Fixed Income Continued

7/31/2019 Fixed Income Continued

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Portfolio Wealth Management: Roleof Fixed Income Securities

Suman BanerjeeMarch 2009


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Life Cycle Hypothesis (LCH)

Young people and retirees are unable to save very much.

Most saving is done by middle-aged people.

Income follows a humped-shaped pattern over an

individual's lifetime. There would be no problem is spending patterns followed

the same path with low consumption needs at beginningand end of life and higher needs during peak earningperiods.

However, individuals prefer to spend at a steadier rateover their lifetimes.


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Diagram: LCH


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Example 1: Bonds Total Return

Suppose an investor is considering purchasing a 7-year semi-annual coupon bond selling at par and having a coupon rate of 9%.

Investor assumes that

The coupon re-investment rate is 9.4%, and

He is going to hold the bond for 5 years (his investment horizon).

Step 1: Total future earnings from this bond includes

1. Coupon payment of $45 every 6 month until the end of investmenthorizon,

2. Interest earned on re-investing the semi-annual coupon paymentsat 4.7% (half of the assumed annual reinvestment rate) until theend of investment horizon, and

3. Expected price of the bond at the end of investment horizon.


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Price at the End of Investment Horizon

Investor assumes that

The market interest rate for similar risk bonds at the end of hisinvestment horizon is 11.2%.

What is the price of the bond then?

Price = PV of coupons + PV of maturity value

=

Remember, it is a 7-year 9% coupon bond!

4

4

11

1 .056 100045 $157.34 $804.16 $961.50

.056 1 .056


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Example continued Step 2: Calculate the future value of coupon interest plus interest on coupon

interest as follows:

Step 3: Compute the total dollar return

Step 4: Compute the following

Step 5: Doubling 4.27% gives the total return of 8.54%Total return on effective rate basis: (1.0427)2 -1 =8.72%.

10(1.047) 1

$45 $45 12.4032 $558.140.047

Coupon interest and interest on coupon interest $ 558.14

Sell price after 5 years (assuming 11.2% required return) $961.50

Total dollar return $1,519.64

1

10$1,519.64 1 0.0427 4.27%$1000


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Example 2 : Total Return

Suppose that an investor has 6 years investment horizon. Theinvestor is considering a 13-year semi-annual coupon bond sellingat par and having a coupon rate of 9%.

The investor expectations are as follows:

The first 4 semi-annual coupon payments can be reinvested from the

time of receipt to the end of the investment horizon at an annualinterest rate of 8%,

The last 8 semi-annual coupon payments can be reinvested from thetime of receipt to the end of the investment horizon at a 10% annualinterest rate, and

The required market interest rate on 7-year bonds at the end of theinvestment horizon is 10.6%.

What is the total return?


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Example 2: We just figure out.

Step 1: Total future earnings from this bondincludes

1. Coupon interest of $45 every six months for 6 years (theinvestment horizon),

2.Interest earned from reinvesting the first 4 semi-annual couponinterest payments at 4% (one-half the assumed annualreinvestment rate) until the end of the investment horizon,

3. Interest earned from reinvesting the last 8 semi-annual couponinterest payments at 5% (one-half the assumed annualreinvestment rate) until the end of the investment horizon, and

4. Expected price of the bond at the end of investment horizon.


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Example 2 continued

Step 2: Calculate the future value of coupon interest plus interest on

coupon interest as follows:

This gives the coupon interest plus interest on interest as of the end ofthe second year (four periods). Reinvesting at 4% until the end of the

investment horizon, 4 years or 8 six-month periods later, $191.09 willgrow to

The coupon interest plus interest on interest for the last eight coupon

payments can be found as follows:

4(1.04) 1

$45 $45 4.2465 $191.090.04

8$191.09 1.04 $261.52

8

(1.05) 1$45 $45 9.5491 $429.71

0.05


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Example 2 continued

Step 2 (continued): The coupon interest and interest on interest from

all 12 coupon interest payments is$691.23 (= $261.52 + $429.71).

Step 3: Compute the expected price of the bond 6 years from today:

Step 4: Compute the total dollar return

14

14

11

1 .053 1000$45 $437.02 $485.29 $922.31

.053 1 .053

Coupon interest and interest on coupon interest $ 691.23Sell price after 5 years (assuming 11.2% required return) $922.31

Total dollar return $1,613.54


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Example 2 continued


Step 5: Doubling 4.07% gives the total return of8.14% on bond equivalent basis.

