1

8. Measuring Interest Rate Risk--

Duration and Convexity

Problems of interest rate sensitivity faced by investors and institutions:

1. Liquidity Management: An institution wants to ensure there was enough liquidity each year to cover its liabilities that come due that year.

• Bank runs and panics

• Assets = Liabilities is often not enough to prevent panics

• Timing of payments matters

2. Portfolio Management and Sensitivity Analysis: An investor recognizes that interest rates will change in the future and wants to measure how susceptible his portfolio is to interest rate movements.

2

8.1 Interest Rate Risk: Overview

Knowing interest rates are not fixed over time, how can we and financial institutions protect ourselves?

We want to know how the value of our assets will change if the interest rate shifts.

Price

Interest Rate (%)

P0

S0

3

8.2 Interest Rate Risk: Overview

For most bonds, the relation between price and interest rates is non-linear.

Duration linearly approximates the relation between price and interest rates.

Convexity estimates the loss from not considering the curvature (non-linearities) between price and interest rates.

Duration is a good measure if the changes in interest rates are expected to be small.

Duration + Convexity is a more accurate measure of the relation between prices and interest rates.

Calculus is an exact measure of estimating the relation between price and interest rates; however, not all financial instruments have smooth relations between price and interest rates, making it difficult to use calculus. Approximations are often used instead.

4

8.3.1 Estimating Dollar Duration using Actual Prices

Calculate the change in price from interest rate changes by calculating a new price for each interest rate.

Notice the changes are always determined from a specific starting point which is today’s expected spot rate or yield, Sn. Notice the 100 is used

to convert interest rates from decimals to percent ie. 0.01 to 1%.

Price

Interest Rate

P0

Sn

nn SS PPP

SnSn

P

S

100*S

PDD

5

8.3.2 Estimating Dollar Duration using today’s spot rates for a Zero-Coupon Bond

For a zero-coupon bond, the price is written as

Therefore, the actual slope between price and interest rates at Sn is

Price

Interest Rate

P0

Sn

SnSn

P

S

slopeS

CFnDD

nn

nperiodnZCB

100*)1(

*10

nn

nperiodnZCB

S

CFP

)1(0

100*)1(

*

100* 1

nn

n

n S

CFnslope

dS

dP

6

8.4 Comparing the Two Methods of estimating Duration

Compare a $1000 15-year ZCB with a $100 15-year ZCB where today’s 15-year spot rate is 8%. What affect does a 100 bp change in interest rates have?

Prices at 7.5%:

$1000 FV Bond $337.97

$100 FV Bond $33.797

Prices at 8.5%:

$1000 FV Bond $294.14

$100 FV Bond $29.414

Duration of $1000 FV 15-Year Bond measured approximately: $43.83

Duration of $100 FV 15-Year Bond measured approximately: $4.383

Duration of $1000 FV 15-Year Bond measured exactly: $43.78

Duration of $100 FV 15-Year Bond measured exactly: $4.378

7

8.5 Problems with Dollar Duration

1. Difficult to interpret and compare dollar duration because it depends on the face value of the bond.

• Divide by price of the bond to normalize dollar duration and make comparable across different bonds.

2. The exact method only applies to zero-coupon bonds, so we need to make it more general.

• Duration is additive

8

8.6 Making Dollar Duration Comparable

Two measures of duration are used to compare across assets with different face value:

Modified duration: Measures the percentage change in price for a change interest rates.

For a zero-coupon bond, modified duration simplifies

Macaulay’s duration: Measures the percentage change in price for a percentage change in interest rates.

For a zero-coupon bond, Macaulay’s duration simplifies

nperiodnZCB

periodnZCB

periodnZCB

Sn

P

DDMD

1100

*

0

00

n

P

SDDMac

periodnZCB

nperiodnZCB

periodnZCB

0

00)1(100

*

100*

100*/

100*rate Interest

Price % 0

S

PPMD

100*)1/(

100*/

rate Interest %

Price %

0

0

SS

PPMac

9

8.7 Measure of Dollar Duration for Coupon Bonds using today’s spot rate

• All measures of duration are additive. If you have half a portfolio of an asset with duration 4 and the other half with a duration of 3, the duration of the portfolio is 3.5.

