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1. Modern Portfolio Theory and
Investment Analysis, 9th Edition
Elton, Gruber, Brown, & Goetzmann
2. Classification: Protected A
Chapter 21: Interest Rate Theory and the
Pricing of Bonds
Until recently, a bond were considered to be easier to
value than a stock because of (1) certain stream of
payments (i.e., coupon and principal) and the bond has a
maximum life (i.e., maturity).
Recently, two factors had let to a change in the difficulty of
valuation:
1. The timing of cash flows became more variable and less
certain as new types of instruments are issued
2. Interest rates become more volatile.
>>> Increased volatility in market value provides both
an opportunity and risk >>> Active bond portfolio
management & PortfolioTheory for Bonds (Ch. 22)
3. Classification: Protected A
Chapter 21: Interest Rate Theory and the
Pricing of Bonds
Before moving to PortfolioTheory for Bonds in the next
chapter (Ch. 21), we must understand the pricing of bonds.
An Introduction to Debt Securities – Major type of bonds
The Many Definitions of Rates
Bond Prices and Spot Rates
Determining Spot Rates
The Determinants of Bond Prices
Factors Explaining Bond Prices
Appendix A: Special Considerations in Bond Pricing
Appendix B: Estimating Spot Rates
Appendix C: Calculating Bond EquivalentYield and Effective Annual
Yield
4. Classification: Protected A
An Introduction to Debt Securities
There are four major categories of long-term fixed
income securities:
Government Bonds
Corporate Bonds
Mortgage Bonds
Municipal Bonds
5. Classification: Protected A
An Introduction to Debt Securities
There are four major categories of long-term fixed
income securities:
Government Bonds
Borrowing of the federal government / largest % of total
debt market / by far the most liquid / considered default
free / simplest to value
Corporate Bonds
Debt obligations for corporations / less liquid / backed by
the credit of the issuing corporation >>> bonds default’s
risk
Investment Grade (i.e., low default risk) or HighYield (Junk)
6. Classification: Protected A
An Introduction to Debt Securities
Corporate Bonds – continued
Debt obligations for corporations / must less liquid /
backed by the credit of the issuing corporation >>>
bonds default’s risk
Investment Grade (i.e., low default risk) or HighYield (Junk)
Different type of options for bonds
Callability – should offer higher return for disadvantageous call
Sinking Fund – gradual retiring of bonds and avoiding a single
large payment / reduces the risk of default / bonds could be
called and so should offer higher return
Convertibility – could be exchanged for common equity / the
option rests with bondholder
7. Classification: Protected A
An Introduction to Debt Securities
Municipal Bonds
Debt obligations of states, cities and state or city
authorities / have default risk / interest on municipal
bonds are exempt from federal taxes (and or from state
taxes)
1. Backed by full faith and credit of the issuing city or state, or
2. Backed by a government agency or authority (called Revenue
Bonds), such as bonds issued by a port authority.
8. Classification: Protected A
An Introduction to Debt Securities
Mortgage Bonds
A group of mortgages are pooled and bonds are issued
against them (e.g., see Mortgage-Backed Securities slide
from Chapter 2, slides 7).
The most liquid publically traded mortgage instrument is
Ginnie Mae (Government National Mortgage Association)
Ginnie Mae insures the payment of principal and interest; then
the default risk has been removed, but there is a risk due to
uncertainty associated with the timing of the payment >>>
investors require higher return than on comparable bonds.
VIDEO: https://www.investopedia.com/terms/m/mbs.asp
9. The Many Definitions of Rates
There are confusing array of terms seemingly related
to interest rates, such as
Spot rates, future rates, yield to maturity and current
yield.
Yield to Maturity:
Internal rate of return earned from holding a bond to maturity.
YTM on three-year bond, annual interest payment of $100,
principal payment of $1,000 and cost of $900
10. Classification: Protected A
The Many Definitions of Rates
Yield to Maturity – if bond is held to maturity
Internal rate of return earned from holding a bond to maturity.
YTM on three-year bond, annual interest payment of $100,
principal payment of $1,000 and cost of $900
11. Classification: Protected A
The Many Definitions of Rates
Yield to Maturity:
C(t) : cash flow at time t ; coupon payment or principal
y : yield to maturity
The frequency of compounding varies
Assume a three-year bond, semi-annual payment of $50,
principal payment of $1,000 and cost $900
12. Classification: Protected A
The Many Definitions of Rates
Assume a three-year bond, semi-annual payment of $50,
principal payment of $1,000 and cost $900
(or 7.1% x 2)
Yield to Maturity or Bond EquivalentYield
This assumes no compounding in converting semiannual yield to
an annual yield.
This ignores the fact that investors can earn interest on the first
coupon received any year for the second half of the year.
15. Classification: Protected A
The Many Definitions of Rates
Yield to Maturity – Example
Assume 30-year Ginnie Maes with 360 (12 x 30)
payments.The price is $1,000,000
C(t) : $8,482
y : yield to maturity or Bond EquivalentYield
Effective AnnualYield
16. Classification: Protected A
The Many Definitions of Rates
Yield to Maturity – Treasury Bills
The yield onTreasury bills is computed differently than on
other bonds.
T bills are US government debts issued with maturities of one
year or less.
There are only two cash flows associated withT bills: (1) when
it is purchased, and (2) when it is matured; there is no interest
payment.
