MANAGING INTEREST RATE RISK: DURATION GAP AND MARKET VALUE OF EQUITY

Chapter 9

Bank ManagementBank Management, 5th edition.5th edition.Timothy W. Koch and S. Scott MacDonaldTimothy W. Koch and S. Scott MacDonaldCopyright © 2003 by South-Western, a division of Thomson Learning

Duration GAP…focus on managing NII or the market value of equity, recognizing the timing of cash flows.

Interest rate risk is measured by comparing the weighted average duration of assets with the weighted average duration of liabilities.

If Asset duration > Liability duration interest ratesMarket value of

equity will fall.

Duration versus maturity

1.) 1000 loan, principal + interest paid in 20 years.2.) 1000 loan, 900 principal in 1 year,

100 principal in 20 years. 1000 + int

|-------------------|-----------------| 0 10 20

900+int 100 + int|----|--------------|-----------------|

0 1 10 20

What is the maturity of each? 20 yearsWhat is the "effective" maturity?

2.) = [(900/100) x 1]+[(100/1000) x 20] = 2.9 yrs

1

2

Duration, however, uses a weighted average of the present values.

Δy

ΔP

y+1ΔyP

ΔP

DUR

Solve for Price: P -Duration x [y / (1 + y)] x P

Price (value) changes Longer maturity/duration larger changes in

price for a given change in i-rates. Larger coupon smaller change in price for a

given change in i-rates.

Duration…approximate measure of the price elasticity of demand

Sec. the ofPV y)+(1(t)CF

y)+(1CF

y)+(1(t)CF

=D

n

1=tt

t

k

1=tt

t

k

1=tt

t

s year2.73 = 951.96

2597.6

(1.12)1000

+ (1.12)

100(1.12)

31000 +

(1.12)3100

+ (1.12)

2100+

(1.12)1100

D 3

1=t3t

332

1

In general notation, Macaulay’s duration (D):

Examples 1000 face, 10% coupon, 3 year, 12% YTM

Measuring duration

1136.16(1.05)

3*1000 +

(1.05)3*100

+ (1.05)

2*100+

(1.05)1*100

D332

1

years2.75 = 1136.16

3127.31D

If YTM = 5%1000 face, 10% coupon, 3 year, 5% YTM

years2.68 = 789.35

2131.95D

If YTM = 20%1000 face, 10% coupon, 3 year, 20% YTM

definition (by 3

(1.12)1000

(1.12)31000

D

3

3

If YTM = 12% and Coupon = 01000 face, 0% coupon, 3 year, 12% YTM

1000|-------|-------|-------|0 1 2 3

Δyy+1

Duration sMacaulay'

P

ΔP

Duration and modified duration

Duration allows market participants to estimate the relative price volatility of different securities:

Using modified duration:modified duration= Macaulay’s duration / (1+y)

We have an estimate of price volatility:%change in price = modified duration x y

)i (iP

P-P duration Effective

-0

i-i

Effective Duration…used to estimate a securities price sensitivity when the security contains embedded options.

Effective duration is

Where Pi- = price if rates fall, Pi+ = price if rates rise; P0 = initial (current) price; i+ initial market rate plus the increase in rate; i- = initial market rate minus the decrease in rate.

Effective duration compares a security’s estimated price in a falling and rising rate environment.

2.54 0.044)- 05$10,000(0.

$9,847.72-$10,000Dur Eff

Effective duration example: Consider a 3-year, 9.4 percent coupon bond selling for $10,000 par to yield 9.4 percent to maturity. The Macaulay’s duration for the option-free version of this bond with semiannual coupons and compounding was calculated to be 5.36 semiannual periods, or 2.68 years at the market rate of 4.7 percent semiannually. The modified duration was 5.12 semiannual periods or 2.56 years. A 30 basis point increase in rate to 5 percent semiannually will lower the price to $9,847.72.

The callable bond’s effective duration for a 30 basis point (0.30 percent) semiannual movement in rates either up or down is 2.54:

Two types of interest rate risk

Reinvestment rate risk... the risk that a bank can not reinvest cash flows from assets or refinance rolled over or new liabilities at a certain rate in the futureCost of funds versus the return on assets

Funding GAP, impact on NII Price Risk

… changes in interest rates will also cause a change in the value (price) of assets and liabilitiesLonger maturity (duration) larger change in value for a given change in interest rates Duration GAP, impact on market value of equity

Reinvestment rate risk and price risk

Reinvestment rate risk: If i-rate , then yield from reinvestment of the cash

flows and holding period return (HPR) increases. Price risk:

If i-rate and you sell the security prior to maturity, the price or value falls, hence HPR falls.

Increases in i-rates will improve HPR from a higher reinvestment rate but reduce HPR from capital losses if the security is sold prior to maturity.

