![Page 1: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/1.jpg)
Fixed – income Securities
![Page 2: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/2.jpg)
Long Term Debt: A Review
Corporate debt can be short-term (maturity less than one year) or long-term.
Different from common stock:– Creditor’s claim on corporation is specified– Promised cash flows– Most are callable
Over half of outstanding bonds are owned by life insurance companies & pension funds
![Page 3: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/3.jpg)
Features of Fixed Income Securities Maturity :
- The length of time until the agreement expires.- Borrowers (Issuers) are committed to meet their obligations over this period
Coupon :- The rate which used for calculating the amount of interest to be paid.
Frequency :- Quarterly : coupon will be paid every 3 months.- Semi-annual : coupon will be paid every 6 months.- annually : coupon will be paid every 1 years.
![Page 4: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/4.jpg)
Features of Fixed Income Securities
Par Value ( redemption or face value)- The amount that borrowers promises to pay
lenders at the maturity.
Remark : Bond’s price depends on movement in interest rates. If market interest rates move above (below) the coupon rate, then a bond will sell below/discount (above/premium) par value.
Bond price is the net present value of bond’s cash flow.
![Page 5: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/5.jpg)
The indenture, a written agreement between the borrower and a trust company, usually lists– Amount of Issue, Date of Issue, Maturity– Denomination (Par value)– Annual Coupon, Dates of Coupon Payments– Security– Sinking Funds– Call Provisions– Covenants
Features that may change over time– Rating– Yield-to-Maturity– Market price
Features of Fixed Income Securities
![Page 6: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/6.jpg)
Issue amount $20 million Bond issue total face value is $20 million Issue date 12/15/98 Bonds offered to the public in December 1998 Maturity date 12/31/18 Remaining principal is due December 31,
2018 Face value $1,000 Face value denomination is $1,000 per bond Coupon interest $100 per annum Annual coupons are $100 per bond Coupon dates 6/30, 12/31 Coupons are paid semiannually Offering price 100 Offer price is 100% of face value Yield to maturity 10% Based on stated offer price Call provision Callable after 12/31/03 Bonds are call protected for 5 years after
issuance Call price 110 before 12/31/08,
100 thereafter Callable at 110 percent of par value through 2008. Thereafter callable at par.
Trustee United Bank of Florida
Trustee is appointed to represent bondholders
Security None Bonds are unsecured debenture Rating Moody's A1, S&P A+ Bond credit quality rated upper medium
grade by Moody's and S&P's rating
Features of Fixed Income Securities
![Page 7: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/7.jpg)
Protective Covenants Agreements to protect bondholders
Negative covenant: They should not:– pay dividends beyond specified amount– sell more senior debt & amount of new debt is
limited– refund existing bond issue with new bonds paying
lower interest rate– buy another company’s bonds
Positive covenant: They should:– use proceeds from sale of assets for other assets– allow redemption in event of merger or spin-off– maintain good condition of assets– provide audited financial information
![Page 8: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/8.jpg)
Yield Curve
Thai Government Bond Yield Curve
1.5000%
2.5000%
3.5000%
4.5000%
5.5000%
6.5000%
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
years
%
16/9/2002
![Page 9: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/9.jpg)
Bond Ratings What is rated:
– The likelihood that the firm will default.– The protection afforded by the loan contract in the
event of default.
Who pays for ratings:– Firms pay to have their bonds rated.– The ratings are constructed from the financial
statements supplied by the firm.
Ratings can change.
Raters can disagree.
![Page 10: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/10.jpg)
Bond Rating and SymbolsS&P Moody's ExplanationAAA Aaa Prime grade, highest safetyAA+ Aa1AA Aa2AA- Aa3A+ A1A A2A- A3
BBB+ Baa1BBB Baa2BBB- Baa3
High credit quality
Upper-medium credit quality
Lower-medium credit quality
Investmen
t grade
![Page 11: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/11.jpg)
Bond Rating and Symbols
![Page 12: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/12.jpg)
Junk bonds Anything less than an S&P “BB” or a Moody’s
“Ba” is a junk bond.
A polite euphemism for junk is high-yield bond.
There are two types of junk bonds:– Original issue junk—possibly not rated– Fallen angels—rated
Current status of junk bond market– Private placement
Yield premiums versus default risk
![Page 13: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/13.jpg)
Types of Bond Fixed rate : the coupon rate constantly fixed.
Floating rate : the coupon rate based on a reference rate (i.e. LIBOR)
Amortizing bond : some partial of principle were scheduled to pay out before the maturity.
Zero-coupon bond : pay no interest and are sold at a discount from their par value. The difference represents the interest costs to the borrower.
