Fixed Income Securities
Professor David McLean
Alberta School of
Business
What We Will Cover
Bond Basics
Valuation
The Term Structure of Interest Rates
Estimating Forward Rates
Corporate Bonds and Default Risk
Interest-Rate
Risk
Duration, Convexity
The Basics
Bond Basics
Bond
Security that obligates issuer to make payments to holder over time
Face Value, Par Value
Payment to bondholder at maturity of bond
Coupon Rate
Bond’s annual interest payment per dollar of par value
Zero-Coupon Bond
Pays no coupons, sells at discount, provides only payment of par value at maturity
Bond Basics
Fixed-Income Market Participants
Bond Basics: Example
Bond
Information:
Par value = $1,000
Coupon rate = 8%
Maturity = 30 years
The investor is entitled to
$40 Semi-annual coupons until maturity
$1,000 at the maturity
Bond Basics: Accrued Interest
Investors pay a different price than
that quoted; accrued
interest is the difference
Example:
40 days since the last semi-annual coupon
Rate is 8%, price is $990
Accrued interest =
$1,000 * 40/365 * .08 = $
Sale Price = $990 + $ = $
Canada
Bonds
Canada Bonds
Issued in $1,000 increments
Make semi-annual payments
Prices are quoted as a percentage of par value
A bid of $ is a bid price of $
OTC market, not very liquid
More Bond Terminology
Yield to Maturity
Discount rate that makes present value of bond’s payments equal to price.
Current Yield
Annual coupon divided by bond price
Premium Bonds
Bonds selling above par value
Discount Bonds
Bonds selling below par value
Valuation and Yield to Maturity
Bond Cash Flows
Bond Cash Flows
Coupons
Principal at Maturity
Cash flow for a
3-year bond with principal of $1,000 and annual coupon payment of 5
%
Bond Valuation
Computing YTM in EXCEL
YTM vs. Yield to Call
Some corporate bonds are
callable
Yield to Call
Calculated like yield to maturity
Time until call replaces time until maturity
C
all price replaces par value
Premium bonds more likely to be called than discount bonds
Why?
Holding Period Return
HPR is
the rate
of return over a
particular investment
holding period
Depends
on market price at end of period
Yield to Maturity measures average rate of return if investment held
until the bond matures
The bond’s price can change during the holding period
Hence, you often get an HPR ≠ Initial YTM
YTM vs. HPR: Example
Consider a 30-year bond with an 8% coupon selling at par value($1,000)
The bond’s YTM is 8%
Suppose rates decrease, so the price increases to $1,050 the next year
HPR = ($80 + $1,050 - $1,000) / $1,000 = 13%
Bond Prices vs. Interest Rates
Prices fall as market interest rate rises
Interest rate fluctuations are a primary source of bond market risk
Bonds with longer maturities are more sensitive to fluctuations in interest rate
Bond Prices vs. Interest Rates
Bond Pricing
When will a
bond’s
price = par value?
When
will a
bond’s
price > par value?
When
will a
bond’s
price < par value?
The Term Structure of Interest
R
ates
Valuation of Discount Bonds
Pure Discount Bond
No coupons, single payment of principal at maturity
Bond trades at a “discount” to
face value
Also known as
zero-coupon bonds
or
STRIPS*
Valuation is straightforward application of PV
What happens
if
expected values
of yearly interest rates over
t
ime?
Denote by
R
t
the one-year
spot rate
of interest in year
t
But we don’t observe future spot rates today…
Valuation of Discount Bonds
Valuation of Discount Bonds
Today’s
T-year spot rate
(
r
) is
an “average” of one-year future spot rates
The T-year spot rate (
r
) is also
referred to as the
y
ield
to
maturity
of a zero coupon bond maturing in year T
Computing YTMs for Zeros
With zeros, if you observe price and maturity, then you can solve for YTM (r).
