Document

Chapter 16

Interest Rate Risk Measurements and

Immunization Using Duration

Copyright © 2000 by Harcourt, Inc.

All rights reserved.

16-1

Interest Rate Risk Defined

The potential variation in returns due to

UNEXPECTED changes in interest rates.

16-2



Fundamental Principles of Financial Asset

Pricing

 A single investor is unable to influence the price of a financial asset.

 The supply and demand for a financial asset also influences price.

 All else equal, the price of a riskier asset will be lower than that of a less risky one because financial market participants are risk averse.

•

Risk aversion causes investors to demand higher expected rates of return for riskier investments.



16-3

Financial Asset Pricing

P

0

 t n 



1





C t 

1

 y







 where: y



= discount rate that makes the sum of the present value of assets’ cash flows (C t

) equal to its price (P

0

)



16-4

 All else equal, prices and yields change simultaneously in the opposite direction.

•

If y

 rises, the price of the security falls.

•

If y

 falls, the price of the security rises.

16-5



Calculating Ex Ante Effective Annual Yields



( y ) for Bonds

 The ex ante effective annual yield ( y



) that an investor expects to receive over the life of the bond is the discount rate that makes the present value of the bond’s future cash flows equal to its current price.

16-6



P

0







Coupon Payment(PV

Maturity

IFA

y



,

Value(PVIF

n

)

y





, n

)



 where:

P

0

= price (market value) of the bond

PVIFA= present value of the annuity factor

[1-(1/(1+ y



) n )/ y



)]

PVIF= present value factor [1/(1+ y



) n ]



16-7

 In order to earn the ex ante effective annual yield, an investor would have to:

• hold the bond until maturity; and

• invest each coupon payment at the y

 rate over the life of the bond.

16-8



Calculate y

 for a bond with 15 years to maturity, a coupon rate of 7.5% and a price of 116% of $1,000 par

16-9 value.

P

0

= 1.160 × $1,000 = $1,160

Coupon Payment = .075 × $1,000 = $75

P

0





Coupon Payment(PV IFA y



, n

)



Maturity Value(PVIF y



, n

)



$1,160 = $75(PVIFA y



, 15) + $1,000(PVIF y



,15) y



= 5.87%

An investor will only receive an annual effective yield of 5.87% if the bond is held to maturity and coupons can be reinvested at a 5.87% rate each year.



Calculating Ex Post (Actual) Effective Annual

Yield

16-10

EAY







FV

P

0





1 n



1 where:

FV = future value of the cash flows received at the end of the life of the bond



16-11

FV





Coupon Payment(FV IFA y



, n

)



Maturity Value

 where:

FVIFA= future value of the annuity factor

[ ((1+ y



) n - 1)/ y



)]



16-12

What is the EAY for a bond with 15 years to maturity, a coupon rate of 7.5% , bought at a price of $1,160 and for which coupons where reinvested at a 5.87% rate?

FV





Coupon Payment(FV IFA y



, n

)



Maturity Value



$75(FVIFA

5.87%,15

) + $1000

$75(23.047) + $1000 = $2,728.50

EAY



 

P

0

1 n



1

($2,728.50/$1,160) (1/15) - 1

(2.35185) 0.0067

- 1 = 0.0587 or 5.87%



If all coupon payments are reinvested at the y

 rate and the bond is held until maturity, the EAY will be equal to the ex ante effective annual rate, y



16-13



16-14

The Two Sides of Interest Rate Risk

 Reinvestment Rate Risk

•

The risk of interest rates falling and having to reinvest coupon payment at a lower rate than y



, resulting in a lower ex-post yield.

 Price or Market Value Risk

•

The risk of rates rising and the market price of the bond falling if the bond must be sold prior to maturity, also resulting in a lower ex-post yield.



Example of Reinvestment Rate Risk

Use the previous example to calculate EAY, assuming rates have dropped to 5% over the life of the bond after the bond was purchased.

16-15

FV of cash flows

$75(FVIFA

5%,15

) + $1000

$75(21.579) + $1000 = $2,618.39

EAY = ($2,618.39/$1,160) (1/15) - 1

= (2.2573) 0.0067

- 1 = 0.0558 or 5.58%

Because of reinvestment rate risk, the investor has a lower ex post yield than the expected y

 of 5.87%.



Example of Market Price Risk

Using the previous example, calculate the EAY for the bond if rates after 5 years rise to 6.5% and remain at that level for the remainder of the life of the bond. Assume that the investor will sell the bond after 5 years.

