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CHAPTER 10Bond Prices and Yields

Chapter Sections:Bond BasicsStraight Bond Prices and Yield to MaturityMore on YieldsInterest Rate Risk and Malkiel’s TheoremsDurationDedicated Portfolios and Reinvestment RiskImmunization

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Bond Yields Bond yield is one of the most important

factors in bond valuation What income is the bond paying? Over the long sweep of time, income is what you

receive from investing in bonds Although there are sometimes opportunities for

capital gains and sometimes risks of capital loss But when the bond is redeemed, you are only going

to get back the par value ($1,000) Given that most all bonds repay their principal

without incident, the valuations calculated using bond yields tend to be very predictable

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Bond Yields The unpredictable factor in bond valuation is

the future direction of interest ratesHowever, for the many people who hold onto their

bonds until maturity (until they get their principal back), the direction of interest rates is unimportant to them They are mostly interested in the income and are not

generally affected by the direction of interest rates since they have no intention of ever selling their bonds before they mature

You are only concerned about changing interest rates if you intend (or are forced) to sell your bonds before they mature

(continued)

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Types of Bond Yields There are several types of bond yields

Nominal Yield (a.k.a. Coupon Yield, Nominal Rate) The stated rate of the bond

Current Yield (a.k.a. Current Rate) The interest rate of the bond given its current price

Yield-to-Maturity The rate if you hold the bond until it matures

Yield-to-Call The rate if you hold the bond until it is called

Taxable Equivalent Yield (Municipal bonds) The rate taking into account no Federal income tax

Double Tax-free Taxable Equivalent Yield The rate taking into account no Federal and no

state income tax

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Nominal Yield Nominal Yield

a.k.a. Coupon Yield, Nominal Rate, Coupon Rate The named interest rate of the bond The bond’s annual interest income divided by its

par value What the bond is paying in absolute dollars

Example: Par value $1,000, $80 interest per year

Nominal Yield = $80 / $1,000 = 0.08 = 8%

6 But the Nominal Yield is not as important as… Current Yield

The amount of current income a bond provides relative to its market price

Yield-to-Maturity The fully compounded rate of return earned by an

investor over the life of the bond Includes current income and price appreciation or

depreciation a.k.a. Promised Yield

Yield-to-Call The yield on a bond assuming it will be called on a

specified date sometime in the future

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Current Yield Annual Interest

Current Yield = ──────────────────── Current Market Price of the

Bond

Example: 8% bond selling at $800

$80Current Yield = ──── = 0.10 = 10%

$800

The nominal yield is 8% but because the bond is selling at a discount, the current yield is actually 10%.

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Current Yield Annual Interest

Current Yield = ──────────────────── Current Market Price of the

Bond

Example: 8% bond selling at $1,200

$80Current Yield = ──── = 0. 0666667 6.67%

$1,200

(continued)

The nominal yield is again 8%, but because the bond is selling at a premium, the current yield is only 6.67%.

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The Yield-to-Maturity takes into account the price appreciation of the bond if the bond is purchased at a discount Or price depreciation of the bond if the bond is

purchased at a premium Two primary methods of calculation (there are others)

The bond pricing formula discussed later combined with an internal rate of return approximation More accurate method but difficult to do manually Remember the spreadsheets from chapter 6?

The YTM formula on the next slide Looks scary but is actually fairly easy to use

Yield-to-Maturity

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Yield-to-Maturity

Par value - Market value

Number of years to maturity

Par value + Market value

2

Example: 8% maturing in 10 years, price $800

$1,000 - $800

10

$1,000 + 800

2

= 0.111111 11.1%

$ Amt Annual Interest +

$80 +

(continued)

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Par value - Market value

Number of years to maturity

Par value + Market value

2

Example: 8% maturing in 10 years, price $1,200

$1,000 - $1,200

10

$1,000 + $1,200

2

= 0.054545 5.45%

$ Amt Annual Interest +

$80 +

(continued)Yield-to-Maturity

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Wait a minute… The current yield on our 8% discount bond selling

at $800 was 10% But the yield-to-maturity was 11.1%

And the current yield on our 8% premium bond selling for $1,200 was 6.67% But the yield-to-maturity was 5.45%

How is that possible? What is going on?

(continued)

It is actually very straightforward. The discount bond was purchased at $800 but will be redeemed at $1,000. The premium bond was

purchased at $1,200 but, again, will be redeemed at $1,000. The yield-to-maturity takes into account the bond appreciation from the

discount price up to the par value or the depreciation from the premium price down to the par value.

