1 chapter 12 bond selection. 2 malkiel’s interest rate theorems u definition u theorem 1 u theorem...
TRANSCRIPT
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
123
456789101112131415161718192021222324252627282930
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$11)5 70 4.00% 1,180.066 70 5.00% 1,115.737 1,070 6.00% 1,055.82
7.00% 1,000.00Value of the bond 1,027.42 <-- =NPV(B2,B5:B11) 8.00% 947.94
9.00% 899.3410.00% 853.9511.00% 811.5112.00% 771.8113.00% 734.6414.00% 699.82
VALUING THE XYZ CORPORATION BONDS
XYZ Bond Value
650
750850
9501,0501,1501,250
1,3501,450
0% 2% 4% 5% 7% 9% 11% 12% 14%
Market interest rate
Bo
nd
val
ue
7
Theorem 3 Higher coupon bonds have less interest rate
risk
Money in hand is a sure thing while the present value of an anticipated future receipt is risky
8
Theorem 4 When comparing two bonds, the relative
importance of Theorem 2 diminishes as the maturities of the two bonds increase
A given time difference in maturities is more important with shorter-term bonds
9
Theorem 5 Capital gains from an interest rate decline
exceed the capital loss from an equivalent interest rate increase
10
Duration as A Measure of Interest Rate Risk
The concept of duration Calculating duration
11
The Concept of Duration For a noncallable security:
• Duration is the weighted average number of years necessary to recover the initial cost of the bond
• Where the weights reflect the time value of money
12
The Concept of Duration (cont’d)
Duration is a direct measure of interest rate risk:• The higher the duration, the higher the interest
rate risk
13
Calculating Duration The traditional duration calculation:
1 (1 )
where duration
cash flow at time
yield to maturity
current price of the bond
years until bond maturity
time at which a cash flow is received
Nt
tt
o
t
o
Ct
RD
P
D
C t
R
P
N
t
14
123456789
10111213141516171819202122
A B C D E F G H
BASIC DURATION CALCULATION
YTM 7%
Year Ct,A t*Ct,A /PA*(1+YTM)t Ct,B t*Ct,B /PB*(1+YTM)t
1 70 0.0654 130 0.08552 70 0.1223 130 0.15983 70 0.1714 130 0.22404 70 0.2136 130 0.27915 70 0.2495 130 0.32606 70 0.2799 130 0.36577 70 0.3051 130 0.39878 70 0.3259 130 0.42589 70 0.3427 130 0.4477
10 1070 5.4393 1130 4.0413Bond price Duration Bond price Duration
1,000$ 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
123456789101112131415161718192021222324252627
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
21
The Meaning of Bond Diversification
Introduction Default risk Dealing with the yield curve Bond betas
22
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
23
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
38
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
40
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.
63
123456789
10111213141516
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