![Page 14: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/14.jpg)
Types of Bond
Callable bond : issuer has right to redeem a bond before maturity at a formula price.
To compensate the call risk, investors usually need additional rate of return which may incur higher cost of borrowing to the issuer.
When market interest rate keeps lowing, issuer might execute call since he upsets with his high cost of borrowing
![Page 15: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/15.jpg)
Types of Bond
Puttable bond : investors have right to redeem the bond before maturity at a formula price.
Put option is a sweetener which help issuer to lower his cost of borrowing.
When market interest keeps soaring, investors might execute put option since they can reinvestment at the higher rate of return.
![Page 16: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/16.jpg)
Types of Bond Convertible bonds :
- Why are they issued? Cheap interest burden- Why are they purchased? Right for converting
- Conversion ratio:Number of shares of stock acquired by conversion
- Conversion price:Bond par value / Conversion ratio
- Conversion value:Price per share of stock x Conversion ratio
- In-the-money versus out-the-money
![Page 17: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/17.jpg)
Reasons for Issuing Warrants and Convertibles Convertible debt carries a lower coupon rate than
does otherwise-identical straight debt.
Since convertible debt is originally issued with an out-of-the-money call option, one can argue that convertible debt allows the firm to sell equity at a higher price than is available at the time of issuance. However, the same argument can be used to say that it forces the firm to sell equity at a lower price than is available at the time of exercise.
Convertible bonds also allow young firms to delay expensive interest costs until they can afford them
![Page 18: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/18.jpg)
Types of Bond Convertible bonds (continue) :
- Conversion ratio: 2 shares per 1 unit
- Conversion price: = (1,000 Baht / unit ) / (2 shares / unit)= 500 Baht / share
- Conversion value:= (40 Baht / share ) x ( 2 shares /
unit)= 80 Baht / unit
![Page 19: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/19.jpg)
Types of Bond
Convertible bonds (continue) :
-The value of a convertible bond has three major components:
1. Straight bond value
2. Option value
3. Conversion value
![Page 20: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/20.jpg)
Types of Bond Convertible bonds (continue) :
Example; Litespeed, Inc., just issued a zero coupon convertible bond due in 10 years.
- The appropriate interest rate is 10%.
- Each convertible is trading at $400 in the market.
– What is the straight bond value?– What is the option value of the bond?
![Page 21: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/21.jpg)
Types of Bond
Convertible bonds (continue) :– What is the straight bond value?
– What is the option value of the bond?
$400 – 385.54 = $14.46
54.385$)10.1(
000,1$10
SBV
![Page 22: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/22.jpg)
Convertible Bond Value
Stock Price
Straight bond value
Conversion Value
= conversion ratio
floor value
floor value
Convertible bond values
Option value
Types of Bond
![Page 23: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/23.jpg)
Types of Bond
Other Convertible bonds (continue) :
Exchangeable bonds– Convertible into a set number of shares of a
third company’s common stock.
![Page 24: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/24.jpg)
Bond Pricing Example : Thai government bond (LB04NA) par 1,000
Baht which will mature on Nov 30, 2002 pay coupon 3.50% semi-annually.
A) What is the amount of interest paid in each coupon date?
Coupon = 1,000 x (3.50%)/2 = 17.50 Baht
B) How many times that investor who bought this bond on Dec8,2002 and hold it until maturity receive the coupon?
The number of period = 4 periods
C) Suppose investor buy this bond on Nov 30,2002 (just after the coupon was paid). How much he need to pay, if the market interest rate for 2 year equals to 3.50%?
![Page 25: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/25.jpg)
Price = PVIA(A=17.5,i = 1.75%,n=4) + PVIF(F=1000,i =1.75%,n=4) = 17.5 / ( 1.0175 )1 + 17.5 / ( 1.0175 )2 + 17.5 / ( 1.0175 )3 + 1,017.5 / ( 1.0175 )4 = (PMT = 17.5; FV = 1,000; n = 4; i = 1.75%; Find PV)= 1,000 It is a par bond.
11/02 05/03 11/03 05/04 11/04
17.5 17.5 17.51,017.5
P = ?
i = 1.75%
Nov-02 May-03 Nov-03 May-04 Nov-04
0 1 2 3 4
0 17.50 17.50 17.50 1,017.50
PV on Nov02 if mkt rate = 3.50%1,000.00 17.20 16.90 16.61 949.29
Date
Period
Coupon and Principle will receive
Bond Pricing
![Page 26: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/26.jpg)
Price = PVIA(A=17.5,i = 2.00%,n=4) + PVIF(F=1000,i =2.00%,n=4) = 17.5 / ( 1.02 )1 + 17.5 / ( 1.02 )2 + 17.5 / ( 1.02 )3 + 1,017.5 / ( 1.02 )4 = (PMT = 17.5; FV = 1,000; n = 4; i = 2%; Find PV)= 990.48 It is a discount bond.