An immunized security is one in which the gain from the higher reinvestment rate is just offset by the capital loss. This point is where your holding period equals the

duration of the security.

There are four steps in duration gap analysis.1. Management develops an interest rate forecast.

2. Management estimates the market value of bank assets, liabilities and stockholders’ equity.

3. Management estimates the weighted duration of assets and weighted duration of liabilities.

The effects of both on- and off-balance sheet items are incorporated. These estimates are used to calculate duration gap.

4. Management forecasts changes in the market value of stockholders’ equity across different interest rate environments.

Duration GAP at the bank

The bank can protect either the market value of equity (MVE) or the book value of NII, but not both.

To protect the MVE the bank would set DGAP to zero: DGAP = DA - u x DL where

DA = weighted average duration of assetsDL = weighted average duration of liabs

1 Par Years Market$1,000 % Coup Mat. YTM Value Dur.

AssetsCash 100 100Earning assets

3-yr Commercial loan 700 12.00% 3 12.00% 700 2.696-yr Treasury bond 200 8.00% 6 8.00% 200 4.99 Total Earning Assets 900 11.11% 900Non-cash earning assets 0 0

Total assets 1000 10.00% 1000 2.88

LiabilitiesInterest bearing liabs.

1-yr Time deposit 620 5.00% 1 5.00% 620 1.003-yr Certificate of deposit 300 7.00% 3 7.00% 300 2.81 Tot. Int Bearing Liabs. 920 5.65% 920Tot. non-int. bearing 0 0Total liabilities 920 5.65% 920 1.59

Total equity 80 80Total liabs & equity 1000 1000

Hypothetical Bank Balance Sheet

700)12.1(3700

)12.1(384

)12.1(284

)12.1(184

3321

D

Calculating DGAP

DA = (700/1000)*2.69 + (200/1000)*4.99 = 2.88

DL = (620/920)*1.00 + (300/920)*2.81 = 1.59 DGAP = 2.88 - (920/1000)*1.59 = 1.42 years

What does 1.42 mean? The average duration of assets > liabilities,

hence asset values change by more than

liability values.

1 Par Years Market$1,000 % Coup Mat. YTM Value Dur.

AssetsCash 100 100Earning assets

3-yr Commercial loan 700 12.00% 3 13.00% 683.47 2.696-yr Treasury bond 200 8.00% 6 9.00% 191.03 4.97 Total Earning Assets 900 12.13% 874.5Non-cash earning assets 0 0

Total assets 1000 10.88% 974.5 2.86

LiabilitiesInterest bearing liabs.

1-yr Time deposit 620 5.00% 1 6.00% 614.15 1.003-yr Certificate of deposit 300 7.00% 3 8.00% 292.27 2.81 Tot. Int Bearing Liabs. 920 6.64% 906.42Tot. non-int. bearing 0 0Total liabilities 920 6.64% 906.42 1.58

Total equity 80 68.08Total liabs & equity 1000 974.5

1 percent increase in all rates.

3

3

1t t 1.13

700

1.13

84PV

Calculating DGAP

DA = (683 / 974) * 2.68 + (191 / 974) * 4.97 = 2.86 yrs

DA = (614 / 906) * 1.00 + (292 / 906) * 2.80 = 1.58 yrs

DGAP = 2.86 - (906 / 974) * 1.58 = 1.36 years

What does 1.36 mean?The average duration of assets > liabilities, hence asset values change by more than liability values.

TA)i(1

ΔyDGAP)(ΔMVE

assets Total

Δy

ΔP

y+1ΔyP

ΔP

DUR

Using the relationship:

We can define the change in the MVE as:

In our case: MVE = (-1.42) x [+0.01 / (1.10)] x 1,000

= -$12.90

Change in the Market Value of Equity

Positive and negative DGAPs

Positive DGAP …indicates that assets are more price sensitive than liabilities, on average. Thus, when interest rates rise (fall), assets will

fall proportionately more (less) in value than liabilities and the MVE will fall (rise) accordingly.

Negative DGAP …indicates that weighted liabilities are more price sensitive than assets. Thus, when interest rates rise (fall), assets will

fall proportionately less (more) in value that liabilities and the MVE will rise (fall).

An immunized portfolio… What is the minimum risk position?

To eliminate the risk of changes in the MVE, how much must DA or DL change by? Change DA = -1.42 Change DL = +1.42/u = 1.54

1 Par Years Market$1,000 % Coup Mat. YTM Value Dur.

AssetsCash 100 100Earning assets

3-yr Commercial loan 700 12.00% 3 12.00% 700 2.696-yr Treasury bond 200 8.00% 6 8.00% 200 4.99 Total Earning Assets 900 11.11% 900Non-cash earning assets 0 0

Total assets 1000 10.00% 1000 2.88

LiabilitiesInterest bearing liabs.

