Leverage adjusted duration gap example

Accounting Fundamentals For Financial Institutions Midterm

Asset and Liability Management for Financial Ins

Ferriter

FIN 6102 – Spring 2018

Interest Rates and Net Worth

FIs exposed to interest rate risk due to maturity mismatches between assets and liabilities

Interest rate changes can have severe impact on net worth

Thrifts, during 1980s

Ch 8-2

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2

US Treasury Bill Rate, 1965 - 2015

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Ch 8-3

3

3

Level and Movement of Interest Rates

Federal Reserve: U.S. central bank

Open market operations influence money supply, inflation, and interest rates

Actions of Fed (December, 2008) in response to economic crisis

Target rate between 0.0 and ¼ percent

Ch 8-4

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Central Bank and Interest Rates

Actions mostly target short term rates

Focus on federal funds rate, in particular

Interest rate changes and volatility increasingly transmitted from country to country due to increased globalization of financial markets

Statements by Jerome Powell can have dramatic effects on world interest rates

Ch 8-5

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5

Repricing Model

Repricing, or funding gap, model based on book value

Contrasts with market value-based maturity and duration models in appendix

Ch 8-6

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Repricing Model Continued

Rate sensitivity means repricing at (or near) current market interest rates within a specified time horizon

Repricing gap is the difference between rate-sensitive assets (RSAs) and rate-sensitive liabilities (RSLs)

Refinancing risk

Reinvestment risk

Ch 8-7

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Maturity Buckets

Commercial banks must report quarterly repricing gaps for assets and liabilities with maturities of:

One day

More than one day to three months

More than three months to six months

More than six months to twelve months

More than one year to five years

More than five years

Ch 8-8

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8

Repricing Gap Example

Cum.

Assets Liabilities Gap Gap

1-day $ 20 $ 30 $-10 $-10

>1day-3mos. 30 40 -10 -20

>3mos.-6mos. 70 85 -15 -35

>6mos.-12mos. 90 70 +20 -15

>1yr.-5yrs. 40 30 +10 -5

>5 years 10 5 +5 0

Ch 8-9

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Applying the Repricing Model

NIIi = (GAPi) Ri = (RSAi - RSLi) Ri

Example 1:

In the one day bucket, gap is -$10 million. If rates rise by 1%,

NIIi = (-$10 million) × .01 = -$100,000

Ch 8-10

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Applying the Repricing Model Continued

Example 2:

If we consider the cumulative 1-year gap,

NIIi = (CGAP) Ri = (-$15 million)(.01)

= -$150,000

Ch 8-11

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Rate-Sensitive Assets

Examples from hypothetical balance sheet:

Short-term consumer loans: Repriced at year-end, would just make one-year cutoff

Three-month T-bills: Repriced on maturity every 3 months

Six-month T-notes: Repriced on maturity every 6 months

30-year floating-rate mortgages: Repriced (rate reset) every 9 months

Ch 8-12

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12

Rate-Sensitive Liabilities

RSLs bucketed in same manner as RSAs

Demand deposits warrant special attention

Generally considered rate-insensitive (act as core deposits), but there are arguments for their inclusion as rate-sensitive liabilities

Ch 8-13

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GAP Ratio

May be useful to express interest rate sensitivity in ratio form as CGAP/Assets, referred to as “gap ratio”

Provides direction and scale of exposure

Example:

Gap ratio = CGAP/A = $15 million / $270 million = 0.056, or 5.6 percent

Ch 8-14

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Equal Rate Changes on RSAs, RSLs

Example 8-1: Suppose rates rise 1% for RSAs and RSLs. Expected annual change in NII,

NII = CGAP ×  R

= $15 million × .01

= $150,000

CGAP is positive, change in NII is positively related to change in interest rates

CGAP is negative, change in NII is negatively related to change in interest rates

Ch 8-15

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Unequal Changes in Rates

If changes in rates on RSAs and RSLs are not equal, the spread changes

In this case,

NII = (RSA × RRSA ) - (RSL × RRSL )

Ch 8-16

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Unequal Rate Change Example

Example 8-2:

RSA rate rises by 1.2% and RSL rate rises by 1.0%

NII =  interest revenue -  interest expense

= ($155 million × 1.2%) - ($155 million × 1.0%)

= $310,000

Ch 8-17

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Weaknesses of Repricing Model

Weaknesses:

Ignores market value effects of interest rate changes

Overaggregative

Distribution of assets and liabilities within individual buckets is not considered

Mismatches within buckets can be substantial

Ignores effects of rate-insensitive runoffs

Bank continuously originates and retires consumer and mortgage loans

Runoff of rate-insensitive asset/liability is rate-sensitive

Ch 8-18

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Weaknesses of Repricing Model Continued