1

121,613.541 0.0407 4.07%

1000


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Analyzing a Callable Bond

An investor is considering bond C: 11% semi-annual coupon, 15years to maturity and the price for this bond is $1,144.88. Then,YTM is 9.2% (Please check!).

Suppose that this bond is callable in 3 years at $1,055 (this iscalled the first call date).

What is the RTC (or YTC) of this bond ? RTC = 7.22%. However, suppose that this investor's investment horizon is 5

years (a period extending beyond the first call date, butshorter than the maturity of the bond).

Also assume that this investor believes that any proceeds can

be reinvested at a 6% annual interest rate.

Compute the total return of the bond on the assumption that thebond is called in three years.


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Cash flow of a Callable Bond

Time 0 1 3 4 5 6 ... ... 29 30

If the

bondmatures

$55 $55 $55 $55 $55 $55 ... ... $55 $55

$1000

If thebond iscalled

$55 $55 $55 $55 $55 $55

$1055


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Callable Bonds: Return to Call

How do you calculate the RTC?

Determine the payoff, if the bond is called on the first call date:Price (or your investment): $1144.88Coupon interest received: 6 payments of $55 eachCall price (you receive): $1055

RTC is that interest rate that solves the following equation

Using the calculator: PV = -1144.88; CP (or FV) = 1055; N = 6; andPMT = 55, compute I/Y = 3.6087% semi-annually => 7.22% annually.

66 )1(

1055

)1(

11

5588.1144

)1()1(

11

RT CRTCRT C

RT C

CP

RT CRTC

cP

mm


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Callable Bonds: Total Return

Step 1: Total future earnings from this bond includes

1. Coupon interest of $55 every six months for 3 years (1st call date),

2. Interest earned from reinvesting the first 6 semi-annual coupon interestpayments at 3% (one-half the assumed annual reinvestment rate of 6%)until the 1st call date, and

3. Proceeds from reinvesting the call price plus (1) and (2) above for the twoyears (end of the investment horizon) at 3% interest rate.

The coupon interest plus interest on interest for the last six couponpayments can be found as follows:

76.355$03.

1)03.1(556


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Callable Bonds: Total Return

Step 2 (continued): This gives the coupon interest plus interest on couponinterest as of the end of the year 3 (six periods) when the bond is called. Addingthe call price of $1,055 gives the total proceeds that must be reinvested at 3%until the end of the investment horizon, for two years or four periods.

Proceeds to be reinvested = $355.76 + $1,055 = $1,410.76.

Reinvesting $1,410.76 for four periods at 3%:


Doubling 3.335% gives the total return of 6.67% in bond equivalent basis.

82.1587$)03.1(76.1410 4

%335.31

88.1144

82.1587 101


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Comparing Coupon and Zero-Coupon Bonds:Break-even Reinvestment Rate

As I have emphasized throughout my lectures, assessing the relativevalue of any two coupon bonds on the basis of YTM may providemisleading conclusions.

A manager of a tax-exempt portfolio who intends to hold a bond until

maturity has the advantage of having an analytical framework necessaryto determine the relative value of a zero-coupon bond and a couponbond.

Suppose that the manager of a tax-exempt portfolio is considering two 5-year bonds: (1) a 9% semi-annual coupon bond selling at par ($1000),and (2) a zero-coupon bond selling at $600. (per $1000 par value and

semi-annual compounding).

Suppose also that the bonds have the same credit quality rating and that theportfolio manager plans to hold either bond until maturity.


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Break-even Reinvestment Rate

What is the total return of the zero coupon bond?

Doubling 5.24% gives the total return of 10.48% in bond equivalent basis.

Note: a zero-coupon bond has no coupon reinvestment risk.

Hence,YTM = Total Return, given that the bond is held until maturity.

What is the total return of the coupon bond?

It depends on the coupon re-investment rate assumed by the portfoliomanager.

Can we determine the coupon reinvestment rate that will equalize the totalreturn of the 9% coupon bond and the zero-coupon bond?

If yes, this reinvestment rate called the Break-even reinvestmentrate.

