39887312 Risk Management Chap 9 Interest Risk II MOD

Interest Rate Risk Interest Rate Risk IIII

Chapter 9

© 2008 The McGraw-Hill Companies, Inc., All Rights Reserved.McGraw-Hill/Irwin

9-2

Overview The weakness of repricing model is its

reliance on book value rather than market value of assets and liabilities.

This chapter discusses a market value-based model for assessing and managing interest rate risk: Duration Computation of duration Economic interpretation Immunization using duration * Problems in applying duration

9-3

Duration Duration is a more complete measure of an

asset or liability's interest rate sensitivity than is maturity because duration takes into account the time of arrival (or payment) of all cash flows as well as the asset's (or liability's) maturity.

9-4

Duration Consider a loan with a 15 % interest rate and

required repayment of half the $100 in principal at the end of six months and the other half at the end of the year. The loan is financed with a one-year CD paying 15% interest per year.

The promised cash flows (CF) received by the Fl from the loan at the end of one-half year and at the end of the year

0 6 m 1 y

CF6m= 53.75 CF1y= 57. 5

9-5

Duration

50.249.8

53.75 57.5

duration is the weighted-average time to maturity on the loan using the relative present values of the cash flows as weights. In present value terms, the relative importance of the cash flows arriving at time t — 6m year and time t = 1 year are as follows:

50.2 0.502 50.2%

49.8 0.498 49.8%

9-6

Duration That is, the Fl receives 50.2 percent of cash flows on

the loan with the first payment at the end of six months (t = 1/2) and 49.8 percent with the second payment at the end of the year (t = 1).

By definition, the sum of the (present value) cash flow weights must equal .502 + .498 = 1

We can now calculate the duration (D), or the weighted-average time to maturity, of the loan using the present value of its cash flows as weights:

D, = X1/2(1/2) + X1(1) = .502(1/2) + .498 (1) = .749 years

9-7

Duration Thus, while the maturity of the loan is one year, its

duration, or average life a cash flow sense, is only .749 years.

The duration is less than the maturity of loan because in present value terms 50.2 % of the cash flows are received at the end of one-half year.

Note that duration is measured in years since we weight the time (t) at which cash flows are received by the relative present value importance of cash flows {X1/2, Xl etc.).

9-8

Duration We next calculate the duration of the one-year, $100, 15 %

interest certificate of deposit. The FI promises to make only one cash payment to depositors at the end of year

CF: = $115, which is the promised principal ($100) and interest repayment ($15) to the depositor.

Since weights are calculated in present value terms. CF1 = $115, and PV1 = $115/1.15 = $100

Because all cash flows are received in one payment at the end of the year, X1 = PV1/PVi = 1, the duration of the deposit is

DD = Xl * 1DD = 1 * 1 = 1 year

9-9

Duration Thus, only when all cash flows are limited to one

payment at the end of the period with no intervening cash flows does duration equal maturity.

This example also illustrates that while the maturities on the loan and the deposit are both one year (and thus the difference or gap in maturities is zero), the duration gap is negative:

gap in maturities = ML – MD=1-1=0 duration gap =DL - DD = .749 - 1 =-.251 years As will become clearer, to measure and to hedge

interest rate risk, the FI needs to manage its duration gap rather than its maturity gap.

9-10

Price Sensitivity and Maturity In general, the longer the term to

maturity, the greater the sensitivity to interest rate changes.

Example: Suppose the zero coupon yield curve is flat at 12%.

Bond A pays $1762.34 in five years. Bond B pays $3105.85 in ten years, and both are currently priced at $1000.

9-11

Example continued... Bond A: P = $1000 = $1762.34/(1.12)5 Bond B: P = $1000 = $3105.84/(1.12)10

Now suppose the interest rate increases by 1%. Bond A: P = $1762.34/(1.13)5 = $956.53 Bond B: P = $3105.84/(1.13)10 = $914.94

The longer maturity bond has the greater drop in price because the payment is discounted a greater number of times.

9-12General formula for Duration Duration

Weighted average time to maturity using the relative present values of the cash flows as weights.

9-13

Duration For bonds that pay interest semiannually,

the duration equation becomes

where t = ½, 1,l 1/2..., N.

9-14

Duration Example1: Eurobonds pay coupons annually. Suppose a Eurobond

matures in 6 years, the annual coupon is 8 percent, the face value of the bond is $1,000, and the current yield to maturity (R) is also 8 percent.