Yield-to-Maturity

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Yield-to-Call The Yield-to-Call takes into account the

possibility of the bond being “called” Only used on premium-priced bonds

(A bond issuer would never call in discount bonds. That would mean they would be refinancing at a higher rate)

Again, two methods of calculation (with some others) The bond pricing formula discussed later combined

with an internal rate of return approximation More accurate method but difficult without a computer

The same formula as Yield-to-Maturity But replace par value with call price and years-to-

maturity with years-to-call

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Call value - Market value

Number of years to call

Call value + Market value

2

8% call in 5 years, price $1,200, call price $1,085

$1,085 - $1,200

5

$1,085 + $1,200

2

= 0.049891 4.99%

$ Amt Annual Interest +

$80 +

(continued)Yield-to-Call

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(continued)Yield-to-Call vs Yield-to-Maturity

The yield-to-call was less than the yield-to-maturity Yes, this is typical This is because if the bonds are called away,

we would have less time to take advantage of the outsized interest income payments

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Taxable Equivalent Yields Taxable Equivalent Yields for Municipal Bonds

a.k.a. “tax-exempt yield” “tax-free yield” Since municipal bonds are free from Federal taxes,

in order to effectively compare municipal bonds to other bonds, we compute the taxable equivalent yields

Federal Taxable Equivalent Yield For municipal bonds free of Federal income tax

Double Tax-free Taxable Equivalent Yield For municipal bonds free of both Federal income

tax and the investor’s state income tax Example: California double tax-free bonds do not

charge income to California resident investors

(review)

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Taxable Equivalent Yield =

Municipal Bond Yield

1.0 – (Your marginal tax rate)

Example: 6% Yield, 25% Tax bracket

Taxable equivalent yield = 0.06

1.0 - 0.25

= 0.08 = 8%

Taxable Equivalent Yields(continued)

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Taxable Equivalent Yield for Both Federal & State – a.k.a. “double tax-exempt” “double tax-free” The formula on the previous slide only takes into

account Federal income taxes If the municipal bond is free of both Federal and

State income taxes (“double tax-free”), then the Taxable Equivalent Yield will be higher The formula on the next slide assumes that you

itemize deductions on your Federal income taxes and deduct State income taxes Which is very typical for municipal bond investors

If you don’t itemize deductions, then the Taxable Equivalent Yield would be a bit higher

Taxable Equivalent Yields(continued)

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Taxable Equivalent Yield for both Fed & State =

Municipal Bond Yield

1.0 – [ Fed rate + (State rate * (1 – Fed rate))]

Example: 6% Yield, 25% Fed rate, 8% State rate

Taxable equivalent yield = 0.06

1.0 – [0.25 + (0.08*(1-0.25))]

= 0.086957 8.7%

Taxable Equivalent Yields(continued)

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Yield Spreads Differences in interest rates that exist among

various sectors of the bond market The shorter the maturity, the lower the rate

The longer the maturity, the higher the rate The higher the rating of the bond, the lower the

interest rate (and vice versa) Treasuries carry the lowest rates Municipal bonds are next

General obligation bonds Revenue bonds

Corporate bonds yield the highest rates Non-callable Callable

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(continued)

Let’s look at the current yield spreads on Yahoo!http://finance.yahoo.com/bonds/composite_bond_rates

2 April 2015, Source: Yahoo!

Yield Spreads

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Yield Spreads Often, investors will speak about the bond

spreads as being “tight” or “wide” A “tight” spread means the interest rates among

the bonds they are evaluating are very close to one another Example: Treasury paying 4.8%, Corporate bond

paying 5.1% A “wide” spread means there is a big difference

between the bond interest rates Example: Treasury paying 3.2%, Corporate bond

paying 8.2%

(continued)

For several years, bond yield spreads were very tight. During the turmoil of 2008 and 2009, the yield spreads widened to levels not seen in decades. They have narrowed significantly over the past few years.

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The Effect of Inflation on Bond Rates

Source: Department of Labor Statistics

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The Yield Curve A graph that represents the relationship

between a bond’s maturity and its yield at a given point in time Also used to make comparisons among types of

bonds Normally, the yield curve is upward sloping

Longer term bonds have higher interest rates than shorter term bonds and bills

Sometimes, the yield curve is downward sloping (a.k.a. “inverted yield curve”) Shorter term bonds and bills have higher interest

rates than longer term bonds

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(continued)The Yield Curve

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(continued)The Yield Curve

Source: Yahoo!, April 2, 2015

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Why do longer term debt securities normally have higher interest rates than shorter term debt securities?

Expectations Hypothesis The shape of the yield curve reflects investors’

expectations of future interest rates Maturity Preference Hypothesis

a.k.a. Liquidity Preference Hypothesis Investors tend to prefer the liquidity of short-term

securities and, therefore, require a premium to invest in long-term securities

Theories re: Yield Curves

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Market Segmentation Hypothesis The market for debt is segmented on the basis of

maturity. Supply and demand within each segment determines the prevailing interest rate.

