dura

tion

Key

rate

t1 t2 t3 t4 t5 t6 t7 t8 ... mat

33

Key rate

Figure 9.4 Key rate duration profile of a coupon-bearing bond

Risk Measurement and Management

• Figure 9 4 shows that the key durations first increase• Figure 9.4 shows that the key durations first increase and than decrease, and to the maturity term is the highest (due to lump sum payment).highest (due to lump sum payment).

– The increase in the cash flow maturity increase theThe increase in the cash flow maturity increase the key rate duration.

– A higher discount due to the longer maturity g g ydecreases the present value of the cash flows, which decreases the key rate duration.

34

Risk Measurement and Management• Using the key rate durations, a portfolio manager can

identify the interest rate risk profile of the portfolioidentify the interest rate risk profile of the portfolio.– A ladder portfolio

It has the similar key rate durations across theIt has the similar key rate durations across the maturity range.=> No specific bets on the shape of the term structure No specific bets on the shape of the term structure movements.

– A barbell (bullet) portfolio( ) pIt has high (low) key rate durations corresponding to the short and long interest rates and low (high) durations for the intermediate rates.=> It is preferred if the short and the long rates fall

35

more (less) than the intermediate rates.

Risk Measurement and Management• Example 9.2

Reconsider the $10 000 initial investment equally inReconsider the $10,000 initial investment equally in Example 9.1.

– This is a ladder portfolio with a traditional duration equal to 2 681 yearsequal to 2.681 years.

– Also consider other two ones with the same initialAlso consider other two ones with the same initial market values and traditional durations, but one with a bullet (contains bonds maturing in years 2 and 4) and the other with a barbell (contains bonds maturing in years 1 and 5) structure.

36


To determine the proportions invested in the bonds inTo determine the proportions invested in the bonds in each portfolio, we solve the following system of linear equations:equations:

2.681short short long longp D p D+ =1

g g

short longp p+ =

where and are the proportions invested in the short-term and the long-term bonds and and

shortp longpshortD longDshort term and the long term bonds and and

are the bonds’ traditional durations.short long

37

Risk Measurement and ManagementThe proportions invested in each bond and the key rate durations of each portfolio are summarized in Table 9 3durations of each portfolio are summarized in Table 9.3.

Ladder Barbell BulletBond 1 0 2 0 479 0 000Bond 1 0.2 0.479 0.000Bond 2 0.2 0.000 0.521Bond 3 0.2 0.000 0.000Bond 4 0.2 0.000 0.479Bond 5 0.2 0.521 0.000KRD(1) 0.268 0.522 0.086( )KRD(2) 0.459 0.080 1.025KRD(3) 0.588 0.113 0.106KRD(4) 0 665 0 141 1 464KRD(4) 0.665 0.141 1.464KRD(5) 0.701 1.825 0.000

T bl 9 3 P ti I t d i E h B d d K R t

38

Table 9.3 Proportions Invested in Each Bond and Key Rate Durations of the Ladder, Barbell, and Bullet Portfolios

Risk Measurement and ManagementFigure 9.5 displays the key rate duration profiles of the three portfoliosthree portfolios.

2.0

1 0

1.5

dura

tion

0.5

1.0

Key

rate

0.01 2 3 4 5

Key rate (years)

Ladder Barbell Bullet

39

Ladder Barbell Bullet

Figure 9.5 Key rate duration profiles

Risk Measurement and ManagementFigure 9.5

The portfolios will yield significantly different returns if– The portfolios will yield significantly different returns if the term structure exhibits nonparallel shifts:Consider the one-year key rate ↑ 50 bps the two-yearConsider the one-year key rate ↑ 50 bps, the two-year rate ↑ 20 bps, the four-year rate ↓ 10 bps, and the five-year rate ↓ 20 bps. (see Figure 9.6)y ↓ p ( g )

6 0

6.5

)

5.5

6.0te

rest

rate

(%)

Figure 9.6 Instantaneous shift in the

4.5

5.0

1 2 3 4 5

Int

term structure of zero-coupon rate

40

Years

Initial curve Shocked curve

Risk Measurement and ManagementGiven the shifts in the term structure, bonds 1 through 5 yield instantaneous returns given as:yield instantaneous returns given as:

1 2 30.499%; 0.408%; 0.075%;R R R= − = − = −

Applying the weights given in Table 9 3 to the above

4 50.233%; 0.660%R R= =

Applying the weights given in Table 9.3 to the above returns, we obtain the following returns on the three portfolios:p

0.2 ︵0.499 ︶0.2 ︵0.408 ︶0.2 ︵0.075 ︶LadderR = × − + × − + × − + 0.2 0.233 0.2 0.660 0.018%

0.479 ︵0.499 ︶0.521 0.660 0.105%0 521 ︵0 408 ︶0 479 0 233 0 101%

BarbellRR

× + × = −= × − + × =

41

0.521 ︵0.408 ︶0.479 0.233 0.101%BulletR = × − + × = −

Risk Measurement and ManagementThe explanations to the change in returns are as follows:

The ladder is the least affected by the shock– The ladder is the least affected by the shockThe losses derived from the increase in the short-term rates are nearly cancelled out by the profitsterm rates are nearly cancelled out by the profits derived from the decrease in the longer-term rates.

– The barbell gives the highest returnIt has high exposure to the five-year rateIt has high exposure to the five year rate.

– The bullet gives a negative return– The bullet gives a negative returnGains from the decrease in the four-year rate with higher losses from the upward movement of the one-

42

higher losses from the upward movement of the oneand two-year rates.


• Key rate durations and convexities can be used in a• Key rate durations and convexities can be used in a variety of portfolio strategies such as index replication, immunization and active trading.immunization and active trading.

• Example 9.3Suppose a manager desires to create an immunizedSuppose a manager desires to create an immunized portfolio over a planning horizon of four years using the model with five key rates.

43


The six immunization constraints can be written using– The six immunization constraints can be written using matrix notation as follows:

11 2 6︵1 ︶︵1 ︶︵1 ︶ 0︵2 ︶︵2 ︶︵2 ︶ 0

pKRD KRD KRDKRD KRD KRD

⎡ ⎤⎡ ⎤ ⎡ ⎤⎢ ⎥⎢ ⎥ ⎢ ⎥

L

21 2 631 2 6

︵2 ︶︵2 ︶︵2 ︶ 0︵3 ︶︵3 ︶︵3 ︶ 0

︵4 ︶︵4 ︶︵4 ︶ 4

pKRD KRD KRDpKRD KRD KRDpKRD KRD KRD

⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⋅ =⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

L

L

41 2 651 2 66

︵4 ︶︵4 ︶︵4 ︶ 4

︵5 ︶︵5 ︶︵5 ︶ 01 1 1 1

pKRD KRD KRDpKRD KRD KRDp

⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥

⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

L

L

L 61 1 1 1p⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦⎣ ⎦

44

Risk Measurement and Management– 6 constraints:

• Match five key rate durations of the portfolio to the• Match five key rate durations of the portfolio to the five key rate durations of a hypothetical zero-coupon bond maturing in four yearscoupon bond maturing in four years

• The sum of the proportions invested is 100%

– Using the matrix calculations, we can obtained the following solution:g

1 2 30.012148; 0.013799; 0.015599p p p= − = − = −4 5 61.422847; 1.274590; 0.893289p p p= = − =

45


Multiplying these proportion by the portfolio value of– Multiplying these proportion by the portfolio value of $10,000 gives short positions in the amounts of $121.48, $137.99, $155.99 and $12,745.90.$121.48, $137.99, $155.99 and $12,745.90.

– The short positions are the investments in bonds 4The short positions are the investments in bonds 4 and 6 must be $14,228.47 and $8,932.89, respectively.

46


Dividing these amounts by the respective bond– Dividing these amounts by the respective bond prices, the immunized portfolio is composed of:

-0.116 number of bonds 10 128 number of bonds 2-0.128 number of bonds 2

-0.140 number of bonds 312 507 number of bonds 412.507 number of bonds 410.962 number of bonds 512 058 number of bonds 612.058 number of bonds 6

47


• Introduction






48

Key Rate Durations and Value At Risk Analysis• VaR is defined as the maximum loss in the portfolio p

value at a given level of confidence over a given horizon.