We show the calculation of its duration in Table. Column 1 lists the time period (in years) in which a cash flow (CF) is

received. Column 2 lists the CF received in time period t. Column 3 lists the discount factor used to convert a future value to a

present value. 1/(1+R)t

Column 4 is the present value of the CF received in each period t (Column 2 times Column 3).

The sum of Column 4 is the present value of the bond: the denominator of the duration equation. Column 5 is the present value of the CF received each period

times the time it takes to receive the CF (Column 4 times Column 1). The sum of Column 5

is the time weighted present value of the bond: the numerator of the duration equation.

As the calculation indicates, the duration or weighted-average time to maturity on this bond is 4.993 years.

9-15

Duration

9-16

Duration Definition of Duration The weighted-average time to maturity on a security. The interest elasticity of a security's price to small interest

rate changes. Features of Duration Duration increases with the maturity of a fixed-income

security, but at a decreasing rate. Duration decreases as the yield on a security increases. Duration decreases as the coupon or interest payment

increases. Risk Management with Duration Duration is equal to the maturity of an immunized security. Duration gap is used by Fis to measure and manage the

interest rate risk of an overall balance sheet.

9-17

Duration Example 2: U.S. Treasury bonds pay coupon interest

semiannually. Suppose a Treasury bond matures in two years, the annual coupon

rate is 8 percent, the face value is $1,000, and the annual yield to maturity (R) is 12 percent.

As the calculation indicates, the duration, or weighted-average time to

maturity, on this bond is 1.883 years.

9-18

Duration if the annual coupon rate is lowered to

6%duration rises to 1.909 years. Since 6 percent coupon payments are lower than 8 percent, it takes longer to recover the initial investment in the bond

9-19

Duration duration is calculated for the original 8 percent bond,

assuming that the yield to maturity increases to 16 percent. The higher the yield to maturity on the bond, the more the investor earns on reinvested coupons and the shorter the time to recover the initial investment

9-20

Duration when the maturity on a bond decreases to 1 year its

duration falls to 0.980 year. Thus, the shorter the maturity on the bond, the more quickly the initial investment is recovered.

9-21

Computing duration Consider a 2-year, 8% coupon bond, with a

face value of $1,000 and yield-to-maturity of 12%. Coupons are paid semi-annually.

Therefore, each coupon payment is $40 Present value of each cash flow equals CFt

÷ (1+ 0.12)t where t is the period number.

9-22 Duration of 2-year, 8% bond:

Face value = $1,000, YTM = 12%

t years CFt PV(CFt) Weight (W) W × years

1 0.5 40 37.7964473 0.040366192 0.0201830962 1.0 40 35.71428571 0.038142466 0.0381424663 1.5 40 33.74682795 0.036041243 0.0540618644 2.0 1040 829.0816327 0.8854501 1.7709002

P = 936.3391936 1 1.883287625D=1.88328763

9-23

Duration Gap Suppose the bond in the previous example

is the only loan asset (L) of an FI, funded by a 2-year certificate of deposit (D).

Maturity gap: ML - MD = 2 -2 = 0 Duration Gap: DL - DD = 1.885 - 2.0 = -0.115

Deposit has greater interest rate sensitivity than the loan, so DGAP is negative.

FI exposed to rising interest rates.

9-24

Duration of Zero-coupon Bond Bond sell at a discount from face value on issue Pay the face value (e.g 1000) on maturity Have no intervening cash flows between issue and maturity. The current price an investor is willing to pay for such a bond is equal to the

present value of the single, fixed (face value) payement on the bond that is received on maturity (here 1000$)

R: is the required annually compounded yield to maturity N: is the number of year to maturity P: the price

Because there are no intervening cash flows between issue and maturity

For a zero coupon bond, duration equals maturity since 100% of its present value is generated by the payment of the face value, at maturity.

For all other bonds: duration < maturity

NRp

11000

BB MD

9-25

Special Case Maturity of a consol bond : M = . Duration of a consol: D = 1 + 1/R Suppose the the Yield curve implies R=5%

annually. The duration of the consol bond:

If interest rate rise the duration of consol bond falls.

yDc 2105.011

yDc 62.0

11

9-26

Features of Duration Duration and maturity:

D increases with M, but at a decreasing rate. Duration and yield-to-maturity:

D decreases as yield increases. Duration and coupon interest:

D decreases as coupon increases

9-27

Economic Interpretation Duration is a measure of interest rate sensitivity or elasticity of the

security’s price to small interest rate changes:

for small changes in interest rates, bond prices move in an inversely proportional fashion according to the size of D.

for any given change in interest rates, long duration securities suffer a larger capital loss (or receive a higher capital gain) should interest rates rise (fall) than do short-duration securities.

where MD is modified duration.