Interest rate on T bills, also called Bankers’ DiscountYield.
P : beginning and ending price
N : Number of days to maturity
17. Classification: Protected A
The Many Definitions of Rates
ComparingYield to Maturity
To compare theYield to Maturity among bonds,
investors should adjust the calculations so that a
common set of assumptions is being used. Most
institution
1. Calculate Effective AnnualYield, or
2. Adjust all instruments for semiannual interest rate
and doubling it (i.e., Bond EquivalentYield)
19. Classification: Protected A
The Many Definitions of Rates
Reinvestment Assumptions in Bonds
In calculating the yield to maturity, the implicit
assumption is that cash flows are reinvested atYield
to Maturity rate.
Theoretically, an investment organization choosing
among bonds with different yield to maturities makes
different assumptions about the reinvestment rate.
In reality, for the organization, there will be some rate
at which funds are (re)invested, no matter which
bond the coupon payments come from.
20. Classification: Protected A
The Many Definitions of Rates
Reinvestment Assumptions & Nonadditivity in Bonds
In portfolios, because of different reinvestment assumptions, yields
are not additive.
The yield to maturity on a portfolio is not a weighted average of
the yields on the bonds that comprise it; weights are the
proportion invested in each bond.
21. Figure 21.1
Graph of yield versus price.
10-year bond with coupon payment of
10% and semi-annual payment.
23. Classification: Protected A
Spot Rates
Spot interest rates are yield to maturity on
loans or bonds that pay only one cash flow to
the investor.
A bond with only one cash flow is called a pure
discount bond or a zero coupon bond.
Spot rates are usually calculated for six-month
intervals and annualized by doubling the six-month
rate. Time will be in six-month intervals.
For example:
Consider a six-month pure discount bond with a principal
of $1,000 and current price/cost of $970.87
24. Classification: Protected A
Spot Rates
For example:
consider a six-month pure discount bond with a principal
of $1,000 and current price/cost of $970.87
The return (yield to maturity) on such a bond is
the six-month spot rate
In S01 : 0 designates today, and 1 designates six months
from now.
In S02 : 2 designates 12 months from now.
26. Classification: Protected A
Spot Rates
Until the early 1980s, only the US government issued
pure discount bonds with maturities of one year or
less (i.e.,T bills).
In the early 1980s, corporations started to issue pure
discount instruments with longer maturities.
Some financial firms put together packages of coupon
bonds and sold off each year’s coupon separately; so
creating pure discount bonds or Stripped Coupon Bonds.
27. Forward Rate
Forward rates are interest rates on bonds where
the date the commitment is made and the date
the money is loaned are different; then the
interest rate on this loan is a forward rate.
Similar to spot rates, forward rates are estimated for
six-month intervals and annualized by doubling it.
Example:
Assume $924.56 is to be lent in 6 months and $1,000 is
to be repaid in one year later >>> the forward rate from
6 months to 18 months is f13
28. Classification: Protected A
Forward Rate
Another example:
Consider a two-year loan to be made in one year. $845.80
is lent and $1,000 is repaid.
As intervals are six-month periods; that is, the loan will start at
period 2 and goes to period 6.
The annualized forward rate f26 is:
29. Spot Rate & Forward Rate
Assume an investor wishing to hold money for two
periods or two six-month intervals:
Options 1: Buying a two-period pure discount instrument.
The ending value per $1 invested is:
Options 2: Buying a one-period pure discount instrument
and simultaneously plan to invest the proceeds at forward
rate from one to two.
The ending value per $1 invested is:
33. Bond Prices and Spot Rates
Assume S01 = 6% and S02 = 7%
As forward rates can be derived from spot rates, so forward rates can also
be used equally to determine bond prices.
35. Classification: Protected A
Determining Spot Rates
This can be continued to derive the other spot rates.
In theory, it should not matter which bonds are used,
because in equilibrium prices are determined using the
same spot rates.
In practice, different set of bonds results in different
estimates for spot rates; due to differences in bonds such
as
Tax treatments,
Callability
Bid-ask spread
Desirably, an average estimate of spot rates should be
calculated; multiple regression.
36. Classification: Protected A
Determining Spot Rates – Appendix B
Consider this discount function:
The price of bond i can be expressed as the present value
of the cash inflows
Pi is the price of bond i - Known
Ci(t) is the cash flow on bond i in period t - Known
dt are the discount functions – like regression coefficients
ei is the random error term, due to all differences in .
…..bond characteristics and etc.
37. Classification: Protected A
Determining Spot Rates
Spot rates are used to
Price private placements;
Private placements can be better matched to the
corporation’s cash generation pattern.
Price strips or zero coupon bonds;
Understand returns in the market for different holding
periods;
Find mispriced bonds, to be purchased or sold.
38. Classification: Protected A
The Determinants of Bond Prices
Yield to maturity on bonds differ for a number of reasons.
Important reasons includes:
The length of time before the bond matures;
The risk of not receiving coupon and principal payments;
The tax status of the cash flows;
The existence of provisions that allow to redeem the
debt before maturity; and,
The amount of coupon.
39. Classification: Protected A
The Determinants of Bond Prices
Term to Maturity andTerm StructureTheory
Segmented Market Theory
Pure ExpectationsTheory
Liquidity Premium Theory
Preferred Habitat
Term Structure and Coupon Bonds
Summary of theTerm Structure of Interest Rates
