ingestriskmgt (2012) - slides - session 3 - v5

7/29/2019 INGESTriskmgt (2012) - Slides - Session 3 - V5

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Risk managementExercises Session 3 Interest Rate Management


2/44

Small Questions

Problems


3/44

Exercise 1

Interest Rate Compounding Question

An investor receives $1,100 in one year in return for an investment of

$1,000 now. Calculate the percentage return per annum with: (a) annual

compounding, (b) semi-annual compounding, (c) monthly compounding,

(d) continuous compounding.

What rate of interest with continuous compounding is equivalent to 15%per annum with monthly compounding?

An interest rate is quoted at 5% per annum with semi-annual

compounding. What is the equivalent rate with (a) annual compounding,

(b) monthly compounding, (c) continuous compounding.


4/44

Exercise 1

Interest Rate Compounding Theory

Your banker proposes you two loans : Loan A : Interest of 12% paid annually

Loan B : Interest of 12% paid monthly.

Which one do you prefer?


5/44

Exercise 1

Interest Rate Compounding Solution


$1,000 now. Calculate the percentage return per annum with:

a) annual compounding

b) semi-annual compounding

1100 1000 1

1100 11000

10%

r

r

r

2

1100 1000 1 2

11001 2

1000

9.76%

r

r

r


6/44

Exercise 1



$1,000 now. Calculate the percentage return per annum with:

c) monthly compounding

d) continuous compounding

12

12

1100 1000 112

11001 12

1000

9.57%

r

r

r

1

lim 1

nN

Nn

n

r

FV P n

rFV P

n

lim 1

n

r

n

re

n

e

1100 1000 e

1100ln

1000

9.53%

rN

r

FV P

r

r


7/44

Exercise 1


What rate of interest with continuous compounding is equivalent to 15%per annum with monthly compounding?

120.15

112

0.1512 ln 1

1214.91%

re

r

r


8/44

Exercise 1


An interest rate is quoted at 5% per annum with semi-annualcompounding. What is the equivalent rate with

a) annual compounding

b) monthly compounding

c) continuous compounding

2

0.051 1

2

5.062%

R

R

2 120.05

1 12 12

4.949%

R

R

20.05

12

4.939%

Re

R


9/44

Exercise 2

Forward Interest Rates Question

Suppose that zero interest rates with continuous compounding are asfollows:

Calculate forward interest rates for the second, third, fourth, fifth, and

sixth quarters.


10/44

Exercise 2

Forward Interest Rates Theory

In discrete time :

In continuous time :

* *

*1 1 1T T T T

r r R

*

*

*11

1

T

T T

T

rR

r

** *

** *

T T RT r Tr

T T RT r Tr

e e e

e e

* *

*

T r Tr R

T T

Forward rates?


11/44

Exercise 2

Forward Interest Rates Solution

Calculate forward interest rates for the second, third, fourth, fifth, andsixth quarters.

Forward rate?

Forward rate for the first quarter:

Forward Rates :

2 2 1 12 1

r T rT F

T T

1 2

6 38.2% 8%

12 128.4%

6 3

12 12

F


12/44

Exercise 3

Interest Rate Term Structure Question

The term structure of interest rates is upward sloping. Put the following inorder of magnitude:

a) The five-year zero rate,

b) The yield on a five-year coupon-bearing bond, and

c) The forward rate corresponding to the period between 5 and 5.25

years in the future?

What is the answer to this question when the term structure of interest

rates is downward sloping?


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Exercise 3

Interest Rate Term Structure Solution

The term structure of interest rates is upward sloping. Put the following inorder of magnitude:


years in the future

a) The five-year zero rate

b) The yield on a five-year coupon-bearing bond


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Exercise 3


C) vs A) ?

http://www.ecb.int/stats/money/yc/html/index.en.html

A) vs B) ?

10 1 2 1 3 2 10 910 1 1 2 2 3 9 101 1 1 1 ... 1T T T T T T T T

r r F F F
http://www.ecb.int/stats/money/yc/html/index.en.htmlhttp://www.ecb.int/stats/money/yc/html/index.en.html


15/44

Exercise 3


What is the answer to this question when the term structure of interestrates is downward sloping?

b) The yield on a five-year coupon-bearing bond

a) The five-year zero rate


years in the future


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Exercise 4

Bond Yield Question & Theory

A three-year bond provides a coupon of 8% semi-annually and has a cashprice of 104. What is the bonds yield?

Example :

Spot rates:

1 year = 10%

2 years = 11%

3 years = 12%

Price of a bond (maturity = 3 years) paying 6% of interest :

Yield to maturity of this bond :

2 3

1 2 3

6 6 106

1 1 1

Pr

r r

2 3

6 6 10685.77

1 10% 1 11% 1 12%

P

2 3

6 6 10685.77 11.941%

1 1 1IRR

IRR IRR IRR


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Exercise 4

Bond Yield Solution

YTM of the bond (discrete time) :

YTM of the bond (continuous) :

0.5 1 1.5 3

4 4 4 104104 ...

