1

Chapter 12Bond Selection

2

Malkiel’s Interest Rate Theorems

Definition Theorem 1 Theorem 2 Theorem 3 Theorem 4 Theorem 5

3

Definition Malkiel’s interest rate theorems provide

information about how bond prices change as interest rates change

Any good portfolio manager knows Malkiel’s theorems

4

Theorem 1 Bond prices move inversely with yields:

• If interest rates rise, the price of an existing bond declines

• If interest rates decline, the price of an existing bond increases

5

Theorem 2 Bonds with longer maturities will fluctuate

more if interest rates change

Long-term bonds have more interest rate risk

6

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A B C D E F G

Market interest rate 6.50%

YearBond

cash flowMarket

interest rateBondvalue

1 70 0.00% 1,490.00 <-- =NPV(E5,$B$5:$B$11)2 70 1.00% 1,403.69 <-- =NPV(E6,$B$5:$B$11)3 70 2.00% 1,323.60 <-- =NPV(E7,$B$5:$B$11)4 70 3.00% 1,249.21 <-- =NPV(E8,$B$5:$B$11)5 70 4.00% 1,180.066 70 5.00% 1,115.737 1,070 6.00% 1,055.82

7.00% 1,000.00Value of the bond 1,027.42 <-- =NPV(B2,B5:B11) 8.00% 947.94

9.00% 899.3410.00% 853.9511.00% 811.5112.00% 771.8113.00% 734.6414.00% 699.82

VALUING THE XYZ CORPORATION BONDS

XYZ Bond Value

650

750850

9501,0501,1501,250

1,3501,450

0% 2% 4% 5% 7% 9% 11% 12% 14%

Market interest rate

Bo

nd

val

ue

7

Theorem 3 Higher coupon bonds have less interest rate

risk

Money in hand is a sure thing while the present value of an anticipated future receipt is risky

8

Theorem 4 When comparing two bonds, the relative

importance of Theorem 2 diminishes as the maturities of the two bonds increase

A given time difference in maturities is more important with shorter-term bonds

9

Theorem 5 Capital gains from an interest rate decline

exceed the capital loss from an equivalent interest rate increase

10

Duration as A Measure of Interest Rate Risk

The concept of duration Calculating duration

11

The Concept of Duration For a noncallable security:

• Duration is the weighted average number of years necessary to recover the initial cost of the bond

• Where the weights reflect the time value of money

12

The Concept of Duration (cont’d)

Duration is a direct measure of interest rate risk:• The higher the duration, the higher the interest

rate risk

13

Calculating Duration The traditional duration calculation:

1 (1 )

where duration

cash flow at time

yield to maturity

current price of the bond

years until bond maturity

time at which a cash flow is received

Nt

tt

o

t

o

Ct

RD

P

D

C t

R

P

N

t

14

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A B C D E F G H

BASIC DURATION CALCULATION

YTM 7%

Year Ct,A t*Ct,A /PA*(1+YTM)t Ct,B t*Ct,B /PB*(1+YTM)t

1 70 0.0654 130 0.08552 70 0.1223 130 0.15983 70 0.1714 130 0.22404 70 0.2136 130 0.27915 70 0.2495 130 0.32606 70 0.2799 130 0.36577 70 0.3051 130 0.39878 70 0.3259 130 0.42589 70 0.3427 130 0.4477

10 1070 5.4393 1130 4.0413Bond price Duration Bond price Duration

1,000$ 7.5152 1,421$ 6.7535

=NPV(B3,B6:B15) =SUM(F6:F15)

Excel formula 7.5152 <-- =DURATION(DATE(1996,12,3),DATE(2006,12,3),7%,B3,1)(need to have the tool "Analysis ToolPak" added in Excel)

15

Calculating Duration (cont’d) The closed-end formula for duration:

1

2

(1 ) (1 ) ( )(1 ) (1 )

where par value of the bond

number of periods until maturity

yield to maturity of the bond per period

N

N N

o

R R R N F NC

R R RD

P

F

N

R

16

Calculating Duration (cont’d)Example

Consider a bond that pays $100 annual interest and has a remaining life of 15 years. The bond currently sells for $985 and has a yield to maturity of 10.20%.

What is this bond’s duration?

