Week07 - Ch09 - Interest Rate Risk - The Duration Model (Part II)

8/21/2019 Week07 - Ch09 - Interest Rate Risk - The Duration Model (Part II)

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Chapter 9

Interest Rate Risk:the Duration Model (Part II)


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Overview

• In this lecture we apply the duration to manage interest

risk of a single investment and a firm.

• Finally, we examine the weaknesses of duration tomeasure and manage interest rate risk

2


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Why duration is useful?

Duration and Risk Management 3

• Recall the definition of duration:

• Predict potential changes in market value/returns in the future based onduration, given any potential change in interest rate

• ∆P/P = -D×∆R/(1+R) = -MD×∆R• Where MD= D/(1+R) - modified duration

• D*P is also called as ‘dollar duration’.• ∆R/(1+R) is also referred as “relative change in interest rate”• When using this relation to calculate change in price,

remember the consistency in (time) frequency of measuring Dand R.

• For example, if you want to estimate the price change for asemi-annual coupon-paying bond, the duration should beexpressed in number of half- years, and R and ∆R should be thelevel and change in interest rate for the six-month period.

• Note on the formulae for semiannual bonds on the textbook:the textbook always expresses the duration (D) in number of

years (but it does not have to)!


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Interest Rate Risk for Portfolios

Duration and Risk Management 4

• The interest rate risk for a portfolio is defined and measured exactlyin the same way as we discussed previously

• The calculation of a portfolio’s duration• Change in the market value of one security i due to change in R

• Change in the market value of a portfolio due to change in R

• Relationship btw change in the portfolio’s value and change in interest

rate

* *1

i i i

RP D P

R

∆ ∆ = − +

1 1 2 2

* *1

,

...

/

N N

i i

RP D P

R

where

D W D W D W D

and

W P P

∆ ∆ = −

+

= + + +

=

1 2 ...

( * ) * ( * ) * *1 1

N

i

i i i

i i

P P P P

R P RP D P D P

R P R

∆ = ∆ + ∆ + + ∆

∆ ∆ ∆ = − = − + +

∑ ∑


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Immunization

• Let’s assume you have some future liabilities to meet,such as paying back tuition loans or foreseeable

investments to make in the future• One way to make sure that you have sufficient funds to

cover the future obligation is to buy a zero-coupon bond whose maturity matches that of the liability

• But in lots of cases, you may have to buy coupon bonds(whose maturity may also differ from that of theliability). Then you will be exposed to changes in theinterest rates

–

Higher interest rates are beneficial for reinvesting interim cashflows (coupons), but bad for liquidating investment.

– So if interest rate changes, you may not have sufficient cashflows to cover your liability.

– And the market value of these bonds will also change.

Interest risk management – Single Investment 5


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Immunization

• You can construct a bond portfolio that balances theseeffects against each other to match our liabilities

• If you set the duration of the bond portfolio equal to yourtargeted horizon, then the value of bond portfolio at yourtarget horizon will be fixed

– Reinvestment risk and liquidity (price) risk will offset each other.

– And thus you will have just enough cash flow to cover yourliability at the target horizon.

• Recall that duration is a measure of the sensitivity of bond value to changes in their yields

–

An alternative way of understand immunization is: by matchingthe duration (and market value) of the bond portfolio with thetarget horizon (and the present value) of your liability, if interestrate changes, the change in market value of the bond portfolio will offset the change in the present value of your future liability.

6Interest risk management – Single Investment


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Immunization - example

• Given a liability with duration DL and two bonds with

durations D A and DB – chose portfolio weights X A and (1 - X A) such that:

– Chose the size of the investment so that the present value of theportfolio is equal to the present value of the liability

( ) L B A A AP

D D X D X D =−+⋅= 1



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Immunization – numerical example

• Suppose we have a liability of $1000 at t = 5

• There are only two bonds on the market: a 10-year zero-coupon bond (bond A) and a 3-year, 10 % coupon bond(bond B). They both have a FV of $100. Let’s assume aflat term structure with y = 5%. The price of Bond A is$61.39.