D) If everything held constant except the market interest rate for 2 year equals to 4.00%. Is this premium or discount bond? And what is the price?
Nov-02 May-03 Nov-03 May-04 Nov-04
0 1 2 3 4
0 17.50 17.50 17.50 1,017.50
PV on Nov02 if mkt rate = 4.00% 990.48 17.16 16.82 16.49 940.01
Date
Period
Coupon and Principle will receive
Bond Pricing
![Page 27: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/27.jpg)
Price = PVIA(A=17.5,i = 1.50%,n=4) + PVIF(F=1000,i =1.50%,n=4) = 17.5 / ( 1.015 )1 + 17.5 / ( 1.015 )2 + 17.5 / ( 1.015 )3 + 1,017.5 / ( 1.015 )4 = (PMT = 17.5; FV = 1,000; n = 4; i = 1.50%; Find PV)= 1,009.64 It is a premium bond.
F) If everything held constant except the market interest rate for 2 year equals to 3.00%. Is this premium or discount bond? And what is the price?
Nov-02 May-03 Nov-03 May-04 Nov-04
0 1 2 3 4
0 17.50 17.50 17.50 1,017.50
PV on Nov02 if mkt rate = 3.00%1,009.64 17.24 16.99 16.74 958.67
Date
Period
Coupon and Principle will receive
Bond Pricing
![Page 28: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/28.jpg)
Bond Pricing Concept checking;
Bond X and Y have the same features except
A) If X pay coupon higher than Y, how should investors value the bonds?
B) If the coupon was equally set and X has better credit rating than Y, how should investors value the bonds?
Bond X will be priced higher than Y.
Investors will require lower rate of return (represent the market rate) on X rather than Y. So, Bond X will be priced higher than Y.
![Page 29: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/29.jpg)
Bond Pricing Example : Thai government bond (LB04NA) par
1,000 Baht which will mature on Nov 30, 2002 pay coupon 3.50% semi-annually. Suppose an investor buy this bond on Jan15,2003. How much he need to pay, if he satisfies with 2% yield? How much for accrued interest and clean price? (Use Actual/365 )
30/11/04
17.5 17.5 17.51,017.5
P = ?
i = 1.00%30/5/0430/11/0330/5/03
15/1/03
30/11/02
![Page 30: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/30.jpg)
Bond Pricing Solution :
= 17.3706 + 17.1986 + 17.0283 + 980.2729
= 1,031.8704 buyer pay dirty price to seller.
7397.37397.27397.17397.0 )01.1(
5.017,1
)01.1(
5.17
)01.1(
5.17
)01.1(
5.17P
30/11/04
17.5 17.5 17.51,017.5
P = ?
i = 1.00%30/5/0430/11/0330/5/03
15/1/03
30/11/02
7397.05.182
135
![Page 31: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/31.jpg)
Bond Pricing Solution :
Dirty Price = 1,031.87
A.I. = (46 / 182.5) x 17.5 = 4.41
Clean Price = Dirty Price – A.I. = 1,027.46
![Page 32: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/32.jpg)
Coupon Rate, Current Yield, and YTM
Annual coupon
Coupon ratePar value
Bond’s coupon rate:
Bond’s current yield:
priceBond
couponAnnualyieldCurrent
The discount rate that equates a bond’s price with the present value of all future cash flows.
Yield to Maturity:
![Page 33: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/33.jpg)
Example: Assume a bond has 12 years to maturity, a 8% coupon (paid semi-annually),and the price is $1039.11.