(Face Value is $1,000)
Term Structure
of
Interest
Rates
The
term structure of interest rates
is the relation between yield to maturity and maturity
The
yield curve
is a graph that displays the relationship between yield and maturity
Expected
short term rates, also known as
forward rates
, can be implied from yield curve
Yield Curves
Estimating Forward Rates
Consider two different investments
Invest in a 2-year zero
Price = $890; Face Value = $100
Invest $890 in a 1-year zero
Assume you buy a proportional amount
A
t maturity, reinvest the proceeds in a second 1-year zero
These two strategies ought to have the same
expected
return
Two 2-Year Investment Programs
Two 2-Year Investment Programs
Investing $890 the 2-year zero results in $1,000 with certainty
The expected value of an $890 investment into a 1-year zero, and then a second 1-year zero, needs to be $1,000 as well
The 1-year zero investment grows $890 to $
$ x (1 + r) = $1,000
r = %
Hence, the year 2 forward rate, or the expected short rate in year 2, is %
f
n
= one-year forward rate for period n
y
n
= yield for a security with a maturity of n
General Formula for Forward Rates
YTM to infer Forward Rates
Assume that the YTM on 1, 2, 3, and 4-year zeros are:
5%, 6%, 7%, 8%
What are the forward rates in years 1, 2, 3, and 4?
Answer: 5
%,
%, %, and %
Can you verify this?
Locking in Forward Rates
Suppose that discount bond prices are as follows:
A customer would like to have a forward contract to borrow $20MM three years from now for one year
Can you (a bank) quote a rate for this forward loan?
All you need is the forward rate
f
4
which should be your quote for the forward loan
Locking in Forward Rates
Strategy:
Buy
20,000,000 of 3-year discount bonds, costing
Finance this by (short
) selling
4-year discount bonds of amount
This creates a liability in year 4 in the amount $21,701,403
Locking in Forward Rates
Example (
cont
):
Cashflows
from this strategy (in million dollars):
The yield for this strategy or
“
synthetic
bond return
”
is given by
:
Theories of the Term Structure
The term structure usually slopes upwards
Why?
What does this tell us about investor’s expectations and preferences?
Theories of the Term Structure
Pure Expectations Theory
No investor preference for long or short-term securities
Hence, long
-term and short-term securities are perfect substitutes
Forward rates are the market consensus of expected future short-term rates
Theories of the Term Structure
The
Pure Expectations Hypothesis:
f
2
= E(r
2
)
(
1+y
2
)
2
=(1+r
1
)(1+f
2
)
(
1+y
2
)
2
=(1+r
1
)(1+E(r
2
))
If
the yield curve slopes upward, it is because investors expect rates to increase
What if investors have short horizons?
Consider 2
consecutive investments
in
1-year zeros
Under pure expectations, we assume E[r
2
] = f
2
Hence, the
expected
return is: (
1+r
1
)(1
+f
2
) = (1+y
2
)
2
This is the same as the YTM on a 2-year zero
Assumer
1
=8% and E[r
2
] = 10%
Then the 2-year bond’s price is:
$
1,000/()() = $
What if
Investors
have
Short
H
orizons
?
What if the investor only wants to invest for 1-year?
Can this investor just buy the 2-year zero and sell it in one year, and get 8%
For
the
1-year investor
to get an 8% return, r
2
has to end up being 10%
Then the bond is worth
$ in Year
2
$ * = $1,000
We do not know r
2
for certain at Time = 0
Hence, for the 1-year investor
the 2-year bond is
riskier
than a
1-year bond
What if
Investors
have
Short Horizons
?
If the investor has a 1-year horizon, then the
2-year bond may sell for less than $
(1+r
1
)[1+E(r
2
)] < (1+r
1
)(1+f
2
)
f
2
> E(r
2
) because investor’s demand a higher return for the 2-year bond
What if Investors Have Long Horizons?
A long-term investor may wish to lock in forward rates
For long-term investors,
short term bonds
are
riskier than long term bonds
In this case, forward
rates may be less than expected short
rates
(1+r
1
)[1+E(r
2
)] > (1+r
1
)(1+f
2
)
Investor Preferences and the Term
Structure
Short term investors will not hold long term bonds unless f
2
> E(r
2
)
Long
term investors will not hold short term bonds unless f
2
< E(r
2
)
If
short term investors are greater in number, then f
2
> E(r
2
)
Liquidity Premium Theory
Liquidity Premium Theory
Investors have short horizons, so long-term bonds are more risky
Investors will therefore demand a premium for the risk associated with long-term bonds
Yield curve has an upward bias built into the long-term rates because of the risk premium
Forward rates contain a liquidity premium and are not equal to expected future short-term rates
Default Risk
Corporate Bonds and Default Risk
Non-Government Bonds Carry Default Risk
A
default
is when a debt issuer fails to make a promised payment (interest or principal)
Credit ratings by rating agencies (., Moody's and S&P) provide indications of the likelihood of default by each issuer.
