7-1
CHAPTER 7Bonds and Their Valuation
Key features of bonds Bond valuation(price) Measuring yield(return) Assessing risk
7-2
What is a bond?
A long-term debt instrument in which a borrower agrees to make payments of principal and interest, on specific dates, to the holders of the bond.
7-3
Key Features of a Bond Par value – face amount of the bond,
which is paid at maturity (assume $1,000).
Coupon rate – stated interest rate (generally fixed) paid by the issuer. Multiply by par to get dollar payment of interest.
Maturity date – years until the bond must be repaid.
Issue date – when the bond was issued.
7-4
The value of financial assets
nn
22
11
r)(1
CF ...
r)(1
CF
r)(1
CF Value
0 1 2 nr
CF1 CFnCF2Value
...
7-5
Bond valuation model
Rd is the required return of the bond. It is also the market rate of interest of the bond, and yield to maturity (YTM) of the bond.
Rd is NOT coupon rate
Nd
Nd
1d
b )r(1
valueface
)r(1
paymentcoupon ...
)r(1
paymentcoupon price) (bondV
7-6
What is the value of a 10-year, 10% annual coupon bond, if rd = 10%?
$1,000 V
$385.54 $38.55 ... $90.91 V
(1.10)
$1,000
(1.10)
$100 ...
(1.10)
$100 V
b
b
10101b
0 1 2 nr
100 100 + 1,000100Vb = ?
...
7-7
Using a financial calculator to value a bond
This bond has a $1,000 lump sum due at t = 10, and annual $100 coupon payments beginning at t = 1 and continuing through t = 10, the price of the bond can be found by solving for the PV of these cash flows.
INPUTS
OUTPUT
N I/YR PMTPV FV
10 10 100 1000
-1000
7-8
An example:Increasing inflation and rd
Suppose inflation rises by 3%, causing rd = 13%. When rd rises above the coupon rate, the bond’s value falls below par, and sells at a discount.
INPUTS
OUTPUT
N I/YR PMTPV FV
10 13 100 1000
-837.21
7-9
An example:Decreasing inflation and rd
Suppose inflation falls by 3%, causing rd = 7%. When kd falls below the coupon rate, the bond’s value rises above par, and sells at a premium.
INPUTS
OUTPUT
N I/YR PMTPV FV
10 7 100 1000
-1210.71
7-10
Semi-annual coupon bond
Majority of bonds pay interest semiannually. Coupon rate=10%/Y Going (nominal) interest=5% 15 year bond
Solution: P semi annual=$1523.26 How: N=15*2 i=5/2 PMT=$100/2 Compared with annual coupon bond,
which should have a higher price? P annual=$1518.98
7-11
What is the YTM on a 10-year, 9% annual coupon, $1,000 par value bond, selling for $887?
Must find the rd that solves this model.
10d
10d
1d
Nd
Nd
1d
B
)r(1
1,000
)r(1
90 ...
)r(1
90 $887
)r(1
valueface
)r(1
paymentcoupon ...
)r(1
paymentcoupon V
7-12
Using a financial calculator to find YTM
Solving for I/YR, the YTM of this bond is 10.91%. This bond sells at a discount, because YTM > coupon rate.
INPUTS
OUTPUT
N I/YR PMTPV FV
10
10.91
90 1000- 887
7-13
Find YTM, if the bond price was $1,134.20.
Solving for I/YR, the YTM of this bond is 7.08%. This bond sells at a premium, because YTM < coupon rate.
INPUTS
OUTPUT
N I/YR PMTPV FV
10
7.08
90 1000-1134.2
7-14
Relationship between interest and bond price (inversely realted)
Interest Rate
Bond Value
Interest Rate
Bond Value
12%
10%
8%
874.50 1,000 1,152.47
7-15
Interest sensitivity and maturity
Annual Coupon rate=10%
% change 1 yr rd 10yr % change
+4.8% $1,048 5% $1,386 +38.6%$1,000 10% $1,000
-4.3% $957 15% $749 -25.1%
The 10-year bond is more sensitive to interest rate changes, and hence has more interest rate risk.