17

Calculating Duration (cont’d)Example (cont’d)

Solution: Using the closed-form formula for duration:

1

2

31

2 30 30

(1 ) (1 ) ( )(1 ) (1 )

(1.051) (1.051) (0.051 30) 1,000 3050

0.051 (1.051) (1.051)

98515.69 years

N

N N

o

R R R N F NC

R R RD

P

18

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A B C D E F G H

EFFECTS OF COUPON AND MATURITY ON DURATION

Current date 5/21/1996 <-- =DATE(1996,5,21)Maturity, in years 21Maturity date 5/21/2017 <-- =DATE(1996+B4,5,21)YTM 15% Yield to maturity (i.e., discount rate)Coupon 4%Face value 1,000

Duration 9.0110 <-- =DURATION(B3,B5,B7,B6,1)

Data table: Effect of maturity on duration9.0110 <-- =B10

5 4.516310 7.482715 8.814820 9.039825 8.788130 8.446135 8.163340 7.966945 7.842150 7.766855 7.722860 7.697765 7.683770 7.6759

Effect of Maturity on Duration Coupon rate = 4.00%, YTM = 15.00%

4.0

5.0

6.0

7.0

8.0

9.0

10.0

0 20 40 60 80Maturity

Du

rati

on

19

31323334353637383940414243444546

A B C D E F G HData table: Effect of coupon on duration

9.0110 <-- =B100% 21.00001% 13.12042% 10.78653% 9.66774% 9.01105% 8.57926% 8.27367% 8.04599% 7.7294

13% 7.370715% 7.259317% 7.1729

Effect of Coupon on Duration Maturity = 21, YTM = 15.00%

5.07.0

9.011.013.0

15.017.019.0

21.0

23.0

0% 5% 10% 15%Coupon rate

Dur

atio

n

20

Bond Selection - Introduction In most respects selecting the fixed-income

components of a portfolio is easier than selecting equity securities

There are ways to make mistakes with bond selection

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The Meaning of Bond Diversification

Introduction Default risk Dealing with the yield curve Bond betas

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Introduction It is important to diversify a bond portfolio Diversification of a bond portfolio is

different from diversification of an equity portfolio

Two types of risk are important:• Default risk• Interest rate risk

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Default Risk Default risk refers to the likelihood that a

firm will be unable to repay the principal and interest of a loan as agreed in the bond indenture• Equivalent to credit risk for consumers

• Rating agencies such as S&P and Moody’s function as credit bureaus for credit issuers

24

Default Risk (cont’d) To diversify default risk:

• Purchase bonds from a number of different issuers

• Do not purchase various bond issues from a single issuer

– E.g., Enron had 20 bond issues when it went bankrupt

25

Dealing With the Yield Curve The yield curve is typically upward sloping

• The longer a fixed-income security has until maturity, the higher the return it will have to compensate investors

• The longer the average duration of a fund, the higher its expected return and the higher its interest rate risk

26

Dealing With the Yield Curve (cont’d)

The client and portfolio manager need to determine the appropriate level of interest rate risk of a portfolio

27

Bond Betas The concept of bond betas:

• States that the market prices a bond according to its level of risk relative to the market average

• Has never become fully accepted

• Measures systematic risk, while default risk and interest rate risk are more important

28

Choosing Bonds Client psychology and bonds selling at a

premium Call risk Constraints

29

Client Psychology and Bonds Selling at A Premium

Premium bonds held to maturity are expected to pay higher coupon rates than the market rate of interest

Premium bond held to maturity will decline in value toward par value as the bond moves towards its maturity date

30

Client Psychology & Bonds Selling at A Premium (cont’d)

Clients may not want to buy something they know will decline in value

There is nothing wrong with buying bonds selling at a premium

31

Call Risk If a bond is called:

• The funds must be reinvested• The fund manager runs the risk of having to

make adjustments to many portfolios all at one time

There is no reason to exclude callable bonds categorically from a portfolio• Avoid making extensive use of a single callable

bond issue

32

Constraints Specifying return Specifying grade Specifying average maturity Periodic income Maturity timing Socially responsible investing

33

Specifying Return To increase the expected return on a bond

portfolio:• Choose bonds with lower ratings

• Choose bonds with longer maturities

• Or both

34

Specifying Grade A legal list specifies securities that are

eligible investments• E.g., investment grade only

Portfolio managers take the added risk of noninvestment grade bonds only if the yield pickup is substantial

35

Specifying Grade (cont’d) Conservative organizations will accept only

U.S. government or AAA-rated corporate bonds

A fund may be limited to no more than a certain percentage of non-AAA bonds

36

Specifying Average Maturity Average maturity is a common bond

portfolio constraint• The motivation is concern about rising interest

rates

• Specifying average duration would be an alternative approach

37

Periodic Income Some funds have periodic income needs

that allow little or not flexibility

Clients will want to receive interest checks frequently• The portfolio manager should carefully select

the bonds in the portfolio

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Maturity Timing Maturity timing generates income as needed

• Sometimes a manager needs to construct a bond portfolio that matches a particular investment horizon

• E.g., assemble securities to fund a specific set of payment obligations over the next ten years