600)1(

1000

)1( 10

YTMYTM

MP

N


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Break-even Reinvestment Rate

Suppose instead of buying 3 coupon bonds by investing $3000 in the 9%coupon bond, an investor invest $3000 in the zero-coupon bond and buy 5bonds.

$3000 invested in the zero-coupon bond will grow to $5000 at the end of 10periods (5 years).

An investor will be indifferent between the choice of the zero-coupon

bond and the 9% coupon bond if the latter produces a total futureamount of $5000.

The total future amount from holding a 12% bond to maturity will beequal to the sum of (1) the coupon payments, (2) the maturity value, and(3) the interest on coupon interest. For the 9% coupon bond, the investorknows for certain that for each $3000 invested, the following future

amounts wi1l be received: Maturity value = $3000.00

We need coupon interest plus interest on coupon interest = $2000 whichimplies that we need $650 (=$2000 - $1350) worth of interest on coupon).


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Break-even Reinvestment Rate Maturity value = 3000.00

We need coupon interest plus interest on coupon interest = $2000

Coupon plus interest on coupon from each bond must equal ($2000/3)

Solving the above equation gives, the annual reinvestment rate i =

16.94%. If the portfolio manager believes at least an 16.94% reinvestment rate can be

realized, the 9% coupon bond will provide more future dollars than the zero-coupon bond. If the reinvestment rate is expected to be less than 16.94%, thenthe zero-coupon bond is a better investment, as it will provide a higher totalreturn.

The analysis assumes that the portfolio manager could invest all the couponpayments at a tax-free rate. If taxes on the coupon payments must be paid,the analysis must take this into consideration.

r

r

r

r

CFV

n

c

11

4567666

11 10

.


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2000-09 Suman Banerjee

Bond Price-Interest Relation:

Convexity

The reason for this property lies in the convex shape of theprice/discount rate relationship. This is illustrated graphicallyin figure below.

Price

discount rate

P1

P

P2

i-2% i i+2%


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Portfolio Managers Problem What market participants would like to have is a measure that can be used to

quantify the price volatility of a bond. For example, suppose a portfolio manager expects that interest rates will fall and is

considering purchasing one of the following two bonds: (1) Bond A, 7%, 22-yearbond selling to yield 8%, and (2) Bond B, a zero-coupon, 15-year bond, selling to give8.40% as required return.

With interest rates expected to decline, the portfolio manager will want to purchase

the bond that offers the greater price volatility. Which has higher price volatility? On the one hand, Bond B has a lower coupon than Bond A, so it would seem that

Bond B has a greater price volatility.

On the other hand, Bond A has a longer maturity than Bond B, so it would seem thatBond A has more price volatility.

Complicating further, the two bonds are trading at different required return.

Next two lectures discuss the measures of bond price volatility appropriate for such asituation.

One of the things we do know is that any measure of price volatility should take intoconsideration the three factors that affect price volatility: coupon rate, maturity, andlevel of discount rate.



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Role of Duration

We explained that the bond return that is realized byinvesting in a coupon security will depend on theinterest rate earned on the reinvestment of the couponpayments.

When interest rates rise, interest on interest from thereinvestment of the coupon payments will be higher, but if theinvestment horizon is shorter than the maturity of the bond, aloss will be realized upon the sale of the bond.

The reverse is true if interest rates fall: price appreciation will be

realized when the bond is sold, but interest on interest fromreinvesting the coupon payments will be lower. Because ofthese two risks, the investor cannot be assured of locking in thereturn at the time of purchase.



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Role of Duration: Important

Question

Because the interest rate risk andreinvestment risk tend to offset each other,

Is it possible to select a bond or a portfolio ofbonds that will lock in the return at the timeof purchase regardless of interest rate changesin the future?

Is it possible to immunize the bond or bondportfolio against interest rate changes?



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Important Question: Answer

Fortunately, under certain circumstances, the answer isYES.

This can be accomplished by buying a bond or byconstructing a portfolio of bonds such that its duration is

equal to the length of the investment horizon of themanager.

Thus, a portfolio manager with an investment horizon of5 years who wants to lock in a return over that timeperiod, should select a portfolio with a Macaulay

duration of 5 years. We will demonstrate this with an example later, but first

we need to learn how to compute duration?.



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Duration of a Bond

Duration is the weighted average time tomaturity, where the weights are therelative PV of cash flows.

It is a measure of the elasticity of an assets orliabilitys value to small changes in the interestrate.