• Coupon bonds are a linear combination of zero-coupon bonds. Therefore, the dollar duration of coupon bond is simply the linear combination of the dollar durations of many zero coupon bonds.

• The idea of additivity applies to Modified and Macaulay’s duration.

n

t

periodstZCBn

tt

t

tperiodsncoupon

DDS

CFtDD

10

110 100*)1(

*

n

t periodncoupon

periodtZCB

tperiodtZCB

periodncoupon

n

tt

t

tperiodsncoupon

P

PMD

PS

CFtMD

10

0

01

10 *100

100*)1(

*

n

t periodncoupon

periodtZCB

periodtZCBn

t periodncoupon

tt

t

tperiodsncoupon

P

PMac

P

S

S

CFtMac

10

00

10

10 *)1(100

*100*)1(

*

10

8.8 Finding expected changes in price using only duration

In general, you can use the estimates of Dollar Duration, Modified Duration, or Macaulay’s Duration to estimate changes in prices (percent changes in price) from changes (percent changes) in interest rates

rate interest %price%

100 rate interest price%

100rate interest price

Mac

MD

DD

11

8.9 Coupon Bonds, Yield-to-Maturity, and Duration

The proper method of accounting for duration uses spot rates (see previous slide) because these weight the cash flows by the appropriate term structure. However, for simplicity, YTM is often used to measure duration.

DISADVANTAGE USING YTM: Using YTM, introduces measurement errors in estimating duration. The measurement errors of using YTM in duration are minimized if the term structure is flat and if spot rates are not volatile. Also, YTM and spot rates can be interchanged for zero coupon bonds.

ADVANTAGE USING YTM: Because the YTM is the same for each time period, it is easier to calculate to duration. Also, calculus techniques are easier to apply since there is only one variable, YTM, that affects the price of the bond, rather than n different spot rates.

12

8.10 Properties of Duration, Yield Curves, and Convexity

Two bonds of the same dollar or modified duration, but different cash flows, may exhibit different relative price changes for large interest rate changes. This is due to differences in the curvature of their price / yield relation, or convexity.

Prices are less sensitive to interest rate changes when interest rates are high.

Price

Interest Rates

Dollar Duration

5Yr ZCB

15Yr ZCB

13

8.10.1 Properties of Duration: Duration vs. YTM

As the yield to maturity increases, all duration measures decrease.

The longer the maturity of the bond, the greater the negative relation between YTM and modified duration.

WHY?

Modified Duration

Yield-to-Maturity

5Yr Maturity

30Yr Maturity Note: Coupon value is held constant for each bond

14

8.10.2 Properties of Duration: Duration vs. Coupon Size

As the coupon value increases, all duration measures decrease.

Increasing coupon size tends to decrease duration more rapidly in high maturity bonds than low maturity bonds.

WHY?

Modified Duration

Coupon

5Yr Maturity

30Yr Maturity

Note: YTM is held constant

15

8.10.3 Properties of Duration: Duration vs. Maturity

As the maturity increases, duration measures tend to increase.

Bond duration increases with maturity at a decreasing rate for coupon bearing bonds. (Low coupon discount bonds even a slightly negative relation between duration and maturity at maturities over 30 years.)

WHY?

Modified Duration

Maturity

Note: YTM is held constant

4% Coupon

10% Coupon

16

8.11 Convexity: Approximate Estimate

Duration is a linear approximation of price changes to changes in the interest rate. Convexity corrects for non-linearities.

Two methods to estimate convexity, or Dollar Gains from Convexity (DGFC): (1) Exact and (2) Approximate

Approximate:

Interest Rates

Price

Approximate DGFC

Sn Sn

S

PS

000,102/

]2/)[(2S

PPPDGFC SSS

17

8.11 Convexity: Exact Measure

If we wanted to find the change in price from changes in interest rate that more exactly accounts for squared non-linearities, we use

Yield-to-maturity is often used instead of the individual spot rates for convenience although the spot rates properly weight cash flows.