Theories re: Yield Curves

Each of these 3 theories makes sense and each has some merit. But how do we account for the times when the yield curve is inverted?

What factors could cause an inverted yield curve to occur?And what can the yield curve tell us about the future of the economy?

(continued)

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Since World War II, every time the yield curve has inverted (short term rates were higher than long term rates), the economy has fallen into a recession The only exception was 1966 The yield curve is currently mostly upward

But for about two years before the beginning of 2008, the yield curve had been inverted!

The Yield Curve & the Economy

The bond market had been predicting a recession for over two years. The stock market, for the most part, didn’t believe them.

It wasn’t until fall of 2008 that the officials charged with tracking the economy acknowledged that we were in a recession. It took

over two years, but the bond “ghouls” were finally proven right.

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Bond Pricing Bonds are normally priced according to the

present value of their future cash flows Semi-annual interest payments, and Repayment of principal

Although other factors will always need to be considered Such as the credit-worthiness of the issuer

If an issuer runs into trouble, the price of their outstanding bonds will fall because investors will be afraid of default

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Bond Pricing Bond price = present value of the interest

income + present value of the repayment Look familiar? It’s the Discounted Cash Flow Model!

Annual versus semi-annual compounding Since bonds pay interest normally every six months,

we really should use semi-annual compounding However, annual compounding is easier to compute

and will give you almost the exact same answer Computations are easily done using the present

value tables Spreadsheets make it even easier

(continued)

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Bond Pricing Example: Avery Dennison Corporation

5.375%, due 15-Apr-2020, priced to yield 2.927% Yield-to-Maturity 2.927%, Current Yield 4.827% 2.927% is close to 3% – Let’s use 3% for 5 years

Bond price = present value of the interest income +

present value of the repayment = $53.75 * 4.580 (present value factor for stream of income) + $1,000 * 0.863 (present value factor for repayment of bond)

= $246.18 + $863.00 ≈ $1,109.18

(continued)

The quoted price on Finra on Apr 2nd was $1,113.49. The bond calculator spreadsheet (on the class web site) gives us $1,113.02 for annual payments

and $1,114.68 for semi-annual payments. Pretty close, eh? Why is the semi-annual payments prediction a bit higher than the annual payments

prediction?

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Reinvestment Risk

Changing interest rates don’t only affect the price of your bonds. They also affect your future income as you need to reinvest the

interest income and bond repayments. If interest rates have fallen, your income level will fall as you reinvest your income

and bond repayments. Likewise, if interest rates have risen, your income level will rise.

The uncertainty about the future value of an investor’s bond investments that result from the need to reinvest bond interest payments and redemptions at yields not known in advance

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Duration

The shorter the duration, the less potential price volatility, and vice-versa.

Measure of a bond price’s sensitivity to changes in interest rates and bond yields Captures both price and reinvestment risk Used to indicate how a bond will react in different

interest rate environments The duration of a bond changes as it approaches

its maturity date and current interest rates change In general…

The longer a bond’s maturity, the longer its duration The higher a bond’s nominal rate and yield-to-

maturity, the shorter its duration

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Duration and Immunization Investors who have a specified time horizon

can use “bond immunization” to increase the probability of successfully achieving their desired goal Keep the average duration of your bond

investments equal to your time horizon You would thus be more protected against

interest-rate induced price swings Requires constant rebalancing of your bond

portfolio since durations of bonds change as interest rates change and bonds get closer to maturity

Mostly used by pension fund & bond mutual fund managers.

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Income Strategy Purchase the bonds simply for the interest income

they produce Capital Gains Strategy

Speculating that interest rates will fall Total Return

Purchasing bonds for both the income and the possibility of capital gains

Bond Investment Strategies

Which of these would be the easiest to implement?Which would be the hardest?

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Bond Laddering Strategy of purchasing bonds with staggering

maturities Purchase some bonds with short-term maturities,

some with intermediate-term maturities and some with long-term maturities

Again, very popular strategy with pension fund and bond mutual fund managers Since they have considerable sums of money to

invest

Bond Investment Strategies

What are the advantages and disadvantages of this strategy?When is it a good time to ladder? When is it not a good time?

(continued)

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(continued)Bond Investment Strategies

As of 2 April 2015

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CHAPTER 10 – REVIEW

Bond Prices and Yields

Chapter Sections:Bond BasicsStraight Bond Prices and Yield to MaturityMore on YieldsInterest Rate Risk and Malkiel’s TheoremsDurationDedicated Portfolios and Reinvestment RiskImmunization

Next: Preferred Stocks & Convertible Securities,Chapter 11, Asset Allocation