drake drake university fin 129 interest rate risk finance 129

DrakeDRAKE UNIVERSITY

Fin 129

Interest Rate Risk

Finance 129

DrakeDrake University

Fin 129

Review of Key Factors Impacting

Interest Rate Volatility

Federal Reserve and Monetary PolicyDiscount Window

Reserve Requirements

Open Market Operations


Fin 129

Review of Key Factors Impacting

Interest Rate Volatility

Fisher model of the Savings MarketTwo main participants: Households and BusinessHouseholds supply excess funds to Businesses who are short of fundsThe Saving or supply of funds is upward slopingThe investment or demand for funds is downward sloping


Fin 129

Saving and Investment Decisions

Saving DecisionMarginal Rate of Time PreferenceTrading current consumption for future consumptionExpected InflationIncome and wealth effectsGenerally higher income – save moreFederal Government Money supply decisionsBusinessShort term temporary excess cash.Foreign Investment


Fin 129Borrowing Decisions

Borrowing DecisionMarginal Productivity of CapitalExpected InflationOther


Fin 129Equilibrium in the Market

Original Equilibrium

SS

S

Decrease in Income

Increase in Marg. Prod Cap

D

D

D

Increase in Inflation Exp.

S

D


Fin 129Loanable Funds Theory

Expands suppliers and borrowers of funds to include business, government, foreign participants and households.Interest rates are determined by the demand for funds (borrowing) and the supply of funds (savings).Very similar to Fisher in the determination of interest rates,


Fin 129Loanable Funds

Now equilibrium extends through all markets – money markets, bonds markets and investment market. Inflation expectations can also influence the supply of funds.


Fin 129Liquidity Preference Theory

Liquidity PreferenceTwo assets, money and financial assetsEquilibrium in one implies equilibrium in otherSupply of Money is controlled by Central Bank and is not related to level of interest rates


Fin 129The Yield Curve

Three things are observed empirically concerning the yield curve:Rates across different maturities move togetherMore likely to slope upwards when short term rates are historically low, sometimes slope downward when short term rates are historically highThe yield curve usually slope upward


Fin 129

Three Explanations of the Yield Curve

The Expectations TheoriesSegmented Markets TheoryPreferred Habitat Theory


Fin 129Pure Expectations Theory

Long term rates are a representation of the short term interest rates investors expect to receive in the future. In other words the forward rates reflect the future expected rate.Assumes that bonds of different maturities are

In other words, the expected return from holding a one year bond today and a one year bond next year is the same as buying a two year bond today. (the same process that is used to calculate forward rates)


Fin 129

Pure Expectations Theory: A Simplified Illustration

LetRt = today’s time t interest rate on a

one period bondRe

t+1 = expected interest rate on a one period bond in the next period

R2t = today’s (time t) yearly interest rate on a two period bond.


Fin 129

Investing in successive one period bonds

If the strategy of buying the one period bond in two consecutive years is followed the return is:

(1+Rt)(1+Ret+1) – 1 which equals

Rt+Ret+1+ (Rt)(R

et+1)

Since (Rt)(Ret+1) will be very small we will

ignore it


Fin 129The 2 Period Return

If the strategy of investing in the two period bond is followed the return is:

(1+R2t)(1+R2t) - 1 = 1+2R2t+(R2t)2 - 1

(R2t)2 is small enough it can be dropped

which leaves


Fin 129

Set the two equal to each other

2R2t = Rt+Ret+1

R2t = (Rt+Ret+1)/2

In other words, the two period interest rate is the average of the two one

period rates


Fin 129

Expectations Hypothesis R2t = (Rt+Re

t+1)/2

When the yield curve is upward sloping (R2t>R1t) it is expected that short term rates will be increasing (the average future short term rate is above the current short term rate). Likewise when the yield curve is downward sloping the average of the future short term rates is below the current rate. (Fact 2)As short term rates increase the long term rate will also increase and a decrease in short term rates will decrease long term rates. (Fact 1)This however does not explain Fact 3 that the yield curve usually slopes up.