1-yr Time deposit 340 5.00% 1 5.00% 340 1.003-yr Certificate of deposit 300 7.00% 3 7.00% 300 2.816-yr Zero-coupon CD* 444.3 0.00% 6 8.00% 280 6.00 Tot. Int Bearing Liabs. 1084.3 6.57% 920Tot. non-int. bearing 0 0Total liabilities 1084.3 6.57% 920 3.11

Total equity 80 80

Immunized portfolio

DGAP = 2.88 – 0.92 (3.11) ≈ 0

1 Par Years Market$1,000 % Coup Mat. YTM Value Dur.

AssetsCash 100 100Earning assets

3-yr Commercial loan 700 12.00% 3 13.00% 683.47 2.696-yr Treasury bond 200 8.00% 6 9.00% 191.03 4.97 Total Earning Assets 900 12.13% 874.5Non-cash earning assets 0 0

Total assets 1000 10.88% 974.5 2.86

LiabilitiesInterest bearing liabs.

1-yr Time deposit 340 5.00% 1 6.00% 336.79 1.003-yr Certificate of deposit 300 7.00% 3 8.00% 292.27 2.816-yr Zero-coupon CD* 444.3 0.00% 6 9.00% 264.94 6.00 Tot. Int Bearing Liabs. 1084.3 7.54% 894Tot. non-int. bearing 0 0Total liabilities 1084.3 7.54% 894 3.07

Total equity 80 80

Immunized portfolio: 1% increase in all rates

Stabilizing the book value of net interest income

This can be done for a 1-year time horizon, with the appropriate duration gap measure:

DGAP* MVRSA(1- DRSA) - MVRSL(1- DRSL) where

MVRSA = cumulative market value of RSAs, MVRSL = cumulative market value of RSLs,

DRSA = composite duration of RSAs for the given time horizon; equal to the sum of the products of each asset’s duration with the relative share of its total asset market value, and

DRSL = composite duration of RSLs for the given time horizon; equal to the sum of the products of each liability’s duration with the relative share of its total liability market value.

If DGAP* is positive, the bank’s net interest income will decrease when interest rates decrease, and increase when rates increase. If DGAP* is negative, the relationship is reversed. Only when DGAP* equals zero is interest rate risk eliminated.

Banks can use duration analysis to stabilize a number of different variables reflecting bank performance.

Market value of equity sensitivity analysis

MVE sensitivity analysis effectively involves the same steps as earnings sensitivity analysis.

In MVA analysis, however, the bank focuses on: the relative durations of assets and

liabilities, how much the durations change in

different interest rate environments, and what happens to the market value of

equity across different rate environments.

Embedded options sharply influence the estimated volatility in MVE

Prepayments that exceed (fall short of) that expected will shorten (lengthen) duration.

A bond being called will shorten duration.

A deposit that is withdrawn early will shorten duration. A deposit that is not withdrawn as

expected will lengthen duration.

Book Value Market Value

Book Yield

Duration*

LoansPrime Based Ln 100,000 102,000 9.00%Equity Credit Lines 25,000 25,500 8.75% -Fixed Rate > I yr 170,000 170,850 7.50% 1.1Var Rate Mtg 1 Yr 55,000 54,725 6.90% 0.530-Year Mortgage 250,000 245,000 7.60% 6.0Consumer Ln 100,000 100,500 8.00% 1.9Credit Card 25,000 25,000 14.00% 1.0Total Loans 725,000 723,575 8.03% 2.6Loan Loss Reserve -15,000 11,250 0.00% 8.0 Net Loans 710,000 712,325 8.03% 2.5InvestmentsEurodollars 80,000 80,000 5.50% 0.1CMO Fix Rate 35,000 34,825 6.25% 2.0US Treasury 75,000 74,813 5.80% 1.8 Total Investments 190,000 189,638 5.76% 1.1

Fed Funds Sold 25,000 25,000 5.25% -Cash & Due From 15,000 15,000 0.00% 6.5Non-int Rel Assets 60,000 60,000 0.00% 8.0 Total Assets 100,000 100,000 6.93% 2.6

ABC's market value of stockholders' equity Market value/duration report as of 12/31/01 most likely rate scenario-base strategy (assets)

Ass

ets

Book Value Market Value

Book Yield

Duration*

DepositsMMDA 240,000 232,800 2.25% -Retail CDs 400,000 400,000 5.40% 1.1Savings 35,000 33,600 4.00% 1.9NOW 40,000 38,800 2.00% 1.9DDA Personal 55,000 52,250 8.0Comm'l DDA 60,000 58,200 4.8 Total Deposits 830,000 815,650 1.6TT&L 25,000 25,000 5.00% -L-T Notes Fixed 50,000 50,250 8.00% 5.9Fed Funds Purch - - 5.25% -NIR Liabilities 30,000 28,500 8.0 Total Liabilities 935,000 919,400 2.0