RMDR

RDPP

RDMDwhen

DR

RPP

1

1

1

9-28

Economic Interpretation To estimate the change in price, we can rewrite

this as:

Duration is a measure of the percentage change in the price of a security for a 1 percent change in the return on the security

Note the direct linear relationship between ΔP and -D.

PRMD

PR

RDP

1

9-29

Economic Interpretation Dollar duration is the dollar value change in the

price of a security to a 1 percent change in the return on the security.

The dollar duration is defined as the modified duration times the price of a security:

Dollar duration = MD X P Thus, the total dollar change in value of a security

will increase by an amount equal to the dollar duration times the change in the return on the security:

ΔP = -Dollar duration X ΔR

9-30

Semi-annual Coupon Payments With semi-annual coupon payments:

RRD

PP

211

9-31

An example: Consider Example 1 for the six-year Eurobond with an 8

percent coupon and 8 percent yield.We determined its duration was D = 4.993 years.

The modified duration is: MD=D/(1+R) MD= 4.993/1.08 MD=4.623 That is, the price of the bond will increase by

4.623 percent for a 1 percent decrease in the interest rate on the bond.

Dollar duration =4.623 x $1,000 = 4623 or a 1 percent (or 100 basis points) change in the

return on the bond would resuit a change of $46.23 in the price of the bond.

9-32

suppose that yields were to rise by one basis point (1/100th of 1 percent) from 8 to 8.01 percent.

Then:

The bond price had been $1,000, which was the present value of a six-year bond with 8 percent coupons and 8 percent yield. The duration model, and specifically dollar duration, predicts that the price of the bond would fall to $999.5377 after the increase in yield by one basis point.

462.01000000462.0

P

9-33

EXAMPLE Consider a consol bond with an 8 percent

coupon paid annually, an 8 percent yield, and calculated duration of 13.5 years

(Dc = 1 + 1/.08 = 13.5). Thus, for a one-basis-point change in the

yield (from 8 percent to 8.01 percent):

9-34

EXAMPLE Recall from Example 2 the two-year T-bond with semiannual coupons

whose duration we derived as 1.883 years when annual yields were 12 percent. The modified duration is:

the price of the bond will increase by 1.776 percent for a 1 percent decrease in the interest rate on the bond

the dollar duration is:

or a 1 percent (or 100 basis points) change in the return on the bond would result a change of $16.53 in the price of the bond.

Thus, a one-basis-point rise in interest rates would have the following predicted effect on its price:

or the price of the bond would fall by 0.01776 percent from $930.70

to $930.5326.

9-35

Duration as Index of Interest Rate Risk An example:

Consider three loan plans, all of which have maturities of 2 years. The loan amount is $1,000 and the current interest rate is 3%.

Loan #1, is a two-payment loan with two equal payments of $522.61 each.

Loan #2 is structured as a 3% annual coupon bond.

Loan # 3 is a discount loan, which has a single payment of $1,060.90.

9-36

t years CFt PV(CFt) 3% Weight (W) W × years1 1 522.61 507.3883495 0.507389163 0.5073891632 2 522.61 492.6100481 0.492610837 0.985221675

1045.22 999.9983976 1 1.492610837D 1.492610837

t years CFt PV(CFt) 2% Weight (W) W × years1 1 522.61 512.3627451 0.504950495 0.5049504952 2 522.61 502.3164168 0.495049505 0.99009901

1045.22 1014.679162 1 1.495049505D 1.495049505

t years CFt PV(CFt) 3% Weight (W) W × years1 1 30 29.41176471 0.028851593 0.0288515932 2 1030 990.0038447 0.971148407 1.942296813

1060 1019.415609 1 1.971148407D 1.971148407

t years CFt PV(CFt) 2% Weight (W) W × years1 1 30 29.12621359 0.029126214 0.0291262142 2 1030 970.8737864 0.970873786 1.941747573