L

Bond

A

Bond

B 10 10 110

100

-1000

T = 3

T = 10

T = 5



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• Bond A carries liquidity risk since we have to sell it at t =

5• Bond B carries reinvestment risk since we have to

reinvest the coupons and FV

• Strategy: Calculate the durations of the liability and

bonds, and choose portfolio weights so that the durationof bond portfolios equals to the duration of liability

– The liquidity and reinvestment risks will cancel out



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• The durations of the liability and bond A are trivially 5and 10 years respectively

• We calculate the duration of Bond B as above:

• Now assume we invest some fraction X A in bond A andthe rest, (1 – X A), in bond B, so that

• We should invest 31% in bond A and 69% in bond B

75.23

62.113

05.1/1102

62.113

05.1/101

62.113

05.1/10

62.11305.1

110

05.1

10

05.1

10

32

32

=⋅+⋅+⋅=

=++=

B

B

D

P

( )

31.075.210

75.25

1

=

−

−=

−

−=

=⋅−+⋅

=−+⋅=

B A

B L

A

L B A B A A

L B A A AP

D D

D D X

D D X D D X

D D X D X D



11/22


• We want to invest enough money in our bond portfolio to

meet the liability – That means the present value of bond portfolio has to be equal to

the present value of liability

– The PV of the liability is straight forward to calculate:

– The present value of bond portfolio should also be $783.53

– 31% should be invested in bond A, and thus number of bond A to buy:

–

69% should be invested in bond B, and thus number of bond B to buy:

( ) 53.78305.1

10005 == LPV

96.339.61

53.78331.053.78331.0=⋅=⋅

AP

76.462.113

53.78369.053.78369.0=

⋅=

⋅

BP



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Manage Interest Rate Risk for Firms

• What are firms? A portfolio of assets and liabilities.

–

Assets (A) = Equities (E) + Liabilities (L) – Risk management is concerned with the value of equities.

• What are the interest rate risks faced by a firm?

–

The change in net firm value due to change in interest rate – ∆ E = ∆ A – ∆ L, due to ∆ R

– If we know the duration of equities, we can calculate the changein firm value using duration definition.

– However, equities are residual claims as defined by Assets net of

Liabilities, and thus could be only inferred from assets andliabilities.

12Interest risk management – The whole firm


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Manage Interest Rate Risk for Firms (Cont.)

• Calculate changes in the market values of assets andliabilities based on durations

• Changes in equity values:

– Assuming the level of interest and expected shock to interestrates are the same for both assets and liabilities, then:

– Where k = L / A (a firm’s leverage measured in market values)

– Important: every variables are market values rather thanaccounting ones

13

( ) )1( R R

Ak D D E L A+

∆

××−−=∆

(1 )

A

A

A

R A D A

R

∆∆ = − × × + (1 )

L

L

L

R L D L

R

∆∆ = − × × +

(1 ) (1 )

A

A

L

A L

L

R R E D A D L

R R

∆ ∆

∆ = − × × − − × × + +

Interest risk management – The whole firm


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Manage Interest Rate Risk for Firms (Cont.)

• Determinants of firm interest rate risk exposure

– ∆E = –(adjusted duration gap) × asset size × interest rate shock

• Generally the objective of risk management is tominimize the effect of interest rate changes on firm value.

– ∆E = 0

– Why not making sure ∆E > 0

14

( ))1( R

R Ak D D E L A

+

∆××−−=∆



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Manage Interest Rate Risk for Firms - Example

• Consider the simplified balance sheet:

• Assume that the average duration of assets is 7 years, while the average duration of liabilities is 4 years. Thecurrent interest rate is 10%, but is expected to increase to11% in the future.

15

Assets Liabilities A = $100 L = $80

E = $20

TA = $100 TL + E = $100



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Manage Interest Rate Risk for Firms – Example (Cont.)

