CHAPTER 10
Bond Prices and Yields
Chapter Sections:
Bond Basics
Straight Bond Prices and Yield to Maturity
More on Yields
Interest Rate Risk and Malkiel’s Theorems
Duration
Dedicated Portfolios and Reinvestment Risk
Immunization
1
2
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
3
Bond Yields

(continued)
The unpredictable factor in bond valuation is
the future direction of interest rates
 However, 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
4
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
5
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%
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
6
7
Current Yield
Annual Interest
Current Yield = ────────────────────
Current Market Price of the Bond
Example: 8% bond selling at $800
$80
Current 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%.
8
Current Yield
(continued)
Annual Interest
Current Yield = ────────────────────
Current Market Price of the Bond
Example: 8% bond selling at $1,200
$80
Current Yield = ──── = 0. 0666667  6.67%
$1,200
The nominal yield is again 8%, but because the bond is
selling at a premium, the current yield is only 6.67%.
9
Yield-to-Maturity

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
10
Yield-to-Maturity
(continued)
Par value - Market value
$ Amt Annual Interest +
Number of years to maturity
Par value + Market value
2
Example: 8% maturing in 10 years, price $800
$1,000 - $800
$80 +
10
$1,000 + 800
2
= 0.111111  11.1%
11
Yield-to-Maturity
(continued)
Par value - Market value
$ Amt Annual Interest +
Number of years to maturity
Par value + Market value
2
Example: 8% maturing in 10 years, price $1,200
$1,000 - $1,200
$80 +
10
$1,000 + $1,200
2
= 0.054545  5.45%
12
Yield-to-Maturity

(continued)
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?
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.
13
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-tomaturity with years-to-call
14
Yield-to-Call
(continued)
Call value - Market value
$ Amt Annual Interest +
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
$80 +
5
$1,085 + $1,200
2
= 0.049891  4.99%
15
Yield-to-Call vs Yield-to-Maturity
(continued)

The yield-to-call was less than the yield-tomaturity
 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
16
Taxable Equivalent Yields

(review)
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
17
Taxable Equivalent Yields
(continued)
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%
18
Taxable Equivalent Yields

(continued)
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
19
Taxable Equivalent Yields
(continued)
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%
20
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
21
Yield Spreads
28 October 2016,
Source: Yahoo
(continued)
Current yield spreads on Yahoo:
http://finance.yahoo.com/bonds/composite_bond_rates
22
Yield Spreads

(continued)
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%
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.
23
The Effect of Inflation on Bond Rates
Source: Department of Labor Statistics
24
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
25
The Yield Curve
(continued)
26
The Yield Curve
(continued)
Source: Morningstar, October 28, 2016
27
Theories re: Yield Curves


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
28
Theories re: Yield Curves

(continued)
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.
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?
29
The Yield Curve & the Economy

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 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.
30
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
31
Bond Pricing

(continued)
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
32
Bond Pricing

(continued)
Example: Goldman Sachs
 5.0%, due 15-Nov-2024, priced to yield 3.011%
 Yield-to-Maturity 3.001%, Current Yield 4.378%
 3.011% is close to 3% – Let’s use 3% for 8 years
Bond price =
present value of the interest income +
present value of the repayment
= $50.00 * 7.020 (present value factor for stream of income) +
$1,000 * 0.789 (present value factor for repayment of bond)
= $352.20 + $789.00 ≈ $1,141.20
The quoted price on Finra on Nov 1st was $1,142.00. The bond calculator
spreadsheet (on the class web site) gives us $1,140.11 for annual payments
and $1,142.42 for semi-annual payments. Pretty close, eh? Why is the semiannual payments prediction a bit higher than the annual payments prediction?
33
Reinvestment Risk

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
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.
34
Duration

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
The shorter the duration, the less potential price volatility, and vice-versa.
35
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 interestrate 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.
36
Bond Investment Strategies

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
Which of these would be the easiest to implement?
Which would be the hardest?
37
Bond Investment Strategies
(continued)

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
What are the advantages and disadvantages of this strategy?
When is it a good time to ladder? When is it not a good time?
38
Bond Investment Strategies
(continued)
As of 28 October 2016
CHAPTER 10 – REVIEW
39
Bond Prices and Yields
Chapter Sections:
Bond Basics
Straight Bond Prices and Yield to Maturity
More on Yields
Interest Rate Risk and Malkiel’s Theorems
Duration
Dedicated Portfolios and Reinvestment Risk
Immunization
Next: Preferred Stocks & Convertible Securities,
Chapter 11, Asset Allocation