Coupon rate = $80/$1000= 8%Current yield = $80/$1039.11=7.7%
24 -1039.11 40 1000N I/YR PV PMT FV
3.75
INPUTS
OUTPUT
2(12) 80/2
YTM=3.75%x2=7.5%
Coupon Rate, Current Yield, and YTM
![Page 34: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/34.jpg)
Now assume a bond has 25 years to maturity, a 9% coupon,and the YTM is 8%. What is the price? (remember all bonds pay semi-annual interest payment)
50 4 45 1000N I/YR PV PMT FV
-1107.41
INPUTS
OUTPUT
More on Bond Prices/Yields
2(25) 8/2 90/2
![Page 35: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/35.jpg)
Now assume a bond has 25 years to maturity, a 9% coupon,and the YTM is 10%. What is the price? (remember all bonds pay semi-annual interest payment)
50 5 45 1000N I/YR PV PMT FV
-908.72
INPUTS
OUTPUT
2(25) 10/2 90/2
More on Bond Prices/Yields
![Page 36: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/36.jpg)
Now assume the same bond has 5 years to maturity (9% coupon & YTM of 8%) What is the price? Is the bond selling at premium or discount?(remember all bonds pay semi-annual interest payment)
10 4 45 1000N I/YR PV PMT FV
-1040.55
INPUTS
OUTPUT
2(5) 8/2 90/2
More on Bond Prices/Yields
![Page 37: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/37.jpg)
Now assume the same bond has a YTM of 10%. (9% coupon & 5 years to maturity) What is the price? Is the bond selling at premium or discount?(remember all bonds pay semi-annual interest payment)
10 5 45 1000N I/YR PV PMT FV
-961.39
INPUTS
OUTPUT
2(5) 10/2 90/2
More on Bond Prices/Yields
![Page 38: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/38.jpg)
Where does this leave us? We found:Coupon Years YTM Price 9% 25 8% $1,107 9% 25 10% $ 908 9% 5 8% $1,040 9% 5 10% $ 961
$900
$950
$1,000
$1,050
$1,100
$1,150
8% 9% 10% 11%
25 years
5 years
More on Bond Prices/Yields
![Page 39: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/39.jpg)
Figure 10.1: Premium, par, and discount bond prices over time (if yield does not change)
70
80
90
100
110
120
130
140
30 25 20 15 10 5 0
Time to maturity (years)
Bo
nd
pri
ces
(% o
f p
ar)
![Page 40: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/40.jpg)
Interest Rate Risk and Maturity
![Page 41: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/41.jpg)
Malkiel’s TheoremsSummarizes the relationship among bond prices, yields, coupons, and maturity:(all theorems hold assuming everything else constant):1) Bond prices move inversely with interest rates.2) The longer the maturity of a bond, the more sensitive is it’s price to a change in interest rates.3) The price sensitivity of any bond increases with it’s maturity, but the increase occurs at a decreasing rate.4) The lower the coupon rate on a bond, the more sensitive is it’s price to a change in interest rates.5) For a given bond, the volatility of a bond is not symmetrical, i.e. a decrease in interest rates raises bond prices more than a corresponding increase in interest rates lower prices.
![Page 42: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/42.jpg)
Duration (Macaulay’s duration) : A widely used measure of a bond's sensitivity to a 1% (1% = 100 bps) changes in interest rate (yield).
The 1st derivative of the bond’s price function with respect to yield. It’s the slope of bond price curve.
The present value-weighted number of years to maturity
Duration
![Page 43: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/43.jpg)
Although modified duration allows us to estimates of price change in a bond’s price for a small change in required yield, it does not provide good estimates of a large price change in required yield. This is because of the convexity in the price/yield relation ship.
Convexity: a measure of the degree of curvature or convexity in the price/yield relationship.
Duration
![Page 44: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/44.jpg)
Duration formula
tPriceBond
CFPVDurationMacaulay
n
1t
t
DurationModified duration
YTM1
2
YTMinChangeDurationModifiedpricebondinΔ%
To compute the percentage change in a bond’s priceusing Modified Duration:
![Page 45: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/45.jpg)
Figure 10.3: Calculating bond duration
Discount Present Years x Present valueYears Cash flow factor value / Bond price
0.5 40 0.96154 38.4615 0.01921 40 0.92456 36.9822 0.0370
1.5 40 0.88900 35.5599 0.05332 40 0.85480 34.1922 0.0684
2.5 40 0.82193 32.8771 0.08223 1040 0.79031 821.9271 2.4658
$1,000.00 2.7259Bond price Bond duration
Example: A 3-year bond with 8% annual coupon selling at par.
Modified duration = (2.7259) / (1.04) = 2.6211 years
Duration Calculation
![Page 46: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/46.jpg)
5.24%2.6211pricebondin Δ%
5.24%
2.08
1
.10)(.082.7259pricebondinΔ%
)10.008.0(
From previous example, if YTM will go to 10%, calculate the percentage change in bond price and the new bond price.
Therefore; Approx. new price = $1,000 x (1 - 5.24%)= $947.60
Duration Calculation
![Page 47: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/47.jpg)
Duration Calculation
![Page 48: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/48.jpg)
Properties of Duration All else the same, the longer a bond’s
maturity, the longer is its duration.
All else the same, a bond’s duration increases at a decreasing rate as maturity lengthens.