Fin 129

Problems with Pure Expectations

The pure expectations theory ignores the fact that there is reinvestment rate risk and different price risk for the two maturities.Consider an investor considering a 5 year horizon with three alternatives:

buying a bond with a 5 year maturitybuying a bond with a 10 year maturity and holding it 5 yearsbuying a bond with a 20 year maturity and holding it 5 years.


Fin 129Price Risk

The return on the bond with a 5 year maturity is known with certainty the other two are not.


Fin 129Reinvestment rate risk

Now assume the investor is considering a short term investment then reinvesting for the remainder of the five years or investing for five years.

Cox, Ingersoll, and Ross 1981 Journal of Finance


Fin 129Local Expectations

Similarly owning the bond with each of the longer maturities should also produce the same 6 month return of 2%.The key to this is the assumption that the forward rates hold. It has been shown that this interpretation is the only one that can be sustained in equilibrium.*


Fin 129

Return to maturity expectations hypothesis

This theory claims that the return achieved by buying short term and rolling over to a longer horizon will match the zero coupon return on the longer horizon bond. This eliminates the reinvestment risk.


Fin 129

Expectations Theory and Forward Rates

The forward rate represents a “break even” rate since it the rate that would make you indifferent between two different maturitiesThe pure expectations theory and its variations are based on the idea that the forward rate represents the market expectations of the future level of interest rates.However the forward rate does a poor job of predicting the actual future level of interest rates.


Fin 129Segmented Markets Theory

Interest Rates for each maturity are determined by the supply and demand for bonds at each maturity.Different maturity bonds are not perfect substitutes for each other.Implies that investors are not willing to accept a premium to switch from their market to a different maturity.


Fin 129

Biased Expectations Theories

Both Liquidity Preference Theory and Preferred Habitat Theory include the belief that there is an expectations component to the yield curve.Both theories also state that there is a risk premium which causes there to be a difference in the short term and long term rates. (in other words a bias that changes the expectations result)


Fin 129Liquidity Preference Theory

This explanation claims that the since there is a price risk and liquidity risk associated with the long term bonds, investor must be offered a premium to invest in long term bonds Therefore, the long term rate reflects both an expectations component and a risk premium.This tends to imply that the yield curve will be upward sloping as long as the premium is large enough to outweigh a possible expected decrease.


Fin 129Preferred Habitat Theory

Like the liquidity theory this idea assumes that there is an expectations component and a risk premium.In other words the bonds are substitutes, but savers might have a preference for one maturity over another (they are not perfect substitutes).However the premium associated with long term rates does not need to be positive. If there are demand and supply imbalances then investors might be willing to switch to a different maturity if the premium produces enough benefit.


Fin 129

Preferred Habitat Theoryand The 3 Empirical

Observations

Thus according to Preferred Habitat theory a rise in short term rates still causes a rise in the average of the future short term rates. This occurs because of the expectations component of the theory.



The explanation of Fact 2 from the expectations hypothesis still works. In the case of a downward sloping yield curve, the term premium (interest rate risk) must not be large enough to compensate for the currently high short term rates (Current high inflation with an expectation of a decrease in inflation). Since the demand for the short term bonds will increase, the yield on them should fall in the future.



Fact three is explained since it will be unusual for the term premium to be so small or negative, therefore the the yield curve usually slopes up.