16-16

Price of the bond in 5 years

$75(PVIFA

6.5,10

) + $1,000(PVIF

6.5,10

$75(7.188) +1000(0.532) = $1,071.89

)

FV of cash flows

$75(FVIFA

5.87,5

) + $1,071.89

$75(5.6225) + $1,071.89 = $1,493.58



16-17

EAY

($1,493.58/$1,160) (1/5) - 1

(1.2876) .2

- 1 = .0519 or 5.19%

Because of market price risk, the annual effective yield received ex post is 5.19% over 5 years, 68 basis points lower than the expected effective yield of

5.87%.



16-18

Bond Theorems

 Theorem I. Bond prices must move inversely to bond yields.

 Theorem II. Holding the coupon rate constant, percentage changes in bond prices are greater the longer term to maturity of a bond.

 Theorem III. The percentage price change for a bond will increase at a decreasing rate as N increases.



16-19

 Theorem IV. (Convexity Theorem)

Starting with a given market yield y



, holding other factors constant, the rise in price with a fall in y

 will be greater than the fall in price with the same absolute value rise in y



.

 Theorem V. Holding N constant and starting from the same y



, the higher the coupon rate, the smaller the percentage change in price for a given change in yield.



Summary

A Bond’s Interest Rate Risk Depends On

16-20

 The current market yield ( y



)

 The coupon rate of the bond

 The maturity of the bond



16-21

Duration Defined

 It is the weighted average time over which the cash flows from an investment are expected.

 Duration is a measure of the effective maturity.

•

For most securities, the cash benefits are received before the maturity date.

 Duration is also a measure of interest rate risk.

•

Securities with higher durations will have a greater change in market value with a change in market rates.



Dur













 t

N 



1 t

N 



1

( 1

C t



 y

 t

( 1

C t

 y



)

)











 where:

C t

= cash flow in time period t t = the time each cash flow comes in y



= current market yield



16-22

Find the duration for the following three bonds:

Bond 1 has an 8% coupon rate, $1000 par, n=6

Bond 2 is a zero-coupon bond with a $1,000 par, n=6

Bond 3 has a 6% coupon rate, $1,000 par, n=6

16-23

Year

BOND 1: 8% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO

MATURITY

Cash Flow PVIF 8% PV Cash Flows t × PV Cash Flow

4

5

6

1

2

3

$80

80

80

80

80

1080

.9259

.8573

.7938

.7350

.6806

.6302

$74.07

68.58

63.50

58.80

54.45

680.62

Sum $1000.00

Duration = $4,992.90/$1,000 = 4.993 years

$74.07

137.16

190.50

235.20

272.25

4,083.72

4,992.90



16-24

BOND 2: ZERO COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO

MATURITY

Year Cash Flow PVIF 8% PV Cash Flows t × PV Cash Flow

3

4

1

2

5

6

$0

0

0

0

0

1000

.9259

.8573

.7938

.7350

.6806

.6302

$0

0

0

0

0

630.20

Sum $630.20

Duration = $3,781.20/$630.20 = 6 years

$0

0

0

0

0

3,781.20

$3,781.20



Year

BOND 3: 6% COUPON RATE, $1,000 MATURITY VALUE, 6 YEARS TO

MATURITY

Cash Flow PVIF 8% PV Cash Flows t × PV Cash Flow

4

5

6

1

2

3

$60

60

60

60

60

1060

.9259

.8573

.7938

.7350

.6806

.6302

$55.55

51.44

47.63

44.10

40.84

668.01

Sum $907.57

Duration = $4,689.97/$907.57 = 5.17 years

$55.55

102.88

142.89

176.40

272.25

4,008.07

4,689.97

Note that duration is a negative function of a bond’s coupon payment. The duration for zero coupon will always be equal to its maturity.

16-25



Alternative Formula for Duration to Use with

Long-term Securities

16-26

Dur



N













Coupon Payment

(P

0

 y



)















N





1

 y





PVIFA y



, N

  where:

PVIFA

 1







1 /( 1

 y



 n

 y





16-27

For Bond 1, with N = 6 years, Coupon

Payment = $80, y



= 8%, P

0

= $1,000 and

PVIFA factor for 8% and 6 years = 4.6229

Dur



N













Coupon Payment

(P

0

 y



)















N





1

 y





PVIFA y



, N

 

Dur = 6 - [($80/($1000×.08)]×[6-(1.08)(4.6229)]

= 6 - (1) × (1.0073) = 4.993 years

The same as was calculated using the regular formula.



Why is duration important for assessing an institution’s interest rate risk?