The percentage in bonds fair value (or PV) fora given change in interest rates can be moredirectly measured by duration.



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Computing Macaulay Duration

Macaulay duration (in periods)

wheret = the period when the cash flow is expected tobe received (t = 1, ...,n);n = number of years until maturity;

PVCF1+PVCF2+..+PVCF2n = PV of all futurecashflows = price of a bond

(1) PVCF1 + (2) PVCF2 + (3) PVCF3 + ...+ (n) PVCFn

PVCF1 + PVCF2 + PVCF3 + ...+ PVCFn



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Duration of a Bond

For an option-free bond with semi-annualpayments, the cash flow for periods 1 to n -1 is one-half the

annual coupon interest.

the cash flow in period n is the semiannual couponinterest plus the maturity value.

Since the bond's price is equal to its cash flowdiscounted at the prevailing YTM, (PVCF1+PVCF2+ PVCF3++ PVCF2n) is the currentmarket price.



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Duration of A Bond: Example 1 Duration of a 4-year bond with a 10% coupon paid semi-annually, selling

at 8% YTM.

Duration =0.5*(7291.19/1067) = 3.42 years

3 4 5

t CFt 1

1 50 0.96154 48.08 48.077

2 50 0.92456 46.23 92.456

3 50 0.88900 44.45 133.349

4 50 0.85480 42.74 170.961

5 50 0.82193 41.10 205.482

6 50 0.79031 39.52 237.094

7 50 0.75992 38.00 265.9718 1050 0.73069 767.22 6137.798

6.733 1067 7291.19

1

1.04t 1.04

t

t

CF

1.04t

t

CF t



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Mathematically, Macaulay duration for a semi-annual paybond can be shown to be equivalent to:

i = semi-annual YTM;

c = semi-annual coupon rate.

n = number of compounding periods

Macaulay Duration: Algebra

1(1 )

i i cDur H n H

i i

bondtheofPrice

paymentscouponallofluePresent vaH


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Duration

Consider a 15-year, 8% semi-annual coupon bond selling at 84.63 to giveYTM=10%. Find the duration of the bond?

PV of all coupons = = $61.49

Hence, H = $61.49/$84.62 = 0.726596

Duration =

=

= 16.90 half years or 8.45 years

11

(1 )nc

i i

1(1 )

i i cH n H

i i

1.05 .05 .04(.726596) 30(1 0.726596).05 .05


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Macaulay Duration: Par Value Bond

For a par value bond, the semi-annual coupon rate is equal to

semi-annual yield; thus c and i are equal. The Macaulay

duration for a par value bond in six-month periods is;

1(1 )

1half-years

1 1years

2

i i cDur H n H

i i

iH

i

iH

i


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Macaulay Duration: ZC Bond

For a zero coupon bond, the coupon rate is zero; that is, c =0.

Hence, H = 0. The Macaulay duration for a par value bond in

six-month periods is;

1 (1 )

(1 0) half-years

half years

= n/2 years

i i cDur H n H i i

in

i

n


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Duration

Consider a 15-year, 8% semi-annual couponbond selling at 84.63 to give YTM=10%. Find theduration of the bond?

Suppose interest rate increases, such that thebond coupons can be reinvested at 11% and thediscount rate for the bond is also 11%. Whathappens to the investors total return? Assume that the investors investment horizon is 8.5

years (or 17 periods) Next, assume that the investors investment horizon

is 7 years (or 14 periods)?