Off-balance-sheet items are not included when considering cash flows

Hedging effects of off-balance-sheet items not captured

Example: Futures contracts

Ch 8-19

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The Maturity Model

Explicitly incorporates market value effects

For fixed-income assets and liabilities:

Rise (fall) in interest rates leads to fall (rise) in market value

The longer the maturity, the larger the fall (rise) in market value for interest rate increase (decrease)

Fall in value of longer-term securities increases at diminishing rate for given increase in interest rates

Ch 8-20

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Maturity of Portfolio

Maturity of portfolio of assets (liabilities) equals weighted average of maturities of assets (liabilities) that make up the portfolio

Principles stated on previous slide regarding individual securities apply to portfolios, as well

Typically, maturity gap, MA – ML, > 0 for most banks and thrifts

Ch 8-21

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Effects of Interest Rate Changes

Size of the gap determines the size of interest rate change that would drive net worth to zero

Immunization

Maturity matching, MA - ML = 0

Note: Doesn’t always protect FI against interest rate risk

Ch 8-22

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Leverage

Leverage affects ability to eliminate interest rate risk using maturity model

Example: $100 million in assets invested in one-year, 10% coupon bonds and $90million in liabilities in one-year deposits paying 10%.

Maturity gap is zero, but exposure to interest rate risk is not zero.

Ch 8-23

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Duration

Matching of maturities can still result in interest rate risk due to the timing of cash flows between assets and liabilities not being perfectly matched

FI can only immunize against interest rate risk by matching average lives of an assets and liabilities

See Chap. 9

Ch 8-24

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Term Structure of Interest Rates

Compares market yields or interest rates on securities

Assumes all characteristics (i.e., default risk, coupon rate, etc.) are the same, except for maturity

Most common shapes of yield curve for Treasury securities

Upward-sloping

Downward-sloping, or inverted

Flat

Ch 8-25

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Unbiased Expectations Theory

At a given point in time, yield curve reflects market’s current expectations of future short-term rates

Long-term rates are geometric average of current and expected short-term interest rates

(1 +1RN)N = (1+ 1R1)[1+E(2r1)]…[1+E(Nr1)]

Ch 8-26

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Liquidity Premium Theory

Weaknesses of unbiased expectations theory

Assumes investors are risk-neutral

Doesn’t recognize that forward rates aren’t perfect predictors of future interest rates

Liquidity premium theory

Allows for future uncertainty

Implicitly assumes that investors prefer short-term securities

Ch 8-27

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Market Segmentation Theory

Investors have specific preferences in terms of maturity

Securities with different maturities are not perfect substitutes

Investors are risk averse to securities that do not meet their maturity preferences

Yield curve reflects intersection of demand and supply of individual maturities

Ch 8-28

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Market Segmentation and Determination of Slope of Yield Curve

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Ch 8-29

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Maturity Model Weaknesses

Two major shortcomings

Does not account for the degree of leverage in the FI’s balance sheet

Ignores the timing of the cash flows from the FI’s assets and liabilities

Ch 8-30

Ch 8-30

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Overview

This chapter discusses a market value-based model for managing interest rate risk, the duration gap model

Duration

Computation of duration

Economic interpretation

Immunization using duration

Problems in applying duration

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Ch 9-31

31

Price Sensitivity and Maturity

In general, the longer the term to maturity, the greater the sensitivity to interest rate changes

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

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Ch 9-32

32

Duration

Duration

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

More complete measure of interest rate sensitivity than is maturity

The units of duration are years

To measure and hedge interest rate risk, FI should manage duration gap rather than maturity gap

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Ch 9-33

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Macaulay’s Duration

where

D = Duration measured in years

CFt = Cash flow received at end of period t

N= Last period in which cash flow is received

DFt = Discount factor = 1/(1+R)t

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Ch 9-34

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Duration

Since the price (P) of the bond equals the sum of the present values of all its cash flows, we can state the duration formula another way:

Notice the weights correspond to the relative present values of the cash flows

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Ch 9-35

Semiannual Cash Flows

For semiannual cash flows, Macaulay’s duration, D, is equal to:

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Ch 9-36

Duration of Zero-Coupon Bond

Zero-coupon bonds: sell at a discount from face value on issue, pay the face value upon maturity, and have no intervening cash flows between issue and maturity

Duration equals the bond’s maturity since there are no intervening cash flows between issue and maturity

For all other bonds, duration < maturity because here are intervening cash flows between issue and maturity

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Ch 9-37

37

Duration of Consol Bonds

A bond that pays a fixed coupon each year indefinitely

Have yet to be issued in the U.S.