As with dollar duration, DGFC is sensitive to the face value of the bond. To control for this, DGFC is divided by the price of the bond as done for moderated duration.

n

tt

t

t

S

CFttDGFC

12)1(

)1(

10000*21

sPDGFC

GFC10000*

18

8.12 Price Changes using Duration and Convexity

10000**100** 2SDGFCSDDP

10000**100**% 2SGFCSMDP

19

8.13 Measuring Duration for Interest Sensitive Cash Flows

Securities that have cash flows that are sensitive (in amount or timing) to interest rate levels should use effective duration (ED) and effective dollar duration (EDD).

Changes in rates is measured as a parallel shift in the expected spot rates rather than a change in today’s spot rates.

Often changes are expressed as a parallel shift in the term structure meaning the forward rates (expected spot rates) move up or down by an equal percent.

Because the relation between interest rates and price is confounded by the dependence of cash flows on interest rates, ED and EDD are estimated using the approximation methods.

100*StructureTermP

EDD

100*%StructureTerm

PED

20

8.14 An Example: Estimating ED and EDD

Suppose you wanted to figure out your exposure to a 2% parallel movement in interest rate (+1% and -1%) for a 3 year coupon bond with a face value of $1000 and a coupon of 10%. This bond is callable if interest rates fall below 7.5%. There is an equal probability of moving up or down.

Below is the expected term structure before the parallel shifts.

Compare to MD and DD (which assume the bond is not callable).

8%

7.65%

11.41%

7%

10.23%

14.96%

21

8.15 Using duration to manage portfolios: Immunization

Given we can measure duration, how is it useful to us?

A. Liquidity Management

MATCHING DURATION OF PAYMENTS AND RECEIPT OF CASH

Example. Suppose a bank has to make payments of $100 at Time 1 and $200 at Time 2. How does it make sure it can make those payments with certainty? Use the following expected term structure, assuming (i) you have access to a 1-yr and 2-yr ZCB with $100 FV each and (ii) you only have access to a 1-yr and 3-yr ZCB with $100 FV and (iii) you only have access to a 1-yr ZCB

8%

7.65%

11.41%

7%

10.23%

14.96%

22

8.15 Using duration to manage portfolios: Price Valuation

Given we can measure duration, how is it useful to us?

A. Hedging changes in asset value of a portfolio

MATCHING PRICE CHANGES OF AN ASSET WITH A HEDGE

Example. Suppose a bank wants to hedge potential changes in the price of an asset for changes in the interest rate. Suppose the five year spot rate is 1.07 and the 15 yr spot rate is 1.08 and the bank wants to use 5 yr ZCBs to hedge price fluctuations in 15 yr ZCBs. Each bond has a face value of $100.

You could add convexity to be more accurate.

314.1332.3

378.4

100***100**

*

50

150

50

150

50

150

YearZCB

YearZCB

YearZCB

YearZCB

YearZCB

YearZCB

DD

DDHedge

rDDHedgerDD

PHedgeP

23

8.16 Duration to minimize interest rate fluctuations

• Price risk and coupon risk offset one another to immunize portfolio.

• Numerous mixtures of bonds possible to achieve a certain duration measure.

• BARBELL strategies: Buy short and long-term assets and no medium-term assets.

• LADDERING strategies: Buy short, medium and long-term assets.

• If an asset doesn’t exist with desired duration, you can always combine assets with larger and smaller duration to create a portfolio with the desired duration. However, convexity can differ for portfolios with the same duration.

• Need more complicated measures of duration to compensate for non-parallel movements in the interest rate.

• Matching duration is not applicable to interest-sensitive investments. Assumes there is positive convexity. Instruments such as callable bonds or other interest sensitive instruments may have more complicated convexities.

24

8.17 Summary

• The concepts of duration and convexity were developed to aid in the measurement of the sensitivity of a security’s value to changes in the interest rate.

• Duration measures for interest sensitive cash flows were also introduced.

• While not perfect, these tools have proven to be useful and are widely employed.