• Given a multivariate normal distribution for the key rate changes, the portfolio return is distributed normally under a linear approximation, with a mean equal to:

︵︶︵9 21 ︶

nKRD iμ μ= − ⋅∑

where is the mean change in the i th key rate

︵︶1 ︵︶

︵9.21 ︶R y ii

KRD iμ μΔ=∑

︵︶y iμΔ• The variance equals to:

2

︵︶︵︶︵︶︵︶︵9 22 ︶

n nKRD i KRD j cov y i y jσ ⎡ ⎤= ⋅ ⋅ Δ Δ⎣ ⎦∑∑

49

1 1 ︵︶︵︶︵︶

, ︵︶︵9.22 ︶Ri j

KRD i KRD j cov y i y jσ= =

⎡ ⎤Δ Δ⎣ ⎦∑∑

Key Rate Durations and Value At Risk Analysis

• Let the dollar return on the portfolio be given aswhere is the initial market value of the portfolio.

0R V×0Vwhere is the initial market value of the portfolio.

• The VaR of the portfolio at a c percent confidence level

0

The VaR of the portfolio at a c percent confidence level is given as:

P R V VaR c⎡ ⎤≤⎣ ⎦

• Using the normal distribution:

0 1 ︵9 . 2 3 ︶P R V VaR c⎡ ⎤× ≤ − = −⎣ ⎦ 　　

g

1 1 ︵9 . 2 4 ︶R c R R c RP R z P R z cμ σ μ σ−⎡ ⎤ ⎡ ⎤≤ + = ≤ − = −⎣ ⎦ ⎣ ⎦

50

⎣ ⎦ ⎣ ⎦

Key Rate Durations and Value At Risk Analysis• Combining equations 9.23 and 9.24, the VaR of the g q ,

portfolio at a c percent confidence level is given as:

︵︶︵9 25 ︶V R V

• If the holding period of the VaR is very small, we may

0 ︵︶︵ 9 . 2 5 ︶c R c RVaR V zμ σ= − −

g p y yignore the expected return and express VaR simply as:

︵9 26 ︶VaR V z σ

• Substituting equation 9.22 in equation 9.26, we obtain

0 ︵ 9 . 2 6 ︶c c RVaR V z σ=

the following solution to VaR:

︵︶︵︶︵︶︵︶︵9 27 ︶

n nVaR V z KRD i KRD j cov y i y j⎡ ⎤= Δ Δ⎣ ⎦∑∑

51

01 1

︵︶︵︶︵︶ , ︵︶︵9 . 2 7 ︶c ci j

VaR V z KRD i KRD j cov y i y j= =

⎡ ⎤= ⋅ ⋅ Δ Δ⎣ ⎦∑∑


• Example 9.4Reconsider the three portfolios in Example 9.2.Reconsider the three portfolios in Example 9.2.

Suppose that monthly changes in the five key rates areSuppose that monthly changes in the five key rates are normally distributed with covariance matrix as follows:

⎡ ⎤

( )

0.076 0.075 0.068 0.062 0.0570.075 0.093 0.092 0.089 0.083

% 0 068 0 092 0 097 0 095 0 091V

⎡ ⎤⎢ ⎥⎢ ⎥⎢ ⎥Δ( )% 0.068 0.092 0.097 0.095 0.0910.062 0.089 0.095 0.095 0.0920 057 0 083 0 091 0 092 0 090

Var y ⎢ ⎥Δ =⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

52

0.057 0.083 0.091 0.092 0.090⎢ ⎥⎣ ⎦


• The one-month VaR at the 95 percent and 99 percent levels for each portfolio can be computed using the following formulas in the matrix form:

95 10,000 1.645 ︵︶

TPORT PORTVaR KRD Var y KRD= × × ⋅ Δ ⋅

99 10,000 2.326 ︵︶︵ 9.28 ︶

TPORT PORTVaR KRD Var y KRD= × × ⋅ Δ ⋅

53

Key Rate Durations and Value At Risk Analysis• Table 9.4 shows the result:

―The bullet portfolio will lose a maximum of $132.58 with 95 percent probability over a one-month horizon.

―The bullet portfolio is expected to incur a loss greater than $132.58 in only 1 out of 20 months.

―The VaR numbers at the 99 percent are greater, they indicate losses exceeded in 1 out of 100 months.