Fin 129Yield Curves Previous Month

0.028

0.033

0.038

0.043

0.048

0.053

0.00 5.00 10.00 15.00 20.00 25.00 30.00

Maturity (Years)

Yie

ld

8/8/2007 8/15/2007 8/22/2007

8/29/2007 9/5/2007 9/12/2007


Fin 129

Yield Curves Previous 6 Months

0.032

0.037

0.042

0.047

0.052

0.00 5.00 10.00 15.00 20.00 25.00 30.00

Maturity (Years)

Yie

ld

6/15/2007 5/15/2007 6/15/2007

7/16/2007 8/15/2007 9/12/2007


Fin 129

Yield Curves Previous 6 quarters

0.03

0.035

0.04

0.045

0.05

0.055

0.00 5.00 10.00 15.00 20.00

Maturity (Years)

Yie

ld

6/15/2006 9/15/2006 12/15/20063/15/2007 6/15/2007 9/12/2007


Fin 129

US Treasury Yields Jan1989 -June 2006

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

12/8/1989 9/3/1992 5/31/1995 2/24/1998 11/20/2000 8/17/2003 5/13/2006

1-mo 3-mo 6-mo 1-yr2-yr 3-yr 5-yr 7-yr10-yr 20-yr 30-yr


Fin 129

US Treas Rates May 1990 – Sept 2007

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

5/7/1990 1/31/1993 10/28/1995 7/24/1998 4/19/2001 1/14/2004 10/10/2006

3-mo 6-mo 10-yr 20-yr


Fin 129

Impact of Interest Rate Volatility on Financial

Institutions

The market value of assets and liabilities is tied to the level of interest ratesInterest income and expense are both tied to the level of interest rates


Fin 129

Static GAP Analysis(The repricing model)

Repricing GAPThe difference between the value of interest sensitive assets and interest sensitive liabilities of a given maturity. Measures the amount of rate sensitive (asset or liability will be repriced to reflect changes in interest rates) assets and liabilities for a given time frame.


Fin 129Commercial Banks & GAP

Commercial banks are required to report quarterly the repricing Gaps for the following time frames

One dayMore than one day less than 3 monthsMore than 3 months, less than 6 monthsMore than 6 months, less than 12 monthsMore than 12 months, less than 5 yearsMore than five years


Fin 129GAP Analysis

Static GAP-- Goal is to manage interest rate income in the short run (over a given period of time)

Measuring Interest rate risk – calculating GAP over a broad range of time intervals provides a better measure of long term interest rate risk.


Fin 129Interest Sensitive GAP

Given the Gap it is easy to investigate the change in the net interest income of the financial institution.


Fin 129Example

Over next 6 Months:Rate Sensitive Liabilities = $120 million

Rate Sensitive Assets = $100 Million

If rate are expected to decline by 1%

Change in net interest income


Fin 129GAP Analysis

Asset sensitive GAP (Positive GAP)RSA – RSL > 0If interest rates NII will If interest rates NII will

Liability sensitive GAP (Negative GAP)RSA – RSL < 0If interest rates NII will If interest rates NII will

Would you expect a commercial bank to be asset or liability sensitive for 6 mos? 5 years?


Fin 129Important things to note:

Assuming book value accounting is used -- only the income statement is impacted, the book value on the balance sheet remains the same.

The GAP varies based on the bucket or time frame calculated.

It assumes that all rates move together.


Fin 129Steps in Calculating GAP

1) Select time Interval

2) Develop Interest Rate Forecast

3) Group Assets and Liabilities by the time interval (according to first repricing)

4) Forecast the change in net interest income.


Fin 129Alternative measures of GAP

Cumulative GAPTotals the GAP over a range of of possible maturities (all maturities less than one year for example).Total GAP including all maturities


Fin 129

Other useful measures using GAP

Relative Interest sensitivity GAP (GAP ratio)

GAP / Bank SizeThe higher the number the higher the risk that is present

Interest Sensitivity Ratio


Fin 129What is “Rate Sensitive”

Any Asset or Liability that matures during the time frameAny principal payment on a loan is rate sensitive if it is to be recorded during the time periodAssets or liabilities linked to an indexInterest rates applied to outstanding principal changes during the interval


Fin 129What about Core Deposits?

Against InclusionDemand deposits pay zero interestNOW accounts etc do pay interest, but the rates paid are sticky

For InclusionImplicit costsIf rates increase, demand deposits decrease as individuals move funds to higher paying accounts (high opportunity cost of holding funds)


Fin 129

Expectations of Rate changes

If you expect rates to increase would you want GAP to be positive or negative?