All else the same, the higher a bond’s coupon, the shorter is its duration.
All else the same, a higher yield to maturity implies a shorter duration, and a lower yield to maturity implies a longer duration.
![Page 49: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/49.jpg)
Figure 10.3: Bond duration and maturity
0
2
4
6
8
10
12
0 5 10 15 20 25 30
Bond maturity (years)
Bo
nd
du
rati
on
(ye
ars
) 0% Coupon
5% Coupon 10% Coupon
15% Coupon
![Page 50: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/50.jpg)
FRN Pricing
Under the assumption of no default risk, the bond should always be 100% on reset dates.
Once the coupon is fixed on a reset date, the bond tends to behave like a ST fixed-coupon bond until the next reset date.
FRN prices are more volatile just after the reset date, because that is when they have the longest fixed-coupon maturity.
![Page 51: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/51.jpg)
Dedicated Portfolios
Dedicated portfolioA bond portfolio created to prepare for a future cash outlay, e.g. pension funds.The date the payment is due is commonly called the portfolio’s target date.
![Page 52: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/52.jpg)
Immunization Interest rate risk: The possibility that changes in
interest rates will result in losses in a bond's value.
Reinvestment rate risk: The uncertainty about target date portfolio value that results from the need to reinvest bond coupons at yields not known in advance.
Price risk: The risk that bond prices will decrease, which arises in dedicated portfolios when the target date value of a bond or bond portfolio is not known with certainty
Immunization: Constructing a portfolio to minimize the uncertainty surrounding its target date value.
![Page 53: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/53.jpg)
Problem 1
CIR Inc. has 7% coupon bonds on the market that have 11 years left to maturity. If the YTM on these bonds is 8.5%, what is the current bond price?
Solution:
$894.16
2.0851
1000
2.0851
11
2) / (.085
35priceBond 2222
![Page 54: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/54.jpg)
Problem 2
Trincor Company bonds have coupon rate of 10.25%, 14 years to maturity, and a current price of $1,225. What is the YTM? The current yield?
Solution:
2828
21
1000
21
11
)2/(
)2/50.102(225,1$
YTMYTMYTM
YTM = 3.805% x 2 = 7.61%
Current yield = $102.50 / $1,225 = 8.37%
![Page 55: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/55.jpg)
Problem 3
XYZ Company has a 9% callable bond outstanding on the market with 12 years to maturity, call protection for the next 5 years, and a call premium of $100. What is the YTC for this bond if the current price is 120% of par value?
Solution:
1010
2YTC1
1100
2YTC1
11
YTC
90200,1$
YTC = 3.024% x 2 = 6.05%
[see next slide for additional information]
![Page 56: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/56.jpg)
What is the YTM, with zero call premium?
Solution:
If interest rates stay at current levels, the bond issuer will likely call the bonds to refinance at the earliest possible time.
Problem 3 (cont’d)
2224
2YTM1
1000
2YTM1
11
YTM
90200,1$
YTM = 3.283% x 2 = 6.57%
[see next slide for additional information]
![Page 57: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/57.jpg)
Problem 3 (cont’d)
What would be the break-even call premium? (If interest rates don’t change, at what level would the call premium have to be to not call the bonds?)
Solution:
1010
203283.1
X1000
203283.1
11
03283.
90200,1$
X = $134.91
The bond will not be called if the call premium is greater than $134.91.
![Page 58: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/58.jpg)
Problem 4
YTM = 5.249 x 2 = 10.498%
66
2YTM1
1000
2YTM1
11
YTM
80937$
What is the Macaulay duration of an 8% coupon bond with 3 years to maturity and a current price of $937.10? What is the modified duration?
Solution:
First calculate the yield:
![Page 59: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/59.jpg)
Calculating bond duration
Discount Present Years x Present valueYears Cash flow factor value / Bond price
0.5 40 0.95 38.00 0.0201 40 0.90 36.11 0.039
1.5 40 0.86 34.31 0.0552 40 0.81 32.60 0.070
2.5 40 0.77 30.97 0.0833 1040 0.74 765.07 2.449
937.06 2.715Bond price Bond duration
Problem 4 (cont’d)
![Page 60: Fixed – income Securities. Long Term Debt: A Review Corporate debt can be short-term (maturity less than one year) or long-term. Different from common](https://reader036.vdocuments.site/reader036/viewer/2022081506/56649e0b5503460f94af3074/html5/thumbnails/60.jpg)
Problem 4 (cont’d)
Mac. Duration = 2.715 years
Modified duration
= 2.715 / (1 + .10498/2) = 2.58 years