 Because for a given change in market yields, percentage changes in asset prices are proportional to the assets’ duration.

%



P

0





P

0

P

0

 

Dur





 y



%

( 1

 y



)





16-28 where:

%



P

0

= percentage change in price

%

 y



= percentage change in market yield y



= the original market rate prior to the change



16-29

What is the percentage change in price for Bond

1 with a duration of 4.993, a P

0 y



= $1,000, and

= 8% if rates increase to 8.5%?

%



P

0





P

0

P

0

 

Dur





 y



%

( 1

 y



)





- 4.99[0.50%/(1.08)] = - 2.31%

Multiplying (0.0231 × $1,000), the bond’s price would fall by approximately $23.10.



 Duration assumes a linear relationship between price changes and changes in market yields.

•

Bond price changes follow a convex pattern with changes in market yields.

 Duration will overestimate the percent fall in price with a rise in rates.

 Duration wil1 underestimate the percent rise in price with a fall in rates.

16-30



Estimating Interest Rate Elasticity

Interest rate elasticity is the percentage price change expected for a 1% change in market yields

.

E



Dur



 y





1

 y









16-31



Calculate the interest rate elasticity for Bonds

1, 2, and 3.

16-32

E



Dur



 y





1

 y









Bond 1: E = -4.993(0.08/1.08) = -0.3699

Bond 2: E = -6.000(0.08/1.08) = -0.4444

Bond 3: E = -5.170(0.08/1.08) = -0.3830



 For Bond 1, for every 1% change in the bond’s yield of 8% , the bondholder could have expected a price change of 0.3699%.

 Bonds 2 and 3 have higher elasticity than Bond 1, which implies greater interest rate risk.

 In this case, elasticities describe the same risk ranking as their relative duration, because each bond has the same market yield of 8%.

 If market yields are different, calculating elasticities provides a better comparison of different securities’ relative interest rate risk.


16-33


Modified Duration

Modified Duration (MD)



Dur



1



y





Multiplying MD by any

 y

 gives the approximate change in value of that fixed-rate asset or liability with a change in y



.



16-34

Limitations of Duration-Based Measures of

Interest Rate Risk

16-35

 Duration assumes that there will be a parallel shift in the yield curve so that for a given level of default risk, yields across the entire term structure change equally.

 Duration assumes a flat yield curve because the same y

 discount rate is used for early and late cash flows.

 As market rate (y

 security’s duration.

) changes, so will a



Portfolio Immunization

 It is a duration based strategy that makes a portfolio immune to interest rate risk over a given holding period.

 Used to achieve a realized annual rate of return at the end of the holding period that is no less than the expected annual yield at the beginning of the period.

16-36



Suppose an insurance company issues a $1 million guaranteed investment contract that guarantees an annual effective yield of 8%. The company is considering two alternatives:

1. Bond 1: six-year bonds with an 8% coupon rate at $1,000 par value which sell for $1,000 per bond, with y



= 8%.

16-37

2. Bond 2: five-year bonds with an 8% coupon rate and $1,000 par value, which sell for $1,000 per bond, with y



= 8%.



16-38

Duration of Bond 1 = 4.993

(from earlier calculations)

Duration of Bond 2

Dur



N













Coupon Payment

(P

0

 y



)















N





1

 y





PVIFA y



, N

 

5 - [(80/80)] - [5 - (1.08)(3.9927)]

5 - 0.68788 = 4.312



16-39

ILLUSTRATION OF DURATION AND PORTFOLIO IMMUNIZATION .

STRATEGY 1: DURATION MATCH; BUY BOND 1 WITH APPROXIMATELY A 5-

YEAR DUATION

Coupon payments reinvested at y

 and bond sold at end of 5 years.

Scenario (1) Market interest rates stay at y



= 8% for 5 years

Ex post effective annual yield for Bond 1:

FV of cash flows per bond at the end of year 5

80(FVIFA 5,8%) + (1080/1.08) = $1,469.33

Ex post effective annual rate is (1469.33/1000) 0.20

- 1 = .

08 or 8%




= 9% for 5 years



80(FVIFA 5,9%) + (1080/1.09) = $1,469.61


- 1




= .08 or 8%

= 7% for 5 years



80(FVIFA 5,7%) + (1080/1.07) = $1,469.41


- 1 = .08 or 8%



STRATEGY 2: MATURITY MATCH; BUY BOND 2 WITH A MATURITY OF 5

YEARS

Coupon payments reinvested at y

 and bond sold at end of 5 years.




= 8% for 5 years



80(FVIFA 5,8%) + 1000 = $1,469.33


- 1 = .