CF P VCF P VCF*


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t CFt P VCF P VCF*t

1 4 0 . 9 5 2 3 . 8 10 3 . 8 10

2 4 0 . 9 0 7 3 . 6 2 8 7 . 2 5 6

3 4 0 . 8 6 4 3 . 4 5 5 10 . 3 6 6

4 4 0 . 8 2 3 3 . 2 9 1 13 . 16 3

5 4 0 . 7 8 4 3 . 13 4 15 . 6 7 1

6 4 0 . 7 4 6 2 . 9 8 5 17 . 9 0 9

7 4 0 . 7 11 2 . 8 4 3 19 . 8 9 9

8 4 0 . 6 7 7 2 . 7 0 7 2 1. 6 5 9

9 4 0 . 6 4 5 2 . 5 7 8 2 3 . 2 0 6

10 4 0 . 6 14 2 . 4 5 6 2 4 . 5 5 7

11 4 0 . 5 8 5 2 . 3 3 9 2 5 . 7 2 6

12 4 0 . 5 5 7 2 . 2 2 7 2 6 . 7 2 8

13 4 0 . 5 3 0 2 . 12 1 2 7 . 5 7 7

14 4 0 . 5 0 5 2 . 0 2 0 2 8 . 2 8 4

15 4 0 . 4 8 1 1. 9 2 4 2 8 . 8 6 1

16 4 0 . 4 5 8 1. 8 3 2 2 9 . 3 19

17 4 0 . 4 3 6 1. 7 4 5 2 9 . 6 6 8

18 4 0 . 4 16 1. 6 6 2 2 9 . 9 17

19 4 0 . 3 9 6 1. 5 8 3 3 0 . 0 7 6

2 0 4 0 . 3 7 7 1. 5 0 8 3 0 . 15 1

2 1 4 0 . 3 5 9 1. 4 3 6 3 0 . 15 1

2 2 4 0 . 3 4 2 1. 3 6 7 3 0 . 0 8 3

2 3 4 0 . 3 2 6 1. 3 0 2 2 9 . 9 5 3

2 4 4 0 . 3 10 1. 2 4 0 2 9 . 7 6 7

2 5 4 0 . 2 9 5 1. 18 1 2 9 . 5 3 0

2 6 4 0 . 2 8 1 1. 12 5 2 9 . 2 4 9

2 7 4 0 . 2 6 8 1. 0 7 1 2 8 . 9 2 8

2 8 4 0 . 2 5 5 1. 0 2 0 2 8 . 5 7 0

2 9 4 0 . 2 4 3 0 . 9 7 2 2 8 . 18 2

3 0 10 4 0 . 2 3 1 2 4 . 0 6 3 7 2 1. 8 9 8

8 4 . 6 3 14 3 0 . 11


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Duration

Duration of the bond =0.5*(1430.11/84.63) = 8.45 years

Next, we will demonstrate theprocess of immunization!



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Duration

What is the FV of coupon + interest on coupons at theend of 8.5 years if the interest rate stays put?

What is the expected sell price at the end of date 8.5after interest rate stays put?

Total dollars earned? $103.36+$90.56=194.47

1717

1.05 1(1 ) 14 $103.36

0.05

n

r

r

iFV c

i

17 13 13

1 1 1001 4 1 $90.56

(1 ) (1 ) (1.05) (1.05)n nM

PV ci i


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Duration

What is the FV of coupon + interest on coupons at theend of 8.5 years after the interest rate changes?

What is the expected sell price at the end of date 8.5after interest rate changes?


1717

1.055 1(1 ) 14 $107.99

0.055

n

r

r

iFV c

i

17 13 13

1 1 1001 4 1 $86.32

(1 ) (1 ) (1.055) (1.055)n nM

PV ci i


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Duration: Horizon 7 years

What is the FV of coupon + interest on coupons at theend of 7 years if the interest rate stayed put?

What is the expected sell price at the end of date 7 afterinterest rate stayed put?


1414

1.05 1(1 ) 14 $78.39

0.05

n

r

r

iFV c

i

14 16 16

1 1 1001 4 1 $89.16

(1 ) (1 ) (1.05) (1.05)n nM

PV ci i


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Duration: Horizon 7 years

What is the FV of coupon + interest on coupons at theend of 7 years after the interest rate changes?

What is the expected sell price at the end of date 8.5after interest rate changes?


1414

1.055 1(1 ) 14 $81.17

0.055

n

r

r

iFV c

i

14 16 16

1 1 1001 4 1 $84.31

(1 ) (1 ) (1.055) (1.055)n nM

PV ci i


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Duration and Change in Price

A securitys price changes for a given change in interestrates is given by:

The approximation works best when there are just smallchanges (i is small) in interest rates.

We call

where i is the semi-annual YTM.

Approximate percentage change in price

1

1

Duration Interest rate Change

i

durationModified

i)(1

Duration


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Duration And Change In Price

Consider a 15-year, 8% semi-annual coupon bond selling at 84.63to give YTM=10%. Duration of the bond is 8.45 years.