Ladder Barbell BulletσR 0.788 0.756 0.806VaR $129 69 $124 42 $132 58VaR95 $129.69 $124.42 $132.58

VaR99 $183.42 $175.97 $187.51

54Table 9.4 Variance of portfolio returns and VaR numbers


• Introduction






55

Limitations of the Key Rate Model

• There are three limitations of the key rate model:

– The choice of the key rates is arbitrary– The unrealistic shapes of the individual key rate shiftsThe unrealistic shapes of the individual key rate shifts– Loss of efficiency caused by not modeling the history

of term structure movements

56


• Limitations of the Key Rate Model

– The Choice of Key Rates

– The Shape of Key ShiftsThe Shape of Key Shifts

– Loss of Efficiency

57

Limitations of the Key Rate ModelThe Choice of Key Rates

• The choice of the risk factor is important, however, the key rate model offers no guidance about how to make the choice of the risk factor.

• When the model was first introduced by Ho (1992), he recommend using as many as 11 key rates.

• Since the number of key rates is still large, the manager ld ill h h i b d h icould still narrow her choices based upon the maturity

structure of the portfolio under consideration.

58






59

Limitations of the Key Rate ModelThe Shape of Key Shifts• Each individual key rate shift has a historically y y

implausible shape.

• Figure 9.7 shows that each key rate shock implies the kind of forward rate saw-tooth shift.

s

Spot

rate

s

Forw

ard

rate

s

Term Term

F

60

Term Term

Figure 9.7 Key rate shift and its effect on the forward rate curve

Limitations of the Key Rate ModelThe Shape of Key Shifts

• In order to address this shortcoming, a natural choice is to focus on the forward rate curve instead of the zero-coupon curvecoupon curve.

J h d M (1989) fi t d thi• Johnson and Meyer (1989) first proposed this methodology and called it the partial derivative approach or PDAapproach or PDA.

• According to PDA the forward rate structure is split up• According to PDA, the forward rate structure is split up into many linear segments and all forward rates within each segment are assumed to change in a parallel way.

61

g g p y


• Under the key rate model, each key rate only affects the present value of the cash flows around the term of the rate.

• However, under the PDA approach each forward rate affects the present value of all cash flows occurringaffects the present value of all cash flows occurring within or after the term of the forward rate.

62


• The continuously compounded zero-coupon rates and instantaneous forward rates are related as:

0

1

︵︶︵︶︵9 . 2 9 ︶

ty t f s ds

t= ∫

• Assuming that the forward rate intervals have a length of i d (f ti i 1 t i) bt ione-period (from time i-1 to i), we obtain:

1

︵︶︵ 1, ︶︵9 . 3 0 ︶

ty t f i i= −∑

Equation 9.30 indicates that zero-coupon rates are i l f th di f d t

1

︵︶︵ 1, ︶︵9 . 3 0 ︶

iy t f i i

t =∑

63

simple average of the corresponding forward rates.


• The present value of a cash flow CF due at time t is:

︵9.31 ︶t tCFP =

1

︵ 1, ︶

t

if i i

e =−∑

According to equation 9.31, the market price of the cash flow is affected by all forward rates preceding the

i dmaturity date.

64


• Example 9.5Reconsider the five-year, $1,000 face value, 10% annual coupon bond and the one-,two-, three-, four-, five-year continuously compounded zero-coupon rates given in Example 9 1Example 9.1.

The forward rate period to one year we obtain the– The forward rate period to one year, we obtain the following forward rates:

︵0,1 ︶︵1 ︶5% ︵1,2 ︶6% ︵2,3 ︶6.25%

︵3,4 ︶6.35 ︵4,5 ︶6.4%f y f ff f

= = = == =

65

︵3,4 ︶6.35 ︵4,5 ︶6.4%f f


• The present value of the bond can be calculated as follows:

︵0,1 ︶︵0,1 ︶︵1,2 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶

100 100 100 100f f f f f f f f f fP e e e e+ + + + + +

= + + + +︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵4,5 ︶

1100 f f f f fe + + + +

0.05 0.11 0.1725 0.236 0.3100 100 100 100 1100e e e e e

= + + + +

$1,162.74

e e e e e

=

66


• PDA durations with respect to the five forward rates are computed as follows:

1

︵1 ︶

PPD ∂= −︵0 1 ︶︵0 1 ︶︵12 ︶︵0 1 ︶︵12 ︶︵2 3 ︶

︵1 ︶

︵0 , 1 ︶1 0 0 1 0 0 1 0 0

f f f f f f

PDP f

+ + +

=∂

⎡ ⎤+ + +⎢ ⎥︵0,1 ︶︵0,1 ︶︵1,2 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶

︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵4,5 ︶

1

1 0 0 1 1 0 0

f f f f f f

f f f f f f f f f

e e eP

e e

+ + +

+ + + + + + +

⎢ ⎥= ⎢ ⎥

⎢ ⎥+⎢ ⎥⎣ ⎦1

e e⎣ ⎦=

67


1

︵2 ︶

PPD ∂

0 1 12 0 1 12 2 3

︵2 ︶

︵1, 2 ︶

1 0 0 1 0 0f f f f f

PDP f

= −∂

⎡ ⎤+ +⎢ ⎥︵0,1 ︶︵1,2 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶

︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵4,5 ︶

1

1 0 0 1 1 0 0

f f f f f

f f f f f f f f f

e eP

e e

+ + +

+ + + + + + +

+ +⎢ ⎥= ⎢ ⎥

⎢ ⎥+⎢ ⎥⎣ ⎦0 . 9 1 8

e e⎢ ⎥⎣ ⎦=

1 P∂1

︵3 ︶

︵2,3 ︶

100 100

PPDP f

∂= −

∂⎡ ⎤

︵0,1 ︶︵1,2 ︶︵2,3 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶

1 0 0 1 0 0

1

1 1 0 0

f f f f f f fe eP

+ + + + +

⎡ ⎤+ +⎢ ⎥

= ⎢ ⎥⎢ ⎥⎢ ⎥

68

︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵4,5 ︶

0 . 8 4 1

f f f f fe + + + +⎢ ⎥⎣ ⎦=


∂1

︵4 ︶

PPD = −∂

⎡ ⎤= +⎢ ⎥

︵4 ︶

︵3 , 4 ︶

1 1 0 0 1 1 0 0

PDP f

+ + + + + + += +⎢ ⎥⎣ ⎦

=︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵4,5 ︶

0 . 7 6 9

f f f f f f f f fP e e

∂= −

∂1

︵5 ︶

︵4,5 ︶PPD

P f

+ + + += ⋅

=

︵0,1 ︶︵1,2 ︶︵2,3 ︶︵3,4 ︶︵4,5 ︶

1 1 1 0 0

0 7 0 1

f f f f fP e= 0 . 7 0 1

The sum of the partial durations measures is 4.229, hi h i l t th t diti l d ti f th b d

69

which is equal to the traditional duration of the bond.

Limitations of the Key Rate ModelThe Shape of Key Shifts• Figure 9.8 shows these measures:

– Partial durations are decreasing in the maturity of the forward rates.

– Changes in short-maturity forward rates have a greater impact in bond.

3

4

1

2

01 2 3 4 5

Sequence of the key rate or the forward rate

70

Sequence of the key rate or the forward rate

PD(i) KRD(i)






71

Limitations of the Key Rate ModelLoss of Efficiency

• Some assert that key rate model is not an efficient one in describing the dynamic of the term structure.

• Because historical volatilities of interest rates provide useful information about the behavior of the different

t f th t t t d th k d lsegments of the term structure, and the key model disregards this information.

72


• Since each key rate change is assumed to be independent of the changes in the rest of key rates, the model deals with movements in the term structure whose probabilities may be too small to worry about.

• As a result, the use of the key rate model for interest rate risk management imposes too severe restrictions onrisk management imposes too severe restrictions on portfolio construction that leads to increased costs and a loss of degrees of freedom.g

73


• A number of variations of the key rate model that try to dealA number of variations of the key rate model that try to deal with this undesirable consequence have gone through the inclusion of the covariance of interest rate changes into the analysis.

– Covariance-consistent key rate hedging (1996)Consists finding the portfolio minimizes the variance of the portfolio returns

– Stochastic immunization (1996)Searches for the portfolio that minimizes a risk measure defined as a weighted average of the portfolio’s return ariance and the orst case risk

74

variance and the worst case risk

Interest Rate Risk ModelingInterest Rate Risk ModelingThe Fixed Income Valuation Course

Sanjay K. Nawalkhaj yGloria M. SotoNatalia A. Beliaeva

Documents

InInterestterest RRateate RiRisksk M Modelingodeling$1,000 face value and a 10% annual coupon rate, but different maturities, as shown in Table 9.1. Bond # Face value Maturity Annual