– Assemble a portfolio that generates income and principal repayments to satisfy the income needs

39

Socially Responsible Investing Some clients will ask that certain types of

companies not be included in the portfolio

Examples are nuclear power, military hardware, “vice” products

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Example: Monthly Retirement Income

The problem Unspecified constraints Using S&P’s Bond Guide Solving the problem

41

The Problem A client has:

• Primary objective: growth of income

• Secondary objective: income

• $1,100,000 to invest

• Inviolable income needs of $4,000 per month

42

The Problem (cont’d) You decide:

• To invest the funds 50-50 between common stocks and debt securities

• To invest in ten common stock in the equity portion (see next slide)

– You incur $1,500 in brokerage commissions

43

The Problem (cont’d)Stock Value Qrtl Div. Payment Month

3,000 AAC $51,000 $380 Jan./April/July/Oct.

1,000 BBL 50,000 370 Jan./April/July/Oct.

2,000 XXQ 49,000 400 Feb./May/Aug./Nov.

5,000 XZ 52,000 270 March/June/Sept./Dec.

7,000 MCDL 53,000 0 --

1,000 ME 49,000 370 Feb./May/Aug./Nov.

2,000 LN 51,000 500 Jan./April/July/Oct.

4,000 STU 47,000 260 March/June/Sept./Dec.

3,000 LLZ 49,000 290 Feb./May/Aug./Nov.

6,000 MZN 43,000 170 Jan./April/July/Oct.

Total $494,000 $3,010

44

The Problem (cont’d) Characteristics of the fund:

• Quarterly dividends total $3,001 ($12,004 annually)

• The dividend yield on the equity portfolio is 2.44%

• Total annual income required is $48,000 or 4.36% of fund

• Bonds need to have a current yield of at least 6.28%

45

Unspecified Constraints The task is meeting the minimum required

expected return with the least possible risk• You don’t want to choose CC-rated bonds

• You don’t want the longest maturity bonds you can find

46

Using S&P’s Bond Guide Figure 11-4 is an excerpt from the Bond

Guide:• Indicates interest payment dates, coupon rates,

and issuer

• Provides S&P ratings

• Provides current price, current yield

47

Using S&P’s Bond Guide (cont’d)

48

Solving the Problem Setup Dealing with accrued interest and

commissions Choosing the bonds Overspending What about convertible bonds?

49

Setup You have two constraints:

• Include only bonds rated BBB or higher• Keep the average maturities below fifteen years

Set up a worksheet that enables you to pick bonds to generate exactly $4,000 per month (see next slide)

50

Setup (cont’d)Security Price Jan. Feb. March April May June

3,000 AAC $51,000 $380 $380

1,000 BBL 50,000 370 370

2,000 XXQ 49,000 $400 $400

5,000 XZ 52,000 $270 $270

7,000 MCDL 53,000

1,000 ME 49,000 370 370

2,000 LN 51,000 500 500

4,000 STU 47,000 260 260

3,000 LLZ 49,000 290 290

6,000 MZN 43,000 170 170

Equities $494,000 $1,420 $1,060 $530 $1,420 $1,060 $530

51

Dealing With Accrued Interest and Commissions

Bond prices are typically quoted on a net basis (already include commissions)

Calculate accrued interest using the mid-term heuristic• Assume every bond’s accrued interest is half of

one interest check

52

Choosing the Bonds The following slide shows one possible solution:

• Stock cost: $494,000

• Bond cost: $557,130

• Accrued interest: $9,350

• Stock commissions: $1,500

Do you think this solution could be improved?

53

BondsSecurity Price Jan. Feb. March April May June

$80,000 Empire 71/2s02

$86,400 $3,000

$80,000 Energen 8s07

82,900 $3,200

$100,000 Enhance 61/4s03

105,500 $3,370

$80,000 Enron 65/8s03

84,500 $2,650

$90,000 Enron 6.7s06

97,200 $3,010

$100,000 Englehard 6.95s28

100,630 $3,470

Bonds subtotal $557,130 $3,000 $3,200 $3,370 $2,650 $3,010 $3,470

Total income $4,420 $4,260 $3,900 $4,070 $4,070 $4,000

54

Overspending The total of all costs associated with the

portfolio should not exceed the amount given to you by the client to invest

The money the client gives you establishes another constraint

55

What About Convertible Bonds?

Convertible bonds can be included in a portfolio• Useful for a growth of income objective• People buy convertible bonds in hopes of price

appreciation• Useful if you otherwise meet your income

constraints

56

Immunization Strategies A portfolio of bonds is said to be

immunized (from interest rate risk) if its payoff at some future date is independent of the future levels of interest rates.

Immunization is closely related to the concept of duration.

57

Immunization consists of matching the duration of the portfolio’s assets and liabilities (obligations).