1060 1000 1 1.970873786D 1.970873786

t years CFt PV(CFt) Weight (W) W × years1 2 1060.9 1000 1 21 2 1060.9 1019.70396 1 2

9-37

Duration as Index of Interest Rate Risk

Yield Loan Value 2% 3% ΔP N D

Equal Payment

$1014.68 $1000 $14.68 2 1.493

3% Coupon $1019.42 $1000 $19.42 2 1.971

Discount $1019.70 $1000 $19.70 2 2.000

9-38

Duration and interest Rate Risk Management on the Whole Balance Sheet of an FI

The Duration Gap for a Financial Institution To estimate the overall duration gap of an FI,

we determine first the duration of an FI's asset portfolio (A) and the duration of its liability portfolio (L). These can be calculated as:

9-39

The Xij's in the equation are the market value proportions of each asset or liability held in the respective asset and liability portfolios.

Thus, if new 30-year Treasury bonds were 1 percent of a life insurer's portfolio and (the duration of those bonds) was equal to 9.25 years, then X1ADf = .01(9.25) = 0.0925.

More simply, the duration of a portfolio of assets or liabilities is a market value weighted average of the individual durations of the assets or liabilities on the FI's balance sheet.

AD1

9-40

Consider an FI's simplified market value balance sheet:

9-41

Immunizing the Balance Sheet of an FI when interest rates change, the change in the FI's equity or net

worth (E) is equal to the difference between the change in the market values of assets and liabilities on each side of the balance sheet.

Since ΔE = ΔA - ΔL, we need to determine how ΔA and ΔL are related to duration.

From the duration model (assuming annual compounding of interest):

9-42

Immunizing the Balance Sheet of an FI These equations can be rewritten to show the dollar

changes in assets and liabilities on an FI's balance sheet

9-43

Immunizing the Balance Sheet of an FI

9-44

Duration and Immunizing

9-45

An example: Suppose DA = 5 years, DL = 3 years and rates are

expected to rise from 10% to 11%. (Rates change by 1%). Also, A = 100, L = 90 and E = 10. Find change in E.

The Fl could lose $2.09 million in net worth if rates rise 1 percent. Since the Fl started with $10 million in equity, the loss of $2.09 million is almost 21 percent of its initial net worth. (2.09/10) The market value balance sheet after the rise in rates by 1 percent would look like this:

9-46

46.21.101.0903

55.41.101.01005

RRLDL

RRADA

L

A

9-47

Duration and Immunizing Even though the rise in interest rates would not push the Fl into economic

insolvency, it reduces the FLs net worth-to-assets ratio from 10 (10/100) to 8.29 percent (7.91/95.45).

To counter this effect, the manager might reduce the FI's adjusted duration gap. In an extreme case, the gap might be reduced to zero:

To do this, the FI should not directly set DA = DL, which ignores the fact that the FI's assets (A) do not equal its borrowed liabilities (L) and that k (which reflects the ratio L/A) is not equal to 1.

But, to set E = 0: DA = kDL

9-48

Immunization and Regulatory Concerns

suppose the manager increased the duration of the FI's liabilities to five years, the same as DA. Then:

The FI is still exposed to a loss of $0.45 million if rates rise by 1 percent. An appropriate strategy would involve changing DL until:

9-49

Immunization and Regulatory Concerns In this case the FI manager sets DL = 5.55 years, or slightly

longer than DA = 5 years, to compensate for the fact that only 90 percent of assets are funded by borrowed liabilities, with the other 10 percent funded by equity.

Note that the FI manager has at least three other ways to reduce the adjusted duration gap to zero Reduce DA. Reduce DA from 5 years to 2.7 years (equal

to kDL or (0.9*3) such that:

Reduce DA and increase DL. Shorten the duration of assets and lengthen the duration of liabilities at the same time. One possibility would be to reduce DA to4 years and to increase DL to 4.44 years such that:

9-50

Immunization and Regulatory Concerns

Change k and DL. Increase k (leverage) from 0.9 to 0.95 and increase DL from 3 years to 5.26 years such that:

9-51

*Contingent Claims Interest rate changes also affect value of

off-balance sheet claims. Duration gap hedging strategy must include the

effects on off-balance sheet items such as futures, options, swaps, caps, and other contingent claims.

9-52

Pertinent WebsitesBank for International Settlements

www.bis.org Securities Exchange Commission

www.sec.govThe Wall Street Journal www.wsj.com