08 or 8%




= 9% for 5 years



80(FVIFA 5,9%) + 1000 = $1,478.78


- 1




= .0814 or 8.14%

= 7% for 5 years



80(FVIFA 5,7%) + 1000 = $1,460.06


- 1 = .0.786 or 7.86%

16-40



 In this example, holding bonds with a duration of 5 years will lock in an annual return of 8% over the holding period.

 Reasoning: With immunization, the expected return on a portfolio is protected from both reinvestment and market value risk.

•

Reinvestment risk exactly offsets market value risk.

16-41



16-42

Immunization Assumptions and Limitations

 Assumes that there is only one parallel shift in the yield curve immediately after the beginning of the period.

•

Market shifts may occur in the middle of the investment period and an investor may be less than perfectly immunized.

 It is difficult to find an investment with a duration exceeding 10 years.



16-43

Differences in Funding Gap and Duration

 A financial institution’s funding gap serves as a measure of reinvestment risk for a given time period.

 Funding gap ignores:

• time value of money; and

• the effects of changes in the market value of assets and liabilities on a financial institution’s balance sheet .

 Duration gap provides a better overall measure of interest rate risk including market price as well as reinvestment risk.



Duration Gap

DGAP



DurA -





DurL





Liabilitie s

Assets

 

16-44 where:

DurA = duration of assets which is the weighted average of the duration of assets on the balance sheet

DurL = duration of liabilities which is the weighted average of the duration of liabilities on the balance sheet



16-45

DURATION GAP AND CHANGES IN THE MARKET

VALUE OF EQUITY

 in Market Value

Position

 y



Assets Liabilities Equity

DGAP > 0 Increase Decrease > Decrease Fall

Decrease Increase > Increase Rise

DGAP < 0 Increase Decrease < Decrease

Decrease Increase < Increase

Rise

Fall

DGAP = 0 Increase Decrease = Decrease Zero

Decrease Decrease = Decrease Zero



Change in the Value of a Depository

Institution’s Equity-to-Asset Ratio

16-46



Value (Equity/To tal Assets)

 

DGAP[(

 y



/( 1

 y



)] where: y



= average rate on assets



16-47

Calculating Excelsior Bank’s DGAP

Assets

Cash

6-month T-bills y



Dur = 0 $10,000

= 6% Dur = .5 years $40,000

10 year Fixed-Rate Loan Dur = ?

$50,000

Total Assets $100,000

Liabilities and Equity

Transactions Deposit

(rate =2%)

1-year CD, rate = 7%

Dur = 0

Dur = 1

Equity

Total Liabilities and Equity

$40,000

$50,000

$10,000

$100,000



16-48

Duration of a Loan

Dur



N













Coupon Payment

(Loan Amount

 y



)















N





1

 y





PVIFA y



, N

 

Coupon Payment (Loan Payment)

Loan Amount/(PVIFA 10%, 10) = $50,000/6.1446 =

$8,137.27 per year

Dur = 10 - [($8,137.27/($50,000×.10)] ×

[10 - (1.10)(6.1446)] = 10 - (1.6275)(3.241) =

4.73 years



16-49

Duration of Assets

($10,000/$100,000)0 + ($40,000/$100,000)0.5 +

($50,000/$100,000)4.73 = 2.565 years

Duration of Liabilities

($40,000/$90,000)0 + ($50,000/$90,000)1 = 0.555 years

DGAP



DurA -





DurL





Liabilitie s

Assets

 

2.565 yrs - ($90,000/$100,000)0.5556 yrs = 2.065 years



16-50

Change in the Value of Equity from a 1% Rise in Rates



Value (Equity/To tal Assets)

 

DGAP[(

 y



/( 1

 y



)] y



= ($10,000/$100,000)0 + ($40,000/$100,000)0.06

+ ($50,000/$100,000)0.10 = 0.074 or 7.4%

- 2.065[0.01/(1.074)] = 0.0192 or -1.92%



Document

•

Coupon Payment(PV

Maturity

IFA

Value(PVIF

)

)

Modified Duration (MD)

Dur

1

y

DGAP

DurA -

DurL

Liabilitie s

Assets

DGAP

DurA -

DurL

Liabilitie s

Assets

Related documents

Products

Support

Document

•

Coupon Payment(PV

Maturity

IFA

Value(PVIF

)

)

Modified Duration (MD)

Dur

1

y

DGAP

DurA -

DurL

Liabilitie s

Assets

DGAP

DurA -

DurL

Liabilitie s

Assets

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