Suppose a firm has a future obligation Q. The prevailing interest rate is r, and the liability is N periods away.

The present value of this liability is denoted by V0=Q/(1+r)N.

58

Now suppose that the firm is currently hedging this liability with a bond whose value VB = V0 and whose coupon payments are denoted by P1,…,PM.

We thus have:

1 (1 )

Mt

B tt

PV

r

59

Suppose now that interest rates change from r to r+r. The new values of the future obligation and of the bond are:

00 0 0 0 1

11

(1 )

(1 )

N

MtB

B B B B tt

dV NQV V V r V r

dr r

tPdVV V V r V r

dr r

60

Rearranging terms and recalling that V0=VB

yields the following expression:

1

1

(1 )

Mt

ttB

tPN

V r

The left-hand side represents the duration of the bond, while the right-hand side represents the duration of the obligation (Since the obligation consisted of only one payment, the duration is its maturity).

61

In conclusion, in order for a portfolio to be immunized, you need to have:

DURATIONASSETS = DURATIONLIABILITIES

Caveat: this works only if the interest rates of various maturities all change in the same manner, i.e. if the yield curve shifts upward or downward in a parallel shift.

62

Immunization Example You need to immunize an obligation whose

present value V0 is $1,000. The payment is to be made 10 years from now, and the current interest rate is 6%. The payment is thus the future value of 1,000 at 6%, therefore it is:1,000(1.06)10 = $1,790.85

The Excel spreadsheet on the next slide shows three bonds that you have at your disposition to immunize the liability.

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A B C D

BASIC IMMUNIZATION EXAMPLE WITH 3 BONDS

Yield to maturity 6%

Bond 1 Bond 2 Bond 3Coupon rate 6.70% 6.988% 5.90%Maturity 10 15 30Face value 1,000 1,000 1,000

Bond price $1,051.52 $1,095.96 $986.24Face value equal to $1,000 of market value 951.00$ 912.44$ 1,013.96$

Duration 7.6655 10.0000 14.6361

=dduration(B7,B6,$B$3,1)

64

When the interest rate increases:

When the interest rate decreases:

THE IMMUNIZATION PROBLEMIllustrated for the 30-year bond.

0

Year 10:Future obligation of $1,790.85 due. 30

Buy $1,014 face value of 30-year bond.

Reinvest coupons from bond during years 1-10.

Sell bond for PV of remaining coupons and redemption in year 30.

Value of reinvested coupons increases.

Value of bond in year 10 decreases.

Value of reinvested coupons decreases.

Value of bond in year 10 increases.

65

Values 10 years later, assuming interest rates do not change

192021222324252627

A B C D E F G H INew yield to maturity, 10 years later 6%

Bond 1 Bond 2 Bond 3Bond price $1,000.00 $1,041.62 $988.53 <-- =-PV($B$19,D7-10,D6*D8)+D8/(1+$B$19)^(D7-10)Reinvested coupons $883.11 $921.07 $777.67 <-- =-FV($B$19,10,D6*D8)Total $1,883.11 $1,962.69 $1,766.20

Multiply by percent of face value bought 95.10% 91.24% 101.40%Product 1,790.85$ 1,790.85$ 1,790.85$

(The goal of getting $1,790.85 is still met)

66

Values 10 years later, assuming interest rates change to 5% right

after we buy the bonds

(The goal of getting $1,790.85 is not met by Bond 1 anymore)

192021222324252627

A B C D E F G H INew yield to maturity, 10 years later 5%

Bond 1 Bond 2 Bond 3Bond price $1,000.00 $1,086.07 $1,112.16 <-- =-PV($B$19,D7-10,D6*D8)+D8/(1+$B$19)^(D7-10)Reinvested coupons $842.72 $878.94 $742.10 <-- =-FV($B$19,10,D6*D8)Total $1,842.72 $1,965.01 $1,854.26

Multiply by percent of face value bought 95.10% 91.24% 101.40%Product 1,752.43$ 1,792.97$ 1,880.14$

67

Observations If interest rates go down to 5%, Bond 1

does not meet the requirement anymore. Bond 3, on the other hand, exceeds the

payment that must be made in year 10. The ability of Bond 2 to meet the obligation

is barely affected. Why? Because its duration is 10 years, exactly matching the duration of the liability. Pick Bond 2.

68

We can compute and plot the bonds’ terminal values in year 10

Immunization Properties of the Three Bonds

$1,550

$1,750

$1,950

$2,150

$2,350

$2,550

$2,750

$2,950

0% 2% 4% 6% 8% 10% 12% 14% 16%

New interest rate

Te

rmin

al

va

lue

Bond 1

Bond 2

Bond 3