• We can calculate the expected change in the FI’s net worth as follows: – ∆ E = –(7 – 0.8 × 4) × $100 × ( 0.01/1.1) = $-3.45. – This means that the FI could lose $3.45 in net worth if interest

rates increase by 1%.

• Potential strategies for the FI to reduce risk exposure: – 1) reduce D A

– 2) increase D L

– 3) increase leverage (k )

16Interest risk management – The whole firm


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Difficulty of Using Duration for Risk Management

• Should FIs always immunize the interest rate sensitivity gapof assets and liabilities? For example, setting duration gap to

zero? – Business specialization

– The cost of adjusting portfolio mix> Immunization is a dynamic process since duration depends on

instantaneous R

> The use of interest rate derivatives and other hedging portfolios tomore efficiently and effectively manage interest rate risk

• Other complexities of using duration – Off-balance sheet assets and liabilities (contingent assets &

liabilities), difficult to estimate duration – Non-parallel interest rate changes

– Large changes in interest rates for which duration is not a correctestimate of risk exposure: Convexity

17Interest risk management


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Duration: Limitations

• Price changes given large interest rate changes are

calculated imprecisely using duration

• why?

– Recall definition of duration – the effect of aminimum (continuous) change in R

– What happens to duration if interest rate changes?

Convexity 18

• ∆P/P = -D×∆R/(1+R) = -MD×∆R


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Duration: Limitations (Cont.)

• Recall the relation between interest rate and duration

– Duration increases if interest rate decreases

– Duration decreases if interest rate increases.

– The same relationship surely holds for modified duration anddollar duration.

• (Dollar) Duration will be constant if the relation between price and

(log) yield, i.e., ln(1+R) is linear. In reality, this relation is convex.

Convexity 19


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Approximation error in estimated price changes

• The (approximate) price change calculated based on duration (∆P~= -MD×∆R×P) is always smaller (either less positive or morenegative) than the actual price changes

– When yield increases (and thus price would decrease), duration willdecrease over the change of yield. Thus the estimated price changeusing the original duration overestimates the loss

– When yield decreases (and thus price would increase), duration willincrease over the change of yield. Thus the estimated price change using

the original duration underestimates the gain.

20Convexity


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Solution to approximation error in using duration

• What give rise to approximation error?• curvature of the price– yield curve, and thus duration is not

constant when interest rate changes• The interest rate moves discretely, rather than continuously• Overlooking the curvature of the price– yield curve would cause

errors in predicting the interest sensitivity of asset and liabilityportfolios.

• Solutions• Breaking down a discrete change in interest rate into numerous

semi-continuous ones, and calculating the price changes in everystep with the continuously updated interest rates

• Duration is continuously updated with every small change in

interest rate (and thus duration will change very little), andas such the approximation error will be very small• OR, taking into account of the curvature of the price-yield curve

and discrete interest rate changes

Convexity 21


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Convexity

• Predicting price changes using both duration and convexity• ∆P/P = -D×∆R/(1+R) + 1/2 × CX × (∆R)2= -MD×∆R + 1/2 × CX ×

(∆R)2

– CX = convexity measure, measuring degree of curvature of price - yield relationship.

– Mathematically, (dollar) duration is the first derivative of price withrespect to interest rate: D*P = - әP/ әR, which measures the slope of

price-yield curve. – (Dollar) Convexity is the second derivative of price with respect to

interest rate: CX*P = ә2P/ әR 2, which measures the curvature ofprice-yield curve. It also measures the change in (dollar) duration asinterest rates change, i.e., -ә(D*P)/әR.

– To be exact, even the above equation is not complete, because CXalso changes with interest rate (just as D does), so higher orders (upto infinity) of derivatives and interest rate changes should be added– which should remind you of Taylor expansion in your advancedmathematic course.

– Note: The exact calculation of CX won’t be required in theassessments – you will be given information related to CX if needed.

Convexity 22

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Week07 - Ch09 - Interest Rate Risk - The Duration Model (Part II)