Corporate Bonds and Default Risk
Source: Federal Reserve
Defaulting Debt Levels
$ Millions
Face value of defaulting debt
Yield Spreads
Yield Spread, %
Corporate Bonds and Default Risk
Decomposition of Corporate Bond Yields
Promised YTM
is the yield if default does not occur
Expected YTM
is the probability-weighted average of all possible yields
Default premium
is the difference between promised yield and expected yield
Risk premium
(of a bond) is the difference between the expected yield on a risky bond and the yield on a risk-free bond of similar maturity and coupon rate
Example:
Suppose all bonds have par value $1,000 and
10-year Treasury Zeros are selling at $, YTM is 8%
10-year zero issued by XYZ Inc. is selling at $
Expected payoff from XYZ's 10-year zero is $
Corporate Bonds and Default Risk
For the 10-year zero issued by XYZ:
Corporate Bonds and Default Risk
Decomposition of Corporate Bond Yields
Default Risk Example
Consider a 1-year zero-coupon corporate bond
Face Value= $1,000
The probability of default is 20%
If there is default, the bondholders will get $500
The 1-year risk-free rate is 5%
The risk premium for this bond is 2%
What is the bond’s Promised and Expected YTM’s?
Default Risk Example
The expected cash flow is:
* $1,000 + * $500 = $900
Price is therefore $900 / = $
Promised YTM is (1,000/) – 1 = %
Expected YTM is (900/) – 1 = 7%
Interest Rate
R
isk
Interest Rate Risk
Look back at the formula to value a bond
Interest rates are in the denominator
Hence, when rates increase, bond prices fall
Can you provide an economic intuition for this relation?
Bond Prices vs. Interest Rates
Bond Prices
vs. Interest Rates
Estimating Interest Rate Risk
Duration
is a measure of the effective maturity of a bond
Duration
is the weighted average of the times until each payment is received
The weights are proportional to the present value of the payment
Computing Duration
Duration of Two Bonds
Duration/Price Relationship
Price change is proportional to
D
uration
Or
, if we denote D
*
=
Modified
D
uration
Summary of Duration Properties
The sensitivity of a
bond’s
price to interest rates is influenced by 3 key
factors
Time to maturity
Coupon rate
YTM
Why?
Interest Rate Risk and YTM
Annual Coupon Bond Prices
Prices of 8% annual coupon bonds
*Equals value of bond at a 9% yield to maturity minus value of bond at (the original) 8% yield, divided by the value at 8% yield.
Zero-Coupon Bond Prices
Prices of zero-coupon bonds
*Equals value of bond at a 9% yield to maturity minus value of bond at (the original) 8% yield, divided by the value at 8% yield.
Why is Duration Important?
It’s a simple summary statistic of the effective average maturity of the portfolio;
It is an essential tool in immunizing portfolios from interest rate risk;
It is a measure of interest rate risk of a portfolio
Immunization
Financial institutions often have a mismatch between the maturities of their assets and liabilities
Bank
liabilities are short-term, while bank assets tend to long-
term
Pension funds can have mismatches as well
Why
might this be a problem?
Immunization
Assume that an insurance company issues a $10,000, 5-year GIC with a rate of 8%
So in 5-years the insurance company will have to pay:
$10,000 x ()
5
= $14, in 5-years
In the interim the insurance company invests the $10,000 in a 6-year annual bond coupon bond, with an 8% coupon rate
The bond’s duration is , about the same as the obligation’s
So even if rates change, the company should have the $14, in 5-years if it holds the bond until year 5, and then sells it
Market Value Balance Sheet at t=0
How things could end up in Year 5
Immunization
In the previous example, the insurance company had “immunized” itself against interest rate risk
Setting
the
dollar-weighted
durations of the assets and liabilities equal to one another accomplishes this
Immunization Issues
The duration of a bond portfolio is equal to the weighted average of the durations of the bonds in the
portfolio:
The
portfolio duration, however, does not change linearly with time
The
portfolio needs be
rebalanced
periodically to maintain immunization
Why?
Convexity
Duration is a
linear approximation
of the price yield relation
The real relation between price and yield is not linear; if plotted, the relation is
curved
This curvature in the price yield relation is called
convexity
Do investors prefer this curvature?
Bond Price
Convexity
30
-Year Maturity, 8% Coupon; Initial Yield to Maturity = 8
%
Correction for Convexity
Correction for Convexity:
Convexity of Two Bonds
Bond B
What you should know
Bond Basics
Valuing Bonds, YTM, YTC
Estimating forward rates
The term structure of interest rates
Default Risk
Interest-rate risk
Duration, convexity