Modified Duration = (8.45/1.05) = 8.05

Suppose the interest rate increases from 10% to 10.10%, then

Interest rate change = 0.1% = .0010

Approximate % price change = - (8.05)(0.0010)

= - 0.81%.

Actual Price change = -0.80%.

For a small change in yield, modified duration provides agood approximation of price change.


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Duration And Change In Price: Large

Interest Rate Change

A 15-year, 8% semi-annual coupon bond selling at 84.63 toyield 10%. Duration of the bond is 8.45 years (Check!). Modified Duration = (8.45/1.05) = 8.05

Suppose the interest rate increases from 10% to 13%, then

Interest rate change = 3.0% = .03

Approximate % price change = - (8.05)(0.03)

= - 24.15%

Actual Price change = -20.41% (Check!)

For a large change in yield, modified duration does notprovides a good approximation of price change.


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Convex Function and Price Change

For a large change in yield, modified duration doesnot provides a good approximation of pricechange.

This is because, duration assumes that the relation between price

and interest rate (or discount rate) is linear. We argued that for a small change in interest rate assuming a

linear relationship does not make a big difference.

The true relation price and interest rate is convex.

Thus, for large change in interest rate, assuming a linear

relationship will give a large error.

We need to add a correction factor

The correction factor is called convexity.


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Price Change: Effect of Convexity

The last term on the RHS is the effect of convexity, where

2))((convexity2

1i)(duration)modified(P% i

Price)(12

1)))(CF(t(tof

Convexity mm mi

PV


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How to Calculate Convexity

oupon rate 8% Initial YTM 10% Price = 92.28

erm (years 5 Par Value 100

Period CF t(t+1) (CF)(t(t+1) PVCF*(t(t+1) PVCF*t

1 4 2 8 7.6190 3.8095

2 4 6 24 21.7687 7.2562

3 4 12 48 41.4642 10.36614 4 20 80 65.8162 13.1632

5 4 30 120 94.0231 15.6705

6 4 42 168 125.3642 17.9092

7 4 56 224 159.1926 19.8991

8 4 72 288 194.9297 21.6589

9 4 90 360 232.0592 23.205910 104 110 11440 7023.1676 638.4698

Total 7965.404669 771.4084


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How to Calculate Convexity

Using the table in the last slide we get

Price)(12

1)))(CF(t(tofConvexity

mmmi

PV

7965.40471)))(CF(t(tofPV

57.1928.92)05.1(2

4047.796522 Convexity


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Adjustment Due to Convexity

A 5-year, 8% semi-annual coupon bond selling at 92.28 toyield 10%. Duration of the bond is 4.55 years and ModifiedDuration = (4.18/1.05) = 3.98. Also, the convexity of thebond is reported to be 19.57.

Annual yield change = 3.0% = .03

Approximate % price change due to duration=-(3.98)(0.03) = - 11.94%

Convexity of the bond = 19.57

% price change due to convexity= (0.5)(19.57)(0.03)2

= .881%


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Price Change: Duration & Convexity

A5-year, 8% semi-annual coupon bond selling at92.28 to yield 10%.

Annual yield change = 3.0% = .03

Actual Price change = -11.11% Total price change (duration & convexity)

= -11.94% + .881%

= -11.06%


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Portfolio Duration

Bond Price YTM Modifiedduration

A: 5 years, 10% coupon $4,000,000 10% 3.861

B: 15 years, 8% coupon $4,231,375 10% 8.047C: 30 years, 14% coupon $1,378,586 10% 9.168

Total $9,609,962

All three bonds are semi-annual coupon bonds.

The portfolio consist of one bond A, one bond B andone bond C.


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Portfolio Duration

The portfolios modified duration is0.416(3.861) + 0.440(8.047) +0.144(9.168) = 6.47 A portfolio modified duration of 6.47 means that for a

100-basis-point change in the yield for all three bonds,the market value of the portfolio will change byapproximately 6.47%.

But the YTM on all three bonds must change by 100basis points for the modified duration of the portfoliomeasure to be useful.


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Managing Interest Rate Risk


53/64


Managing Interest Risk: Assets

The first step in assessing interest-rate risk is for the bankmanager to decide which assets and liabilities are rate-sensitive,that is, which have interest rates that will be re-priced within theyear.

Note that rate-sensitive assets or liabilities can have interest rates

re-priced within the year either because the debt instrumentmatures within the year or because the re-pricing is doneautomatically, as with variable-rate mortgages.