Equity 65,000 82,563 9.9 Total Liab & Equity 1,000,000 1,001,963 2.6

Off Balance Sheet Notionallnt Rate Swaps - 1,250 6.00% 2.8 50,000

Adjusted Equity 65,000 83,813 7.9

ABC's market value of stockholders' equity Market value/duration report as of 12/31/01 most likely rate scenario-base strategy (liabilities)

Liab

ilitie

s

Duration Gap for ABC’s MVE

1. Mkt value of assets = 1,001,963Duration of assets = 2.6

2. Mkt value of liabilities = 919,400Duration of liabilities = 2.0

Dur Gap= 2.6 – (919,400 / 1,001,963) * 2.0

= 0.765 yrs Example:

A 1% increase in rates would reduce MVE by 7.2 million

= 0.765 (0.01 / 1.0693) 1,001,963Recall that the average rate on assets is 6.93%

Sensitivity of economic value of equity (MVE) versus most likely (zero shock) interest rate scenario

2

(10.0)

20.0

10.0 8.8 8.2

(8.2)

(20.4)

(36.6)

13.6

ALCO G uide lineBoard Limit(20.0)

(30.0)

Ch

ang

e in

EV

E (

mill

ion

s o

f d

olla

rs)

(40.0)-300 -200 -100 +100 +200 +3000

Shocks to Curre nt Rates

Sensitivity of Economic Value of Equity measures the change in the economic value of the corporation’s equity under various changes in interest rates. Rate changes are instantaneous changes from current rates. The change in economic value of equity is derived from the difference between changes in the market value of assets and changes in the market value of liabilities.

Effective “duration” of equity

By definition, duration measures the percentage change in market value for a given change in interest rates

Hence, a bank’s duration of equity measure the percentage change in MVE that will occur with a 1 percent change in rates:

Effective duration of equity 9.9 yrs. = 8,200 / 82,563

Asset / liability sensitivity and DGAP

Funding GAP and Duration GAP are not directly comparable. Funding GAP examines various “time

buckets” while DGAP represents the entire balance sheet.

Generally, if a bank is liability (asset) sensitive in the sense that net interest income falls (rises) when rates rise and vice versa, it will likely have a positive (negative) DGAP suggesting that assets are more price sensitive than liabilities, on average.

Advantages of DGAP over Funding GAP DGAP analysis has the advantage of

focusing on all cash flows from the underlying assets and liabilities and not just cash flows that are expected to arise over short time intervals.

Interest rate risk can be summarized in one measure for the entire portfolio.

Speculating on DGAP

It is difficult to consistently alter either GAP or DGAP on the balance sheet and increase earnings or the market value of stockholders' equity.

Whenever management chooses to change asset and liability maturities and/or durations in anticipation of rate changes, it is placing a bet against forward rates from the yield curve.

Gap and DGAP management strategies: what are your bets? Cash flows from investing $1,000 either in a 2-year security

yielding 6 percent or two consecutive 1-year securities, with the current 1-year yield equal to 5.5 percent.

0 1 2|-----------|-------------|

2-year security: ----$60----|----$60----- $120 at 6% per yr1-year security: ----$55----|----- ? ------ $120 another 1-yr sec. It is not known today what a 1-year security will yield in one

year. For the two consecutive 1-year securities to generate the same

$120 in interest, ignoring compounding, the 1-year security must yield 6.5% one year from the present.

This break-even rate is a 1-year forward rate, one year from the present:

6% + 6% = 5.5% + ? with ? = 6.5 percent

DATE WHEN 1-YEAR RATE FIRST EXCEEDS 10-YEAR RATE

LENGTH OF TIME UNTIL START OF NEXT RECESSION

Apr. ’68 20 months (Dec. ’69)Mar. ’73 8 months (Nov. ’73)Sept. ’78 16 months (Jan. ’80)Sept. ’80 10 months (July ’81)Feb. ’89 17 months (July ’90)Dec. ’00 15 months (March ’01)

The inverted yield curve has predicted the last five recessions

In expansionary stages rates rise until they reach a peak as the Federal Reserve tightens credit availability.

In contractionary stages rates fall until they reach a trough when the U.S. economy falls into recession.

Interest Rates and the Business CycleThe general level of interest rates and the shape of the yield curve appear to follow the U.S. business cycle.

Time

ExpansionContraction

Expansion

Long-TermRates

Short-TermRatesPeak

Trough

MANAGING INTEREST RATE RISK: DURATION GAP AND MARKET VALUE OF EQUITY

Chapter 9

Bank ManagementBank Management, 5th edition.5th edition.Timothy W. Koch and S. Scott MacDonaldTimothy W. Koch and S. Scott MacDonaldCopyright © 2003 by South-Western, a division of Thomson Learning