1 1 1 1IRR IRR IRR IRR

6.406%IRR

0.5 1 3104 4 4 ... 104

6.4%

IRR IRR IRRe e e

IRR


18/44

Exercise 5

Duration Question & Solution

What does duration tell you about the sensitivity of a bond portfolio tointerest rates?

Duration provides information about the effect of a small parallel shift

in the yield curve on the value of the bond portfolio.

The percentage decrease in the value of the portfolio equals the

duration of the portfolio multiplied by the amount by which interestrates are increased in the small parallel shift (replace duration by

modified duration for non-continuous compounding).

What are the limitations of the duration measure?

Its limitation is that it applies only to small parallelshifts in the yield

curve.


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Exercise 5

Duration Theory

Consider a zero-coupon with t years to maturity:

What happens ifrchanges?

For given P, the change is proportional to the maturity.

As a first approximation (for small change ofr):

trP

)1(

100

Pr

t

rr

t

rtdr

dPtt 1)1(

100

1)1(

1001

rr

t

P

P

1

Duration = Maturity


20/44

Exercise 5

Duration Theory

Duration for coupon bonds (discrete time) : Consider now a bond with cash flows: C1, ...,CT

View as a portfolio ofTzero-coupons.

The value of the bond is: P = PV(C1

) + PV(C2

) + ...+ PV(CT

)

Fraction invested in zero-coupon t: wt= PV(Ct) / P

Duration : weighted average maturity of zero-coupons

D= w1 1 + w2 2 + w3 3++wt t++ wTT

Duration for coupon bonds (continuous time) :

1

irtni

i

i

C eD t

P


21/44

Exercise 6

Duration Theory

Where does the following relation come (discrete time)?

Firstly

Because:

rr

Duration

P

P

1

22

2 2

3

1 2

1

Cd

rdPV C C

dr dr r

)(1

...)(1

2)(

1

1

)(...

)()(

21

21

T

T

CPVr

TCPV

rCPV

r

dr

CdPV

dr

CdPV

dr

CdPV

dr

dP


22/44

Exercise 6

Duration Theory

Which gives :

By definition duration is equal to:

Which means that:

r

Duration

Pdr

dP

1

1

P

CPVT

P

CPV

P

CPVDuration T

)(...

)(2

)(1 21

))(

...)(

2)(

1(1

11 21

P

CPVT

P

CPV

P

CPV

rPdr

dP T

1

dP Durationdr

P r


23/44

Exercise 6

Duration Theory

Where does the following relation comes (continuous time)? We know that :

By definition, duration is equal to :

The change in the value of the bond due to a change in interest rate is

1

i

nrt

i

i

P C e

1

i

nrt

i i

i

t C e

DP

dPP r

dr

1

i

nrt

i i

i

t C eP

rP P

PD r

P


24/44

Exercise 6

Duration Question

A five-year bond with a yield of 11% (continuously compounding) pays an8% coupon at the end of each year.

a) What is the bonds price?

b) What is the bonds duration?

c) Use the duration to calculate the effect on the bonds price of a 0.2%

decrease in its yield.

d) Recalculate the bonds price on the basis of a 10.8% per annum yield

and verify that the result is in agreement with answer (c).

What if the yield is compounding annually and if we use modified

duration?


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Exercise 6

Duration Solution

What is the bonds price?

What is the bonds duration?

Use the duration to calculate the effect on the bonds price of a 0.2%


New price : 86.8+0.74=87.54

0.11 0.11 2 0.11 3 0.11 4 0.11 58 8 8 8 108

86.8

P e e e e e

86.80 4.256 0.2%

0.74

B BD y

B


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Exercise 6

Duration Solution

Recalculate the bonds price on the basis of a 10.8% per annum yield andverify that the result is in agreement with answer (c).

What if the yield is compounding annually and if we use modified

duration?

What is the bonds price?

0.108 0.108 2 0.108 3 0.108 4 0.108 58 8 8 8 108

87.54

P e e e e e

2 3 4 5

8 8 8 8 108

1 0.11 1 0.11 1 0.11 1 0.11 1 0.11

88.91

P


27/44

Exercise 6

Duration Solution

What is the bonds duration?

Use the duration to calculate the effect on the bonds price of a 0.2%


New price : 88.91+0.68 = 89.59 Recalculate the bonds price on the basis of a 10.8% per annum yield and

verify that the result is in agreement with answer (c).

4.2660.68 88.91 0.02

1 0.11

2 3 4 5

8 8 8 8 108

1 0.108 1 0.108 1 0.108 1 0.108 1 0.108

89.60

P


28/44

Exercise 6

Duration Theory

0,00

50,00

100,00

150,00

200,00

250,00

300,00

350,00

400,00

450,00

0% 1% 2% 3% 4% 5% 6% 7% 8% 9% 10% 11% 12% 13% 14% 15% 16% 17% 18% 19% 20%

Bond

price

Interest rate

5-Year

10-Year

15-Year


29/44

Exercise 7

Partial Duration Question

When the partial durations are as in the following table below, estimatethe effect of a shift in the yield curve where the 10-year rate stays the

same, the 1-year rate moves up by 9e and the movements in the

intermediate rates are calculated by interpolation between the 9e and 0.