For many assets and liabilities, deciding whether they are rate-sensitive is straightforward.

In our example, the obviously rate-sensitive assets are securitieswith maturities of less than one year ($5 million), variable-ratemortgages ($10 million), and commercial loans with maturitiesless than one year ($15 million), for a total of $30 million.


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Managing Interest Risk: Assets

However, some assets that look like fixed-rate assets whoseinterest rates are not re-priced within the year actually have acomponent that is rate-sensitive.

For example, although fixed-rate residential mortgages mayhave a maturity of 30 years, homeowners can repay theirmortgages early by selling their homes or repaying themortgage in some other way.

This means that within the year, a certain percentage of thesefixed-rate mortgages will be paid off, and interest rates on thisamount will be re-priced.

From past experience the manager knows that 20% of the fixed-rate residential mortgages are repaid within a year, whichmeans that $2 million of these mortgages (20% of $10 million)must be considered rate-sensitive.

YOU have to be careful in estimating your exposure!


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Managing Interest Risk: Liabilities Similarly, determine the total amount of rate-sensitive liabilities.

Obvious rate-sensitive liabilities are money market deposit accounts($5 million), variable-rate CDs and CDs with less than one year tomaturity ($25 million), federal funds ($5 million), and borrowingswith maturities of less than one year ($10 million), for a total of $45million.

Checkable deposits and savings deposits often have interest ratesthat can be changed, although financial institutions often like tokeep their rates fixed for substantial periods.

Thus these liabilities are partially but not fully rate-sensitive. Thebank manager estimates that 10% of checkable deposits ($1.5million) and 20% of savings deposits ($3 million) is considered rate-sensitive.

Adding the $1.5 million and $3 million to the $45 million figureyields a total for rate-sensitive liabilities of $49.5 million.


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Managing Interest Rate Risk

Now the manager can analyze what will happen if interest ratesrise by 1 percentage point, say, on average from 10% to 11%.

The income on the assets rises by $320,000 (= 1% x $32 million ofrate-sensitive assets), while the payments on the liabilities rise by$495,000 (= 1% x $49.5 million of rate-sensitive liabilities).

The First National Banks profits now decline by $175,000 =($320,000 $495,000).

Conversely, if interest rates fall by 1%, similar reasoning tells usthat the First National Banks income rises by $175,000 and its netinterest margin rises by 0.175%.

Thus, if a financial institution has more rate-sensitive liabilitiesthan assets, a rise in interest rates will reduce the net interestmargin and income, and a decline in interest rates will raise thenet interest margin and income.


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IRM: Income Gap Analysis


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Example: Gap Analysis

The manager of notices that the bank balance sheet produces amore refined maturity bucket that allows him to estimate thepotential change in income over the next one to two years.

Rate-sensitive assets in this period consist of $5 million of securitiesmaturing in one to two years, $10 million of commercial loans

maturing in one to two years, and an additional $2 million (20% offixed-rate mortgages) that the bank expects to be repaid.

Rate-sensitive liabilities in this period consist of $5 million of one-to two-year CDs, $5 million of one- to two-year borrowings, $1 .5million of checkable deposits (the 10% of checkable deposits that

the bank manager estimates are rate-sensitive in this period), andan additional $3 million of savings deposits (the 20% estimate ofsavings deposits).

For the next one to two years, calculate the gap and the change inincome if interest rates rise by 1%.


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Example continued


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More to Interest Rate Risk Mgmt.

The gap analysis we have examined so far focuses only on theeffect of interest- rate changes on income.

Clearly, owners and managers of financial institutions care notonly about the effect of changes in interest rates on income butalso about the effect of changes in interest rates on the market

value of the net worth of the financial institution. An alternative method for measuring interest-rate risk, called

duration gap analysis, examines the sensitivity of the marketvalue of the financial institutions net worth to changes in interestrates.

Duration analysis is based on Macaulays concept of duration,which measures the average lifetime of a securitys stream ofpayments.


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Duration Gap Analysis


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Duration of Assets and Liabilities


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Example: Duration Gap

The manager wants to know whathappens to the institutions net worthwhen interest rates rise from 10% to11%. The total asset value is $100million and total liability value is $95million Note that total liabilities excludes buffer

capital and are thus, $95 million.


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Example continued

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