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CHAPTER 7Bonds and Their Valuation

Key features of bonds Bond valuation(price) Measuring yield(return) Assessing risk

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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.

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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.

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The value of financial assets

nn

22

11

r)(1

CF ...

r)(1

CF

r)(1

CF Value

0 1 2 nr

CF1 CFnCF2Value

...

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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

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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 = ?

...

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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

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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

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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

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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

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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

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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

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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

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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

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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.

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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

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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

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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

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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.

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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.

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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

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Short-term AND/OR

High-coupon Bonds

Long-term AND/OR

Low-coupon Bonds

Interest rate risk Low High Reinvestment rate risk High Low

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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.

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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)

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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)

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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)

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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?

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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

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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.

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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%

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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)