Fin 129

Unequal changes in interest rates

So far we have assumed that the change the level of interest rates will be the same for both assets and liabilities.If it isn’t you need to calculate GAP using the respective change.Spread effect – The spread between assets and liabilities may change as rates rise or decrease


Fin 129Strengths of GAP

Easy to understand and calculate

Allows you to identify specific balance sheet items that are responsible for risk

Provides analysis based on different time frames.


Fin 129Weaknesses of Static GAP

Market Value EffectsBasic repricing model the changes in market value. The PV of the future cash flows should change as the level of interest rates change. (ignores TVM)

Over aggregationRepricing may occur at different times within the bucket (assets may be early and liabilities late within the time frame)Many large banks look at daily buckets.


Fin 129Weaknesses of Static GAP

RunoffsPeriodic payment of principal and interest that can be reinvested and is itself rate sensitive.You can include runoff in your measure of rate sensitive assets and rate sensitive liabilities.Note: the amount of runoffs may be sensitive to rate changes also (prepayments on mortgages for example)


Fin 129Weaknesses of GAP

Off Balance Sheet ActivitiesBasic GAP ignores changes in off balance sheet activities that may also be sensitive to changes in the level of interest rates.

Ignores changes in the level of demand deposits


Fin 129Other Factors Impacting NII

Changes in Portfolio CompositionAn aggressive position is to change the portfolio in an attempt to take advantage of expected changes in the level of interest rates. (if rates are have positive GAP, if rates are have negative GAP)Problem:


Fin 129Other Factors Impacting NII

Changes in VolumeBank may change in size so can GAP along with it.

Changes in the relationship between ST and LT

We have assumes parallel shifts in the yield curve. The relationship between ST and LT may change (especially important for cumulative GAP)


Fin 129Extending Basic GAP

You can repeat the basic GAP analysis and account for some of the problemsInclude

Forecasts of when embedded options will be exercised and include them


Fin 129The Maturity Model

In this model the impact of a change in interest rates on the market value of the asset or liability is taken into account.The securities are marked to marketKeep in Mind the following:

The longer the maturity of a security the larger the impact of a change in interest ratesAn increase in rates generally leads to a fall in the value of the securityThe decrease in value of long term securities increases at a diminishing rate for a given increase in rates


Fin 129Weighted Average Maturity

You can calculate the weighted average maturity of a portfolio. The same three principles of the change in the value of the portfolio (from last slide) will apply

ininiiiii MWMWMWM 2211


Fin 129Maturity GAP

Given the weighted average maturity of the assets and liabilities you can calculate the maturity GAP


Fin 129Maturity Gap Analysis

If Mgap is + the maturity of the FI assets is longer than the maturity of its liabilities. (generally the case with depository institutions due to their long term fixed assets such as mortgages).This also implies that its assets are more rate sensitive than its liabilities since the longer maturity indicates a larger price change.


Fin 129

The Balance Sheet and MGap

The basic balance sheet identity state that:Asset = Liabilities + Owners Equity orOwners Equity = Assets - Liabilities

Technically if Liab >Assets the institution is insolvent

If MGAP is positive and interest rate decrease then the market value of assets increases more than liabilities.Likewise, if MGAP is negative an increase in interest rates would cause


Fin 129Matching Maturity

By matching maturity of assets and liabilities owners can be immunized form the impact of interest rate changes.However this does not always completely eliminate interest rate risk. Think about duration and funding sources (does the timing of the cash flows match?).


Fin 129Duration

Duration: Weighted maturity of the cash flows (either liability or asset)Weight is a combination of timing and magnitude of the cash flowsThe higher the duration the more sensitive a cash flow stream is to a change in the interest rate.


Fin 129Duration Mathematics

Taking the first derivative of the bond value equation with respect to the yield will produce the approximate price change for a small change in yield.