Maturity of a consol (perpetuity):

Mc = 

Duration of a consol (perpetuity):

Dc = 1 + 1/R

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Ch 9-38

Features of Duration

Duration and maturity

Duration increases with maturity of a fixed-income asset/liability, but at a decreasing rate

Duration and yield

Duration decreases as yield increases

Duration and coupon interest

Duration decreases as coupon increases

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Ch 9-39

Economic Interpretation

Duration is a direct measure of interest rate sensitivity, or elasticity, of an asset or liability:

[ΔP/P]  [ΔR/(1+R)] = -D

Or equivalently,

ΔP/P = -D[ΔR/(1+R)] = -MDdR

where MD is modified duration

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Ch 9-40

Economic Interpretation Continued

To estimate the change in price, we can rewrite this as:

ΔP = -D[ΔR/(1+R)]P = -(MD) × (ΔR) × (P)

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Ch 9-41

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Dollar Duration

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

Dollar duration = MD × Price

Using dollar duration, we can compute the change in price as

ΔP = -Dollar duration × ΔR

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Ch 9-42

Semi-annual Coupon Bonds

With semi-annual coupon payments, the percentage change in price is calculated as:

ΔP/P = -D[ΔR/(1+(R/2)]

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Ch 9-43

Immunization

Matching the maturity of an asset with a future payout responsibility does not necessarily eliminate interest rate risk

Matching the duration of a fixed-interest rate instrument (i.e., loan, mortgage, etc.) to the FI’s target or investment horizon will immunize the FI against shocks to interest rates

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Ch 9-44

Balance Sheet Immunization

Duration gap is a measure of the interest rate risk exposure for an FI

If the durations of liabilities and assets are not matched, then there is a risk that adverse changes in the interest rate will increase the present value of the liabilities more than the present value of assets is increased

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Ch 9-45

Immunizing the Balance Sheet of an FI

Duration Gap:

From the balance sheet, A = L+E, which means E = A-L. Therefore, DE = DA-DL.

In the same manner used to determine the change in bond prices, we can find the change in value of equity using duration.

DE = -[DA - DLk]A(DR/(1+R))

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Ch 9-46

Duration and Immunizing

The formula, DE, shows 3 effects:

Leverage adjusted duration gap

The size of the FI

The size of the interest rate shock

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Ch 9-47

Example 9-9

Suppose DA = 5 years, DL = 3 years and rates are expected to rise from 10% to 11%. (Thus, rates change by 1%). Also, A = 100, L = 90, and E = 10. Find DE.

DE = -[DA - DLk]A(DR/(1+R))

= -[5 - 3(90/100)]100[.01/1.1] = - $2.09.

Methods of immunizing balance sheet.

Adjust DA, DL or k.

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Ch 9-48

Immunization and Regulatory Considerations

Regulators set target ratios for an FI’s capital (net worth) to assets in an effort to monitor solvency and capital positions:

Capital (Net worth) ratio = E/A

If target is to set (E/A) = 0:

DA = DL

But, to set E = 0:

DA = kDL

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Ch 9-49

Difficulties in Applying Duration Model

Duration matching can be costly

Growth of purchased funds, asset securitization, and loan sales markets have lowered costs of balance sheet restructurings

Immunization is a dynamic problem

Trade-off exists between being perfect immunization and transaction costs

Large interest rate changes and convexity

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Ch 9-50

Convexity

The degree of curvature of the price-yield curve around some interest rate level

Convexity is desirable, but greater convexity causes larger errors in the duration-based estimate of price changes

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Ch 9-51

Basics of Bond Valuation

Formula to calculate present value of bond:

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Ch 9-52

Impact of Maturity on Security Values

Price sensitivity is the percentage change in a bond’s present value for a given change in interest rates

Relationship between bond price sensitivity and maturity is not linear

As time remaining to maturity on bond increases, price sensitivity increases at decreasing rate

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Ch 9-53

Incorporating Convexity into the Duration Model

Three characteristics of convexity:

Convexity is desirable

Convexity and duration

All fixed-income securities are convex

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Ch 9-54

Modified Duration & Convexity

DP/P = -D[DR/(1+R)] + (1/2) CX (DR)2, or DP/P = -MD DR + (1/2) CX (DR)2

Where MD implies modified duration and CX is a measure of the curvature effect

CX = Scaling factor × [capital loss from 1bp rise in yield + capital gain from 1bp fall in yield]

Commonly used scaling factor is 108

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Ch 9-55

Calculation of CX

Example: convexity of 8% coupon, 8% yield, six-year maturity Eurobond priced at $1,000

CX = 108[(DP-/P) + (DP+/P)]

= 108[(999.53785-1,000)/1,000 + (1,000.46243-1,000)/1,000)]

= 28

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Ch 9-56

Contingent Claims

Interest rate changes also affect value of (off-balance sheet) derivative instruments

Duration gap hedging strategy must include the effects on off-balance sheet items, such as futures, options, swaps, and caps, as well as other contingent claims

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