7-16
Interest sensitivity and coupon rate
10 year annual bond
% change 5%c rd 10%c % change
+44% $1,000 5% $1,386 +38.6%$693 10% $1,000
-28% $498 15% $749 -25.1% Low coupon bond has CF concentrated at the
end of maturity from repayment of principal. The bond with lower coupon rate is more
sensitive to interest rate changes
7-17
Calculation details
N I(%) PV=? PMT FV1 5 -1048 100 10001 10 -1000 100 10001 15 -957 100 100010 5 -1386 100 100010 10 -1000 100 100010 15 -749 100 100010 5 -1000 50 100010 10 -693 50 100010 15 -498 50 100010 5 -1386 100 100010 10 -1000 100 100010 15 -749 100 1000
10Y, 10% coupon bond
10Y, 5% coupon bond
1Y,10% coupon bond
10Y, 10% coupon bond
7-18
Summary of Factors that Affect bond Prices and Price sensitivity when Interest Rates Change
Interest Rate negative relation between interest rate
changes and bond price increasing interest rates correspond to bond
price decrease (at a decreasing rate) Time Remaining to Maturity
longer time to maturity corresponds to larger price change for a given interest rate change
Coupon Rate the lower the coupon rate, the bigger the
price change for a given change in interest rates
Interest Rate negative relation between interest rate
changes and bond price increasing interest rates correspond to bond
price decrease (at a decreasing rate) Time Remaining to Maturity
longer time to maturity corresponds to larger price change for a given interest rate change
Coupon Rate the lower the coupon rate, the bigger the
price change for a given change in interest rates
7-19
Reinvestment risk
The risk that bondholders have to reinvest future cash flows (coupon and principal when expires)at lower interest rates if general interest level declines
EXAMPLE: Suppose you just won $500,000 playing the lottery. You intend to invest the money and live off the interest.
7-20
Reinvestment Rate Risk Example
You may invest in either a 10-year bond or a series of ten 1-year bonds. Both 10-year and 1-year bonds currently yield 10%.
If you choose the 1-year bond strategy: After Year 1, you receive $50,000 in income and have
$500,000 to reinvest. But, if 1-year rates fall to 3%, your annual income would fall to $15,000.
If you choose the 10-year bond strategy: You can lock in a 10% interest rate, and $50,000 annual
income for 10 years.
7-20
7-21
Conclusions about Interest Rate and Reinvestment Rate Risk
CONCLUSION: Nothing is riskless! Interest risk is a big concern in low interest environment, and
reinvestment risk is a big concern in high interest environment
7-21
Short-term AND/OR
High-coupon Bonds
Long-term AND/OR
Low-coupon Bonds
Interest rate risk Low High Reinvestment rate risk High Low
7-22
Default Risk
If an issuer defaults, investors receive less than the promised return.
Influenced by the issuer’s financial strength and the terms of the bond contract.
7-22
7-23
Evaluating Default Risk:Bond Ratings
Investment Grade Junk Bonds
Moody’s
Aaa Aa A Baa Ba B Caa C
S & P AAA AA A BBB BB B CCC C
Bond ratings are designed to reflect the probability of a bond issue going into default.
High risk demands high yield. (PIGS 4 are paying high yield on their bonds)
7-23
7-24
Optional: Bond investing in real world
Find bond information Where to buy? Don’t buy individual bond (unless
risk-free T bond) Risk-free could be illusion, price
does change Bond ETF (HYG,JNK, BSV, SCPB)
7-25
Factors to consider when investing in bonds Yield
For single bond: current yield( annual interest payment/current price) and yield to maturity(consider principal payment).
For bond ETF or mutual fund: Yield (current yield) and SEC yield (consider ytm and expense)
Default risk and rating Interest risk and duration
Duration: the effective maturity of a bond (average maturity for a bond ETF)
7-26
Optional: practical questions
Is Treasury bond really risk-free? Risk free bond price changes
Is it good time to buy bonds now? Interest changes LT vs. ST bond
Should I buy individual high-yield corporate bond?
7-27
Example of treasure bond
Yahoo-Finance-Market data-Bond Comparing LT bond yield with ST
yield Expectation of interest rate
Default risk? Interest risk and time to maturity
7-28
Examples of bond portfolio
HYG (bond ETF) 12-month Yield: the sum of a fund’s
total trailing 12-month interest and dividend payments divided by the last month’s ending share price (NAV) plus any capital gains distributed over the same period.
SEC yield: based on 30-day period ending on the last day of previous month, reflects the deduction of the fund’s expenses.
7-29
Examples of bond portfolio
Rating(default risk): Duration(interest risk)
Duration: Average years to maturity Longer duration is more sensitive to interest
changes When interest rate increases from 3% to
4%, a bond with duration of 4 years will see price decline of approximate 4*(4%-3%)=4%
7-30
Examples of bond portfolio
Portfolio holdings Weight, maturity and coupon rate Risk of each individual issue and the benefit
of diversification Compared with other bond ETF
BSV (ST T bond+ corporate,yield,grade, duration)
SCPB (corporate A bond) JNK-high yield (similar to HYG)