1n1n432 r)(1

(-n)MV

r)(1

(-n)CP

r)(1

(-3)CP

r)(1

(-2)CP

r)(1

(-1)CP

r

P

nn32 r)(1

MV

r)(1

CP

r)(1

CP

r)(1

CP

r)(1

CPP

nn32 r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

1

r

P

The approximate price change for a small change in r



nn32 r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

1

r

P

To find the % price change divide both sides by the originalPrice

PPr

P 1

r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

11nn32

The RHS is referred to as the Modified DurationWhich is the % change in price for a small change in yield


Fin 129

Duration MathematicsMacaulay Duration

Macaulay Duration is the price elasticity of the bond (the % change in price for a percentage change in yield).Formally this would be:

P

r)(1

price Original

yield Original

Yieldin Change

Pricein Change

yield originalyieldin change

price originalpricein change

DMAC

r

P


Fin 129

Duration MathematicsMacaulay Duration

P

r)(1

price Original

yield Original

Yieldin Change

Pricein Change

yield originalyieldin change

price originalpricein change

DMAC

r

P

nn32 r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

1

r

P

substitutesubstitute

P

r)(1

r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

1nn32

MACD


Fin 129

Macaulay Duration of a bond

N

1tNt

N

1tNt

r)(1MV

r)(1CP

r)(1N(MV)

r)(1t(CP)

MACD

P

1

r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CPnn32

MACD


Fin 129Duration Example

10% 30 year coupon bond, current rates =12%, semi annual payments

periods 3895.17

)06.1(1000$

)06.1(50

)06.1()1000($60

)06.1()50($

60

160

60

160

tt

tt

MAC

t

D


Fin 129Example continued

Since the bond makes semi annual coupon payments, the duration of 17.3895 periods must be divided by 2 to find the number of years.17.3895 / 2 = 8.69475 yearsThis interpretation of duration indicates the average time taken by the bond, on a discounted basis, to pay back the original investment.


Fin 129

Using Duration to estimate price changes

P

r)(1DMAC

r

P

r)(1

rDMAC

P

PRearrange

% Change in Price

Estimate the % price change for a 1 basis point increase in yield

000776.012.1

0001.69925.8

r)(1

rDMAC

P

P

The estimated price change is then

-0.000776(838.8357)=-0.6515


Fin 129Using Duration Continued

Using our 10% semiannual coupon bond, with 30 years to maturity and YTM = 12%Original Price of the bond = 838.3857If YTM = 12.01% the price is 837.6985

This implies a price change of -0.6871Our duration estimate was -0.6515


Fin 129Modified Duration

PPr

P 1

r)(1

nMV

r)(1

nCP

r)(1

3CP

r)(1

2CP

r)(1

1CP

r1

11nn32

From before, modified duration was defined as

Macaulay Duration

r)(1

DurationMacaulay

Duration

Modified


Fin 129Modified Duration

000776.012.1

0001.69925.8

r)(1

rDMAC

P

P

r)(1

DurationMacaulay

Duration

Modified

Using Macaulay Duration

000776.0)0001(.12.1

69925.8

rDrr)(1

D

r)(1

rD MODIFIED

MACMAC

P

P


Fin 129Duration

Keeping other factors constant the duration of a bond will:Increase with the maturity of the bondDecrease with the coupon rate of the bondWill decrease if the interest rate is floating making the bond less sensitive to interest rate changesDecrease if the bond is callable, as interest rates decrease (increasing the likelihood of call) duration increases


Fin 129Duration and Convexity

Using duration to estimate the price change implies that the change in price is the same size regardless of whether the price increased or decreased.The price yield relationship shows that this is not true.