30/44

Exercise 7

Partial Duration Question

How could your answer be calculated from the results for a rotationpresented below?

0

1

2

3

4

5

6

7

0 2 4 6 8 10 12

Maturity (yrs)

ZeroRate(%)


31/44

Exercise 7

Partial Duration Solution

Partial duration? Reminder : a positive partial duration means that a change in the

interest rate will have a negative impact on the value of your

portfolio.

The change is equal to - 0.283/100 (dont forget the minus before the

duration in the formula giving the change of value).

1Partial Duration i

i

P

P x


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Small Questions

Problems


33/44

Problem 1

Portfolio Duration Question

Portfolio A consists of a 1-year zero-coupon bond with a face value of$2.000 and a 10-year zero-coupon bond with a face value of $6.000.

Portfolio B consists of a 5.95-year zero-coupon bond with a face value of

$5.000. The current yield on all bonds is 10% per annum (continuously

compounded).

a) Show that both portfolios have the same duration.b) Show that the percentage changes in the values of the two portfolios

for a 0.1% per annum increase in yields are the same.

c) What are the percentages changes in the values of the two portfolios

for a 5% per annum increasing in yields?

d) What are the convexities of the portfolios?

e) To what extent does duration and convexity explain the difference

between the percentage changes calculated in (c)?


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Problem 1

Portfolio Duration Solution

Show that both portfolios have the same duration. Compute the weights of the bonds in the portfolio :

The duration of the portfolio 1 is the weighted average of the

durations :

Portfolio 2s duration

1 1 2 2Portfolio 1's duration =w Duration w Duration

0.1 1 0.1 10Portfolio 1's value = 2000 6000 4016.95e e

0.1 1

1

200045%

4017

ew

0.1 10

2

600055%

4017

ew

Portfolio 1's duration = 45% 1 55% 10 5.95

Portfolio 2's duration 5.95


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Problem 1


Show that the percentage changes in the values of the two portfolios for a0.1% per annum increase in yields are the same.

Portfolio 1s change in value :

Classical way

With the duration

0.1 1 0.1 10Portfolio 1's value (10%) = 2000 6000 4016.95e e

0.101 1 0.101 10Portfolio 1's value (10.1%) = 2000 6000 3993.18e e

3993.18 4016.950.59%

4016.95Change

0.001 5.95 0.595%Change


36/44

Problem 1


Show that the percentage changes in the values of the two portfolios for a0.1% per annum increase in yields are the same.


Classical way

With the duration

0.10 5.95Portfolio 2's value (10%) = 5000 2757.81e

0.101 5.95Portfolio 2's value (10.1%) = 5000 2741.45e 2741.45 2757.81

0.59%2757.81

Change

0.001 5.95 0.595%Change


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Problem 1


What are the percentages changes in the values of the two portfolios for a5% per annum increasing in yields?


Classical way

With the duration



3060.20 4016.9523.82%

4016.95Change

0.05 5.95 29.75%Change


38/44

Problem 1


What are the percentages changes in the values of the two portfolios for a5% per annum increasing in yields?


Classical way

With the duration

0.15 5.95Portfolio 2's value (15%) = 5000 2048.15e

0.10 5.95Portfolio 2's value (10%)= 5000 2757.81e

2048.15 2757.8125.73%

2757.81Change

0.05 5.95 29.75%Change


39/44

Problem 1



40/44

Problem 1


What are the convexities of the portfolios? Convexity ?

Portfolio 1s convexity :

Portfolio 2s convexity :

2

2

1

2

1i

nrt

i i

i

c t eP

CP r P

2 2 0.1 2 0.15 10

1

1 2000 10 6000 222537.34in

rt

i i

i

c t e e e

0.1 1 0.1 10Portfolio 1's value = 2000 6000 4016.95e e

2

1 222537.34 55.404016.95

i

nrt

i ii

c t e

CP

2 2

21 1 5.95 35.40

i i

Pn n

rt rt

i i i i

i i

c t e t c e

C

P P


41/44

Problem 1


To what extent does duration and convexity explain the differencebetween the percentage changes calculated in (c)?

Change in the bonds price taking convexity into account :

Portfolio 1s

Portfolio 2s

21 2

Taylor:22

1 2Thus: ( )2

P PP r r

r r

PD r C r

P

2

1 2( )2

122.83% 5.95 5% 55.4 5%

2

PD r C r

P

2125.32% 5.95 5% 35.4 5%2


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Problem 2

Portfolio Duration Question

Suppose that the change in a portfolio value for a 1-basis-point shift in the3-month, 6-month, 1-year, 2-year, 3-year, 4-year, and 5-year rates are (in

$ million) +5, -3, -1, +2, +5, +7 and +8 respectively. Estimate the delta of

the portfolio with respect to the first three factors in the table of factors

loadings for US treasury data (The interest rate move for a particular

factor):

Note that:

PC1: Roughly parallel shift in the yield curve.

PC2: twist, change of slope of the yield curve.

PC3: bowing of the yield curve.


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Problem 2



44/44

Problem 2

Example PCA

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