Fin 129Duration and Convexity

0

500

1000

1500

2000

2500

3000

0 0.05 0.1 0.15 0.2

Interest Rate

Bo

nd

Val

ue


Fin 129Basic Duration Gap

Duration Gap


Fin 129Basic DGAP Conintued

iasset ofDuration Macaulay DaAssets All of ValueMarket

Asset wwhere

DawDAPortfolioAsset of

Duration Weighted$

i

ii

i

N

1ii

jLiability ofDuration Macaulay DlsLiabilitie All of ValueMarket

Asset wwhere

DlwDLPortfolioLiability of

Duration Weighted$

j

jj

j

N

1jj


Fin 129Basic DGAP

If the Basic DGAP is +If Rates in the value of assets > in value of liab

If Rate in the value of assets > in value of liab


Fin 129Basic DGAP

If the Basic DGAP is (-)If Rates in the value of assets < in value of liab

If Rate in the value of assets < in value of liab


Fin 129Basic DGAP

Does that imply that if DA = DL the financial institution has hedged its interest rte risk?

No, because the $ amount of assets > $ amount of liabilities otherwise the institution would be insolvent.


Fin 129DGAP

Let MVL = market value of liabilities and MVA = market value of assetsThen to immunize the balance sheet we can use the following identity:


Fin 129DGAP and equity

Let MVE = MVA – MVLWe can find MVA & MVL using duration

From our definition of duration:

formula theApplying Pi)(1

ΔiDΔP


Fin 129

MVAy1

Δy-DGAPΔMVE

MVAy1

Δy

MVA

MVL(DL)- (DA)-

y1

Δy(DL)MVL-(DA)MVA -

MVLy1

ΔyDL--MVA

y1

Δy-DA

ΔMVL-ΔMVA ΔMVE


Fin 129DGAP Analysis

If DGAP is (+)An in rates will cause MVE to An in rates will cause MVE to

If DGAP is (-)An in rates will cause MVE to An in rates will cause MVE to

The closer DGAP is to zero the smaller the potential change in the market value of equity.


Fin 129Weaknesses of DGAP

It is difficult to calculate duration accurately (especially accounting for options)Each CF needs to be discounted at a distinct rate can use the forward rates from treasury spot curveMust continually monitor and adjust durationIt is difficult to measure duration for non interest earning assets.


Fin 129More General Problems

Interest rate forecasts are often wrongTo be effective management must beat the ability of the market to forecast rates

Varying GAP and DGAP can come at the expense of yield

Offer a range of products, customers may not prefer the ones that help GAP or DGAP –


Fin 129Duration in Practice

Impact of convexityShape of the yield curveDefault RiskFloating Rate InstrumentsDemand DepositsMortgagesOff Balance Sheet items


Fin 129Convexity Revisited

The more convexity the asset or portfolio has, the more protection against rate increases and the greater the possible gain for interest rate falls.The greater the convexity the greater the error possible if simple duration is calculated.All fixed income securities have convexityThe larger the change in rates, the larger the impact of convexity


Fin 129Flat Term Structure

Our definition of duration assumes a flat term structure and that the all shirts in the yield curve are parallel. Discounting using the spot yield curve will provide a slightly different measure of inflation.


Fin 129Default Risk

Our measures assume that the risk of default is zero. Duration can be recalculated by replacing each cash flow by the expected cash flow which includes the probability that the cash flow will be received.


Fin 129Floating Rates

If an asset or liability carries a floating interest rate it readjusts its payments so the future cash flows are not known.Duration is generally viewed as being the time until the next resetting of the interest rate.


Fin 129Demand Deposits

Deposits have an open ended maturity. You need to define the maturity to define duration. Method 1

Look at turnover of deposits (or run). If deposits turn over 5 times a year then they have an average maturity of 73 days (365/5).

Method 2Think of them as a puttable bond with a duration of 0

Method 3Look at the % change in demand deposits for a given level of interest rate changes.

Simulation


Fin 129Mortgages

Mortgages and mortgage backed securities have prepayment risk associated with them. Therefore we need to model the prepayment behavior of the mortgage to understand the cash flow.


Fin 129Off Balance Sheet Items

The value of derivative products also are impacted by duration changes. They should be included in any portfolio duration estimate or GAP analysis.

drake drake university fin 129 interest rate risk finance 129

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